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MATCHED INTERFACE AND BOUNDARY (MIB)
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APPLICATION TO MOLECULAR BIOLOGY AND
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i

MATCHED INTERFACE AND BOUNDARY (MIB)
METHOD FOR GEOMETRIC SINGULARITIES AND ITS
APPLICATION TO MOLECULAR BIOLOGY AND
STRUCTURAL ANALYSIS

By
SINING YU

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment Of the requirements

for the degree Of
DOCTOR OF PHILOSOPHY

Department. Of h-‘lathematics

2007

ABSTRACT

MATCHED INTERFACE AND BOUNDARY (MIB) METHOD FOR
GEOMETRIC SINGULARITIES AND ITS APPLICATION TO MOLECULAR
BIOLOGY AND STRUCTURAL ANALYSIS

By

Sining Yu

This dissertation describes the development of high-order matched interface and
boundary (MIB) method [158—161] for geometric singularities and its applications to the
implicit model of biomolecules and the vibration analysis of plates.

The matched interface and boundary (MIB) method [163, 168—170] was previously
devised for solving elliptic problems with curved interfaces to shape-edged interfaces and
boundaries. However, the previous MIB method suffers fi'om the loss of its designed
accuracy at geometric singularities. In this work, flexible strategies are developed to
maintain the accuracy of the MIB method near the geometric singularities according to
the local topology of the irregular point around sharp-edged interfaces. New ideas, such
as using two sets of physical jump conditions around sharp-edged interfaces, are
proposed to systematically determine fictitious values near the geometric singularities.

This dissertation also reports the three-dimensional (3D) generalization of the
previous 2D higher-order MIB method. The resulting 3D MIB schemes are of second
order accuracy for arbitrarily complex interfaces with sharp geometric singularities, of
fourth-order accuracy for complex interfaces with moderate geometric singularities, and
of sixth-order accuracy for curved smooth interfaces. A systematical procedure is
introduced to make the MIB matrix optimally symmetric and banded by appropriately
choosing auxiliary grid points. Consequently, the new MIB linear algebraic equations can
be solved with smaller number of iterations. The new MIB method is extensively
validated in terms of the order of accuracy, the speed of convergence, the number of
iterations and CPU time. Numerical experiments are carried out to complex geometries
and interface conditions.

Furthermore, this dissertation presents a new MIB method based Poisson—Boltzmann
(PB) solver, denoted as MIBPB-II. The MIBPB-II solver is systematical and robust in

treating geometric singularities, and delivers second order convergence for arbitrarily

Sining Yu

complex molecular surfaces of proteins. The MIBPB-II solver is extensively validated by
the molecular surfaces of few-atom systems and a set of twenty four proteins. Converged
electrostatic potentials and solvation free energies are obtained at a coarse grid spacing of
0.5 A and are considerably more accurate than those obtained by established methods,
such the PBEQ and the APBS, at finer grid spacings.

Finally, this dissertation provides the extension of the MIB method to solve boundary
value problems for free vibration analysis of rectangular plates. Fictitious values are
obtained iteratively using three types of boundary conditions, namely simply supported,
clamped and free ones. It is shown that the present method provides convergent and
accurate solutions for different combinations of these boundary conditions. Moreover, it
is an extremely challenging problem if the plate has at least one fiee comer, which means
two free boundaries next to each other. The present work provides much more stable and
consistent results comparing to the established GDQ method [133] in this case. This
demonstrates that the present method is a suitable method for the vibration analysis of

plates with free edges and free comers.

Acknowledgments

This is a summary Of work done over a five year period while in the Department
of Mathematics at the Michigan State University. I wish to thank Professors Guowei
Wei, Gang Bao, Changyi Wang, and Keith Promislow in this department for being
on my dissertation committee, along with Professor Shanker Balasubramaniam in the
Department of Electrical and Computer Engineering at the Michigan State University,
who was the fifth committee member.

The one I would like to thank the most is my Ph.D advisor, Prof. Guowei Wei.
He brought me to this research area five and half years ago. In my early stage of the
Ph.D study, he guided me through all kinds of computational trainings, which built a
strong base for my further Ph.D research. He is the captain who always brings us one
and another great ideas that eventually lead us to a whole new world. He is also the
greatest partner who I could discuss with about any challenge I’ve faced at any time.
The most importantly, it is his passion that gives me strength to move foward in my
research. This dissertation does not come out easily. At the first time I mentioned
about developing new algorithms for several irregular points where we had troubles in
Obtaining fictitious values, I wasn’t sure how much it would be worth. Prof. Wei drove
me to follow my thought, gave me plenty Of time to try different ideas out and pointed
out that these new algorithms would be a breakthrough in solving interface problems
with geometric singularities. We share our faith and dream together in all these years.
Once the first stone has been set for the dissertation, we start rolling our ideas quickly.
We moved from two dimensional geometric singularities tO three dimensional ones.
We successfully realized the second-order accuracy on arbitrary interface and applied
the method to solve for protein solvation energy, which becomes the only second-order
method in this area by far. Besides researches, we also share the greatest joy in our
lives. He is the witness of my marriage, and we almost celebrate all the holidays and
important days in his house. He is always the most generous host and a great toaster.

I would like also thank to Prof. Shan Zhao, who just started his tenure track in
University Of Alabama last year. He is the one who taught me how to use Fortran for
my very first code five and half years ago. In the past several years, I have learned
numerous techniques from him. He is always an open resource for my questions and an
effective detector for any invisible bugs in my code. His initial work on developing the
high-order MIB method in one dimension is the base Of all chapters Of my dissertation.
When I was developing the MIB method for plate vibration analysis, I stopped by his
Office almost every day to discuss the troubles I have met for hours. Those valuable
discussions helped me deeply understand the MIB method so that I could overcome
many Obstacles that I met later.

Dr. Yongcheng Zhou is another brilliant person who helps me make this dis-
sertation possible. His significant work in developing the MIB method for smooth
interface Opens all my imaginations and is the starting point of most Of my disserta-
tion. His pioneer work in applying the MIB method to solve protein solvation energy
shorten the path Of building the MIB—PB II solver. He gave me many useful tOOls and
packages, and kept helping me out even when he has started his postdoc research in

iv

UCSD. I have to thank Weihua Geng, who is also a Ph.D candidate in Prof. Wei’s
group, for his support in the past several years. We reproduced the results Of the
IIM method and the MIB method for smooth interface together. We went through
the whole process side by side, but still independently. It has become an enjoyable
memory to have such a great research partner. I wish to thank the rest students in
Prof. Wei’s group, Yuhui Sun, Zhan Chen, Duan Chen and Li Yang, for providing
valuable suggestions and helps in my research and everyday life. I wish to thank
a wonderful couple, Fangfang Sun and Yongfeng Li, who are studying in GIT now.
They have been very supportive to me in my research and personal life just like my
family. I wish to thank Prof. Yibao Zhao in Singapore for his valuable suggestions
on solving plate vibration problems. I wish to thank Prof. Songming Hou for our
numerous discussions and his valuable suggestions on developing interface method for
geometric singularities in 2D.

I would like to thank Prof. Gang Bao. I have been taking his class for more than
three years. His kindness and encouragement build up my confidence and help me
recognize my abilities. I would like to thank Prof. Changyi Wang for inspiring me by
many interesting tOpics and vivid examples in his class.

I would like to acknowledge the continuous support and encouragement of my
parents, Guiqin and Wuyi. I could never repay the kindness and support they have
given me during my life, and especially over the last five years. They were without
exception always willing to talk to me on the phone at any time day or night if I had
a problem. They encourage me to pursue my interests, and were always there to offer
help or advice.

Finally, to Yichong, my husband and loved one, I thank you for supporting me
during my graduate studies, and for putting up with all the plane trips, the phone
calls, and the seemingly endless separations. We can now look forward to a wonderful
life together.

Table Of Contents

Chapter 1 Introduction to Mathematical Methods for Elliptic Interface

Problems 1
1.1 Elliptic equations with discontinuous coefficient and singular source: status,

challenges and thesis outline .......................... 1

1.2 Introduction of 1-D MIB method ........................ 6

1.3 Introduction of 2-D MIB method for smooth interface ............ 9

1.3.1 Off-interface scheme .......................... 12

1.3.2 On—interface scheme ........................... 15

Chapter 2 Two dimensional matched interface and boundary (MIB) method

for treating geometric singularities 21

2.1 Theory and algorithm .............................. 22

2.1.1 Off-interface schemes for geometric singularities ........... 22

2.1.2 Orr-interface scheme for geometric singularities ............ 26

2.1.3 A pseudo—code of MIB ......................... 33

2.1.4 The critical value of the acute angle .................. 35

2.2 Numerical study ................................. 36

2.3 Conclusion .................................... 53
Chapter 3 Three-dimensional matched interface and boundary (MIB)

method for treating geometric singularities 55

3.1 Theory and algorithm .............................. 56

3.1.1 General setting of the MIB method for elliptic interface problems . 56

3.1.2 Treatment of smooth interfaces in 3D — an illustration ....... 61

3.1.3 Matrix Optimization .......................... 64

3.1.4 Type I auxiliary points ......................... 67

3.1.5 Type II auxiliary points ........................ 69

3.2 Results and Discussion ............................. 74

3.2.1 Case 1: Matrix optimization ...................... 76

3.2.2 Case 2: Validation on interfaces with geometric singularities . . . . 80

3.2.3 Case 3: A missile interface ....................... 83

3.3 Conclusion .................................... 85
Chapter 4 Three-dimensional fourth and sixth order matched interface

and boundary (MIB) method 91

4.1 Theory and algorithm .............................. 92

4.2 Results and Discussion ............................. 99

4.2.1 Case 4: Fourth—order MIB scheme ................... 99

, 4.2.2 Case 5: Variable diffusion coefficients ................. 105

4.2.3 Case 6: Non-zero linear term ...................... 106

4.2.4 Case 7: Sixth—order MIB scheme ................... 107

4.3 Conclusion .................................... 109

vi

Chapter 5 Treatment of geometric singularities in implicit solvent models 114

5.1 Theory and algorithm .............................. 119
5.1.1 Poisson Boltzmann Equation ...................... 119

5.2 Results and Discussion ............................. 120
5.2.1 Validation ................................ 121

5.2.2 Applications ............................... 129

5.3 Conclusion ...... . ............................. 133
Chapter 6 MIB method for the nonlinear Poisson-Boltzmann equation 138
6.1 Inexact-Newton methods ............................ 139
6.1.1 Inexactness and superlinear convergence ............... 140

6.1.2 Inexactness and global convergence .................. 141

6.1.3 Damped-inexact-Newton algorithm for PBE ............. 142

6.2 Results and Discussion ............................. 144
6.2.1 Case I: Spherical interface ....................... 145

6.2.2 Case II: van de Waals surface of two atoms .............. 145

6.2.3 Case III: Molecular surface of protein lajj .............. 146

6.3 Conclusion .................................... 146
Chapter 7 MIB Method for the vibration analysis of plates 150
7.1 Governing Equations .............................. 153
7.2 Theory and Algorithm .............................. 155
7.2.1 Second order MIB scheme for plate vibration analysis ........ 155

7.2.2 High order MIB schemes ........................ 164

7.2.3 Discretization matrix .......................... 169

7.3 Results and Discussion ............................. 171
7.3.1 Uniform meshes ............................. 171

7.3.2 Non-uniform meshes .......................... 177

7.4 Conclusion .................................... 179
Chapter 8 Thesis Achievement and Future Work 182
8.1 Thesis Achievement ............................... 182
8.2 Future work ................................... 184
References 186

vii

List Of Tables

2.1
2.2

2.3

2.4
2.5
2.6

2.7
2.8

2.9

2.10

3.1

3.2

3.3

3.4

Numerical efficiency tests of the 2D Poisson equation (Case 1). . . . 37

Comparison of numerical solutions involving the left and right flux
jump conditions at the tip in the geometry of Case 1(c). ....... 39

Numerical efficiency tests of the pentagon star interface (Case 2(a)-

(d)). .................................... 41
Numerical efficiency tests Of the pentagon star interface (Case 2(e)-(f)). 44
Numerical accuracy tests of the missile interface case (Case 3) ..... 44

Numerical accuracy tests Of the chess board interface case (Cases 4(a)

and 4(b)). ................................ 48
Numerical accuracy tests Of Case 5. ................... 48
Numerical accuracy tests of the thin layer case (Case 6). ....... 50

8
Numerical accuracy tests Of the imaginary and real parts Of w(:.) = 23
(Case 7) ................................... 51

Numerical accuracy tests Of large discontinuity ratio Of ;‘3‘ and {3+
(Case 7) ................................... 53

Comparison Of different choices made for Step 1 (QT or Q“) and Step
2 (mesh line sets as shown in Fig. 3.3). Auxiliary point sets (i)(ii) as
shown in Fig. 3.4 are used for Step 3 whenever they are available. . 86

Comparison of different choices made for Step 2 (mesh line sets as
shown in Fig. 3.3) and Step 3 (auxiliary point sets as shown in Fig.
3.4). Subdomain 0+ is used ........................ 87

Comparison of different choices made for subdomains, mesh line sets
and auxiliary point sets. The interface is defined by the molecular
surface of protein 2pde, shown in Fig. 3.8. The exact solutions are
given by Eqs. (3.23) and (3.24). ..................... 88

Order Of convergence test on selections Of type I auxiliary points using
different combinations of Step 1 and Step 3. Mesh line set (a) is used in
all the calculations. The interface is defined by the molecular surface
of protein 2pde, shown in Fig. 3.8. The exact. solutions are given by
Eqs. (3.23) and (3.24). .......................... 89

viii

3.6

4.1

4.2

4.3

4.4

4.5

4.6

4.7

5.1

5.3
5.2

5.4

6.1

Convergence test on interface Of intersecting spheres. Case 2(a): two
intersecting spheres; Case 2(b): three intersecting spheres; Case 2(c):
18 intersecting spheres. .........................

Convergence test on missile interface shown in Fig. 3.11. The exact
solutions of Case 3(a) and Case 3(b) are respectively given by Eqs.
(3.25) and (3.23), with coefficients defined in Eq. (3.24). .......

Convergence test of the fourth-order MIB scheme (Case 4). Three

interfaces are shown in Fig. 4.3 and 4.4. The exact solutions are given
by Eqs. (4.11) and (3.24). ........................

Convergence test of the fourth-order MIB scheme (Case 4(d) and (e)).
Two interfaces are shown in Figs. 4.3(a) and 4.4(e). The exact solu-
tions are given by Eqs. (3.23) and (3.24) .................

Convergence test on variable diffusion coefficients (Case 5). Two in-
terfaces are shown in Figs. 4.3(a) and 4.4(e). The exact solutions are
given by Eqs. (4.11) and (4.14). .....................

Convergence test on the non-zero linear term with the acorn interface
shown in Fig. 4.4(e)(f). The exact solutions are given by Eqs. (4.11)
and (3.24) with rs:(:c, y, z) redefined by Eqs. (4.15) and (4.16) for Case
6(a) and Case 6(b), respectively ......................

Convergence comparison Of the second-, fourth- and sixth-order MIB
schemes on a spherical interface (Case 7(a)). The exact solutions are
given by Eqs. (4.17) and (3.24). .....................

Convergence comparison Of the second-, fourth- and sixth-order MIB
schemes on an ellipsoid interface (Case 7(b)) as shown in Fig. 4.3(a)(b).
The exact solutions are given by Eqs. (4.18) and (4.19) .........

Convergence comparison of the second-, fourth- and sixth-order MIB
schemes on a diatomic interface (Case 7(0)) as shown in Fig. 3.7. The
exact solutions are given by Eqs. (4.11) and (3.24) with k = 3 .....

Electrostatic solvation energy AG in kcal/mol and the error in the
surface potential for a sphere with a centered unit charge. The exact
solvation energy is -81.98 kcal/mol. ...................

Accuracy tests on the molecular surface .................

Numerical errors Of three methods for the Poisson equation. Case I:
Sphere of radius 2.0A; Case II: Diatomic molecular surface with cusp
singularities; Case III: Four-atom molecular surface with cusp and self—
intersecting surface singularities ......................

90

90

104

105

106

110

111

112

113

122
126

136

Electrostatic solvation energies calculated by using the MIBPB-I, MIBPB-

II, APBS and PBEQ. CPU times used by MIBPB-II at h = 0.5A and
by APBS at h ~ 0.2A are given. ....................

Numerical errors for the nonlinear Poisson—Boltzmann equation. Case
I: Sphere Of radius 2.5A; .........................

ix

137

147

6.2

6.3

7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
7.9

Numerical errors for the nonlinear Poisson-Boltzmann equation. Case

II: Van de Waals Surface of two atoms; ................. 148
Numerical errors for the nonlinear Poisson—Boltzmann equation when

h = 0.5. Case III: Protein lajj; ..................... 149
Convergence of the 8888 plate. ..................... 172
Convergence of the CCCC plate ...................... 173
Convergence of the FSSS plate. ..................... 173
Convergence Of the F CCC plate ...................... 174
Convergence of the FFSS plate. ..................... 175
Convergence of the FFSC plate ...................... 175
Convergence Of the FFCC plate ...................... 176
Convergence of the F FFS plate. ..................... 176
Convergence of the FFFF plate ...................... 17 7

7.10 The numerical results Obtained with adaptive grids (N = 13, M =

5, L = 5). The relative errors Obtained with uniform grids is denoted
as L2.uni and are listed for a comparison. ............... 178

List Of Figures

1.1

1.3
2.1

2.2
2.3

2.4

2.5

Illustration of MIB method for 1-D problem. The dash line indicates
the position Of interface :1: = a, where u(r) has a discontinuity. Real
solutions u(:r) at grid points are marked in black and the fictitious
values f (:r) are marked in green. Fictitious values fi+1 and fi+2 are the
smooth extension of the solution in left subdomain; f,_1 and f, are the
smooth extension Of the solution in right subdornain. These fictitious
values are solved by matching two interface conditions simultaneously. 7

Situation handled by Off-interface Scheme for smooth interface. “0”
indicates a pair of irregular points (i, j), (i + 1, j) and two auxiliary
points (i — 1, j), (i + 2, j) with respect to the present irregular points. 13

Situation handled by On-interface scheme 1. ............. 16

Situation (a) handled by Off-interface scheme 2(a); Situation (b) han-
dled by Off—interface scheme 2(b). ................... 23

Situation handled by On—interface scheme (a) for geometric singularities 26

Four different situations handled by On—interface scheme (b) for geo-
metric singularities. ........................... 30

(a) The critical value Of the smallest angle that MIB can handle; (b)
An angle smaller than the critical value. ................ 35

Interfaces on 20 x 20 meshes (top row) and the computed solution
(bottom row) for the 2D Poisson equation (Case 1(a) and Case 1(b)). 38

2.6 Interface on a 20 x 20 mesh (left) and the computed solution (right)

for the 2D Poisson equation (Case 1(c)). ................ 38
2.7 The pentagon star interface in a 20 x 20 mesh (Case 2) ......... 41
28 Computed solution for the pentagon star interface (Case 2(a), 2(b),

2(e), 2(d)), 2(e) and 2(f)). ........................ 42
2.9 The configuration of a rnissile(Case 3). ................. 45
2.10 The missile interface on a 20 x 20 mesh (left) and computed solution

(right) (Case 3). ............................. 45
2.11 The chess board interface on a 20 x 20 mesh (Case 4). ........ 47

2.12 Computed solution for the chess board interface (Cases 4(a) and 4(b)). 47

xi

2.13

2.14

2.15

3.1

3.2

3.3

3.4
3.5

3.6

3.7

3.8

The interface Of Case 5 on a 20 x 20 mesh (left) and computed solution
ona80x80mesh (right). . . . . . . . . . . . . . ...........

The thin layer interface on a 20 x 20 mesh (left) and computed solution
(right) (Case 6). .............................

Domain with reentrant corner on a 20 x 20 mesh (left) and computed
solution (right) (Case 7). ........................

Illustration Of a smooth interface. The kth y mesh line intersects the
interface at point ($0,310, :0). A pair of irregular points (2', j, k) and
(i, j + 1, k), in red 0, is on the kth y-mesh line. Type I auxiliary points
are in blue A. Type II auxiliary points are in blue 0. Auxiliary lines,
in red dashed lines along the :r- and z-directions, are used to calculate
ug, uj‘, u; and uj’. ...........................

(a) Type I auxiliary points A typically selected in MIB 1; (1)) Type I
auxiliary points A typically selected in MIB II ..............

Three different sets of mesh lines from which type I auxiliary points
are selected. Set (a): mesh lines zk_1 and 2),“; Set (b): mesh lines
zk+1 and Zk+2§ Set (e): mesh lines zk_1 and zk_2 ............

Four sets of type I auxiliary points on a single mesh line 3k+1~ . . . .

Type I auxiliary points and other three points on mesh line 3),, all
in blue 0, are used together to derive a? Situation (a): Type I
auxiliary points locate on mesh lines zk_1 and zk+1; Situation (b):
Type I auxiliary points locate separately on mesh line zk_1; Situation
(c): Type I auxiliary points locate on nonconsecutive mesh lines zk_2

and Zk+1 . .................................

Interface singularities and type II auxiliary points. Type II auxiliary
points and two irregular points, all in blue 0, are used in all discretiza—
tion Of jump conditions about interface point (330, yo, 20) and an addi-
tional jump condition about the second interface point. Situation(a):
Type II auxiliary point (i, j — 1, k) and irregular point (2', j, k) are in dif-
ferent subdomains, which creates a second interface point (2:3, 31:), 25,);
Situation(b): Type II auxiliary point (i, j + 2, k) and irregular point
(i, j +1, k) are in different subdomains, which creates a second interface
point (173,313.23)- .............................

Surface maps Of the exact solution, (a), and numerical errors, (b), for
the molecular surface Of a diatom of atomic radius 1.5 A with atoms
centered at (—-1.52, 0,0) and (1.35.0, 0), respectively. .........

Surface maps of the exact solution, (a), and numerical errors, (b), for
the molecular surface Of protein 2pde. .................

xii

62

64

65
66

68

71

3.9

3.10

3.11

4.1

4.2

4.3

4.4

4.5

5.1

Interface Of intersecting spheres. (a) two intersecting spheres Of ra-
dius 1.5 with centers (—1.52,0,0) and (1.35,0,0). (b) three inter-
secting spheres Of radius 1.5 with centers (—1.52, 0, 0), (135,0, 0) and
(0,1.62,0). (c) 18 intersecting spheres. The coordinates of the center
and radius of each spheres are given in the form of (x,y,z,r) as follows:
(—2.0270, 0.9540, -0.6510, 1.7), (-1.6690, 0.2340, 0.6650, 1.7), (-0.4530,
-0.6870, 0.4410, 1.7), (0.7510, 0.1480, -0.0400, 1.7), (0.3930, 0.8680,
-1.3560, 1.7), (-0.8230, 1.7880, -1.1320, 1.7), (~2.2840, 0.2080, -1.4180,
1.2), (-2.8880, 1.6170, -0.4830, 1.2), (-2.5270, -0.3680, 0.9970, 1.2), (-
1.4260, 0.9800, 1.4350, 1.2), (—0.1960, -1.1890, 1.3850, 1.2), (-0.7010,
-1.4410, -0.3200, 1.2), (1.0070, 0.8940, 0.7270, 1.2), (1.6120, -0.5150,
-0.2080, 1.2), (0.1490, 0.1210, -2.1260, 1.2), (1.2510, 1.4700, -1.6880,

1.2), (-1.0810, 2.2910, -2.0760, 1.2), and (.0.5750, 2.5430, .0.3710, 1.2).

Cross section views of interface generated by intersecting spheres. (a)
A cross section Of Fig. 3.9(a); (b) A cross section Of Fig. 3.9(b). . . .

A missile geometry with solution and error maps. Numerical errors are
represented in seven scales. (a) exact solution mapped on the surface
of a missile; (b) numerical error obtained at h = 0.5; (c) numerical
error obtained at h = 0.25; (d) numerical error Obtained at h = 0.125.

Illustration of a fourth-order MIB scheme. The y-mesh line intersects
the interface at (Io, ya, .20). A set Of irregular points (i,j—1,k), (i, j, k),
(z',j -l- 1,12) and (2',j + 2,13), in red 0, is on the y-mesh line. Type I
auxiliary points are in blue A. Type II auxiliary points are in blue 0.
Auxiliary lines, in red dashed lines along the :r- and z—directions, are
used for computing uJD and uz. .....................

Illustration Of iterative schemes. The y-mesh line intersects the in-
terface at (11:0,yo, 20). The fictitious values at 0 points are generated
pairwisely. The order of generating fictitious values is indicated by the
numbers on the upper right corner Of o ..................

The surface maps of exact solutions (left column) and numerical errors
(right column). Numerical errors are represented in seven scales. (a)(b)
Ellipsoid; (c)(d) Cylinder. ........................

The surface maps Of exact solutions (left) and numerical errors (right).
Numerical errors are represented in seven scales. (c)(f) Oak acorn . .

The surface maps Of exact solutions (Top row) and numerical errors
(Bottom row). Numerical errors are Obtained at h. = 0.2 for the aplle
and at h = 0.1 for the flower. All images are plotted in seven scales.

(a) Aplle; (b) Flower ............................

Molecular surface singularities. Radius Of carbon atoms is 1.5 A in
both cases. Left: Centers of atoms are (—3.62,0,0), (362,0,0), and

the probe radius is 5.1 A; Right: Centers of atoms are (0,4.2,0),

(0, —4.2, 0), (5, 0, 0) and (—5, 0, 0), and the probe radius is 4.9 A.

xiii

81

81

84

96

100

101

103

123

5.2

5.3

5.4

5.5

7.1

7.2

The surface projections Of numerical errors in MIB I and MIB II. (a)
Numerical errors Obtained by MIB II for the diatomic system; (b) Nu-
merical errors Obtained by MIB I for the diatomic system; (c) N umeri-
cal errors Obtained by MIB II for the four-atom system; ((1) Numerical
errors obtained by MIB I for the four-atom system. .......... 124

Accuracy test on molecular surface of twenty four proteins. (a) Loo
errors Obtained by MIB II; (b) Loo errors Obtained by the accelerated
MIB I. ................................... 129

Comparison of solvation free energies of proteins, which are listed in the

order of gyration radii as shown in Table 5.4. (a) Solvation free energies

of AGMIBPB—Il(h = 0-5A)~ AGMIBPB—IU‘ = 054). AGAPBS(h ~
0.2A) and AGPBEq(h = 0.25A); (b) Differences of solvation free en-
ergies between coarse mesh and fine mesh, i.e., A(AGMIBPB—II(h =
0-5Al'AGMIBPB-IIU‘ = 0.25A)), A(AGPBEQ(h = 0.5A)-AGPBEQ(h 2‘
0.25:3», and A(AGAp}33(h ~ 0.4A)-AOApgs(h ~ 0.2A)); (C) Differ-

ence of solvation free energies between MIBPB II and other methods,

i-e-a A(AGMIBPB-—Il(h = 0-5Al'AGMIBPB—IIUI = 035A». A(AGMIBPB—I(h =
0-5A)'AGMIBPB—Il(h = 0254)), A(AGPBEQ(h = 0-25Al-AGMIBPB—mh =
0.25A)), and A(AGAsz(h ~ 0'2A)'AGMIBPB—II(h = 0.25230); ((1)

CPU time used by APBS at about 02.4 and MIBPB-II at 0.5/t for 18
proteins ................................... 131

Comparison Of surface electrostatic potentials Of cytochrome C551 at
h = 0.54- (a) AGMIBPB—11; (b) A(AG'MIBPB—II - AGMIBPB—Ili (C)

A(AGMIBPB—II — AGPBEQ). ...................... 134
The distribution of first two layers of fictitious grid points (N3; = Ny =
7, M = 2). The solid line ‘-—’ indicates the boundary of the plate. . . 156

The distribution of fictitious points: (a) p = 3, (b) p = 4; ‘V’ indicates
the next layer Of fictitious points to be extended, ‘0’ indicates the
fictitious points have already been found by previous steps ....... 165

xiv

Chapter 1

Introduction to Mathematical
Methods for Elliptic Interface

Problems

1.1 Elliptic equations with discontinuous coefficient
and singular source: status, challenges and
thesis outline

The numerical solution Of elliptic equations with discontinuous coefficients and sin-
gular sources has drawn much attention in recent years [15, 16, 23, 45, 53, 61, 80,
81, 84, 122, 138, 144]. Simple Cartesian grids are preferred in these studies since the
complicated procedure of generating unstructured grid could be bypassed, and well
developed fast algebraic solvers could be utilized. The importance of elliptic interface
problems has been well recognized in a variety of disciplines, such as fluid dynamics
[48, 57, 77, 115], electromagnetics [65, 66] and material science [73]. However, to
construct highly efficient methods for these problems is a difficult task due to the low
global regularity Of the solution. Traditional numerical methods that are constructed
with the assumption of smooth solutions cannot perform at designed accuracy, and

might even diverge.

A pioneer work in this field was due to Peskin, who proposed the immersed bound-
ary method (IBM) inn the 1970-19805 [122—124] to model complicated time-varying
geometric boundaries via a singular source on the interfaces. The success of the IBM
method is due to its flexibility, efficiency and robustness. Tornberg and Engquist[138]
extended Peskin’s method by using the high order approximation to the delta func-
tion. However, the IBM approaches typically smear the interface.

Many high order boundary schemes have been developed. Gibou and Fedkiw[61]
constructed a fourth-order boundary scheme to solve the Laplace and heat equations.
Linnick[103] and Fasel developed another fourth-order boundary scheme for flow past
cylinders. These high order schemes have a potential to be generalized for elliptic
problems.

The immersed interface method (IIM), proposed by LeVeque and Li [94] is a
remarkable second order sharp interface scheme. The IIM has been regarded as a
major advance in this field. It has been made robust and efficient over the past
decade [1, 42, 95, 96, 131]. The IIM formulations in polar coordinates[99] and using a
finite element method [97] were presented. A fast IIM was constructed [95] for interface
problems with piecewise constant coefficient.

The ghost fluid method (GFM)[50], proposed by Osher and his coworkers, is a
relatively simple and easy-to—use approach. It is developed to treat contact discon-
tinuities in the inviscid Euler equations. The GFM is typically first-order accurate
for interface problems, including the elliptic one[104] and could be of second-order
accuracy for elliptic irregular domain problems. It directly incorporates jump condi-
tions into the numerical discretization such that the symmetry of the FD coefficient
matrix is maintained, allowing the use Of standard fast solvers. In high dimension,
the jump in the normal derivative is correctly captured but the jump in the tan—
gential derivatives is neglected [104] so that. the GFM can be applied dimension by
dimension.

The finite element method, proposed by BabuSka[-l] and many other researchers[21,

97], is a nature approach to construct a solution for irregular interfaces. In particu-

lar, the discontinuous Galerkin technique is proposed by Guyomarch[64]. A relevant,
while quite distinct approach is the integral equation method for complex geom-
etry, proposed by Mayo, Greengard and coworkers [111, 113]. In this approach,
the ghost cell is introduced outside the domain as a fictitious domain. Aforemen—
tioned methods have found much success in scientific and engineering applications
[3, 15, 16,45, 53,61, 72, 73, 75, 76, 80,84, 90, 98, 99, 103, 105, 130, 131, 142, 144, 154].

One of the most challenging problems in the field is the solution of elliptic equa-
tions with sharp—edged interface, i.e., non—smooth interfaces. Numerical solutions to
this class of problems have widespread applications in science and engineering, such
as electromagnetic wave scattering and propagation [24, 58, 121], wave-guides analy-
sis [114], plasma-surface interaction [110], friction modeling [139], and turbulent-flow
[6]. To the best Of our knowledge, none of the aforementioned methods proposed for
elliptic interface problems have been directly applied to the treatment of sharp—edged
interfaces. Essentially, as the gradient near the tips Of sharp-edged interface is not
well defined, some earlier interface methods might not work. Most existing results on
this class of problems are obtained by using finite element methods [114, 121]. How-
ever, finite element methods might exhibit a reduced convergence rate when used for
the analysis of geometries containing sharp edges [72, 121]. Consequently, dramatic
local mesh refinement is required in the vicinity of sharp edges [25], and leads to
severe increase in computational time and memory requirement. In particular, local
mesh refinement does not work if the solution is highly oscillatory due to the so called
pollution effect [5], which is a common situation in dealing with electromagnetic wave
scattering and propagation. Hou and Liu proposed a finite element formulation [72]
for solving elliptic equations with sharp—edged interfaces. Remarkably, these authors
have achieved about 0.8th order convergence with non-body-fitting grids.

Another challenging problem is to realize the higher-order accuracy in high dimen-
sion, which is particularly desirable for problems involving both material interfaces
and high frequency waves where conventional local adaptive refinement approaches

do not work well. Typical examples are the interaction of turbulence and shock,

and high frequency wave propagation in inhomogeneous media[11]. Simple Cartesian
grids are preferred in these situations because of the bypass Of the mesh generation,
better temporal stability and the availability of fast algebraic solvers. Up to sixth
order MIB schemes for two dimensional elliptic problems with smooth interface has
been constructed[163, 170].

The objective of the present dissertation is five-fold. First, we extend the matched
interface and boundary (MIB) method previously devised for smooth interfaces to
interfaces with geometric singularities. The MIB method was proposed by Zhao and
Wei [163] as a systematic higher-order method for electromagnetic wave propagation
and scattering in dielectric media. Recently, it has been generalized for solving elliptic
equations with curved smooth interfaces by Zhou et al [170]. But the earlier MIB
scheme does not maintain the designed accuracy due to the presence of geometric
singularities or large curvature[168—170]. The MIB approach makes use of fictitious
domains so that the standard high order central finite difference (FD) method can
be applied across the interface without the loss of accuracy. But it is very difficult to
obtain the fictitious values around geometric singularities and interfaces with large
curvature because the number Of unknowns around these areas is usually greater than
the number of interface jump conditions. In this work, the aforementioned challenge
is overcome by making use of additional set of jump conditions and iteratively solve
fictitious values around geometric singularities. The MIB schemes with the 2nd order
accuracy for arbitrarily complex interfaces with arbitrary geometric singularities are
therefore constructed and demonstrated by numerous numerical tests in both 2D and
3D.

Second, we extend the earlier 2D higher—order MIB schemes [163, 170] to 3D. For
straight interfaces, MIB schemes of up to 16th order have been constructed [163, 169].
For lightly curved interfaces, up to 6th order schemes have been demonstrated [169].
In this work, we report the first known fourth- and sixth-order 3D results for elliptic
problems with smooth interfaces. We demonstrate that 3D 4th-Order MIB schemes

can be constructed for interfaces with moderate geometric singularities. Such schemes

are particularly efficient for problems involving both material interfaces and high
frequency waves.

Third, a systematic strategy is developed to optimize the structure of the MIB ma-
trix, which dramatically accelerates the convergence rate of the linear system. The
representations of fictitious values are improved by selecting different set of points
around irregular points. This improvement makes the MIB matrix optimally sym-
metric and banded, consequently reduces the computational time. It is shown that
the Optimized MIB method converges fast even for large system where the earlier
MIB method converges 100 times slower or even diverges.

Fourth, a specific MIB method is developed to solve boundary value problems with
fourth—order partial differential equations in plate vibration analysis. The fictitious
values are calculated by using combinations of simply supported (S), clamped (C) and
free(F) boundary conditions along each side of the rectangular plate. The detailed
comparisons between the MIB method and the GDQ method[133] are performed for
different plate configurations. It is found that the MIB method provides uniformly
accurate and stable numerical results in all plate configurations. This feature has
the greatest advantages in the case there is at least one free corner when the GDQ
method reduces its accuracy dramatically.

Finally, the MIB method is applied to calculate the electrostatic potential and free
energy of biomolecules. The interface problem in the Poisson-Boltzmann equation
(PBE) was first addressed by an MIB based (MIBPB-I) solver to explicitly consider
the flux continuity condition in the finite difference framework [168]. However, the
earlier MIBPB-ll solver does not maintain the 2nd order convergence for molecular
surfaces of proteins because of the existence of cusps, sharp edges, sharp wedges or
self—intersecting surfaces. Moreover, due to the asymmetric matrix of the interface
method, the MIBPB—I requires a large number of iterations in solving linear equations.
In fact, its matrix does not converge for large proteins or small proteins with dense
grids. The present approach, called MIBPB-II, overcomes these difficulties. Apart

from its ability to maintain the 2nd order convergence under the presence Of geometric

singularities, the MIBPB-II has an optimally symmetrical matrix, which dramatically

reduces the number Of iterations.

1.2 Introduction of 1-D MIB method

First proposed by Zhao and Wei [163] for the solution Of the Maxwell's equation with
straight material interface, the MIB method implicitly enforces both interface con—
ditions by utilizing fictitious values near the dielectric interface. A major ingredient
of the MIB method can be illustrated with the following 1-D example. Consider a
uniform mesh with grid spacing h and let the interface be located at :1: = a. where
1:,- g a g n+1 for some i, as in Figure 1.1. Suppose the diffusion coefficient is 3—
in the left subdomain and 3+ in the right subdomain. For a one-dimensional elliptic

equation (1.1) with interface conditions

(1311.132: = (1(37): (1-1)
[u] = 'u+(a) — u"(a), (1.2)
[3113;] = 3+u$(a) — 3_u$(a) (1.3)

a straightforward application of the finite difference scheme says at r,-

flzi+1 2“i+1 - 2(r’3r+1 2 +/3i—1 2M +(3r—1 2"1'i—1
/ / 122 / / = (1(Ii)a (1-4)

 

where (3141/2 and ffi—l/Q are the values Of diffusion coefficient at positions .11," +1 /2 and
r,_1/2, respectively. This discretization, based on Taylor expansions of smooth func-
tions, renders a second-order convergence only if both u(1‘) and 3(1) are continuous,
which is not the case for the interface problems.

The MIB method first smoothly extend the solution of two subdomains. Two
fictitious values f,- at grid point r, and fi+1 at grid point. .'l”,‘+1 are found by solving

two discretized interface conditions in Eq.1.5 and Eq.1.6.

 

1 1 1 1 ’_ x

3 1 1 1 1
i-3 i-2 i-l i i+l i+2 i+3 i+4

Figure 1.1: Illustration of MIB method for 1-D problem. The dash line indicates
the position of interface a: = a, where u(:c) has a discontinuity. Real solutions 21(3)
at grid points are marked in black and the fictitious values f (x) are marked in green.
Fictitious values fi+1 and fi+2 are the smooth extension of the solution in left subdo-
main; fz-__1 and f,- are the smooth extension of the solution in right subdomain. These
fictitious values are solved by matching two interface conditions simultaneously.

+ .+,. .+. ,— . .-. .—.

= ['11-] (1-5)
3+(u1fifg+'uf1+:i+lui+1+ wfi+2-u,-+2)—.3_('u!ii_111;_1+ min, + w1—.i+lfi+1)

= [HUI], (1.6)

where (wfiiifl + “’Ji+1"‘i+1 + U7(_;:i+2ui+2) is the interpolation Of 11+(a) by using
11,4 1, 11.,-+2 and fictitious value f,- with corresponding interpolation weights 1126} +1, 1113:, +2
and may (urn-If, + wit-+111,“ + wit-+211H2) is the approximation of rifle.) using the
same set of nodes via finite difference. Similar approximations to u" (a) and u; (a) are
conducted by using the nodes u,_1, u, and fictitious value f,+1 with respective inter—
polation weights or finite difference coefficients, which are calculated with Fornberg's
algorithm [55].

The fictitious values f,- and fi+1 are solved to be the linear combinations Of real

solution values u,_1, 11,-, 11;“, 11,212 and the prescribed jumps [11], [311$]:

fi ‘-‘—" Cllli_1+ C211,“ ‘I' C21L1+1 + C4lti+2 + Ci" O[U-] ‘I‘ CG[{3UJ-,]
f1+1 = Ci+1u,_1 + C§+1ui+C§+1u,-+1 + cyst-,2 + Cg+1[u.] + Cg+1[.3u,j,

where the vectors are

T-S—wl—JHKI + u'fTi+lK2

 

 

C"- =
D
_,+ + ,+
CH1 _ 3 w122K1-l—u/022-K2
I)

_ — — _ ,+ _. +

K1 — (wO,i—1’wO,i’ “0,241, “’0.i+2=110)
_ —. — —, — _.+ + __../+ +

K2 '— (5 “113—1’5 101,11 5 wo.1+1= 3 u'O,z'+2’0’1)

and D— — —3 u12 2+1111001 + 3+ u102+1wfr2 These two fictitious values are supplied to

the standard central difference schemes at 1',- and .1:,-+1 such that

2-1/2711—1 - 2C3. -1/2 + 3z‘+1/2)uz‘+ 131+1/2f1+1
h~

321+1/2fz‘ — 2f.-321+1/2 + ,3,+3/2)u, + 181+3/2ui+2
21.2 = (1(431—1), at Izfi+1

 

= (Ami): at $1.1

 

are actually conducted on smooth subdomains, and thus have a better convergence
property than (1.4).

More fictitious values can be solved to support higher order discretization of the
equation near the interface via iterative enforcing of the interface conditions as only
two fictitious values can be solved from two interface conditions. Following the above

procedure, we first solve f ,- and _f,-+1 from

i+4 2'
+ _ _
(112022], + Z Illakltk) — ( 2: 1110.12,“); + "'0.2’+1f1'+1) : [u] (1.7)
i+4 i
(u 1. +2-f1 + :"11'13— 3 ( Z U'I—Jt'uk + "'Lz'+1fi+1) = [3am] (1.8)
k=i+1 k=i—3

which apjnoximate the interface conditions at fifth-order accuracy. We then solve

f;_1 and fi+2 from

2'

( Z u’ikuk + “1'6,,f+1fi+1 ‘1’ u’(i.;+2fi+2)

kzi—3

3+(w:i_1fi—1 + “’1':z'fi+ Z wikuk)

z'

.3_( Z u’ikuk + u’i;’+1f~i+l + wii+2fi+2)

k=i—3

= [u],

= [3111;].

(1.9)

(1.10)

Using two fictitious values at either side of the interface is consistent with a fourth-

order central finite difference in the vicinity of the interface, for example,

 

 

 

 

3h2 _ 12h.2

 

 

3h? '_ 12h2
4fi—1 _ fi—2

 

 

4'Hz'+1 __ f2"

 

22m : _fi+l fi_ .

' 121.2 3112 4122
22” : _f2'+2 4fi+1_ 'uz'

' 1‘.th2 3h2
22” _ _ “i+2 47"z'+2 _ “i+1
12h.- 3112
22” = __'”z'+4 4'";+3 __ 111-g

12h2 3h?

to achieve fourth-order convergence.

 

at 1‘1—1,
172',
at 1:14—11

at n+2,

(1.11)
(1.12)
(1.13)

(1.14)

1.3 Introduction of 2-D MIB method for smooth

interface

The 2-D MIB method for smooth interface is first proposed by Zhou et al. [169, 170].

Conside an Open bounded domain Q C R2. Let F be the interface which divides 51

into disjoint. open subdomains, 9+ and 9‘, hence Q = (PLUS)- UP. Assume that the

boundary 89 and interface F are Lipschitz continuous and there is a piecewise smooth

level-set function 6‘) on D, which F 2 {c5 = 0}, Q— : {(3 < 0} and (2+ = {d} > 0}.

We seek solutions Of the 2D elliptic equation with variable diffusion coefficient 3(17. y)

away from the interface F given by

(,3(I~y)11(1‘~.y)1-)x + (13(1‘, y)U(I«, y)y)y = qtr» y). .2: E MP (MS)

Consider a case where the interface intersects the jth mesh line at a point which is
between (i. j) and (2' + 1, j). A direct calculation of um at (i j) using the second-
order central difference scheme involves grid points uz-_1‘j, um and ui+1,j and will
lead to the reduction in the convergence order. A solution to this problem is to
replace 111-+1.]- at the irregular point ('13 + 1, j) by a fictitious value fi+1,ja which is a
smooth extension of function values from the left hand side of the interface. A jump
condition is required to uniquely determine fi+1,j. For the same reason, (2' + 1, j) is
also an irregular point and fig-+1 is a fictitious value required for the central finite
difference scheme at (i + 1,‘j).

For elliptic equations (1.15), the solution might admit a prescribed jump at an
interface point. Moreover, the gradient along the normal direction can also be pre-

scribed. Therefore, two jump conditions can be given

[it] = 71+ — u— = as, (1.16)

[[31171] = [31-71; — [3—11; :1)”: (1.17)

where normal vector fi = (cos (Lsinfi) can be defined a.e. on F and points from 12—
to 0+, while 0 S 6 < 27r is the angle between positive x- direction and the vector ii.
We assume that both [u] and [3211",] are C1 continuous along the Lipschitz continuous
interface I‘.

When considering the interface which is not always aligned with the .T- or y— mesh
line, one more interface condition can be attained by differentiating Eq. (1.16) along

the tangential direction of the interface, 7'" = (— sin 9, cos 6). Hence for a point. ($0,310)

10

on the interface, we have three jump conditions,

[11] = u+ - u— = 99 (1.18)
[117] = (—u.}f sin6 + uy+ cos 6) — (-u; sin6 +1157 cos 6) = p (1.19)
[1311"] = 3+(u: 0056 + u; sin 6) — 1'3—(11; cos6 + u; sin 6) = 1!). (1.20)

+

To enforce these jump conditions, we must compute 11$ , 11;, 11+

y and u; . For a given

interface geometry, it can be very difficult to compute some of these four partial
derivatives. One way to avoid this difficulty is to use only two of three jump conditions

Eq. (1.18), Eq. (1.19) and Eq. (1.20). Therefore we can eliminate one of four partial

+

a u ‘ + I __
derivatives, 111., (11., “y

and 11.3; , and result in the following sets of jump conditions:

[11] = 11+ — u“, and [flan] — 13— tan 6[uT] = (7:11; — (71:11; + C3213“), (1.21)

where C; = {3+ c086 + J— tan6sin 9, Cz— = (3- €059 + [3- tan6sin6 and C;- :

3"” sin 6 — )3" sin 6;

[u] = 11+ — ’11—, and [311"] — 13+ tan 6(117] = Cf“: — (71711; — 031—1137, (1.22)

where C: = {3+ cos6 + 3"" tan6sin 6, C; = ,3— cos6 + [3+ tan6sin6 and Cy— :

{3‘ sin6 — [3+ sin6;

[u] = 11+ — '11—, and [dun] + [3— cot ()[uT] = (73-11;- + C1711; — Cy— 11;, (1.23)

where C: = (3+ —fl—) cos 6, C; 2 1’3- cos 6 cot 6+3+ sin 6 and Cy— = ,‘3— (cos 6 cot. 6+

sin 6), and

[u] 2 11+ — u— and [311”] + 13+ cot6[uT] = (31721; + C3113" — Cy—ug, (1.24)

where C; = (3+ —[3—) cos 6, Cy— : 3+ cos 6 cot 6+1’3_ sin 6 and ('3' = t3+(cos 6 cot. 6+

sin 6). For given local environment, only one of these four combinations is needed to

11

determine two fictitious values. We choose an appropriate combination such that its
involved partial derivatives can be conveniently computed.

To restore the order of convergence of the discretization at irregular points, we need
to treat (3111),; and (fizzy)y to the designed order of convergence near the interface.
Since these two terms are considered separately, we only need to illustrate how to
locally recover the second order accuracy of the standard 3-point central FD scheme
for (13113;) I. The treatment of (31131)!) can be carried out similarly. We classify irregular
points as off—interface ones and on—interface ones. Different schemes are required to

deal with these cases.

1.3.1 Off-interface scheme

Off-interface scheme considers the situation that a pair of adjacent irregular points
are separated by the interface and neither of them stays on the interface. As shown in
Fig. 1.2, the intersection of the interface I‘ and the :r mesh line is (3:0, yo). We discuss
the interface schemes at irregular point (13, j) E S2+ and irregular point (i+1, j) E (2".

Define auxiliary point as the point that is next to the irregular point and stays on
the mesh line that connects two irregular points and crosses the interface. Therefore,
there are two auxiliary points for each pair of off-interface irregular points. When
curvature is relatively small or there is no sharp edge, both auxiliary points belong
to a ‘proper’ region. A proper region refers to a neighborhood where inside each
subdomain, (1+ or Q”, at. least one auxiliary point can be found next to each irregular
point. This situation is shown in Fig. 1.2. To discretize (3113);. by standard second
order FD central scheme at irregular point (i, j), primary fictitious value fi+1,j is
needed. Similarly, primary fictitious value fij is needed for the discretization at
irregular point (27 + 1, j). Since both auxiliary points belong to the ‘proper’ region,
standard MIB scheme can be used to solve for these two primary fictitious values
without use of a secondary fictitious value. We consider jump conditions at the
intersection between the interface and the jth mesh line. In this particular case, it

+

is convenient to use jump conditions ( 1.22) since they involve u. , u“, 11+

I , 11.x and

12

+

11.37 at the intersection point ($0.110). Here u and u— are obtained by interpolations

from information in (2+ and {2‘ res ectivelv. 117*.” is com uted from u-_ .-, u- .- and

a P s ,1 P z 1,} 2,]
fi+1.j- Similarly, u; is computed from f“, u,+1'j and 11,412,} These expressions are
explicitly given as follows:

T
+ = (1001—1916032 1100.141) ' (Uzi—1.j,'tt1,jafi+1,j) (135)

U
. - _ . . . . . . . . . T
u — (u’0,21w0,2+12w0,2+2) ' (fig: uz+l,J- uz+2.y)
T
U: = (u’l.i—1-u’1.ie’w1.i+1)'('u'i—1.je"i,jsfi+1.j)

~ T
“x = ('U-’1,i,'w1.i+11w1.z‘+2l ' (fi,jauz'+1,jaui+2.j) :

where 10an denote finite difference (FD) weights. The first subscript n represents
either interpolation n = 0 or first order derivative 72. = 1 at interface point (;ro,y0),

while their second subscript is for the node index.

 

 

 

 

   
 

i+2 \ 5
i+1 I
j G ' t

 

 

 

 

 

 

 

 

i-t i i+1 i+2

u )3

Figure 1.2: Situation handled by Off-interface Scheme for smooth interface. 0
indicates a pair of irregular points (1', j ), (i + 1, j ) and two auxiliary points (i — 1, j),
(i + 2, j) with respect to the present irregular points.

To approximate 11,7 , we need three u values in the y-direction. These values are
located at intersection points between the dash line and j th. (j + 1)th and (j + 2)th
mesh line, see Fig. 1.2. Unfortunately, these values are unavailable and have to be

approximated from interpolation schemes along the I—direction. This means six more

auxiliary points are involved. In practice, one of these two can be easily found. In

13

the situation shown in Fig. 1.2, u.“ can be represented as follows:

3}
“J = lwl.jew1,j+lawl,j+2l' (1-25)
11:0,, wO,-i+1 'wo.z'+2 0 0 0 0 0 0 l
X 0 0 0 “’62: 106.241 “fl-i+2 0 0 0
0 0 0 0 0 0 111614 “"62” rival-+1

 

 

X

[fije Ui+1.j, ui+2,je 'Ui,j+1: uzt+1.j+1 . Ui+2,j+1a 'U-i-1,j+2: ui,j+2: ui+1,j+2]

Substituting Eqs. (1.25) and (1.26) into jump condition Eq. (1.22), we have

T
[U] = (u’O,i——law0,i:w0,i+l)'(“i—l,jaui,jefi+1,j) (1-27)

T
— (11:0,),2110J1H, u’0,-i+2) ° (fa). U-'i+1,ja ui+2,j)

 

 

 

 

 

 

 

 

 

 

.— .. T (- _
“’1,i—1 Tie—1,)
[81171] — 5+ tall 6[UT] = C: QUI‘Z' uz'j (1.28)
101.241 fi+1.j
' l T c " _ - T
wrz‘ fi,j wrj
“ C; 'U’1.z'+1 ‘ “i+1,j — Cy— "’1,j+1 (1'29)
wl,i+2 j Ui+2.j ( w1.j+2 1
WM wO,i+1 wont+2 0 0 0 0 0 0
x 0 0 0 “'61 “'6 i+1 "”().i+2 0 0 0
0 0 0 0 0 0 “6.1—1 “’61 “"5 1+1

 

 

X

[fi,ja"’«z'+1.j~"’-i+‘2.j1“2T,j+1-. "i+1.j+1‘-ui+2.j+lv7"i-l,_j+221"i.j+2171i+1.j+2]

’7

Solving for fiJ and fi+1,j from Eqs. (1.21) and (1.28), then the representation of

these fictitious values in terms of function values and the physical jumps can be

14

written as

fw- = Ci - U (1.30)

f,+1,,- = Ci+1 - U (1.31)

where Cf = ( ’i, 05’, - ~ ,Cj3) and Ci+1 = (Ci+1,C§+1,- -- ,Cigl) are the expansion
coefficients of two fictitious values with respect to 10 function values and 3 jumps
which are also given by the vector U = ( Uzi—1J1 uifij, ui+1,j, 22,+2’j, 21,3341, 21,414“,
ui+2,j+1- Ui—1,j+2s “1,342, ui+1,j+21 [Illa [131121]: lurllT [170]-

With these two expansions of f2,j and fi+1,j: one could discretize (,1321.;,;)J,~ at irreg-

ular points (2, j) and (2' + 1, j) as if at a regular point:

1

——2(é3.+1 .1-1’3.+ 1 .—..3.+1 .1131] J.) ' (It-i—1,j,uz',j,fi+1,j)T (1-32)
21

321 ,=
( 1:)! A]: 2—2,] #2,] 2+2.] 2..

at (M)
1

311, =—. ’3‘ ,——fl— —,-’3_ , _ - --,u- -,u- -T
( I)£L‘ Ax2(/i+%,j 2413.3. .2430. fi2+%,j) (fw 2+1,] 2+2,])

at (i+1,j).

To find the fictitious values at a pair of irregular points (i, j ) and (i+1, j), auxiliary
points (2 — 1, j) and (2', j ) must be on the same of side of the interface, and auxiliary
points (i. + 2, ,j) and (2+1,j) must also be on the same side of the interface. However,
when the interface has a large curvature or a sharp corner, or one of the auxiliary
points is out of computational domain, these conditions can no longer be satisfied.

The new MIB schemes for geometric singularities are desired.

1.3.2 On—interface scheme

In this situation, there are a total of three irregular points, one on interface, one
in the :r-direetion and the other in the y—direction. Oil-interface scheme requires
two auxiliary points and two off-interface irregular points all belong to the same

subdomain. Fig. 1.3 shows a typical situation that can be resolved by Oil-interface

15

 

 

j+1 \ e

 

 

 

] O fl 0
1‘1 * \\
j-2 e

 

 

 

 

 

 

 

i-2 i-1 i i+1

Figure 1.3: Situation handled by On-interface scheme 1.

scheme 1. Note that when [21] 7A 0 the function value on the interface is not well
defined. From the computational point of view the interface itself can be regarded
either as part of the interior or as part of the exterior. Here we regard 21(2, 3') = u+(2', j)
if a grid point (2,1) is on the interface.

Two primary fictitious values are required in order to formulate the difference
scheme for (6211.)! + (321g)y at point (i, j). As indicated in Fig. (1.3), if points
(2—2,j), (2—1.j), (2',j— 1) and (2',j—2) belong to subdomain Q’, the approximations
for all derivatives, 21:, 21;, 21+ and “3; are available. Therefore Eqs. (1.19) and (1.20)

y

exactly provide two approximate equations for fig j_1 and f,_1‘j

[217] = (— sin 6, cos 6, sin 6, — cos 6) - (21;, 11;”, 21;, 213;)T (1.33)

[321”] 2 (3+ cos 6, 3+ sin6, —3_ cos 6, —,{3— sin 6) - (2117,2131 21;,21;)T, (1.34)
with
+_ ,. .,.,—. ,. .T 135
“1: — (u‘1.2—1~u~1,2awl,2+1) (f2—1.]eu-2,Jauz+1,]) ( - )
— T
“I = (wit-2a wit—1a 101,2.) ' (111-2,)..ur—1JJHJ — [“6

16

U; = ('1.Ul.j—1~.u’l.j=w1,j+1)'(fz’.j—1~.“'2T,jyuz’.j—1)T (1.36)

I I I T
11.1 = (1U11j_2,'u,’1,j_1. (1213].).(2,1i’j_23-u.i‘j_1,uitj — [11]) .

Substituting Eqs. (1.35)-(1.36) into jump conditions (1.33) and (1.34), and solving
for fig-4 and fi_1,j. the expansions of fig—1 and f1;_1,j in terms of real unknowns
and jump conditions at point (i, j) can be obtained [170]. Then the difference scheme
of the Poisson equation at point (2', j) can be written in 9+:

1
.311, + 311 = —— + ,—3+ — 3+ ,5+ 1.37
(. I)l‘ ( y)y A12 (Ji—%.j . Pi]. 314%.]. i+%,j) ( )

T
'(fi—1.jeu2,jauz'+1,j)

 

(3+ 11_I/3+ — 5+ 3+

+ . . . . a, . .
Ag? l-J-Q sz—Q 103% 2:136

'(fij-lauijv ui,j+1)

In some situation, auxiliary points (2' — 2,j) and (2,j — 2) do not belong to 51—.
Instead, we have points (2+2, j) and (2', j +2) in (2+. The other two primary fictitious
values fi+17j and fiJ+1 need to be considered instead. The approximations for all

derivatives, 11$, 26,21; and 21,; are given as

T
“2+?- = (“’11, 'w1.i+1: 'Ut’1,z'+2) ' (111.), U241). “i+2,j) (1-38)
_ I I I T
”I = (‘w1,I—12w1,i=’wi,z‘+1)'(ui—ljv'utj— ll‘lrfiHJ)

T
U; = (“"1.je'w1,j+1auv’1,j+2)'(“z’.jaui,j+1aui.j+2) (1-39)
— I I ,I T
“y = (wIJ—lr 101.)» 111.141) ° (l‘i..I'-1~'“'-i-j — fut fzzj+1l

Substituting Eqs. (1.38) and (1.39) into Eqs. (1.33) and (1.34), and solving for fi+1,j
and f21j+11 the expansions of the primary fictitious unknowns can be obtained and

the difference scheme of the Poisson equation at point (2', j) can therefore be written

17

in Q‘:

1

(,3111‘)1' + (3Uy)y : A—r2

(6‘

1%)”?— —3— .6‘1) (1.40)

2—%j 'i+%j'i+2)
'(“i—l,j1ui.j - lul» £41,le

+—— 3‘ .—3- -0- ,3-
Ay‘2(' 2.1—é ”LI—5 4.3-+1

T
(Haj—1,ULj-hdifhj+1)

1)

I 7213+?

One of two conditions must be satisfied in this scheme: points (i — 2, j) and
(2',j— 2) E Q‘, or points (2' +2,j) and (2',j+2) 6 0"". In a situation that meets none
of these two conditions, On-interface scheme for geometric singularities are needed.

It has been proved [12] that the solution has uniform 0(122) accuracy if the trun-
cation error is O(h) at the irregular points and 0(h2) at the regular points. The
theorem is provided below for the case 3+ = ,1‘3_. When (3+, [3- are unequal positive
constants, the elliptic problem in free space can be treated by first solving an integral
equation on S for [(9,121] and then proceeding as before.

Without loss of generality, consider the interface problem with 3+ = 1’3- = 1. The

elliptic equation can be written as

Alli = fi in Sli (1.41)
[21] = 1,9, [0,121] = w on F

’(L = 210 on US?
To discuss discretization, we write the region Q as
Q={1‘ERdl 0<Ik<Ak, 151:3(1) (1.42)

For simplicity we assume ratios of the lengths Ak are rational. so that the domain

can be partitioned by grid cubes of size It for arbitrarily small h. We assume h is

18

chosen so that Ak = Nkh with integer N), for each 1:. The computational domain is
9),: {jhehzdz 13),,3 Nk—l, 151-91} (1.43)
with boundary
(IQ), = {_jh: 0 3,1,, g Nk, 1 g k g (I; jk = 0 or Nk for some k} (1.44)
and the partial boundary
3012,, = {jhz 0 g j), g N,C — 1, 15 k g d; j), = 0 for some 1:}. (1.45)
At each regular point, the truncation error is given as follows:
AthCUh) = fi(jh) + Th(jh), lTh(jh)l S Ch.2 (1.46)
The truncation error at the irregular points is given as follows:
A,,u(’()'h) = fi(jh.) + Th(jh) + Thu/l), lrh(jh)l 3 Ch (1.47)

where Th(jh) is determined by the jumps.

Theorem 1.3.1. [12] Let a" be the exact solution of the problem (1.41) with F at
least Cl. Suppose Aha“ has the form given by (1.46),(1.47), with. lrh(jh)| 3 Ch at
irregular grid points and |T;,_(jh)| _<_ Ch2 at regular grid points. Let uh be the solution

0f

Ahuh = fh 2n Qh, uh = 210 on 8(2), (1.48)

19

where

f:(jh) + Th(jh), jh irregular

f*(jh)== (149)
[i.(jh). jh regular
Then
(21h(jh) — 06m)! 5 00112, jh. e 9), (1.50)
andfor 1 S l S d,
(Dy-Mn.) — Dfue(jh)l s Clh210g(1/h), (1.51)

jh e e), u 309,,

with C0. C1 dependent on ue but independent of h.

20

Chapter 2

Two dimensional matched interface
and boundary (MIB) method for

treating geometric singularities

Although there have been many methods successfully applied to elliptic equations
with smooth interfaces , little work has been done on the mathematical treatment
of geometric singularities, such as sharp—edged and/or sharp-wedged interfaces, and
sharp tips. This class of problems are ubiquitous in science and engineering, such as
wave-guides analysis [114], electromagnetic wave scattering and propagation [24, 58,
121], friction modeling [139], plasma-surface interaction [110] and turbulent-flow [6].
The presence of geometric singularities, such as cusps and self-intersecting surfaces, is
a major obstacle to the accuracy, convergence and stability of the numerical solution
of the Poisson-Boltzmann equation for the electrostatic analysis of biomolecules [158].
Technically, it is much more challenging to construct efficient numerical methods for
this class of problems due to the lack of regularity in the interface. Because the gra-
dient near the tip of a sharp—edged interface is not well defined, the aforementioned
interface methods might not maintain the designed accuracy. A standard approach
to these problems is the use of local adaptive approaches [114, 121], which exhibit.
reduced convergence rates and require dramatic mesh refinements in the vicinity of

geometric singularities [25]. These approaches fail to work if the solution is highly os-

21

cillatory due to the so called pollution effect [5]. A remarkable result, due to Hon and
Liu [72], was of 0.8th order convergence obtained with a two-dimensional (2D) finite.
element formulation for elliptic interface problems with geometric singularities. Their
method is convenient for geometric singularity induced weak solutions. Recently, we
have developed a 2D second-order MIB treatment of sharp-edged interfaces [161].
Unlike the Galerkin formulation, our MIB scheme does not specifically handle the ge-
ometric singularity induced weak solutions. A standard approach that multiplies the
solution with an appropriate polynomial [121] was incorporated in our MIB method

to deal with geometry induced weak solutions.

2.1 Theory and algorithm

At each irregular point, we need to construct a primary fictitious value. A primary
fictitious value is a value that will be used in the standard finite difference discretiza-
tion. While a secondary fictitious value is one that helps resolving a primary fictitious
value. However, the secondary fictitious value can also be a primary fictitious value
for the discretization of other irregular points. The concept of secondary fictitious
values is essential for overcoming the geometric difficulties created by sharp-edged

interfaces.

2.1.1 Off-interface schemes for geometric singularities

In sharp-edged interfaces, it. is very common that. one cannot locate one auxiliary
point for each irregular point inside a subdomain. For example, as depicted in Fig.
2.1. irregular point (i, j) has no auxiliary point associated with it inside the same
subdomain. This ‘improper’ region cannot be eliminated by refining the mesh. The
same situation will occur if the mesh is doubled in both the :r-direction and the y
direction. This difficulty is not resolvable by the previous MIB scheme because the
auxiliary points in the ‘improper’ region leads to inconsistence between the number

of unknowns and the number of jump conditions. In this case, we introduce the

22

 

 

j+1>—— *7<>—-——-——&~———¥.—r—w__ f——

 

 

 

 

 

 

 

j+1 .
1
i g i i
j 1 \L—‘j ‘ 1
1-1 ] j
1—1 L
l

 

 

 

 

 

 

 

 

 

 

 

 

 

l
i-1 i i+1 i+2
(a) (b)
Figure 2.1: Situation (a) handled by Off-interface scheme 2(a); Situation (b) handled
by Off-interface scheme 2(b).
secondary fictitious values, fi_1,j and fi+2,j, to replace the role of the auxiliary

points in the ‘improper’ region. Therefore the number of unknowns can be reduced,

and the resulting unknowns can be solved by Off-interface scheme 2(a) or 2(b).

Off-interface scheme (a)

Le‘ F = [Ftp Fi+1,j~ Fi—1,j-. F2+2,lea Where F14 = In and Fi+1.j = fi+1,j
are. primary fictitious values at irregular points (213') and (2 + 1,j). If (i — 1,3)
and (2, j) belong to the same subdomain, Fi—lj represents function value 11,-19j.
Otherwise, F,_1_J- is the secondary fictitious value f,_1_j at. point (i — 1, j). Similarly,
FHQJ- = 21,423]- if (i + 1,3) and (i + 2,3) belong to the same subdomain. Otherwise,
Fi+29j is the secondary fictitious value fi+2J at point (I + 2,j). The expansion (1.30)

could be rewritten as

B-Fzé-E' (2.1)

23

where

 

1 0 —C] —C,'[
B = . 1 . 1 (2.2)
2+ 2+
2 2' 2'. 2'
5‘ = 02 C3 CS 013 (2 3)
2+1 2+1 i+1 2+1 ’
_C2 C3 Cs 013
U = [II-2;), “ma:'uz‘.j+1euir+1.j+1, ui+2.j+1:ui—1.j+2 (2.4)

, T
~“i..j+2a “i+1.j+2~. [u]. [Jun], [Uri]

If two of F, F m and F l~ are function values or secondary fictitious values that have
already been expressed in terms of function values, while the other two components
of F, F,- and F j, are the fictitious values whose expansions are to be determined, Eq.

(2.1) can be rewritten as

31.2? Bl,j F2 _~ ~,_ 31.1 31.111 _ F1 (25)
I321 I32’j Fj I323] 82‘,” Fm

with 2',j,l,'m E {1, 2,3,4}. Therefore,

_1 I. F - _]
Ff ___ BM 81,3 . 5, . L7 _ I 81.1 81.222 . Fl I (2 6)
F3 ”23 832d i 82,! 82m Fm

gives the expression of F,- and F j. W'ith obtained primary fictitious values, (3211);;
can be discretized at point (i,_j) and (i + 1,_j) as Eq. (1.32).

Notice that the expansion of fictitious value f’i-j is not. unique. For example, as
shown in Fig. 2.1(a), points (i.j + 1), (i + 1,j) and (I — 1,3) are all in Q‘. and
f'iJ should have three expansions by considering three different pairs of points, (i, j)
and (Lj + 1), (2',j) and (i + 1,j), and (22.)) and (2' -— 1,j). Each expansion can
be generated by the jump condition at the intersection point of the interface and
the grid line segment between a pair of points. These three expansions should be

stored separately if they are all available. In the present. work, different. expansions

24

of a fictitious value are labeled by directions, {:r+, T—, 11+, 21—}. Here l‘+ means the
expansion is generated by the jump condition involving points (I , j) and (2' + 1, j), :r—
involving (23,3) and (2' — 1,j), y+ involving (2,3) and (i,j+1), and y— involving (2,3)
and (i, j — 1). When the primary fictitious value is needed for the standard five point
FD scheme, the expansion generated by the nearest pair of points should be used.
For example, the five point FD scheme at point (2', j + 1) should select the expansion
of fig- from y+ direction, while the five point FD scheme of point (2 + 1, j) should
choose the expansion of fid' from I+ direction. If this cannot be done, the expansion
from y+ direction can be carried out instead. A simple rule is to always select the

expansion generated by the nearest pair of points to achieve the best accuracy.

Off-interface scheme (b)

If only one of F component, F,- (i E {1,2,3,4}), is the fictitious value whose
expansion is to be determined and the other three are function values or have expan-
sions in term of function values, then the jump condition of 22(1‘, y) is enough for the

determination of 17,-:
a+ — 22— = [u]. (2.7)

' + —- - + 1+ 1+ - I_ — _ — 1.— ' . 2
Let 1120 —— (200,4, 210,2... u'U,z‘+1) and 21,0 — (woww0.2‘+1‘u’0,i+2)' Then Eq. (2.7) can

be written as

T _ . T
— 2210 ° (Fi,j2ui+1,j2 Fi+2,j) = [u]. (2.8)

.+.
100 '(Fi—ljvuiijHLj)
Rearranging the above equation, we have

B’ - F = C’ - U’, (2.9)

I _ _, ,— + + _ — I _ _ + — I _
where B — ( ILIOJ,2L’01i+1,’lU0J-_1, wO,i+2)’ C — ( 200i, 2110],“,1) and U —

('uz',j=1121+1,j» [21])T.

 

 

 

 

 

 

 

 

l___ m . _.

 

i-1 i i+1 i+2
Figure 2.2: Situation handled by On-interface scheme (a) for geometric singularities

Solving for F, from the above equation, we have

4
I-1 I I I
17,: B,— C -U — 2: 8,5) . (2.10)
1:12942' I
Eq. (2.10) gives the expansion of F, in terms of function values since Fl are function
values or have expansions in terms of function values. W ith this expansion, (321.17%;

can be discretized at point (2,1) as in Eq. (1.32).

2.1.2 On—interface scheme for geometric singularities

As mentioned in Chapter 1, the MIB On-interface scheme for smooth interfaces re—
quires that two auxiliary points and two irregular points all belong to the same
subdomain. However. for concave or multiply connected geometry, this requirement.

is hardly satisfied.

On—interface scheme (a)

On-interface scheme (a) can handle a situation that has two irregular points while
without auxiliary points on the same subdomain. The present scheme utilizes formu-

las in the :r-direction and in the y-direction separately to resolve secondary fictitious

26

values first. Then the primary fictitious values are solved. The situation is illustrated
in Fig. 2.2, where auxiliary point (2',j —2) E Q7 while (i+2,j) 6 9+. The secondary
fictitious values fi+13j and fi,j_1 can be obtained. However, since the fictitious values
fig-+1 and f,_1,j are not available, the Poisson equation at point (i, 3') can neither
be solved in 9+ by Eq. (1.37) nor in Q" by Eq. ( 1.40). Therefore, the secondary
fictitious values f2‘+1,j and fig--1 are used to solve primary fictitious values fig-+1
and f,_1‘J-.

First, in order to obtain the secondary fictitious values fi+1.j and fig-.4, Eqs.

+

(1.38) are used to obtain a; and 21; . Eq. (1.36) is used to obtain 21,,

and 21,7 .
Substituting this equation into Eqs. (1.33) and (1.34) and solving for [iii-10' and
f,“ j_1, the representation of secondary fictitious values in terms of real values can be
written as

.- .- ci+1
3+1" = . U, (2.11)

fig—1 [Cf—1

where (7H1, (971 are the expansion coefficient vectors, and

r r T
L’ = (ui,ja“i—1.j:712',j—1~.'u2+1.j:ui,j+12in]: ["2]: [1311-01) - (2-12)

Second, the primary fictitious value fig-+1 and fi_1.j are to be obtained. To
consider a more general situation, let F = (fi_1‘j,fi.j_1.fi+1‘j,fi2j+1)T. If two
secondary fictitious value, F1 and Fm in vector F , can be obtained from Eq. (2.11).

In this particular case, F1 2 fi+1sj and Fm = fig—1-

Fl (1'1
= . (I, (2.13)
Fm. Cm

then the other two values, Fp and Fq, can be solved by the following procedure. In

this particular case, F1) = fig“ and F, = f2—1,j- First, the derivatives 21;, 21;, 11,7

27

and 11.7 can be represented as follows:

Substituting Eq.

obtained:

where

T
= (u’1,2—1aw1,1ew1,2+1) ' (fr—1,): Haj. uzfi+1,j) (2.14)
= (wl,i—1»w1,i~.w1,i+1)‘(“i-ljauij'7 l“l:f1+1,j)T

T
= (u’l,j-1:w1,j~u’l.j+1)’(fi.j—1~Ui.j~,ui,j+1)

= (“’1.j—1ew1,jau’1,j+1)'(UI'J—‘lauij‘ iulsfij-i-llT-

(2.14) into Eqs. ( 1.33) and ( 1.34), the following equations can be

 

A-F=B-U, (2.15)
— sin 6 cos 6 sin 6 — cos 6
= (2.16)
3+ cos 6 (3+ sin 6 —,13_ cos 6 —/3‘ sin 6
1012—1 0 0 0
0 20131-4 0 0
X 1
0 0 lULi+1 0
0 0 0 aim-+1 ]

 

28

and

00000010

= (2.17)
0 0 0 0 0 0 0 1

— sin 6 cos 6 sin 6 —— cos 6

+
3+ cos 6 3+ sin 6 +37 cos 6 —1’3— sin 6
—wl,,: 0 0 "'Ill1,.i+1 0 0 0 0
—wl.j 0 0 0 -—'wl,j+1 0 0 0

x .
—201,i -w112‘_1 0 0 0 2121‘, 0 0
—’w1.j 0 —w1,j_1 0 0 ww- 0 0

 

 

Combining Eqs. (2.13) and (2.15), the expansions of Fp and E, can be easily obtained:

—1

. l
F}; = ALI) ALq ' B _ A” ALm . C . U (2 18)
Fq .41], .4241 .42.) A2,,” Cm

Now all four primary fictitious values around (2', j) are availal')le. Either Eq. (1.37)

or Eq. (1.40) can be used as the difference scheme at point (2., j).

On—interface scheme (b)

On-interface scheme (a) requires two off-interface irregular points on the same subdo-
main. When three out of four surrounding points around on-interface irregular points
belong to the same subdomain and one surrounding point is left alone in the other
subdomain. On-interface scheme (1)) is needed. The situation is illustrated in Fig. 2.3
(a). Points (i—1,_j), (1,j — 1), (27,.) + 1) 6 0+, Point (1+ Lg) 6 0‘. To discretize the

Poisson equation at point (2', j) in 9““, the primary fictitious value fi+1~j is required.

29

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 2.3: Four different situations handled by On-interface scheme (b) for geo-
metric singularities.

30

At point (2', j), a3”, '11; and 21$ can be represented as follows:

T
”if = (u’1.j—1,W1,jau’1,j+1)' (uri,j—1a“i,jwui.j+1) . (2.19)
— I I T
111. = (101,27.- 'w1,i+1’w,1,i+2) - (“2,7 — [21].21.,-+1'j,21,-+22J-) ,
T
u; = (u’i.i—1vu’i,1e 101-i+1) ' (“i—1.j~ u-i.jaf27+1,j)

Substituting Eq. (2.19) into Eq. (1.21), then solving for f,+1,j, the expansion of

fi+1,j is obtained as

 

 

 

 

,- ' . _., ' . + .
f'+1 ' — 1 Q all” “1* 1m + C“ ”’14 Cy 101.] (2 20)
2 {I _ , + ,+ 2— '1,2'.—12_-'1,i «.+ _ w+ I '
“*’1,21+1 CI ("1‘ C4"
v— ,I I - I _
(I “*1,2'+1 _CJU’IJ—i _CJIULJ'H _CI “"11 _I3 tel-n6 1
.+ 1 .+ 2 w+ ’ + ’ ~+ ’ 1+
c, c, c, CI c, c,

'(7‘i+2.ja 111—1.): "1,3: "’2'+1.ja Haj—1, '“2',j+1~ [ill [21,], [I321,,])T.

The difference scheme of the Poisson equation at point (2', j) can be given in {2+ as
Eq. (1.37).

For the situation in Fig. 2.3(b), points (2' — 1,3), (2,j — 1), (2',j + 1) 6 9‘, point
(i + 1, j ) 6 (1+. The Poisson equation at point (i, j) should be solved in Q“ instead.

Therefore, the expansion of f,_1,j in terms of function values is to be determined.

Here 2137,2571" and 21; can be represented as:
—- T
Uy = (u’1,j—1vw1,j~u'1,j+1)'(7‘2.j—1211'2.j - l”l“~i,j+1) ~ (2.21)
I I I T
“if = (WU—g}.71’1,2'_1:u’1.2’)'(“i—2g»ui—l,j~.ui.j)
_ , T
“I, = (wit—r u’i,1-u’i.1+1) ' (fr—1.]: “ij ‘ [urlyu1:+1,j)

Substituting Eq.(2.19) into Eq. (1.22). then solving for f,_1.J-. the expansion of [111,}-

31

is obtained as

 

 

 

+ I '+ I + I, _ 2 .
ft—lJ — '1. _ ' _ f—ulj—i— _ _ _ 2 ( ° )

C,7 'U’1,j 13+ tan 6 1

 

 

_(717 “’1-2-1 (717 “II-1+1 ...
—u'1,i+12 ”fa—T.” ’2' C'— 1 C'_ . (,_
Ix '11. I x ./I
,, T
'('UI—2,j.U-I—1,j.u1,jeU1+1,j.uzf,j—1.U-z'.j+1=l'U-llUrlvlflunll 0

The difference scheme of the Poisson equation at point (2', j) can be given in Q7 as

Eq. (1.40).

Similarly, for the situation in Fig. 2.3(c), 21,7, 21;; and 21,7 can be represented as

_ T
“y = (""1,j»“’1.j+1a“’1,j+2) ' ("1.) — ["111'U-1,j+1. u1,j+2) 1 (233)
T
21;- = (2I1’1J._1,21,2’1,i,211’,,2.+1)-(21.,_1,j,21,-,j,u,'+1,j) ,
“’37 : (Mild—11“liji'u’ijfi-I).("lJ—liuijafig-+1)

The expansion of fig-+1 can therefore be obtained by using Eqs. (1.23) and (2.23)

+ ,I _
C, ”1.1 CI) w1. j

 

 

- 1 CJTU1J+2 ... ,,
.l1,j+1 = “7* . C'+ , —“J1.j—1?—“’1,j’ T + 7?, (2.24)
._ 1+. I +, ,I _ _
Cy u’l-J'H CI ”’1.i—1 Car u"1.2'.+1 Cy U’lg‘ ,3 eot6 1
.+ ’ _ -.+ ’ _ -.+ ’ _ .+ ‘ + ’ .+
Cy Czy _/y C/y Cy C’y

'("2'.j+2s “21.j—1a 11.1,). 711,j+1-. “HI—1.)» “-2+1.j> [U]: [“2]. [67121])

The difference scheme of the Poisson equation at point (i, j) can be given in 12+ as

Eq. (1.37).

For the situation in Fig. 2.3(d), 21,7L , a; and 21g— can be represented as

T
u;- = (111131-42,aim-4,1111.”.(21,,j_2,u.,-,]~_1,u,-,j) , (225)
- T
“.1: = ('u-’i.1—1-“’i,1~u’i,1+1)‘(UI—ijdtzzj-['U].U1+1,j) .
- T
“y = (U’i.j—1-u’igauigfll thy—1.111,) — l‘ul- 'Ui.j+1)

32

The expansion of fi_j_1 can therefore be obtained by using Eqs. (1.24) and (2.25)

 

 

 

+ "‘ ’ +.
1 CU U-’1j—2 * (’1‘ ml 'i C w] -
. . ... . ' __ , ,. y J __ all
f1.]—1 — u7* . _ , lL’lqj + '_ 'i‘ (Y— , MIL-i+1, (2.26)
— I — I V— _/
Cg; wl.i—1 CI ”n+1 CJ'u’1,j_1 w,“ (’1' u’Li 3+ Got. 0 1

1— , _ l _ ’ 1. i — — ,_ y— 3— _
y .y c, J C, c, (7,,

‘(ui.j—2s “Lj: “i,j+1~ 71-1—14, ui+1.ja uz‘,j—1- lulu [UT], [31172])?

The difference scheme of the Poisson equation at point (i, j) can be given in Q— as
Eq. (1.40).

In practice, if not all of three points are on the same side of the interface, secondary
fictitious values can be used instead. For example, in Fig. 2.3(a), if uigj+1 6 (2',
while the secondary fictitious value fi,j+1 is already resolved, Eq. (2.20) can still be
used for the primary fictitious value fei+1,j by replacing rim-+1 with the secondary

fictitious value fig-+1.

2.1.3 A pseudo-code of MIB

1. If ((i, j) is a regular point), call standard five-point FD scheme
2. If ((i,j) is an irregular point and (Lg) 6 (2+) then

0 If ((i + 1,j) E (2‘) then

— If (both auxiliary points are in ‘proper’ region and enough points for
+ — a
try or uy) thtn
* call Off-interface scheme for smooth interface

-- else store the unsolved point (1,.j)

o If any other surrounding point is in 52—, a similar procedure is applied.
3. If ((i,j) E P) then

0 If (exact. two surrounding points stay on the same side of the interface)

— If (two auxiliary points on :r. and y directions are on the same side of

the interface) then

33

a: call Orr-interface scheme for smooth interface
— elseif (two auxiliary points are. on the different side of the interface)
then
* call On-interface scheme (a) for geometric singularities
— else (cannot find two auxiliary points)

* store the unsolved point (13]. l

0 else store the unsolved point (2', j)
4. Revisit stored unsolved point (2', j ), use secondary fictitious values

a If ((‘i,j) 6 9"") then
— If ((i + 1,3) 6 (2‘) then

* determine n, the number of unknowns, after introducing secondary
fictitious values among Fifi,jsFi+1,jaFi—1.j and 17,429]-

* If (n == 0) then
- all surrounding points are real unknowns or have primary ficti—

tious value now, call standard five-point. FD scheme
4: If (n. == 1) then

- call Off-interface scheme (b) for geometric singularities;

* elseif (n. == 2) then
- call Off-interface scheme (a) for geometric singularities;
* else

- pause
— If (any other surrounding points are in 0‘). similar procedure is ap-
plied.
o If ((i,j) E P) then

— If (all surrounding points are real unknowns or have primary fictitious

value now), call the standard five-point FD scheme.

34

1+3
1+2

1+1

j-1

 

 

1+3 ‘ /

 

 

 

 

J
~—<i.>

/ A i+2 \‘J ‘
( i+1 I r I

 

 

 

 

 

 

 

 

j-1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 2.4: (a) The critical value of the smallest angle that MIB can handle; (b)
An angle smaller than the critical value.

— If (three surrounding points are on the same side or have secondary

fictitious value on the same side), then
a: call On-interface scheme 3

— else pause

5. If ((i,j) is an irregular point and (i,j) E 52‘), use primary fictitious values and

call the standard five-point FD scheme.

2.1.4 The critical value of the acute angle

The standard MIB method can always handle a C1 irregular interface even if it
c O

has large curvature, since enough auxiliary points can always be found by refining

mesh. When the interface is Lipschitz continuous but not C 1, refining mesh usually

cannot. provide required points around a kink. The present generalized MIB method

overcomes the difficulty by using secondary fictitious values to substitute the function

of unavailable auxiliary points.

Fig. 2.4(a) shows the smallest acute angle that the generalized MIB can handle

when a vertical or horizontal line bisects the angle. The number 1,2,3 labels the

sequence of the fictitious values that were resolved. The fictitious values at points

labeled 1 were found with On—interface scheme 1. Then the fictitious values at points
labeled 2 were found by using points labeled 1 with On—interface scheme 3. Finally
the fictitious values at points labeled 3 were found by using secondary fictitious values
at points labeled 1 and ‘2 with Off-interface scheme 3. If the angle is smaller than
this one, the fictitious values at (i - 1, j + 2) and (2' + 1, j + 2) cannot be founded,
and the fictitious value at (71,.7' + 2) is also not available. Therefore, 2 tan—1Q.) is the
critical value of the smallest angle that the present MIB can handle with 2nd order
accuracy. Any angle that is larger than the critical value can always be treated with
the designed 2nd order accuracy.

In most cases, the mesh line that bisects the angle is not vertical or horizontal.
Fig. 2.4(b) shows a situation in which the angle is smaller than the critical value and
it can still be handled by the MIB. The fictitious values at points labeled 1 were found
with On-interface scheme 1 and Off-interface scheme 1. Then the fictitious values at
points labeled 2 were found with On-interface scheme 3. Finally the fictitious values
at points labeled 3 were found by using secondary fictitious values at points labeled
1 and 2 with Off-interface scheme 3. However if the angle is very small such that
(1? — 1, j + 3) and (7', j + 3) are both in 82—, the present MIB method will not be able
to capture the shape of the corner. In the case of Fig. 2.4, the present method has
211d order accuracy if and only if the third horizontal mesh line above the tip contains
at least two points inside the acute angle.

This limitation could be avoid by using the jump conditions of two interfaces
from the both side of an irregular point around the tip to obtain fititious value of
three irregular points simultanuously. The detailed discussion of the new strategies

is provided in Chapter 3.

2.2 Numerical study

In this section, we examine the performance of the proposed MIB scheme for solving
the Poisson equation with sharp-edged interfaces, thin-layered interfaces, and inter-

faces that intersect with the geometric boundary. W'e consider six different. interface

36

Table 2.1: Numerical efficiency tests of the 2D Poisson equation (Case 1).

 

 

 

Case 1(a) Case 1(b) Case 1(c)
n3; x ny Loo Order Loo Order Loo Order
20 x 20 1.02e-2 9.12e-3 9.75e—3

 

40 x 40 2828-3 1.85 2.22e—3 2.03 2.39e—3 2.03
80 X 80 7.33e—4 1.94 5.64e—4 1.98 5.89e—4 2.02
160 x 160 1.86e—4 1.98 1.428—4 1.99 1.48e-4 1.99

 

 

 

 

 

 

 

 

 

 

 

geometries coupled with various boundary conditions and solution behaviors. In the
first test case, we examine capability of the present MIB method in treating the criti-
cal sharp edge. We also test the present scheme for grid points exactly at the tip of the
sharp edge, for which the interface jump conditions are not unique due to undefined
derivatives. The second case is designed to test the present method for handling mul-
tiple sharp edges and oscillatory solutions. Case 3 is a missile geometry. Case 4 has
multiply connected domains. In Case 5, the level set. function is not piecewise linear.
In Case 6, we consider a thin-layered interface geometry. Finally, we demonstrate the
proposed method with a problem having weak solution. In all the cases, numerical
results are compared to analytical ones. The standard Loo norm error measurement
is employed in this section.
Case 1

we consider the 2D Poisson equation
(fluids + (flux/lg = (1(I~ y) (227)
defined in a square domain [—-1, 1] x {—1, 1]. The exact solution is designed to be
u.+(.-I;, y) = :er + 3/2 + 1, "_(gr, y) = sin(7r;1:) sin(7ry) (2.28)
with piecewise continuous coefficients

133-(fry):(.172 — y“2 + 3)/7, [3_(;r,y) = (.173; + 2)/5. (2.29)

37

  
 

    
  

 
   
  

WWW
[W

5:

[ll ' i: .

l

-1 —1 0 -1 -1

Figure 2.5: Interfaces on 20 x 20 meshes (top row) and the computed solution
(bottom row) for the 2D Poisson equation (Case 1(a) and Case 1(b)).

   

-1 -1
X

Figure 2.6: Interface on a. 20 x 20 mesh (left) and the computed solution (right) for
the 2D Poisson equation (Case 1(0)).

38

Table 2.2: Comparison of numerical solutions involving the left and right flux jump
conditions at the tip in the geometry of Case 1(c).

 

 

 

 

 

 

Left flux jump condition Right flux jump condition
11;; x 129 L00 Order Loo Order
20 x 20 1.16e-2 1.15e—3
40 x 40 2.67e—3 ‘ 2.11 2.66e-3 2.11
80 X 80 6.69e—4 2.00 6.68e-4 2.00
160 x 160 1.68e-4 2.00 1.69e—4 1.98

 

 

 

 

 

 

 

As shown in Figs. 2.5 and 2.6, interfaces with both obtuse and acute angles are

considered. Level-set functions (,6 of the interfaces are given as:

0 Case 1(a)
y + :13/2 I < 0
y — 21'. :r 2 0
0 Case 1(b)
_ y + 3x I < 0
6507. y) = . (2.31)
y — 3.17 1‘. Z 0
0 Case 1(c)

y+312 a3<0

¢(;I;, y) = (2.32)

y—111: 1‘20

Numerical results are presented in Table 2.1. Second order convergence is observed
in all the three cases.

As the derivative at. the tip of the sharp edge is undefined, it. is interesting to
examine the performance of the present. method for treating irregular points ‘on the
tip’. To this end, we place the tip on a grid point during the mesh refinement and

computer the error rate. Since the jump condition involves the derivative on the

39

interface, we need to compute derivatives at the tip if it is on a grid point. As shown
in Fig. 2.6, the derivative at the tip of the sharp edge can be computed as asymptotic
limits from either the left edge or the right edge shifting toward the tip. These limits
are not equal, and leads to two flux jump conditions. We need using only one of
these flux jump conditions. An interesting question is what happens to the solution if
different flux jump conditions are used. Our numerical test is based on the interface
geometry of Case 1(c). We modify the solution from u_(;r, y) = sin(7r:r) sin(7ry) to
u_(:r,y) = cos(7r:r)sin(7ry) to avoid zero at :r = 0. The errors computed by using
two flux jump conditions are presented in Table 2.2. It is seen that the errors in
two computations are both of second order, and differ very little. A further error
comparison is conducted at the tip point (0,0.1) using a 20 x 20 mesh. The solution
error from using the left flux jump condition is 1.02845419680633 X 10‘2 and that
from using the right fiux jump condition is 1.02845419680637x10‘2. Similar behavior
is found in other examples. Therefore, we conclude that the proposed MIB method
works well for grid points at the tip of sharp edges.

Case 2

In this case, we test the presented method with a more complicated geometry. As

shown in Fig. 2.7, the level—set function 6.5 of a pentagon star is given as:

RSlIl(9t/2)
sin(6t/2 + 6’ —- 6,. — 27r(i—1)/5)

 

—T Hr+7r(2i—2)/5£ (9

< 9,. + 77(221—1)/5
95016) = . (2.33)

Rsin(9t/2) _
~ -‘ 6- 2—3 r<0
sin(a,/2_9+9,+2,,(i_1)/5) 1 . +7r(2 )/.)_

 

< 0,. + «(22' — 2)/5

with (it = ,6,- = and 2' = 1,2,3,4. 5. The discontinuous coefficients are

on:

given by

{3+(.1:,y) = 1, fi—(Jr, y) = 2 + sin(.~r + y). (2.34)

40

 

X

Figure 2.7: The pentagon star interface in a 20 X 20 mesh (Case 2).

Table 2.3: Numerical efficiency tests of the pentagon star interface (Case 2(a)-(d)).

 

 

 

 

 

 

 

 

 

 

 

Case 2(a) Case 2(b) Case 2(e) Case 2(d)
n3 X ny Loo Order Loo Order Loo Order Loo Order
20 X 20 6.118—4 6.11e—4 5.266-2 9.728-1
40 X 40 6.076—5 3.33 6.07e—5 3.33 8.51e-3 2.62 1.94e—1 2.32
80 X 80 1348-5 2.18 1.34e—5 2.18 2.39e—3 1.83 5.49e-2 1.82

160 X 160 4.15e-6 1.69 4.15e-6 1.69 6.64e—4 1.85 1.48e—2 1.89

 

 

 

 

 

Four different solutions are designed with variable amount of oscillations:

0 Case 2(a)

'll.+(.‘I.‘, y) = 8,

0 Case 2(b)

0 Case 2(e)

u+(.r, y) = 6 + sin(27r.1:)sin(27ry),

ii—(T, y) = .772 + 3/2 + S111(.‘I.’+ y).

u+(:r, y) = 5 + 5(11:2 + 312),

u_(;r, y) = 172 + 312 + sin(.r + y).

41

(2.35)

(2.36)

u_(;r., y) = 1‘2 + y2 + sin(1‘ + y). (2.37)

 

_L
O

  
 
   

   

’llllllll lllllll I -

: 1 jmll
5" llllllllllll Lllllllllll _1 llllllllllll I‘ll“

  
 

   

   
  
 

 

 

-1(a-)-1
”.2 . lllllml lllll“
y -1 —1 x 1
(C)

 

(e)

(0
Figure 2 8x Computed solution for the pentagon star interface (Case 2(a) 2( )
2(e), 2(d)), 2(e) and 2(f)).

42

a Case 2(d)

u+(:r., y) = 6 + sin( 7.11:) sin(677y), u_(r, y) = 3:2 + y2 + sin(.r + .11). (2.38)

In each case, the jumps in u and un along the interface can be evaluated from the
solution. Table 2.3 gives the numerical results of the second order MIB method for
this difficult problem. These results are also illustrated in Fig. 2.8. Obviously, Case
2(d) is a very challenging numerical example. It consists of three difficulties, i.e.,
material interface, sharp edge and oscillatory solution. Similar problems described
by the Helmholtz equation has much impact to computational electromagnetics.

We also discuss the following two cases with the same geometries but with homo—

geneous source terms (i.e., q(.r, y) = 0 in Eq. 2.27). We set [i E 1.

0 Case 2(e)

u+(r, y) = 0, u“(:r.,y) = exp(.r) sin(y). (2.39)

0 Case 2(f)

u+(:r,y) = 0, u_(r, y) = exp(7r;r) sin(7ry). (2.40)

The numerical errors of the homogeneous cases are listed in Table 2.4. The numerical
solutions of the homogeneous cases are also illustrated in Fig. 2.8. Comparing the
computed solutions in Cases 2(a)-(d) to those in Case 2(e)-(f), there is no significant
difference between homogeneous cases and non-homogeneous cases. The accuracy of
the computed solutions strongly depend on the natural of the solution itself. The
solutions that vary rapidly with respect to spatial coordinates are more difficult to
capture than those that vary slowly. Therefore, case 2(d) and 2(f) have relatively
larger errors than others. However, it can be seen that all the above cases have
achieved second order convergence. The convergence is not quite uniform because

the relative location of tips with respect to mesh lines varies as the grid is refined.

43

Table 2.4: Numerical efficiency tests of the pentagon star interface (Case 2(e)-(f)).

 

 

 

Case 2(e) Case 2(f)
n; X ny Loo Order Loo Order
20 X 20 3.84e-4 6.74e—2

 

40 X 40 7.34e—5 2.39 1.02e-2 2.72
80 X 80 1.94e—5 1.92 4.60e-3 1.15
160 X 160 3.23e-6 2.59 7.43e—4 2.63

 

 

 

 

 

 

 

 

 

Case 3

Here we solve the Poisson equation for a missile geometry. The geometric pa-
rameters is given in Fig. 2.9. In the present computation, the missile is rotated 60°
counterclockwise. All the parameters but the angular ones are divided by 100 to fit in
[—1,1] X [—1, 1] computational domain. Fig. 2.10(a) shows the missile interface on a
20 X 20 mesh. The coordinate of center C, which is labeled in Fig. 2.9, is (—0.4, -—0.4).

In this case, the exact solution is given by:
u+(;r., y) = 1, “LL-(:13, y) = cos(7r.r) sin(7ry) (2.41)

with discontinuous coefficients

:rg—y2+3 _ry+2

7 . (flay) 5

 

s+(.2:, y) = (2.42)

Numerical results in Table 2.5 verify the convergence of the proposed method.

Table 2.5: Numerical accuracy tests of the missile interface case (Case 3).

 

n3 X n,y LOO Order
20 X 20 5.68e—3
40 X 40 1.71e-3 1.73
80 X 80 4.33e-4 1.98
160 X 160 1.09e-4 1.99

 

 

 

 

 

 

 

 

 

Case 4

Here we solve the Poisson ecuation on a iece of twisted chess board, see Fir.
l

44

171.2597

116.8263 _. ‘ 26.8266

 

 

 

 

7

 

——

 

 

 

 

 

 

 

22.2222

19,3317

 

15,1916

 

Figure 2.9: The configuration of a missile(Case 3).

    

-1
-1 —o.5 o 0.5 1

Figure 2.10: The missile interface on a 20 X 20 mesh (left) and computed solution
(right) (Case 3).

45

2.11. The level set function a) of the interface is given as

1 1 :r 1 :r 1
65.. =— -—‘—— — — —-—— - — 2.43
(my) (y x 3m; x+ 3>(y+ 3 2>(y+ 3 + 2) < )
The discontinuous coefficients are given by
fl+(rr. :11) = 1. (Vt-13.31) = 2 + sin(-E + 9) (2-44)
Two different solutions are designed.
0 Case 4(a)
u+(.~r, y) = 8, u-(r, y) = 3:2 + y2 + sin(;r + y). (2.45)
0 Case 4(b)
u+(r, y) = 8 + 31y, u-(r, y) = 1‘2 + y2 + sin(:r + y). (2.46)

In this case, there are several irregular points near the domain boundary whose
primary fictitious values can not be solved by interface scheme because the auxiliary
points of the irregular points on the boundary are not available. As shown in 20 X 20
mesh in Fig. 2.11, the primary fictitious value at (1.0, 0.6) of point (0.9, 0.6) can not
be determined by the interface scheme mentioned before. Under Dirichlet boundary
condition, we could discretize (Busch at (0.9, 0.6) by using a one-sided finite difference
scheme, which involves three points (0.7, 0.6), (0.8, 0.6) and (0.9, 0.6). Other boundary
conditions will not be discussed in the present chapter.

Numerical results in Table 2.6 verify the convergence of the proposed method.

Case 5

In this example, we considered a problem with multiply connected curved edges,

46

 

-0.5

1
X

Figure 2 11 The chess board interface on a 20 x 20 mesh (Case 4)

   
       

     
  
     

  

 
  
    

 

     
    
  

ill M

;:—_;- .

l
M l l‘

y -1 -1

   
 

Hillliilll'll

          

Figure 2 12 Computed solution for the chess board interface (Cases 4(a) and 4(b))

47

Table 2.6: Numerical accuracy tests of the chess board interface case (Cases 4(a)
and 4(b)).

 

 

 

Case 4(a) Case 4(b)
nx x ny Loo Order Loo Order
20 x 20 5.92e—4 5.97e—4

 

40 x 40 1.50e-4 1.98 1.50e—4 1.99
80 x 80 4.26e—5 1.82 4.26e-5 1.82
160 x 160 9.40e—6 2.18 9.40e-6 2.18

 

 

 

 

 

 

 

 

 

see Fig. 2.13. The level set function o(.z:, y) is given by
2 1 2 1
am. y) = —(y — 4.17 + 1m; + 4.1? — §)- (2.47)

In this case, the interface F is consist of six pieces of smooth curves. The exact

solution is given by:
u+(1:, y) = 8 + 3173/, u—(I. y) = 2:2 + y2 + sin(;r. + y) (2.48)
with discontinuous coefficients
[3+(.1:,y) = 1, {3_(:17,y) = 2 + sin(;r + y). (2.49)

Numerical results are presented in Table 2.7. Second order convergence is achieved.

Table 2.7: Numerical accuracy tests of Case 5.

 

nx x my Loo Order
20 x 20 6.71e—4
40 x 40 1.7le-4 1.97
80 x 80 5.71e—5 1.58
160 x 160 8.32e-6 2.78

 

 

 

 

 

 

 

 

 

Case 6
We consider a thin-layered interface problem which is in11.)ortant to evaluating
thin—layered coatings, see Fig. 2.14. In this case, the whole domain is divided into

three subdomains, thin layer 9+, outside domain (2;, and inside domain 0;. The

48

 

-0.5 0 0.5 1

Figure 2.13: The interface of Case 5 on a. 20 x 20 mesh (left) and computed solution
on a 80 x 80 mesh (right).

interface between 9; and 9+ is To and the interface between Qi— and 0+ is I}. There
are two types of irregular points. Type I irregular points are near F0 and their five-
point FD scheme involves the points from both 9+ and $2; . Type II irregular points
are near I“,- and their five-point FD scheme involves the points from both 0“” and
Q; . Both types irregular points can be solved by the present MIB method directly.
However, when we consider the practice problems, the material within (2; is usually
perfect conductor, therefore we have u(.r, y) E 0 within (217. In this case, the scheme
for Type II irregular points can be significantly simplified by letting all the 11;, u;
equal zero.

The level set function ¢o(rr, y) of F0 is given by

_y_1/13 x<—11/23
w. y) = - (250)
—y — 3(.r +11/23)/4 — 1/13 1- 2 —11/23

The level set function (inf-”c, y) of l",- is given by

y+1/3 :c<—11/23
Mr, 31) = ' (2'51)

y + 3(.r+11/23)/4+1/3 :c 2 ~11/23

49

    

-1 _
—o.5 o 0.5 1 Y 1 x

-1
-1

Figure 2.14: The thin layer interface on a 20 x 20 mesh (left) and computed solution
(right) (Case 6).

The exact solution is given by:

u+(:c,y) = 1+ $31. (2.52)
u;(:r, y) = cos(m:) sin(ny),
"i— (33,11) = 0

with discontinuous coefficients

2_ 2
[3+(x.y) = x 3 +3, (253)
fiJt'uy) = —Iy,+2.

v.)
[ii—(2.33;) = 0

Numerical results in Table 2.8 verify the convergence of the proposed method.

Table 2.8: Numerical accuracy tests of the thin layer case (Case 6).

 

50

 

 

 

 

'1—1 —o:s A o ‘o.5 H 1
x

 

Figure 2.15: Domain with reentrant corner on a 20 x 20 mesh (left) and computed
solution (right) (Case 7).

Case 7

8
Table 2.9: Numerical accuracy tests of the imaginary and real parts of w(z) = 23
(Case 7).

 

It is interesting to discuss a weak-solution problem whose solution has unbounded
first order derivative on sharp edge [20]. Consider a domain (2 = [—1 1] x [—1 1]. Let

us define

4
ST ={(1‘7y)E§R2; I2+y2< 5. $<Oory>0}, (2.54)

with reentrant corner (See Fig. 2.15). The solution on the domain (2+ = Q \ Q'—

vanishes. The solution in Q' is a harmonic function “(1: y) = Im(z2/3), solving the

51

boundary-value problem

Au 2 0 in Q, (2.55)
-. 2 2 /
'u.(eu") = gsin(§<p) 4 for 0 S Lp S £321,
a = 0 elsewhere on 09".

The solution is C262) 0 CWT). Finite element (Galerkin) formulations can handle
this problem directly [20, 72]. However, this problem can not be directly solved by
collocation formulations. A standard technique is to introduce an algebraic factor.

Let us define a function
8 .
'u,i(z) = 23, (2.56)

where 10(3) satisfies Aw = 0 with appropriate boundary condition. When 112(2) has
been obtained from solving the harmonic equation, it is simply an algebraic operation
to resolve the solution u

f 1
Im(w3) for 0 3 up < 3—473

. 1
U(:r,y) = 1111(32/3) = Im(i'u,2:f) for Q47: 3 1,9 < 3275 (2.57)
1
K Im(—w:l) elsewhere

 

Both the imaginary and real parts of function 10(2) are harmonic function, there-
fore they can be easily solved by the present MIB method. The errors of both the
imaginary and real parts of function w(z) are given in Table 2.9. Numerical results
of Im(w) and Re(w) verify the convergence of the proposed method. The numerical
solution of u(x, y) obtained by Eq. (2.57) is shown in Fig. 2.15.

We next. test accuracy and convergence on a sharp-edge case that has large contrast
ratios of diffusion coefficients. The solution is v = Re(w(z)) in 52‘ and 'v = 0 in 52+,
where Q‘ and (2+ are defined above. Here, 12(1‘, y) satisfies V - (3+Vv) = 0 in 9+ and
V - (,B—Vv) = 0 in Q“. We fixed the diffusion coefficient ,3— to be 1. The numerical

errors for [3+ = 1 have been given in columns 4 and 5 in Table 2.9. Table 2.10 shows

52

Table 2.10: Numerical accuracy tests of large discontinuity ratio of [3‘ and {3+ (Case

 

 

 

7).
0+ = 10000 0+ = 0.001
71;; x ny Loo Order Loo Order
20 x 20 8.00e—4 3.48e—3

 

40 x 40 1.33e—4 2.59 1.46e-3 1.25
80 x 80 2.19e-5 2.60 2.67e—4 2.45
160 x 160 4.97e—6 2.14 9.21e-5 1.54

 

 

 

 

 

 

 

 

 

the numerical errors for 3+ = 10000 and [3+ = 0.001, respectively.

We found that the present method is not entirely robust with respect to large
contrast ratios of diffusion coefficients. In the case [3+ >> ,8‘, the numerical solution
converges faster than 2nd order. While in the case 13+ << 6‘, the numerical solution
converges slower than 2nd order. This behavior will be explored further in our future

work.

2.3 Conclusion

A wide variety of scientific and engineering problems involve sharp—edged material
interfaces, and their governing equations are of elliptic type. These problems call
for new efficient methods that do not depend on massive local mesh refinement,
which does not work for highly oscillatory waves due to the pollution effect [5]. The
present work provides a solution to this class of problems on the Cartesian grid by
extending the marched interface and boundary (MIB) method [163, 170] previously
designed for straight or curved interfaces. The concept of secondary fictitious values
is introduced to deal with difficult topology where primary fictitious values cannot
be solved directly. The present MIB method is extensively validated by problems
with sharp-edged interfaces, as well as problems with a combination of sharp edges
and oscillatory solutions, which pose a severe challenge to most existing numerical
methods. The designed second order convergence has been confirmed for all the
test problems, which has significantly improved the previous best result, 0.8th order,

reported by Hou and Liu [72] using a finite element formulation for similar problems.

53

The main ideas of the present MIB method are follows. First, simple Cartesian
grids are used even if the problem is defined on an irregular domain and/or with
sharp-edged interfaces. Second, the standard (higher-order) central finite difi'erence
(FD) schemes are utilized so that the condition number of the discretization matrix
is relatively low, and efficient linear algebraic equation solvers can be used. Third,
primary fictitious values are created on the irregular points near the interface to fa-
cilitate the use of the central FD schemes. Secondary fictitious values are introduced
to resolved primary ones. Fourth, the fictitious values are determined by using phys-
ical jump conditions whose proper enforcement warrants the convergence of the FD
discretization. Fifth, physical jump conditions are enforced “on the interface” to en-
sure an efficient restoration of accuracy. Sixth, the lowest order jump conditions are
used to avoid the possible involvement of cross derivatives in the higher-order jump
conditions and higher dimensional polynomials. Finally, to achieve higher—order con-

vergence, the lowest order jump conditions are implemented repeatedly.

54

Chapter 3

Three-dimensional matched
interface and boundary (MIB)
method for treating geometric

singularities

The objective of this chapter is two-fold. First, we generalize an earlier 2D 2nd-
order MIB method for geometric singularities [161] to 3D. The three dimensional
MIB scheme is developed with the second order accuracy for smooth interface in
[167, 168]. But the MIB scheme does not maintain the designed second accuracy due
to the presence of non—smooth interfaces in bimoleeules. In this work, we demonstrate
the 2nd order convergence for arbitrarily complex three dimensional interfaces with
arbitrary geometric singularities. In addition, the critical acute angle restriction in
our earlier 2D MIB scheme in Chapter 2 is resolved by proposing a new algortihm,
which makes use of two interface intersecting points for sharp-edged interfaces, is
proposed to resolve this problem. Finally, in the earlier MIB method proposed by
Zhou and Wei, a large number of iterations is required to solve the linear algebraic
equations due to the asymmetric matrix of the interface scheme. In this work, we
have addressed this problem by making the MIB matrix optimally symmetric and

diagonally dominant by an appropriate selection of auxiliary grid points, which dra-

55

matically improves the convergence property. An essential feature of the proposed
MIB method is that it locally simplifies a 2D or a 3D interface problem into lD-like
ones. As a result, the MIB method is able to systematically achieve higher-order
convergence and to efficiently deal with geometric singularities. The proposed MIB
algorithm, particularly its matrix acceleration and treatment of singularities, provides
the crucial technical foundation to a new generation of Poisson-Boltzmann equation
solvers for the electrostatic analysis of biomolecules [158, 168].

In the next two sections, the theoretical formulation and the computational al-
gorithm are given to the 3D MIB method for elliptic interface problems. Detailed
consideration of topological variations is given, particularly for 3D geometric singu-
larities. The MIB matrix is made optimally symmetric and diagonally dominant.
The proposed MIB method is extensively validated by benchmark tests, such as the
surfaces of missile, protein and intersecting spheres. This chapter ends with a con-

clusion.

3.1 Theory and algorithm

3.1.1 General setting of the MIB method for elliptic interface

problems

Consider a given interface F which divides an open bounded domain 9 E R3 into
disjoint open subdomains, domain f2” and domain (2+, i.e., f2 = Q" U 9+ U F. we

solve the following 3D elliptic equation
V - (6(23, y, z)Vu(.r, y, z)) — H(1‘,g, z)u(;r.y, z) = q(.r,y, z). 1*,y,z E Q (3.1)

with variable coefficients [3(1', y, z) and h:(.r, y, 2:) which may admit jumps at the in—

terface I‘. To make the problem well-posed, this equation should be solved with

56

appropriate boundary conditions and two interface jump conditions

in] = u+ — u“. (3.2)

[Sun] = 6+Vu+ - n — fi‘V u_ - n, (3.3)

where n = (72.13, ny, n3) is the normal direction of the interface. To achieve higher-
order convergence, the interface jump conditions are to be rigorously enforced at
each intersecting point of the interface and the Cartesian mesh lines, which means
the incorporation of subgrid information for overcoming the staircase phenomena.
In the MIB approach, we seek to reduce the 3D interface problem into 1D-like ones
locally so that the standard higher-order finite difference schemes can be implemented
for the discretization.

As the interface normal direction varies on the interface, it is necessary to define
a set. of local coordinate at each intersecting point of the interface and the Cartesian
mesh. At a specific intersecting point, it is convenient to denote the local coordinates
(5, 77, C) such that 5 is along the normal direction and n is in the .1: — y plane. The

coordinate transformation can be given as

 

 

 

 

-6- .r.
.4- _Zl

where the transformation matrix p has the form

r- -i

sin (,0 cos 9 sin (p sin (7’ cos 0")

P = — sin 0 cos 9 0 - (3.5)

 

 

—cosq0cos€ —cos¢')sinl9 sings]

h

Here 6 and (z) are the azimuth and zenith angles with respect to the normal direction

11, respectively. In the new coordinates, the jump condition (3.3) can be discretized

57

[“16] = 6+(SII1 0') cos 6n: + sin 6 sin 6213' + cos on?) (3.6)

—6_ (sin 90 cos Bu; + sin 0") sin 611.; + cos (021;).

Without creating higher—order jump conditions, we can generate two additional jump
conditions by differentiating Eq. (3.2) with respect to the tangential direction 7) and

the binormal direction C

[21.7,] = (— sin 6n: + cos 621;) — (— sin Bu; + cos Hug) (3.7)
[*ug] = (— cos (,0 cos 011; — cos 0‘) sin 6113'] + sin 0211?) (3.8)

— (— cos 4’) cos 0n; —— cos (1') sin 6a,; + sin en; ).

In principle, we can generate more jump conditions by further differentiating these low
order jump conditions. However, a byproduct of such a procedure is the creation of
higher-order derivatives and cross derivatives, whose evaluation often involves larger
stencils and is unstable for constructing higher-order interface schemes [163]. The
proposed MIB method makes use of only lower-order jump conditions. Therefore, we
end up with four interfacejump conditions, i.e., Eqs. (3.2), (3.6) (3.7) and (3.8), which
can be used to determine four desired unknown quantities related to the intersecting
point of the interface and the mesh. In the MIB method, we extend the computational
domains with fictitious values on both sides of the interface so that the standard finite
difference scheme can be applied on a smooth domain near the interface without the
loss of designed convergence. This is done along a mesh line at a time, so that our
MIB method is locally a lD-like scheme for a higher dimensional interface. Fictitious
values can be determined by the aforementioned four jump conditions. However,
in association with four jump conditions there are six partial derivatives, uj, 11;,
ill], u,” , uj", and 11:, which have. to be computed right on the intersecting point
between the interface and the meshline. N umerically, these computations are carried

out. within appropriate subdomains near the interface, i.e., MI, 11,] and u: in 0+, and

58

11;, try- and u; in 9‘. For slightly curved smooth interfaces, it is usually quite easy
to calculate all the six partial derivatives. However, for non-smooth and / or arbitrarily
complex interfaces, it is often very difficult to compute all of these partial derivatives
due to geometric constrains. To make the present MIB scheme applicable to real
world problems, we eliminate two partial derivatives that are the most difficult to
compute by using two appropriate jump conditions. The selection of two eliminated

partial derivatives is pursued as follows.

1. All fictitious values on all irregular points will be determined. At an intersecting
point between the interface and an r-mesh line, two fictitious values on the :1:-

mesh line will be determined.

2. Two derivatives along this mesh line must be kept. In the above case, 11.; and

a; must be kept.

3. We therefore select two partial derivatives from 113'], ug', a; and u; We first
check if the interface local geometry allows us to easily compute a; and u"; in

0+. If so, we will eliminate uy

and u; — it is done. If one of a; and it:
cannot be computed, say 113]", we will check whether a,“ can be compute in Q".

If yes, it is done. If u; cannot be computed either, we will check whether uy

can be computed. If yes, it is done. If no, the scheme fails at this intersecting
point. We will try to determine the nearby fictitious values from other nearby

intersecting points.

The mathematical operation of the elimination is derived below. From Eq. (3.4),

we have
f "(Lg Um
(1,7 = P uy ( 3 9)
NC U~ J

 

 

 

 

59

Therefore, Eqs. (3.6), (3.7) and (3.8) can be rewritten as follows:

 

 

 

 

’P23

“P33

11:]?
’ ' “u;
[531%] u+
y
[7111)] =C a
1 1 "f
”C
a?
u:
L ..
where
C1 P111"3+ —P1n‘3_ P12.3+ -P12fi“ P1313+ —P13/3‘
C: C2 = P21 -P21 P22 -P22 P23
1 C3 P31 ~P31 P32 —P32 P33

 

 

 

q

 

J

(3.10)

(3.11)

Here pij is the ijth component of the transformation matrix p and Ci represents the

17th row of C. After the elimination of the lth and mth elements of the array (11;,

— + - + — .
um, uy , uy , uz , u; ), Eq. (3.10) becomes

a [flag] + b [11,7] + 0 [11C] 2 (aC1 + bCQ + CC3) '

 

where

a = 02103771 _ C31 C27"
b = (731017); " C'1103171.

‘3 = CIIC2m — (72101771-

60

11;]

IQ
I" .
<2+ 8|

Q

{:1
61+ ‘CI

1

 

(3.12)

(3.13)

Therefore, two remaining jump conditions, Eqs. (3.2) and (3.12), are to be used
to determine two fictitious values near the interface along a specific mesh line at
a time. To construct higher order interface schemes, the above procedure can be
systematically repeated to determine fictitious values along other two mesh lines
and at other interface locations. A more explicit and alternative description of this
elimination procedure can be found in Ref. [168].

Alternatively, for very simple geometry, one may determine three or four fictitious
values at a time by using three or four jump conditions. However, such a method
will have to approximate more partial derivatives and involve more grid values. Con-
sequently, the resulting matrix will be less symmetric and more iterations will be
required to solve the linear algebraic equation.

In the present work, we propose a novel algorithm to deal with sharp-edged inter-
faces. Specifically, when there is a very narrow interface or a thin region enclosed by
two interfaces, two sets of interface jump conditions at two intersecting points will be
used to determine a few fictitious values together, see Section 3.1.5. This algorithm is
developed to remove the acute angle constraint in our earlier 2D MIB method [161].
It gives us flexibility to handle arbitrarily complex geometry and geometric singu-
larities. In summary, we determine fictitious values along a mesh line at time, and

locally reduce a 3D interface problem into 1D—like ones.

3.1.2 Treatment of smooth interfaces in 3D — an illustration

Considering a geometry illustrated in Fig. 3.1, the interface intersects the kth y-mesh
line on the y— 2 plane at point (so, yo, 20). Two fictitious values f1, fl, and fi,j+1,k are

to be determined on irregular grid points. We denote the left domain as 9+ and the

+ +

right one as 9‘. Here, 1.1 , 11‘, lty and 11.37 are easily expressed as interpolations and

61

 

 

 

 

 

 

 

 

 

 

/

 

h
ha

4.
A

Figure 3.1: Illustration of a smooth interface. The kth y mesh line intersects the
interface at point (:50, yo, 20). A pair of irregular points (i, j, k) and (i, j + 1, k), in red
0, is on the kth y-mesh line. Type I auxiliary points are in blue A. Type II auxiliary
points are in blue 0. Auxiliary lines, in red dashed lines along the :r- and z-directions,
are used to calculate 11;, 11;", u; and 11;".

standard finite difference FD schemes from information in (2‘ and (2+, respectivel
Y

11+ = (um—1»“1’0.j1111'0.j+1) ‘ (“i.j—lfi» 21.1.1313: fi.j+1,k)T (3-14)
V = (100.33 w0.,j+1= 1L’0.j+2) ' (fi,j.k- ”21,3413: ui,j+2.k)T
"I; = (101,341, 101.)» “1’1,j+1) ' (“i,j—1.k» ”1.33112 fi,j+1.k)T
u; = (1010': u’1.j+1»w1,j+2) ' (fair: 'uvzi,j+1.k»ui.j+2.k)T.
where wmm denote FD weights, which are generated by using the standard Lagrange
polynomials [5:3]. The first subscript. n represents either the interpolation (n = 0)
or the first order derivative (71 = 1) at interface point (1:0, ya, 20), while their second
subscript is for the node index. We only need to compute two of the remaining four
interface quantities. If u; and u; can be conveniently computed, then uj," and u:
will be eliminated by using Eqs. (3.12) and (3.13) with l = I and m = 5.

To approximate 11: or 11;, we need three n values along the auxiliary line y = yo
on the y—z plane, see Fig. 3.1. Unfortunately, these values are unavailable on the grid
and have to be approximated by interpolation schemes along the. y-direction. This

means six more auxiliary points are involved. In practice, one of these two derivatives

62

can be easily found. In the situation shown in Fig. 3.1, u; can be represented as

 

 

follows
11; = [rump "1-’1.k+11“’1,k+2] (3.15)
U"0.j “’0.j+1 “"0.j+2 0 0 0 0 0 0
,I ,I ,r
* * *
_ 0 0 0 0 0 0 wOj w02j+1 w0,j+2 3
X [fi.j,ka ui.j+1,kr Uzi,j+2.k» Uzi.j.k+1-. “i,j+1,k+1

T
, “i,j+2,k+1-“i,j.k+21“i,j+1,k+21“i.j+2,k+2] ,

where we use superscripts on w to denote different sets of FD weights [55]. Similarly,
u; can be computed along the auxiliary line y = ya on the .1: —;1/ plane, the dashed line
shown in Fig. 3.1. By choosing l = 1 and m = 5 in Eq. (3.13), we can eliminate 11;."
and 112‘. Then, by solving equations (3.2) and (3.12) together, two fictitious values
fi,j,k and fi,j+1.k can be easily represented in terms of 16 function values around
thenr

Note that the calculation of fictitious values strongly depends on whether auxil-
iary points can be found in proper subdomains. At the second order accuracy, this
calculation is quite easy for smooth interfaces and relatively dense meshes. However,
it is a challenge to determine all the fictitious values for higher-order schemes and
with the presence of geometric singularities. In order to discuss this issue clearly in
the following sections, we define type I auxiliary points as the auxiliary points that
are only used to interpolate '11 values on auxiliary lines passing through the intersect-
ing point. Type II auxiliary points are defined as the auxiliary points that are in the
same line with two irregular points. For instance, in the situation depicted in Fig.
3.1, there are six type I auxiliary points used to interpolate 11 value at (20.310, 2;) and
(.r0. yo, 35,), which are used to calculate 11;. Another six type I auxiliary points are
used to calculate u; in a similar way. Here. (i, j — 1, A?) and (i, j + 2. k) are two type

II auxiliary points.

63

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1 z: 1 *7 r (.1 ‘ 1
+E—~Ak——»AL——i ———4 [ :] 1 -1
k+1+~—- : \——]—AL v.13. [ f k+1‘ X—i—l—va ].—E
52*] o‘ 1 [ 12*] t
k ] ,’I’ [ = . k ]
ledyozo) y ]
k-1L—7‘ I 4 1 k-1 X. __‘.____
’ 4 : [ l 1 I
i ] m1 ]
I 1+1 I

 

A
g:
V

Figure 3.2: (a) Type I auxiliary points A typically selected in MIB 1; (b) Type I
auxiliary points A typically selected in MIB 11.

3.1.3 Matrix optimization

In the calculation of each pair of fictitious values, 12 type I auxiliary points are in-
volved. In most situations, the locations of type I auxiliary points are not unique.
The utility of this non-uniqueness was not explored in our previous MIB schemes. In
fact, the selection of type I auxiliary points has a substantial impact to the conver-
gence property of the MIB matrix. Essentially, it is important to make the matrix
optimally symmetric and banded so as to accelerate the speed of the convergence of
linear algebraic solvers. However, in practice, it is not an easy task to choose type
I auxiliary points because of the complex geometric constraints. In our earlier MIB
method, denoted MIB 1, our consideration was dictated by geometric complexities
and little attention was paid to the matrix optimization. Since the locations of type
I auxiliary points are very flexible, a systematical strategy is introduced to select. op—
timal type I auxiliary points so as to maximize the speed of matrix convergence and
minimize the numerical error of the interface scheme. This new approach. denoted
as MIB II, is described as follows.

A typical interface topology is depicted in Fig. 3.2. In this case, 6 type I auxiliary

+

)oints are re uired to calculate it? or '11—. Here u. re( uires 6 tv )e I auxiliarv oints
~ Z a l v u

64

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

l ,1 Z] l . . [3 z} [ '12 ‘ ]
k+2 — [_"I: ‘ ] [r—q‘ k+250oo+ ....... 4...]III‘IIIHIIII k+2 1 : iv — .
H ' f ' - i ' 1 , 1 l- . ]—]
k+1huaob ...... {Int-gt. ....... q k+1hooupoochulo<uu uyidonnq-o-I k+1[ __,‘.__._. ..:_..—4]—71_.__
f' 1: ] f 1 [ 1 1: .'-] (xo,yo,.on ;: . ' ‘ ] [
k ] Wk f L y : k 1" y‘4 k" . \3.a' ] AX__.
‘ l1 I T I | | v
I",I[: (Xo'yp,zo) ] 1 E”,'f:\, L ”,1 : ! ]
k—1r-oob03'fiou-q'o-I]ooou ccccccc - k—1) 7#——o+—onay ~20, k-‘I IIIrI:,h...].4DIOII01IIOIOOO1
/ . : -? -, i . x 1: , x : 1 , i
“-2 A L¥ ]: + ‘ i k-Z 7X ][ ;: j k-Z...u‘-x-gunned-Io-uldl-IOWOIIJI
] 1? l ' ._ n 1 . . . .1 .‘
I"? I“ i I” 1‘2 1+3 1'2 l'1 1 1+1 1+2 1‘3 1‘2 l’1 1 1+1 1‘2 1+3

(a) (b) (C)

Figure 3.3: Three different sets of mesh lines from which type I auxiliary points are
selected. Set (a): mesh lines zk_1 and zk+1; Set (b): mesh lines zk+1 and zk+2; Set
(c): mesh lines zk__1 and zk_2.
all from {2+ while 11:— requires 6 type I auxiliary points all from (2‘. Since normally
only one of these two derivatives are computed, one could select either (2+ or Q‘.

The next step is to decide which two mesh lines should be used to accommodate for
these 6 type I auxiliary points. The present MIB II method prefers using mesh lines
zk_1 and 3k+1 together, while the earlier MIB I method typically does not choose
this kind of combinations. The strategic difference of MIB I and MIB II in this step
is indicated in Fig. 3.2. In the present work, the essential idea is to select type I
auxiliary points as close to the intersecting point as possible. The CPU time can be
significantly reduced in MIB II by using the new strategy, especially when the MIB
method are applied to large computational domains and/ or complicated interfaces.

In order to optimize the proposed MIB II method for accelerating the matrix
convergence and minimizing numerical errors, the selection of type I auxiliary points

is separated into three steps:

0 Step 1: Select. a subdomain, 9+ or (2‘. In solving the Poisson-Boltzmann
equation, there is a tendency to select the out subdomain such that the inter-
ference with the charge singularity can be avoided. In general, an appropriate

subdomain is selected to optimize the matrix property.

0 Step 2: Select a set. of mesh lines, 1. e., a pair of mesh lines, as shown in Figs.
3.3(a), (b) and (c). In normal cases, the scheme in Fig. 3.3(a) is preferred in

our MIB II methods.

65

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

] f z ] ] I f z I ] ] ]
k+2———-¢—~—1i —- -——*——4 k+2 ————-1—-~fi ——+—— .~——.
: X 1 : ,’
k+17~«]——4—&1—]L7’—'A——]-—— k+1 T 1 ,4] ] ]
' I x ' l x’ ' ' i
k [ ] i} ' lay k] ] “\ ] ”A
I | rl . ‘ - ‘
k-1 Fir f E (xo'yo zo)—+—— k-1]‘ ' wkow
x ; [ ; j .
k-2 ‘]x f 15% ]-~ k-2 ‘x E 1 [
1'2 1‘1 1 1+1 i+2 1+3 1‘2 1'1 1 1+1 #2 H
(i) (ii)
] ] [I z [
k+2 ' :~ .2 , 1
k+1 ] ] é , .-F
i I ' a" l
k [ l 'L’ Ly
. ] , 1:\
k-1 I "i,’ I; (X vYo’zp) ]:
H Ml, 15 l
1 ‘ [i
] L: f .
1‘2 1'1 1' 1+1 1+2 1+3 l‘2 l'1 I 1+1 1+2 1+3

(iii) (iv)

Figure 3.4: Four sets of type I auxiliary points on a single mesh line zk+1.

0 Step 3: Select two sets of grid points from 4 different options shown in Figs.
3.4(i), (ii), (iii) and (iv). Note that an option can be used twice. In normal
cases, choice of 3.4(i) or (ii), or their combination is preferred. However, the

selected grid points must be within the same selected subdomain.

The process of selecting type I auxiliary points is described in details in the following

pseudocode:

A pseudocode for the selection of Type I auxiliary points
1: SD = 0+; D Select a subdomain (SD) to which all 6 points should belong
2: M L. = (a); 1> Select a preferred mesh line set (M LS)
3: Search for 3 points in SD on each mesh line in M LS, following the order of (i),
(ii), (iii), and (iv).
4: if all 6 points are found on these two mesh lines then

Return;

0"!

66

6: else
7: MLS = (b);
8: Search for 3 points in the SD on each mesh line in the M LS, following the

same order described above.

9: if all 6 points are found on these two mesh lines then

10: Return;

11: else

12: MLS = (c);

13: Search for 3 points in the SD on each mesh line in the M LS, following

the same order described above.

14: if all 6 points are found on these two mesh lines then
15: Return;

16: else

17: Goto 1, choose SD = Q- and repeat the rest;

18: end if

19: end if

20: end if

3.1.4 Type I auxiliary points

Indeed, the proposed selection strategy of type I auxiliary points not only leads to
optimally symmetric MIB matrices, but also introduces additional flexibility to the
MIB method. For instance. in the situation depicted in Fig. 3.5(a), type I auxiliary
points can only be (1',j — 2, k —1),(z',j —— 1,13 —1),(z',j,k — 1) on mesh line 2 = zk_1
and (2,)" — 2, k+ I), (2',j — 1, 19+ 1), (i,j,k+1) on mesh line 2 = ck“. Therefore, the
present MIB II allows one to use mesh lines from both upper and lower quadrants,
i.e., z = zkfll and z = zk+1 in this situation, simultaneously.

In dealing with complex interface geometry, there are a few cases in which type
I auxiliary points cannot all be found by following the aforementioned schemes. For

example, in the situation depicted in Fig. 3.5(b), type I auxiliary points can only

67

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

4 0
1-]2 i-]1 ] i+1

 

 

 

...—.__._,

1

 

 

P--—---*

N
‘ "r——i

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

j-2

Figure 3.5: Type I auxiliary points and other three points on mesh line zk, all in blue
0, are used together to derive 112'. Situation (a): Type I auxiliary points locate on
mesh lines zk_1 and zk+1; Situation (b): Type I auxiliary points locate separately
on mesh line zk_1; Situation (c): Type I auxiliary points locate on nonconsecutive

mesh lines zk_2 and zk+1.

68

m- a A, ' 1” --
[ 9+ [ l1» .yo.zo)\
\ I’% y

y .
\qJ-J '
.’, : . +
x; ./ Q
1-1 1' 1+1 1+2

be (i,j — 1,k -1), (i,j + 1,1; — 1), (i,j + 2,]: — l) on mesh line 3 = 311—11 and
(2T,j - 2,11: + 1), (2,.) — 1,k +1) and (2,),117-1— 1) on mesh line 2 = 2k+1- Similarly,
there is another kind of cases in which type I auxiliary points cannot be found in
consecutive mesh lines. One mayneed to use 2 = zk_1 and z = zk+2 together for
the calculation of u; or 212' instead of mesh lines 2: = zk_1 and z = Zk+1~ To resolve
these two kinds of difficulties, we first check whether primary fictitious values can be
computed from other directions or other intersecting points. If this is true, then we
just compute the primary fictitious values from another direction. If it is not true,
then a new scheme is introduced to allow the use of non—consecutive type I auxiliary
points as in Fig. 3.5(b) or non-consecutive z-mesh lines as in Fig. 3.5(c). Note that
when non-consecutive type I auxiliary points or non-consecutive z-mesh lines are
used in interpolations, the matrix becomes less symmetric and the accuracy of the
interpolations would be slightly reduced. However, our numerical tests indicate that
the accuracy of the fictitious values solved by using the new scheme is still comparable
to that of other fictitious values solved by our previous schemes. An explanation is
that the interpolation by type I auxiliary points only plays a minor role in the whole
calculation of fictitious values. In addition, the situation depicted in Figs. 3.5(b) and
(c) often disappears when the mesh is refined. Nonetheless, schemes similar to those

in Figs. 3.5(b) and (c) should be avoided whenever possible.

3.1.5 Type II auxiliary points

Unlike type I auxiliary points which are only used in Eq. (3.12) and have many
options as described above, type II auxiliary points are directly involved in all the
jump condition equations and have unique locations once the corresponding irregular
points are selected. Furthermore, in the above scheme for smooth interfaces, type II
auxiliary points and the irregular points next to them must. be in the same subdomain.
For instance, to find the fictitious values at irregular points (2, j, k) and (2', j + 1, k),
auxiliary point (2., j — l, k) and irregular point (2, j, k) must be in the same subdomain,

and similarly, auxiliary point (27, j + 2, A?) and irregular point (2, j + 1,1.) must be in

69

the same subdomain too.

However, when the interface has large curvatures, sharp edges, sharp wedges, tips
or corners, there are ‘improper’ regions [161] where the required auxiliary points can—
not be found in the same subdomain. For example, as depicted in Fig. 3.6, irregular
point (2, j, k) has no auxiliary point associated with it inside the same subdomain.
This ‘improper’ region cannot be eliminated by refining the mesh. The same situation
occurs in the neighboring mesh lines if the mesh is doubled. The auxiliary points in
the ‘improper’ region lead to inconsistence between the number of unknowns and the
number of jump conditions.

The MIB schemes for sharp-edged interfaces [161] were developed to deal with
this situation. In Ref. [161], the secondary fictitious points were introduced to re-
place the role of type II auxiliary points in the ‘improper’ region. Secondary fictitious
values are computed by appropriate interpolations and are used as auxiliary points
to determine primary fictitious values. Consequently, the number of unknowns can
be reduced, and the resulting unknowns can be solved successfully. This MIB scheme
gives satisfactory results and has been used in the present work for geometric singu-
larities. However, it has a critical acute angle limitation because the scheme depends
on a priori calculation of secondary fictitious values. When edges become very sharp,
it is likely that no secondary fictitious value can be computed around the primary
fictitious point. Here, we present a new scheme that deals with sharp-edged problems
by using an additional interface condition to solve primary fictitious value directly.
This new scheme therefore removes the critical acute angle limitation in our earlier

MIB method [161].

Second order scheme

Consider the situation depicted in Fig. 3.6(a), type II auxiliary point (2, j — 1, Ir) and
irregular point (2, j, k) are not on the same. subdomain. Fictitious values cannot be
calculated directly from Eqs. (3.2) and (3.12). Assuming that point (2,j -— 1.1:) is in

(2‘ with secondary fictitious value fi.j—1.k- Then, Eq. (3.2) can be discretized with

70

 

 

----5

 

1
k+1 __L_,._ _L’:___]_'._]
(X’ .y '

V
O O

 

z.
'N.
b 0"
\
M ~.
A s
x s
O _
~<
0'
N
O
v
0— —
"<
H

 

 

 

 

 

EV
bI

 

 

 

 

 

 

 

 

 

p--------

 

 

 

1—1 1 141 1+2
(a) (b)

Figure 3.6: Interface singularities and type II auxiliary points. Type II auxiliary
points and two irregular points, all in blue 0, are used in all discretization of jump
conditions about interface point (220, yo, 20) and an additional jump condition about
the second interface point. Situation(a): Type II auxiliary point (2, j — 1, k) and
irregular point (2, j, k) are in different subdomains, which creates a second interface
point (235,434,); Situation(b): Type II auxiliary point (2', j + 2,12) and irregular
point (2, j + 1,12) are in different subdomains, which creates a second interface point
(133. 313, ZS)-

prirnary fictitious values, f2.j.ka fi~j+1~k and secondary fictitious value fig-4,),

['11] = (w0,j—1. 1110,), w0,j+1) ' (fry—1,2, 711,312: fi,j+1,k)T (3-15)

—(w0.j= w0,j+11 waj+2l ' (12,31,112)‘U'2',j+1.kaui.j+2,k)T-

Eq. (3.12) can be discretized in the same manner by simply replacing all the u.,:,j_1’k
with the secondary fictitious value f,7j_1.k.

To solve for three unknowns, f2“,j,ka fi.j+1,k and fi,j—1.k1 one more equation
is needed. For this purpose, we make use of another interface intersecting point
(12,218, 22,) located between (2, j — 1.10) and (2, j, 1;). Note that this interface intersect-
ing point always exists in the situation that (2. j — 1, k) and (2, j,k) are not on the
same subdomain. we should use the jump condition of the function, which, unlike

the flux jump condition, does not introduce any additional derivative. The jump

71

condition at the intersecting point (23,0, y[,, 25,) can be discretized as

I T T
[U] = (“’0,j—1=U~’0.j»w0,j+1)' [UM—1.1;»ui,j.kafi,j+1,k) — (Haj—1,12»f1,j,kauz',j+1.k) ] -

(3.17)

Combining Eqs. (3.16), (3.17) and modified Eq. (3.12), the primary fictitious values
fiJ-k and fi,j+1,k can be represented in terms of u,,j_12k, “2.211%: “2,141,162 "i.j+2,k and
other 12 auxiliary points located in :1: — y and y — 2 planes.

In the situation depicted in Fig. 3.6(b), auxiliary point (2,,j + 2, k) and irregular
point (2, j + 1,12) are not on the same subdomain. The secondary fictitious value
fi,j+2,k is used to replace aid-4,2,), in the discretization of Eqs. (3.2) and (3.3). By use
of the jump condition on the interface intersecting point (1731313135) located between

(2,j + 1,12) and (2,j + 2,117), the third equation can be discretized as

T T
[111* = (ufifirwajfl,226d”)- [furjwafafirka111.j+2,kl - (f2,j.kaui,j+l,kaf2,j+2,k) ] -

(3.13)

Eq. (3.2) can then be discretized with primary fictitious values, fi,j,ka f2,j+1,k and

secondary fictitious value fi.j+2’k

[U] = (Hing—1,2004, 1110.341) ' (Um—1,1.» 111,332, fi,j+1,k)T (319)

-(u"0,]a u’O,]+19“',0]+2) . (f’I-JJCT u'7'9]+11k’ [2.14-23k) '

Eq. (3.12) can be discretized in the same manner by replacing all the ui.j+2,k by the
secondary fictitious value f2», j+21 11:-

Combining Eqs. (3.18), (3.19) and the modified Eq. (3.12), the primary fictitious
values fi,j,k, and f2,j+1_,k can still be represented in terms of-11.,j,j_1.k, 2114*, "i.j+1,ke
2nd-+2), and other 12 auxiliary points located in r — y and y — 2. planes.

Note that the above scheme assumes that either Ui.j+1,k and “2.3411.- locate on
the same subdomain as shown in Fig. 3.6(a), or ”id—1.}: and “1',ch locate on the same

subdomain as shown in Fig. 3.6(b). That is, there is only one secondary fictitious

72

value needed, which is fi.j—1,k in Fig. 3.6(a) or fiVj+2gk in Fig. 3.6(b). Therefore, one
more equation, Eq. (3.17) or Eq. (3.18), is enough to resolve the situation depicted
in Fig. 3.6(a) or Fig. 3.6(b), respectively. This is generally true in our second-order

MIB scheme.

Third order scheme

The main obstacle in constructing higher-order interface schemes for arbitrarily com-
plex interfaces is the presence of geometric singularities. It is important to under-
stand what high order interface schemes one can construct for a sharp-edged and
sharp-tipped interface depicted in Fig. 3.6(a). To this end, we discuss the feasibility
of a third order scheme. However, we have to relax some rules in the MIB method
in order to construct a third order one. Specifically, we need to admit asymmetric
or one-sided discretization schemes near the interface, instead of purely central finite
differences as employed in our earlier MIB method. To discretize Eq. (3.1) up to the
third order accuracy at a typical point (2, j, k) around the singularity depicted in Fig.
3.6(a), three fictitious values are required for a non-central finite difference scheme.
Let us denote them as figj+27k2 f2.j+1,k and f,,j_1,k. In addition, fictitious value
fi,j,k is required for the third-order non-central discretizations at points (2, j — 1, k)
and (2, j + 1, k). Therefore, around the typical singular point (2, j, k), four primary
fictitious values are to be solved. Consequently, we need to discretize four jump con-
ditions around two interface intersecting points, (any, 2) and (1],, y’o, z' ). The first

two equations are

,__,
~
N

L—J
Is‘
5
~
N

(3.20)

73

which can be discretized with function values and four unknowns. fiJ-yg, fi.j+1.ka

fi,j—1.k and f1“,j+2.k

11+ = (“’0.j—11“’0,j.~“’0.j+la“’0.j+2) ° (fry—1,2, "1.)“,2-1 f1.j+1,1.~, fi.j+2,k)T(3-21)
"— = ("10.33“10.1417"7’0,j+2»'U’0,j+3) ° (f2.j,k1 111.3412» “72.3422, “1,3”.le
“1+ = (“’f),j—1vwf),j=“’f).j+1v“"f),j+2) ' (f2.j—1,k1 “1411-7 fi.j+1,kv f2,j+2,le
“ _ = (“10.3422 “’0.j—l» “0.), “’0,j+1) ' (Haj—2.1;, “i,j—1.k1f2,j.k1 “27¢“,le

The other two jump conditions are given by Eq. (3.12) at points (1:0, yo, 20) and

(25,, y5,, 25,), respectively. Two pairs of partial derivatives, 21; , 21;, 213+, and 222+ at two

2+

intersecting points are eliminated. Here, 21+ 22,7 , 21,,

y , and u5,‘ can be discretized in a

+, 21‘, uH' and 21.". The discretizations of 212'

similar way as that in Eq. (3.21) for 12
(or 22;) and a; (or 21;) involve 2 x 2 x 3 type I auxiliary points around (230,310, 20).
Similarly, the approximations of 212+ (or 222‘) and 225,1“ (or 225,") take another 2 x 2 x 3
type I auxiliary points around (2:5,, 3,15,, 35,). After obtained all the fictitious values, the
third order discretization of the elliptic equation can be achieved without introducing
additional difficulty. The fourth-order central finite difference discretization should
be used away from the interface. While near the interface, third-order non-central
finite difference discretization can be used. This scheme is as feasible as the second-
order MIB scheme for geometric singularities. However, it is not clear at this point
whether a truly fourth-order MIB scheme can be constructed for arbitrarily complex
interfaces with sharp geometric singularities. In the following subsection, we describe

3D higher-order MIB schemes for smooth interfaces and for interfaces with moderate

geometric singularities.

3.2 Results and Discussion

In this section, we examine the convergence order, test the accuracy, validate the
matrix optimization and demonstrate the capability of the proposed MIB II schemes.

The impact of the proposed algorithm acceleration is studied with the molecular

74

surfaces of a diatom and a protein. Molecular surfaces are generated with the MSMS
program [129] with probe radius 1.4 and density 10. Reported CPU time is recorded
on an AMD Turion 64TM MT -30 laptop with 1.6GHz clock speed. The ability of the
proposed MIB II for handling geometric singularities is tested with systems of two,
three and eighteen intersecting spheres (van der Waals surfaces). Both molecular
surfaces and van der Waals surfaces are very important to the theoretical modeling of
biophysics and structural biology. Comparison is made to our earlier MIB technique,
the MIB I [168], in which the matrix is not optimized and interface singularities are
not treated. A challenging test case with a missile interface is provided to demonstrate
the robustness of the MIB II on severe geometric singularities. The numerical tests on
the fourth-order MIB method are carried out. Five different interfaces are used and
oscillatory solutions are compared. Moreover, cases with variable diffusion coefficients
and non-zero linear term are studied. Finally, we demonstrate the sixth-order MIB
scheme with three test cases. A detailed comparison on the performance of second-,
fourth- and sixth-order MIB schemes has been provided.

To analyze the numerical performance of our MIB schemes, we use two error
measurements, the maximum absolute error Loo and the surface maximum absolute

error E1

Loo = msgx [21(17, y, z) — uex(r, y, z)] (3.22)

E1 = mlax [22(2, y, z) — uex(r, y, 2)]

where u. and 220x are numerical and exact solutions, respectively. Here E1 is computed
over all the irregular points near the interface F where the modified difference schemes
are used. In many surface error maps, the difference, 21(.-2:, y, z)—u.Cx(:r, y, z), is plotted.
The standard preconditioned biconjugate gradient (BiCG) is used for solving the
linear algebraic equation system and the tolerance of the BiCG iterations is set to be
1.0 x 10‘6 in all test cases of second-order accuracy, and 1.0 X 10‘14 in all test cases

of higher-order accuracy. The diagonal of the matrix is used as the preconditioner.

75

The number of BiCG iterations, NBiCG: is listed for some test cases to access the
speed of convergence. The order of convergence is reported for the Loo errors in many
cases. In this section, we denote the subdomain enclosing the subjects of interest,

such as molecules and missiles, as ,Q‘ and the rest as 9+.

3.2.1 Case 1: Matrix optimization

The different selections of subdomains, sets of mesh lines and sets of auxiliary points
are compared in the following test case. In this study, a smooth interface as shown
in Fig. 3.7 is used to avoid the effect of interface singularities. The exact solutions

as shown in Fig. 3.7(a) are given by

u‘ = 10 cosrcosycos : + 20, 22+ =10(:2: + y + 2) +1 (3.23)

with coefficients

,3" = 1, 3+ = 80, and 1: = 0. (3.24)

The source term q(;r, y, z) and interface conditions [22], [1322"] can be easily derived from
the exact solutions. The computational domain is set to [—6, 6] x [-4, 5] x [—4, 4].
Table 3.1 gives a comparison of numerical errors, convergence orders, number
of iterations and CPU time for six different combinations of Step 1 and Step 2 as
illustrated in Section 3.1.3. The preferred auxiliary point sets (i) and (ii) are used in
this comparison whenever possible. The Loo orders of convergence are quite similar
in all combinations and the designed 2nd order is numerically confirmed. In terms
of numerical errors, different choices of mesh line sets in Step 2 do not affect much
the accuracy. However, the selection of Q‘ in Step 1 leads to slightly larger errors,
probably because of the geometric constraints of Q‘. The most significant difference
is in the number of biconjugate gradient iterations and CPU time. The proposed
mesh line set (a) requires 30-50% fewer NBiCG and less CPU time than other mesh

line sets, indicating the impact of proposed optimization schemes. Moreover, the

76

24
22
20

in.

 

0.07.
. 7 (MN
0.00
-0-01
i002

 

(b)

Figure 3.7: Surface maps of the exact solution, (a), and numerical errors, (b), for
the molecular surface of a diatom of atomic radius 1.5 A with atoms centered at
(—1.52,0, 0) and (1.35, 0, 0), respectively.

77

selection of {2‘ in Step 1 results in fewer NBiCG in most cases, possibly because 9‘
is the subdomain enclosed by the interface, and the auxiliary points selected from Q‘
is therefore more compactly distributed than those selected from 52+.

Table 3.2 compares six different combinations of Step 2 and Step 3 as illustrated
in Section 3.1.3 with a fixed subdomain (2+. It is seen that both iteration number
NBiCG and numerical errors increase slightly when non-compact auxiliary points are
selected in Step 3. However, no significant difference is observed. Comparing Table
3.1 and Table 3.2, the selection of Step 2 is the dominant factor for the speed of
convergence. In order to achieve the best result in terms of both numerical accuracy
and convergence speed, the combination of subdomain 9+, mesh line set (a) and
auxiliary point sets (i)(ii) is preferred. Fig. 3.7(b) illustrates the surface map of the
numerical errors obtained from this preferred selection at h = 0.25. It is evident that
most errors are distributed around the extrema of the solution.

Due to the simple geometry, the difference of various selections is not very obvious.
To better test our method, we consider a more complex system, the molecular surface
of a protein whose coordinates are obtained from the Protein Data Bank (PDB ID:
2pde). To obtain a full all-atom model for the protein, all attached water molecules
are cleaned and hydrogen atoms were added. Atomic van der Waals radii defining
the dielectric boundary were taken from the CHARMM22 force field [82], and the
molecular surface is generated by the MSMS [129]. It is well-known that the molecular
surface definition admits cusps and sharp self-intersecting surfaces [129]. This test is
motivated by the electrostatic analysis of biomolecules in the implicit solvent model in
structural biology. A detailed account of this aspect can be found in Ref. [158]. The
exact solutions for this system are still given by Eqs. (3.23) and (3.24), and illustrated
in Fig. 3.8(a). It is seen that the molecular surface of protein 2pde with 667 atoms
has a very irregular interface with geometric singularities. The computational domain
is set to [—16,16] x[—20, 15] x[—17.19] A3, which is slightly larger than the domain
of the molecular surface. All calculations are performed at the mesh size of ft = 0.5A.

Table 3.3 shows 12 different. combinations of subdomains, mesh line sets and auxiliary

78

 

I 13.03

 

Figure 3.8: Surface maps of the exact solution, (a), and numerical errors, (b), for
the molecular surface of protein 2pde.

79

point. sets. The improvement by using mesh line set (a) over that of (b) or (c) is more
obvious in this test case. Mesh line sets (b) and (c) require up to 30 times more
NBiCG than that required by mesh line set (a) given the same combination of Step 1
and Step 3. Moreover, mesh line set (a) is consistently more accurate than mesh line
sets (b) and (c). Table 3.3 also confirms that the combination of subdomain 0+, mesh
line set (a) and auxiliary point sets (i)(ii) is the best choice for faster convergence
and higher accuracy. A good alternative is the combination of Q“, mesh line set (a)
and auxiliary point sets (i)(ii).

The LoO order of convergence is investigated for this test case in Table 3.4. Mesh
line set (a) is used in all the cases in Table 3.4. It can be seen that second order
convergence is achieved in all four combinations although the numerical errors vary
in different combinations. The combination of subdomain (2+ and auxiliary point
sets (i)(ii) is still the best choice while the combination of subdomain Q" and point
sets (iii)(iv) gives similar numerical errors at h. = 0.25.

The exact solution is mapped onto the interface in Fig. 3.8(a). The numerical
errors obtained with the best combination and h = 0.5A are mapped on the interface
in Fig. 3.8(b). It can be seen that the numerical errors are about 0.3% of the original
solution at h = 0.5A (note the difference in scale). Another interesting point is
that there is a good correlation in the color distribution in two figures, indicating
that numerical errors are mostly induced by the amplitude variation of the solution
instead of the geometric variation of the interface. The matrix optimization and the
treatment of geometric singularities proposed in this work underpin our recent success

in solving Poisson-Boltzmann equation for electrostatic analysis of biomolecules [158].

3.2.2 Case 2: Validation on interfaces with geometric singu-

larities

The performance of the present MIB method on interface singularities is tested by
using the van der Waals surfaces of three different systems shown in Fig. 3.9. These

interfaces are composed of spheres with their intersecting sections removed. Singu-

80

   

(a) (b) (C)

Figure 3.9: Interface of intersecting spheres. (a) two intersecting spheres of radius
1.5 with centers (—1.52,0,0) and (1.35, 0,0). (b) three intersecting spheres of radius
1.5 with centers (—1.52,0,0), (135,0, 0) and (0,1.62,0). (c) 18 intersecting spheres.
The coordinates of the center and radius of each spheres are given in the form of
(x,y,z,r) as follows: (-2.0270, 0.9540, -0.6510, 1.7), (-1.6690, 0.2340, 0.6650, 1.7), (-
0.4530, -0.6870, 0.4410, 1.7), (0.7510, 0.1480, -0.0400, 1.7), (0.3930, 0.8680, -1.3560,
1.7), (0.8230, 1.7880, 4.1320, 1.7), (2.2840, 0.2080, -1.4180, 1.2), (2.8880, 1.6170,
-0.4830, 1.2), (-2.5270, -0.3680, 0.9970, 1.2), (-1.4260, 0.9800, 1.4350, 1.2), (-0.1960,
-1.1890, 1.3850, 1.2), (0.7010, 4.4410, 0.3200, 1.2), (1.0070, 0.8940, 0.7270, 1.2),
(1.6120, -0.5150, —0.2080, 1.2), (0.1490, 0.1210, -2.1260, 1.2), (1.2510, 1.4700, -1.6880,
1.2), (—1.0810, 2.2910, -2.0760, 1.2), and (-0.5750, 2.5430, -0.3710, 1.2).

 

(a) (b)

Figure 3.10: Cross section views of interface generated by intersecting spheres. (a)
A cross section of Fig. 3.9(a); (b) A cross section of Fig. 3.9(b).

81

larities are generated at the intersecting wedges and tips as shown in Fig. 3.10, which
gives the cross-section view of the interfaces. Fig. 3.10(a) indicates a sharp wedge
generated by the intersecting of two spheres. Fig. 3.10(b) illustrates a sharp tip and
sharp wedges generated by the intersection of three spheres.

The same exact solutions as those in Case 1, given by Eqs. (3.23) and (3.24),
are employed in the present test examples. The computational domains are [—5,4]
x [—3,3] x [—3, 3], [—5,4]x[—3,5]x[—3,3] and [—6,4]x[—4,5]x[—5, 4] for three test
cases whose interfaces are given in Figs. 3.9 (a), (b) and (c), respectively. The
numerical errors are computed at mesh sizes h = 0.5, 0.25 and 0.125 by using the
present MIB method (MIB II) and our earlier MIB method (MIB I), in which the
surface singularities are treated by simply replacing unknown fictitious values with
the values at neighbor points. Table 3.5 lists the overall maximum absolute error
Loo and the surface maximum absolute error E]. It is seen that the MIB 11 method
achieves the designed second order convergence in all three test cases. In fact, the
accuracy of the MIB II is independent of the geometric complexity. At a given mesh
size, the same level of accuracy is achieved by the MIB II in all three interfaces. In
contrast, the MIB I scheme does not converge because of the geometric singularity.
Moreover, Loo and E1 errors are the same in MIB I, indicating that the largest error
occurs at the interface. Since our 3D MIB I is of second-order convergence for smooth
interfaces, these Loo and E1 errors must be from geometric singularities. These
results demonstrate the importance of special treatments for geometric singularities
in practical applications. Ordinary interface methods forfeit their power at sharp-
edged and sharp-tipped interfaces.

In addition, Figs. 3.9(a) and 3.10(a) show clearly that the intersecting wedges are
very sharp. The MIB II method handles this situation very well and is not subject
to the critical angular limitation [161] any more. The following test case is designed

to further confirm this point.

82

3.2.3 Case 3: A missile interface

To further verify the ability of the proposed MIB scheme for treating arbitrarily
complex interface singularities, we consider a missile geometry which was reported
by Wang and Srinivasan [145] as a challenge for grid generation. The missile tip is
very sharp and the missile fins are very thin compared to the size of the geometry,
see Fig. 3.11. These features normally take enormous amount of grids to reduce the
local numerical errors. The computational domain is set to [—5,49] x {—12, 11] x
[—12,11]. Two sets of solutions are tested on this interface. The set of solutions of

Case 3(a) is given by

u— = cosxcosycos 2, 11+ = 0. (3.25)
While Case 3(b) adopts the solution set given by Eq. (3.23). The coefficients .13 and
a in these two cases are the same as those given in Eq. (3.24).

The numerical errors of both cases are listed in Table 3.6. For both Case 2(a) and
Case 2(b), their convergence order from h = 0.5 to h = 0.25 is slightly lower than
what is designed because it is much more computationally challenging at h = 0.25
than at h = 0.5 in the following sense. At 11 = 0.5, there are a fewer irregular grid
points located near the sharp edges of the missile. In other word, much sharp feature
of the missile has not been recognized by a coarse mesh yet. While when the mesh is
refined to h. = 0.25, the sharp feature captures more grid points and there are more
irregular points across the interface. When the mesh is refined further to h = 0.125,
the number of irregular grid points increases again. It is seen that the proposed MIB
scheme is of second order convergence in the last mesh refinement from h = 0.25 to
h. = 0.125.

Fig. 3.11 provides the exact solution of Case 3(b) and errors obtained at three
different mesh sizes. An important feature of our results is that, the errors are dis-
tributed around the extrema of the solution. This feature suggests that the thin

fins and the sharp tip are properly treated and do not cause additional errors in our

83

14161820222420

41.09 -0.06 0.03 0 0.03 0.06 0.09

   

a) (b)

”3: :2
q o

,. 0 pg
HS '6]
° 8

§ :5

a' 8

8.

O 0

g o

a 3.
9 9

IO N

‘i’ 9‘

e

8

°.

   

(C) ((0

Figure 3.11: A missile geometry with solution and error maps. Numerical errors
are represented in seven scales. (a) exact solution mapped on the surface of a missile;
(b) numerical error obtained at h = 0.5; (c) numerical error obtained at h = 0.25; (d)
numerical error obtained at h = 0.125.

84

scheme. Indeed, there is little error on the tip and the fins due to our special treat-
ment of singularities. The level of accuracy in Case 3(b) remains the same as that
in Table 3.5, confirming that interface complexity and geometric singularities do not

compromise the accuracy of the proposed MIB method.

3.3 Conclusion

This chapter introduces the three-dimensional (3D) matched interface and bound-
ary (MIB) method for solving elliptic equations with discontinuous coefficients and
geometric singularities. The 2D second-order MIB scheme [161] for solving elliptic
equations with discontinuous coefficients and interface singularities is extended to a
3D MIB scheme of up to fourth-order convergence for interface singularities. Non-
smooth interfaces are notoriously challenging and the best result in the literature is
of 0.8th order convergence by Hou and Liu [72]. New algorithms, particularly the
use of jump conditions at two interface intersecting points, are proposed to effectively
remove the critical acute angle restriction of our earlier MIB scheme [161]. The result-
ing MIB schemes are able to attain second-order convergence for arbitrarily complex
interfaces with sharp edges, wedges, and tips. Furthermore, we propose a systematical
procedure to make the MIB matrix optimally symmetric and diagonally dominant.
Consequently, our new MIB algorithm requires a fewer number of iterations to con-
verge by using a standard preconditioned biconjugate gradient solver. The proposed
3D MIB schemes are extensively validated in terms of the speed of convergence, the
order of accuracy, the number of iterations and CPU time. A large variety of numer-
ical examples, including a protein molecular surface, a missile interface, and systems
of multiple intersecting spheres, are utilized to validate the proposed MIB method.
First known results of second order accuracy for arbitrarily complex interfaces with

sharp geometric singularities are obtained for 3D elliptic equations.

85

 

 

 

 

 

 

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89

Table 3.5: Convergence test on interface of intersecting spheres. Case 2(a): two
intersecting spheres; Case 2(b): three intersecting spheres; Case 2(6): 18 intersecting

 

 

 

 

 

spheres.
Case 2(a) Case 2(b) Case 2(e)

h MIB II MIB I MIB II MIB I MIB II MIB I
0.5 Loo 2.576 — 1 1.55 2.146 — 1 1.05 2.386 — 1 2.46
E1 2.576 — 1 1.55 2.146 - 1 1.05 1.096 — 1 2.46

0.25 Loo 3.216 — 2 1.00 3.736 - 2 1.21 5.836 — 2 2.86
E1 2.286 — 2 1.00 2.576 - 2 1.21 4.966 — 2 2.86

Order 3.0 0.6 2.5 —0.2 2.0 —0.2

0.125 Loo 7.216 — 3 1.21 8.666 — 3 1.16 1.426 — 2 1.29
E1 5.236 — 3 1.21 5.586 — 3 1.16 3.826 — 3 1.29
Order 2.2 —0.3 2.1 0.1 2.0 1.2

 

 

 

 

 

Table 3.6: Convergence test on missile interface shown in Fig. 3.11. The exact
solutions of Case 3(a) and Case 3(b) are respectively given by Eqs. (3.25) and (3.23),
with coefficients defined in Eq. (3.24).

 

 

 

 

 

 

Case 3(a) Case 3(b)
h Loo E1 Order Loo E1 Order
0.5 1.636 — 2 6.016 — 3 1.646 — 1 6.016 —- 2
0.25 4.806 — 3 2.526 — 3 1.8 4.956 - 2 2.626 - 2 1.7
0.125 1.196 — 3 4.446 — 4 2.0 1.196 — 2 4.446 - 3 2.1

 

 

90

Chapter 4

Three-dimensional fourth and sixth

order matched interface and

boundary (MIB) method

A new trend in this field is the construction of higher-order (interface) methods
[61, 112, 138, 163, 169, 170] that are particularly desirable for problems involving
both material interfaces and high frequency waves where conventional local adap—
tive refinement approaches do not work well. Typical examples are the interaction
of turbulence and shock, and high frequency wave propagation in inhomogeneous
media. Simple Cartesian grids are preferred in these situations because of the by-
pass of the mesh generation, better temporal stability and the availability of fast
algebraic solvers. Recently, a systematic higher-order method, the matched interface
and boundary (MIB) method, for treating electromagnetic wave propagation and
scattering in dielectric media has been proposed[163]. In particular, the use of high
order jump conditions, which require large stencils and are numerically unstable in
higher-order schemes, is avoid. By repeatedly using only the lowest order jump con-
ditions, an arbitrarily higher-order interface scheme can be constructed in principle
[163]. More recently, the MIB method is generalized for solving elliptic equations with
curved interfaces [169, 170]. The MIB method makes use of simple Cartesian grids,

standard finite difference schemes, lowest order physical jump conditions and ficti-

91

tious domains. The physical jump conditions are enforced at each intersecting point
of the interface and the mesh lines. Because of the use of the sub-grid information
at the intersecting points, the MIB method overcomes the staircase approximation
of ordinary finite difference schemes for complex geometry and can be of arbitrarily
higher-order convergence in principle when the lowest order physical jump conditions
are repeatedly enforced. MIB schemes of up to 16th order accuracy have been con-
structed for straight interfaces [163, 170]. For two—dimensional curved interfaces, 4th-
and 6th-order MIB schemes have been demonstrated [169, 170].

The objective of the present chapter is to extend the earlier 2D higher-order MIB
schemes [163, 170] to 3D. We report some first known sixth-order 3D results for elliptic
problems with smooth interfaces. Such schemes are particularly efficient for problems
involving both material interfaces and high frequency waves. We demonstrate that
3D 4th-order MIB schemes can be constructed for interfaces with moderate geometric
singularities.

In the next two sections, the theoretical formulation and the computational algo-
rithm are given. A systematic procedure is introduced to construct 3D higher-order
MIB schemes. The proposed MIB method is extensively validated by benchmark

tests. This chapter ends with a conclusion.

4.1 Theory and algorithm

The MIB method was originally proposed as an arbitrarily higher order interface al-
gorithm [163]. The fourth- and sixth-order MIB schemes in 2D have been developed
for curved interfaces [169, 170]. The 3D generalization of higher-order MIB schemes
is described in this work. In brief, an nth-order MIB method means the use of the
nth-order finite difference discretization. Therefore, more irregular points, and thus
more fictitious values are required in our fictitious domain formulation, although an
equivalent interpolation formulation can be constructed to avoid the use of fictitious
values. On the other hand, because we avoid using higher-order interface jump con-

ditions, the number of interface conditions remains the same. Therefore, lower-order

92

l
,
l

] s
/ 1
‘./.-i-------------

] ' Iz—
k+3 D

 

 

 

k+2 + \

/

 

k+1

1, -— 1»— IF— —-
‘\
\

\
_‘<_

 

 

 

 

 

A
X

 

 

 

 

 

 

 

 

 

._. -__ I»-

 

 

.g____,_-__

h—

h—

+
..s

Figure 4.1: Illustration of a fourth-order MIB scheme. The y-mesh line intersects
the interface at (1:0, yo, 20). A set of irregular points (i,j — 1, k), (i,j, k), (i,j + 1, k)
and (i, j + 2, k), in red 0, is on the y-mesh line. Type I auxiliary points are in blue
A. Type II auxiliary points are in blue 0. Auxiliary lines, in red dashed lines along
the x- and z-directions, are used for computing um and uz.

jump conditions are repeatedly used as proposed in the earlier paper [163]. The basic
procedure is to first generate a pair of fictitious values of required nth order accuracy
around an interface intersecting point with given lower-order interface jump condi-
tions. By repeatedly using the same set of jump conditions, we then add two more
fictitious values at a time until all the required fictitious values on a mesh line are
determined. This procedure is repeated for other interface intersecting points and

other mesh lines. Finally, the elliptic equation is solved by the nth-order central

finite difference scheme with appropriate fictitious values.

Generation of the first pair of fictitious values

Let us consider a 3D fourth-order MIB scheme. The first step is to calculate the first
pair of fictitious values by using fourth-order finite difference stencils to discretize
interface jump conditions.

Unlike in the second-order MIB scheme 11+ 11‘, 11+ and u.’ are now re resented
7 , y y p

93

by five-point finite difference schemes

+ _ ., . . . . .

“U - (wily—31w(),_']—2sw0,j—lvw0.]1w0.j+l) (4-1)
( . . . . . . . . . . T
”Ly-3.1mHaj—2k:“1.3—1k:“2,3,k1f2.3+1,k)

"— = (100,}. u'0.j+1, w0,j+2. u’0,j+31w0,j+4)

(f. . . . . . . . . . T
2,],kv“z.y+1,k=uz.]+2,k~“'z.]+3,k1uz,]+4.k)

+ — < - - - - 1 - - >

“y — 1U1.]-39w1,]—2?w1,]—19u1,]3w1,_]+1
(”2.3—3.};91‘24—2151 uz,]—l.k1 u-z,_].k1 f2.,j+1,k)

“,7 = ('W1,j.w1,j+1,w1,j+2-.u’1.j+31w1,j+4)

, T
(fi,j,keui,j+1,k1ui,j+2,keui,j+3.k1ui,j+4,k)

To approximate u: or u; with the fourth-order accuracy, we need five 21 values

along the auxiliary line 3; = ya on the y -— 2 plane, see Fig. 4.1. These values are
approximated by 20 more auxiliary points. In the situation shown in Fig. 4.1, u;

can be represented as follows

Uz— = (lec— 11 “’11, 101.1411 'w1.k+2~ u’1,1.~+3) (42}

T
(”1,0,111—11ui,0,k1 ui.0,k+11“i.e.k+21“‘i,o,k+3) .

94

where

“1.0,k—1 = (u’O,j+1=w0,j+211U0,j+3-'u’0.j+4aw0.j+5)

' (“1,j+1.k—11ui,j+2,k—1>ui.j+3,k—1aui.j+4.k—11“-1',j+5,k—1)T

”1,0,1: = (1004': U’O.j+1a U’O,j+2: wO,j+31 'w0,j+4)

' (fi,j,k1ui,j+1,k1ui.j+2,kaui,j+3,k1ui,j+4,k)T
ui,o.k+1 = (u’0,j+1eu’0,j+21u’O.j-+—31u’0,j+41wO,j+5)

' ("i.j+1,k+11ui,j+2.k+11ui,j+3,k+11ui,j+4,k+11ui.j+5,k+1)T
“13,0,k+2 = (way we, 1+1, H’0,j+2: 1170.343, w0,j+4)

' ('u'i,j,k+21u‘i,j+1,k+2=ui,j+2,k+21“'z',j+3,k+21ui,j+4,k+2)T
ui,o,k+3 = (u’0,j—1-u’0,ja u"0,j+1: w0,j+21u’0.j+3)

T
' (“17, j,k+3~ ui,j,k+31 U1, j+1,k+3a “2T, j+2,k+31 ui.j+3,k+31 U1,j+4,k+3)

Similarly, a; can also be computed along the auxiliary line y = ya on the :1: — y
plane, the dashed line shown in Fig. 4.1. Then, by solving Eqs. (3.2) and (3.12) with
l = 1 and m = 5, two fictitious values fi,j,k and fi,j+1,k can be easily represented in

terms of the function values of 48 nearby grid points.

Iterative generation of other fictitious values

Let us consider a typical case as shown in Fig. 4.1, in order to discretize am) at
point (2', j, k) by central finite difference schemes with nth-order accuracy, % fictitious
values on the right side of point (i, j, k) are required. An iterative scheme is applied
to determine the rest of fictitious values. The order of generating fictitious values is
illustrated in Fig. 4.2.

+

Here, n , u‘, 11+

y and u; are represented by six-point. finite difference schemes,

95

.1;-

1' 1+1

 

Figure 4.2: Illustration of iterative schemes. The y-mosh line intersects the interface
at ($0,110, 20). The fictitious values at 0 points are generated pairwisely. The order
of generating fictitious values is indicated by the numbers on the upper right corner
of 0.

which include the second pair of fictitious values fi,j—1,k and f1”,j+4,k

+ . . . . .
u = (wo,j~3,Way—2.14044,w0,].'wo.;+1.u'0,g+2) (4-3)
("1,1—3Jw"i,j-2,kvut.j—1,kr“1,3,kaf2,]+1.kaf1,j+2,k)
if = (WOJ—l)w0,j1u’0,j+11w0,j+21w0,j+31w0,j+4)
T
(fi,j—1,k~fi.j,k1“id—(1,19'll-z',j+2,k1“i,j+3,k1”1i,j+4,k)
+ .
"y = (”’1,j—3=“’1,j—217”1,j—11“’1,j1'u'1,j+11w1,j+2)
, , T
(“'1',j—3,ks“'i,j—2,k1“i,j-1,ka“2’,j,k1fi,j+1,k1f-1,j+2,k)
u; = (“11,j—11“’1,j1"01,j+11“-‘1,j+21‘wl.j+31U’1,j+4)

T
(fi,j—1,k1f1i,j,kv”i,j+1,k1ui,j+2,k1“z',j+3,kv“1',j+4,k) -

In these equations, fi.j+1,k and fi,j,k are resolved earlier, and are used as sec-
ondary fictitious values, i.e., the fictitious values that are used to resolve other ficti-
tious values.

The approximation of 11;]L or u; is still given by Eq. (4.2). The. only difference is

96

that here “-17.0,k is given by the following equation

“2104' = (WM—1» “10.11 ”0.141, 1110.342» U’O.j+3a w(),j+4)

. T ,
' (fi,j—1.k1fi,j,k1ui,j+1.kaui,j+‘2,k1“i.j+3,k1ui,j+4,k) - (4-4)

Similarly, u; can be computed along the auxiliary line y = yo in the :1: — y plane.
Then, by solving Eqs. (3.2) and (3.12) with l = 1 and m = 5 again, two more fictitious
values fi,j—1,k and fi,j+2,k can be easily represented in terms of the function values
of 48 points around them and the secondary fictitious values fink/C and fi,j+1,k- In
addition, it may be noticed that fi,j,k and fi,j+1,k are also functions of the same 48
surrounding points. Therefore, the newly found fictitious values fi,j—1,k and fi,j+2,k
can be discretized in terms of the function values of 48 surrounding grid points.

All the other fictitious values labeled in Fig. 4.2 can be systematically found in a
similar manner. It is important to note that the accuracy of the ith pair of fictitious
values is restricted by the discretization accuracy of the first pair of fictitious values.
Therefore, in order to obtain 6th-order schemes, a seven-point finite difference stencil

should be used to generate the first pair of fictitious values.

High order discretization of elliptic equations

The fourth-order discretization of Eq. (3.1) at point. (1,], k) is then given by

1333111 +1311,” + 53yuy + flayy + 5311,: + .3112; — Ku(i,j, k) = q(i,j, k) (4.5)

97

where

Ux==(Una—2auur—1au041u0¢+2auu¢+3l 016)
.~. . ~. . .. .~. . ~. . T
' (“z—2,],k1,uz—l,],k1 u2,],k‘ ”2.+1,_-),k=“2+2.],k)
Urx==(Hes—2aHes—lvueseu2¢+2aues+3)
.(~..~....~..~..T
.u't-2.],h" uz—1,],k1uz,J.k1 uz+l,j,kv “2.4-2,3.“
uy==(”NJ-2:“Hg—lsufljsuflj+2fw1j+3)
.(~.. ~.. .. ~.. ~.. T
Haj—21-UaJ—1ws”aakvuay+rk1uay+2kl
Uyy = (1121-2 1024—11 102.1: w2.j+2a w2.j+3)
.~.. ~.. .. ~.. ~.. T
' (um—2.1.“-“2.3—1.161uz,],k1u2,]+1,kauz.y+2,k)
Uz==wak—2JULk—1JULkflULk+2flvrk+3l
. (11.. 1].. u-- 1].. 2].. )T
2,],k—21 2,],k—11 2,].ka "I-,],k+1‘ 2,],k+2
“z: = (10218—21 w2,k— 1- 1021a u’2.k+2= w2.1-+3)

~ ~ ~ ~ T
' (“1“,j,k—2~.Ui,j,k—1=Ui,j,ka Ui,j,k+1. “‘i,j,k+2)
and

u] m n if (1,771, n) and (i,j, k) in the same subdomain,
17,13,721": , a (4'7)
fl.-m,n otherwise.
Obviously, the 3D MIB procedure outlined here is systematic and is of arbitrarily
high order accuracy, in principle. in practical applications, complex interface and
geometric singularities give rise to the difficulty of finding required auxiliary points.
A wide variety of MIB techniques developed in our earlier work [161, 169, 170] and
in the present work will alleviate this difficulty.
As shown in an earlier section, it is possible to slightly alleviate the difficulty of
finding required auxiliary points by using non-central or one-sided finite difference
discretizations near the interface. This is similar to the MIB treatment of boundary

conditions [164]. However, the difficulty persists for sharp-edged and sharp—tipped

98

interfaces. In the following section, we explore the capability and examine the limi-

tation of the proposed MIB method.

4.2 Results and Discussion

In this section, the numerical tests on the fourth-order MIB method are carried out.
Three different interfaces are used and oscillatory solutions are compared. Moreover,
cases with variable diffusion coefficients and non-zero linear term are studied. Fi-
nally, we demonstrate the sixth—order MIB scheme with three test cases. A detailed
comparison on the performance of second-, fourth- and sixth-order MIB schemes has
been provided.

The standard preconditioned biconjugate gradient (BiCG) is used for solving the
linear algebraic equation system and the tolerance of the BiCG iterations is set to be
1.0 x 10"14 in all test cases of higher-order accuracy. The number of BiCG iterations,
NBicg, is listed for some test cases to access the speed of convergence. The order of

convergence is reported for the L00 errors in many cases.

4.2.1 Case 4: Fourth-order MIB scheme

Higher—order methods are particularly valuable for problems with high wavenumbers.
Normally, it is difficult to construct higher-order methods for complex geometry.
Moreover, the presence of interfaces will lead to low regularity in solutions and reduce
the convergence order of common higher-order schemes. Furthermore, the presence
of interface singularities will reduce the convergence order of traditional interface
methods that are not designed for sharp edges and sharp tips, as shown in Table 3.5.
Consequently, the best result in the literature was of 0.8th order [72] for 2D elliptic
equations with interface singularities before year 2005, to our knowledge. In the
present study, we demonstrate our fourth MIB method for elliptic interface problems
with moderate interface singularities.

To examine the proposed method, we first employ three typical and relatively

99

I
14101820222420

. ‘ IE-
0.002 0.004 0.000

0

-o.ooo 41.004 0.002—

   

(a) (b)
I: '8.

n O

I: g

g

I:

q

   

(c) (a)
Figure 4.3: The surface maps of exact solutions (left column) and numerical errors

(right column). Numerical errors are represented in seven scales (a)(b) Ellipsoid;
(c)(d) Cylinder.

100

   

V ('9
I“, ., E
a g
.— Q
N O
In .-
g‘ 8.
0
5‘3
'0. o
2
2 §.
°.
N
8
=.1
I.
q
q
(6) (0

Figure 4.4: The surface maps of exact solutions (left) and numerical errors (right).
Numerical errors are represented in seven scales. (e)(f) Oak acorn

simple test interfaces, an ellipsoid, a cylinder and an oak acorn.

0 Case 4(a): Ellipsoid

(312411242121 (....
20 25

25
where a = 7, b = 1—4 and c = fi' The computational domain is set to

[—5,5]x[—5, 5]x[—5, 5].

0 Case 4(b): Cylinder of height 2% and diameter 7r. The computational domain
is set to [—4,4]x[—4,4]x[-2,8.4].

0 Case 4(c): Oak acorn

101

and

:1:2+y2+(::—g)2=R2 ifzSO (4.10)

6 1 ' 15 R2 -— 2
where q = —5, g = 5, = 7 and d = ——9—g—. The computational domain
(1“
is set to [—5,5]x[—5,5]x[—5,5].
We set solution as
u‘ = 10 cos kr cos ky cos k2 + 20, 11+ =10(.T + y + z) + 1, (4.11)

with coefficients prescribed by Eq. (3.24). These three interfaces are depicted in
Fig. 4.3 and 4.4 with surface maps of exact solutions and numerical errors computed
at k = 3. The cylinder has two corners and the oak acorn has an edge and a tip.
However, these singularities are quite mild. The solution becomes oscillatory when
a large k is used. As in earlier cases, there is a good correlation between the exact
solutions and the numerical errors, indicating that the extrema of the surface errors
are induced by the large amplitude of the solutions, rather than by the geometric
features.

Table 4.1 shows the numerical errors given by the second- and fourth-order MIB
schemes. It can be seen that the designed second- and fourth-order convergences are
achieved in all the test cases. Under the same mesh size, the numerical errors obtained
by the fourth-order method are up to 1000 times smaller than those obtained by the
second-order method.

It remains to know whether the proposed MIB method delivers fourth—order ac-
curacy for more complex interface geometry and interfaces with more challenging
geometric singularities. To this end, we consider two more test cases, an apple and a

flower.

102

H 3505

2505
18%

ans
I 4506
4505

 

a 35115
25115
- - 12.115
413-05
-2505
45115

   

(a) (b)
Figure 4.5: The surface maps of exact solutions (Top row) and numerical errors

(Bottom row). Numerical errors are obtained at h = 0.2 for the aplle and at h = 0.1
for the flower. All images are plotted in seven scales. (a) Aplle; (b) Flower.

103

Table 4.1: Convergence test of the fourth-order MIB scheme (Case 4). Three inter-
faces are shown in Fig. 4.3 and 4.4. The exact solutions are given by Eqs. (4.11) and

(3.24).

 

 

Second-order

F ourth—order

 

Order

 

 

 

 

 

 

 

h Loo E1 Order Loo E1

Ellipsoid k = 1 0.4 1.256 — 1 2.306 — 2 2.236 — 3 2.236 — 3
0.2 3.426 — 2 3.546 — 3 1.9 1.446 — 4 1.136 — 4 4.0
0.1 8.706 - 3 5.706 — 4 2.0 1.046 -- 5 3.506 -— 6 3.8

k = 3 0.4 1.026 — 0 8.296 — 1 3.186 — 1 3.186 —1
0.2 3.326 — 1 1.016 — 1 1.6 1.876 — 2 1.876 — 2 4.1
0.1 8.456 — 2 1.786 — 2 2.0 1.066 — 3 7.016 — 4 4.1

Cylinder k = 1 0.4 1.136 — 1 3.856 — 2 2.116 - 3 6.176 — 4
0.2 3.016 — 2 6.286 — 3 1.9 1.286 - 4 7.566 — 5 4.0
0.1 7.686 — 3 5.686 — 4 2.0 9.016 - 6 2.796 — 6 3.8

k = 3 0.4 1.086 — 0 9.486 — 1 2.576 - 1 2.576 - 1
0.2 3.176 — 1 1.556 — 1 1.8 1.716 - 2 1.716 — 2 3.9
0.1 7.976 — 2 2.506 — 2 2.0 8.956 — 4 4.776 — 4 4.3

Acorn k = 1 0.2 2.246 — 2 7.596 — 3 1.056 — 4 7.326 — 5
0.1 5.696 - 3 1.076 — 3 2.0 6.846 — 6 2.336 — 6 3.9
0.05 1.436 — 3 1.566 — 4 2.0 4.426 — 7 1.216 — 7 4.0

k = 3 0.2 3106—1 1686— 1 1516—2 1.066-2
0.1 7.816 — 2 2.576 — 2 2.0 9.596 — 4 4.746 - 4 4.0
0.05 2.016 — 2 3.526 — 3 2.0 5.946 — 5 2.226 — 5 4.0

0 Case 4(d): Apple

p = 1.9(1— cos <0) (4.12)

 

where p = \/:r2 + y2 + 22 and (b = cos—1(1). The computational domain is set
,0

to [—5,5] x[—5,5] x[—8,4].

0 Case 4(6): Flower

T: —+—Sin56,
7

2

WIN

3::

ODIN)

(4.13)

1 . . .
where T = V12 + y2 and 6 = tan—1(1). The computational domain 18 set to
I

{—5.5} ><[—5.5] ><[—2,2].

we adopt Eqs. (3.23) and (3.24) for exact solutions and parameters, respectively.

104

Table 4.2: Convergence test of the fourth-order MIB scheme (Case 4(d) and (6)).
Two interfaces are shown in Figs. 4.3(a) and 4.4(e). The exact solutions are given
by Eqs. (3.23) and (3.24).

 

 

h Loo E1 Order
0.4 9.836 — 3 9.836 - 3
Apple 0.2“ 1.006 — 4 8.116 - 5 6.6
0.1 7.156 —- 6 7.156 — 6 3.8
0.1 1.306 — 5 1.306 — 5
Flower 0.05 5.736 — 7 _5.736 — 7 4.5
0.025 1.796 — 8 6.486 — 9 5.0

 

 

 

 

 

These two interfaces are depicted in Fig. 4.5 with surface maps of exact solutions
and numerical errors. The apple interface admits a singularity at the origin which is
computationally challenging. The flower interface is complex and non-smooth. A 2D
interface that is similar to the flower shape has been considered in the previous work
[169] to test the 2D fourth-order MIB scheme for complex interface geometry. Table
4.2 gives the convergence studies of the proposed MIB method for these two cases.
Due to complexity and large curvature of the flower interface, a relatively dense grid
is required to ensure the availability of auxiliary points and thus the designed fourth-
order accuracy. In both cases, the designed fourth—order convergence is achieved.
For the apple interface, LOO and E1 are the identical at h = 0.1, indicating that
the largest absolute error occurs at the origin i.e., the singular point. For the flower
interface, large errors occur where there are large curvatures. It is to point out that
the construction of 3D fourth—order interface methods for arbitrarily complex interface

with geometric singularities is still an open problem.

4.2.2 Case 5: Variable diffusion coefficients

The previous 2D MIB method has been tested for variable diffusion coefficients and for
large contrast in diffusion coefficients. Here, we show that the present 3D MIB method
works well for variable diffusion coefficients, which are piecewise smooth function of
positions. In order to test the performance of the MIB method on different ,6(.r, y, z),

the following diffusion coefficients are used together with the solution given in Eq.

105

Table 4.3: Convergence test on variable diffusion coefficients (Case 5). Two inter-
faces are shown in Figs. 4.3(a) and 4.4(e). The exact solutions are given by Eqs.

(4.11) and (4.14).

 

 

 

 

 

 

 

 

 

Second-order F ourth—order
h Loo E1 Order Loo E1 Order

Ellipsoid k = 1 0.4 2.006 — 1 1.526 — 1 1.766 — 2 1.766 — 2
0.2 4.766 —— 2 3.026 - 2 2.1 1.036 — 3 1.036 — 3 4.1
0.1 1.146 — 2 5.256 — 3 2.1 4.926 — 5 4.926 — 5 4.4

k = 3 0.4 2.396 — 0 2.396 — 0 1.036 — 0 1.036 — 0
0.2 4.576 — 1 4.136 - 1 2.4 1.256 — 1 1.256 — 1 3.0
0.1 9.226 — 2 6.876 — 2 2.3 5.196 — 3 5.196 — 3 4.6

Acorn I. = 1 0.2 3.796 — 2 2.706 — 2 4.826 — 4 4.826 - 4
0.1 9.176 — 3 5.816 — 3 2.1 2.526 — 5 2.526 — 5 4.3
0.05 2.326 — 3 1.396 - 3 2.0 1.826 — 6 1.826 — 6 3.8

k = 3 0.2 4.016 — 1 3.106 — 1 7.096 — 2 7.096 — 2
0.1 9.746 — 2 6.276 — 2 2.0 3.676 — 3 3.676 — 3 4.3
0.05 2.436 — 2 1.276 — 2 2.0 2.006 — 4 2.006 — 4 4.2

(4.11)

1r = $0 + y) + 10, 0+ = z +15, (414)

where we set h: = 0.

The interfaces of ellipsoid and acron shown in Figs. 4.3( a) and 4.4(6) are employed.
The computational domains are set to [—5, 5] x [—5, 5] x {—5, 5] for both test cases. The
numerical errors are given in Table 4.3. The computational results are of designed
LOO order of convergence. The magnitudes of numerical errors are very similar to
those in Table 4.1, indicating that the proposed method is robust against variable

coefficients.

4.2.3 Case 6: Non-zero linear term

It is very important to include non-zero K.(;r, y, z) in many practical problems. The

performance of the present MIB method is tested in the following two cases

106

0 Case 6(a):

K—(.r,y, z) = 10exp(——2:), n+(17,y, z) = 20y2 (4.15)

0 Case 6(b):

1c_(-1:,y, z) = 1000 exp(—;L‘), n+(x,y, z) = 20003,;2 (4.16)

The exact solution u(;r, y, z) is given in Eq. (4.11) with k = 1 and the diffusion
coefficient fi(:1:,y,z) is given in Eq. (3.24). The source term q(:r,y,::) and jump
conditions can therefore be easily derived. The second— and fourth-order MIB schemes
are applied to solve these two cases with the acorn interface given in Eq. (4.9). The
computational domain is set to [—5. 5]>< [—5, 5]x [—5, 5].

Table 4.4 shows that the present MIB scheme achieves the designated order of
convergence in both cases. It is interesting to note that when a. is increased 100 times
from Case 6(a) to Case 6(b), the MIB method converges in Case 6(b) twice as fast
as it does in Case 6(a), which is due to the fact that the matrix in Case 6(b) is more

banded.

4.2.4 Case 7: Sixth-order MIB scheme

Finally, we test the convergence property of our sixth-order MIB scheme. Three
different interfaces, a sphere, an ellipsoid and a diatomic molecule are considered.
Different solutions are prescribed for these interfaces to further validate the present

MIB method.

0 Case 7(a): Spherical interface. The interface is a. sphere of radius 2. The

diffusion coefficients are given by Eq. (3.24). The solutions are given as

u— = cos k1“ cos Icy cos 1.2, 11+ = 0. (4.17)

The computational domain is set to [—5, 5] x [—5, 5] x {—5, 5].

107

0 Case 7(b): Ellipsoidal interface. The ellipsoidal interface is given by Eq. (4.8).

The exact solution and variable diffusion coefficients are given as the following

. 2 2 .,2
'11- = cos liar cos Icy cos k2, 11+ = exp <—f—:F%OL) (4.18)
,8‘ = (a: + y)/2 +10, ,6“ = 2: +15, 14 = 0. (4.19)

The computational domain is set to [—5, 5] x [—5, 5] x {—5, 5].

0 Case 7(c): Molecular surface of a diatom. This interface is defined by the
molecular surface of a diatom, shown in Fig. 3.7. The exact solutions are
given by Eqs. (4.11) and (3.24) with k = 3 (Note that these exact solu-
tions are not the ones shown in Fig. 3.7(a)). Unlike two other cases, this
interface contains concave surface curves. The computational domain is set to

{—6, 6.16]>< [—4,4.96] x {—4, 4].

Table 4.5 provides the results of convergence tests for Case 7(a). It is seen that the
designed Loo orders of convergence are achieved in these MIB schemes. The sixth-
order method is up to 107 times more accurate that the second one, demonstrating the
advantage of higher-order schemes. The Loo and E1 errors in the sixth-order scheme
are identical, indicating that maximum errors are generated from the interface.

The convergence results for Case 7(b) are listed Table 4.6. Similar to the last case,
the designed orders of convergence are numerically confirmed for all MIB schemes.
It is noted that Loo and E1 errors become identical in both fourth- and sixth—order
schemes. Therefore, the interface errors are the major error source in this case.

Finally, Table 4.7 shows the comparison of three MIB schemes for the molecular
surface of the diatom studied in Case 1. This problem is more difficult than the
last two test cases because of the presence of concave curves in the interface. The
mesh sizes are chosen slightly smaller than those used in the earlier examples to avoid

the (_lifficulty of finding insufficient number of auxiliary grid points. The number of

108

iteration NBiCG is also provided for each computation. It is seen that the fourth- and
sixth-order MIB schemes require slightly larger NBiCG than the second—order scheme
does, because more auxiliary points are involved to represent a fictitious value in
higher order schemes. Under the same mesh size h, the CPU times required for all
iterations are about. the same since the size of the matrix is the same for a given h.
The major factor that determines NBiCG is the matrix size, instead of the order of the
scheme. Therefore, the proposed higher order MIB schemes do not cause additional
convergence problems. At h = 0.16, the sixth-order scheme delivers the same level
of accuracy as the second-order scheme does at h = 0.04, indicating the efficiency of
higher order methods. Similar to the last two cases, the Loo and E1 errors in the sixth—
order scheme are identical. This study suggests that the construction of sixth-order
3D interface methods for elliptic equations with arbitrarily curved smooth interfaces

remains an open problem.

4.3 Conclusion

This chapter introduces the three-dimensional (3D) higher-order matched interface
and boundary (MIB) method for solving elliptic equations with discontinuous coeffi-
cients. The previous higher-order MIB methods in 1D [163] and 2D [169, 170] have
been generalized to 3D. The present MIB method is of fourth-order convergence for
complex interfaces with moderate geometric singularities, and of six-order conver-
gence for smooth interfaces. The proposed 3D MIB schemes are extensively validated
in terms of the speed of convergence, the order of accuracy, the number of iterations
and CPU time. First known results are obtained for 3D elliptic equations in the fol-
lowing two categories: solutions of fourth-order accuracy for complex interfaces with
moderate singularities and solutions of sixth-order accuracy for smooth curved inter-
faces. Some new open problems, such the construction of fourth-order 3D interface
schemes for arbitrarily complex interfaces with sharp geometric singularities and the
construction of sixth order 3D interface schemes for arbitrarily curved interfaces, are

identified.

109

 

 

 

 

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Table 4.7: Convergence comparison of the second-, fourth- and sixth-order MIB
schemes on a diatomic interface (Case 7(c)) as shown in Fig. 3.7. The exact solutions
are given by Eqs. (4.11) and (3.24) with k = 3.

 

 

 

 

 

Scheme h Loo E1 Order NBiCG

0.16 2.11e — 1 8.276 — 2 296

2nd-order 0.08 5.32e - 2 1.081: -— 2 2.0 529
0.04 1.34e - 2 2.34e — 3 2.0 2072
0.16 1.12e—2 1.12e-2 329

4th-order 0.08 4.146 - 4 2.046 — 4 4.8 670
0.04 2.626 — 5 1.05e — 5 4.0 2613
0.16 5.20e — 3 5.201: — 3 409

(Sm-order 0.03 2.561: — 5 2.561: — 5 7.7 824
0.04 1.13e — 6 1.132 — 6 4.5 3575

 

 

 

113

Chapter 5

Treatment of geometric
singularities in implicit solvent

models

The electrostatics of biomolecules plays an important role in their structure, func-
tion, stability, and dynamics [87, 141]. Accurate evaluation of electrostatics has
therefore always been a major task in molecular/structural biology. The natural
form of biomolecules mostly involves solvent. Explicit electrostatic calculations of
biomolecules in the solvent remain extremely expensive, despite of the development
of efficient Ewald summation methods, and alternative approaches based on reaction
field theory, periodic images and Euler summations. The continuum dielectric implicit
solvent models [7, 51], in which the explicit interactions between molecules and sol-
vent are represented by a mean-field formulation, have been very popular in structural
biology and biochemistry [39, 56, 71, 128, 132, 143, 162] since the pioneering work by
Warwicker and Watson in the early 19803 [148], and Honig in the 19908 [71]. Recent
efforts focus on free energy estimation [54, 88, 136], pKn analysis [60, 118, 147], and
applications in molecular dynamics [107, 125]. The implicit. solvent theory retains a
microscopic treatment of biomolecules, while adopting a macroscopic description of
the solvent. In such an approach, the Poisson—Boltzmann equation (PBE), or Poisson

equation (PE) if no salt is present, needs to be rigorously solved. An alternative while

114

much fast approach is the generalized Born (GB) formalism. Nevertheless, the GB
approach relies on the PBE solution as a calibration or reference standard [52], and
has an unknown validity when it is applied to new problems.

Implicit solvent models require a biomolecular surface definition and a prescription
of dielectric functions both inside and outside the biomolecule. The solution of the
PBE is very sensitive to the discontinuous dielectric function. Earlier biomolecular
surfaces, such as van der Waals surfaces solvent accessible surfaces [89], are not smooth
and cause instability in numerical simulations. The molecular surface (MS) as pro-
posed by Lee and Richards [89] are designed to provide smoother interfaces. In fact,
most PB calculations have used the MS. However, in certain geometric situations, the
MS definition admits cusp and self-intersecting surface singularities [37, 47, 62, 129],
which still cause numerical instability. Smoothly varying functions were proposed
to avoid this problem [63, 78, 107]. Nonetheless, McCammon and coworkers [137]
have shown that some atomic centered dielectric functions may lead to unphysical
interstitial high dielectric regions in implicit solvent models. The solvent exclusion
property of the MS is crucial to implicit solvent models.

Few analytical solutions of the Poisson-Boltzmann equation (PBE) or Poisson
equation exist for realistic biomolecular geometries and charge distributions. There—
fore, these equations are usually solved numerically by a variety of computational
methods. The computational tools available can be broadly categorized into (1) fi-
nite difference method (F DM) [56, 71, 106—108, 132, 148], (2) finite element method
(FEM) [9, 10, 39, 68], and (3) boundary integral method [17—19, 83, 100, 143, 162].
All methods are subject to certain inherent advantages and limitations, which are
closely tied to the associated underlying formulation. Finite element methods are
optimal for applications that. require the rapid adaptation of grid points to account
for structural variation of biomolecules. However, generating unstructured grids for
complex biomolecular interfaces is time—consuming, especially for large biomolecules.
Boundary integral methods enjoy several intrinsic advantages such as fewer unknowns,

exact. far-field treatment, and accurate representation of surface geometry and charge

115

singularity. However, boundary integral methods are not very efficient in dealing with
the nonlinear term in the PB model. Finite difference methods are the main workhorse
for solving the PBE in computational structural biology and computational genetic
engineering for the following reasons. (1) Using 3D Cartesian grids, finite difference
methods avoid the time-consuming grid generation step. (2) 3D Cartesian grids are a
standard option in the most widely used software packages in computational biology,
such as DelPhi [86, 127], UHBD [40], MEAD [127], APBS [8, 67, 68] and CHARMM
[22] codes; therefore, finite-difference based PB solvers naturally fit into these simu-
lation packages. (3) Finite difference methods are relatively simple, and particularly
in conjunction with multigrid linear algebraic solvers, can offer the best combination
of speed, accuracy, and efficiency, making this the most popular approach [7]. In the
past decade, the main developments with respect to the PB methodology have been
focused on acceleration of these numerical methods for the PBE.

Molecular surfaces and discontinuous dielectric functions are commonly employed
in the solution of the PBE. Mathematically, the electrostatic potential field becomes
non-differentiable whenever the dielectric constant is discontinuous, which causes nu-
merical instability. In general, the lack of smoothness leads to slow convergence
in solving the PBE. In the worst scenario, the standard numerical methods do not
converge at all for complex irregular solvent-protein interfaces. Computationally, to
improve the convergence, the continuities of both electrostatic potential 03‘ = 6+,
and its flux 6’ 239'”; = 6% should be explicitly enforced across the dielectric inter-
face, where superscript — indicates the potential is inside the biomolecules (protein)
and the + means the potential is in the solvent. The partial derivative is defined in
the normal direction of the interface as n is the unit outer normal vector. Among
many existing PB solvers, errors induced by discontinuous dielectric functions at the
solvent-molecular interface are alleviated in the adaptive discretization of some finite
element methods [38, 39, 46] and controlled in the posteriori error estimates of other
finite element methods [67]. However, explicit enforcement of the flux continuity

condition in the context of geometric singularities has not been considered in the lit-

116

erature. Consequently, no PB solver of 2nd—order convergence, which means the error
is reduced by a factor of four when the grid is halved, has ever been reported in the
context of molecular surfaces of biomolecules. Irregular solvent-solute interface and
geometric singularities reduce the accuracy and slow down the convergence of most
existing PB solvers. Therefore, stability enhancement and convergence acceleration
are pressing practical issues in developing the next generation PB solvers.

In the mathematical context, Peskin [122] pioneered the treatment of elliptic equa-
tions with discontinuous coefficients and singular sources. Recently, a number of other
elegant methods have been proposed. Among them, the immersed interface method
(IIM), proposed by LeVeque and Li [94] is a remarkable second order sharp interface
scheme. The ghost fluid method (GFM) [50] was proposed as a relatively simple and
easy to use approach. For irregular interfaces, it is natural to construct a solution
in the finite element formulation [4]. A relevant while quite distinct approach is the
integral equation method for complex geometry [111]. Recently, we have proposed
the matched interface and boundary (MIB) method [163] as a systematic higher-order
method for electromagnetic wave propagation and scattering in dielectric media. More
recently, we have generalized the MIB for solving elliptic equations with curved in-
terfaces with 4th and 6th order convergences [169, 170]. The MIB approach makes
use of fictitious domains so that the standard high order central finite difference (FD)
method can be applied across the interface without the loss of accuracy. The fictitious
values on fictitious domains are determined from enforcing the flux continuity condi-
tions at the exact position of the interface. Nevertheless, none of the aforementioned
interface techniques is able to maintain its designed order of convergence and accu-
racy in the presence of geometric singularities. Indeed, technically, achieving higher
order convergence at geometric singularities is extremely challenging, despite of great
desire for doing so in practical applications. The best result in the literature is of
0.8th order convergence reported by Hou and Liu [72], achieved with a finite element
formulation in 2D. Very recently, we have developed the first 2nd-order convergent

MIB scheme in 2D for solving elliptic equations with geometric singularities [161].

117

Most recently, we have addressed the interface problem in the PBE by proposed
an MIB based Poisson-Boltzmann (MIBPB-I) solver to explicitly consider the flux
continuity condition in the finite difference framework [168]. The MIBPB-I solver is
of 2nd order convergence in conjunction with smooth molecular surface and discon-
tinuous dielectric functions. It has been validated with the exact Kirkwood solution
[85] and tested by solvation energies of twenty four proteins. Comparison with other
existing methods indicates the potential of the MIBPB—I method. However, the ear-
lier MIBPB-I solver does not maintain the 2nd order convergence when the molecular
surface includes cusps, sharp edges, sharp wedges or self-intersecting surfaces. More-
over, due to the asymmetric matrix of the interface method, the MIBPB-I requires
a large number of iterations in solving linear equations. In fact, its matrix does not
converge for large proteins or small proteins with dense grids. The objective of the
present work is to overcome these difficulties. The present approach, called MIBPB-
II, is a generalization of our new two-dimensional (2D) MIB method developed for
handling sharp-edged interfaces [160, 161]. Apart from its ability to maintain the 2nd
order convergence under the presence of geometric singularities, the MIBPB-II has an
optimally symmetrical matrix, which dramatically reduces the number of iterations.

In the next two sections, the theoretical formulation and the computational algo-
rithm are given for the MIBPB-II solver. The treatment of molecular singularities
and the optimization of the MIB matrix structure are developed. Then, the validation
and application of the proposed MIBPB-II solver are presented. The accuracy and
the order of convergence of the proposed MIB II method is validated by cusp singu-
larity in a diatomic system, and by the cusp and self-intersecting surface singularities
in a four-atom system. The molecular surfaces of twenty four proteins generated by
the MSMS [129] are employed to further test the convergence of the proposed Poisson
solver. Detailed comparison on the accuracy and speed of convergence of the present.
method is given to the previous MIBPB-I and two other established methods. This

work ends with a brief conclusion summarizing the main points.

118

5.1 Theory and algorithm

5.1.1 Poisson Boltzmann Equation

Consider an open boundcr domain 9 E R3. Let F be the interface which divides Q into
disjoint open subdomains, the biomolecular domain (2’ and the solvent domain (2+,
i.e., f2 = (2‘ U {2+ U P. The Poisson-Boltzmann equation arises under the assumption
of the Boltzmann distribution for the solvent ions and leads to a hyperbolic sine term
(sinh(u)) for salt effect. Such a nonlinear term can be approximated by 11, under the

weak potential approximation, and the linearized PBE can be given as:
—v ~ (c(r)Vu(r)) + 1.2005(1) = f(r), (5.1)

where e is the dielectric coefficient, u = eC‘LLI/KBT is the dimensionless electrostatic
potential, 1,0 is the electrostatic potential, K,(I‘) is the Debye—Hiickel screening function
describing ion strength. The source term represents the charge contribution f =
2,278; 221:1 2,6(1' — r3), with k B the Boltzmann constant, co the electron charge, and
z,- the charge fraction at position r,. The PBE satisfies the far field boundary condition
lim]rl_,oo u(r) = 0, although the Dirichlet boundary condition is often used in a finite

domain. With discontinuous dielectric coefficient at the solute-solvent interface, the

PBE should be solved with the following interface jump conditions:

[11] = 21+(r)-—u—(r), (5.2)

[611”] = 6+(r)Vu+(r) -n — €_(r)Vu— (r) - n (5.3)

where n = (le, ny, 71;) is the normal direction of the molecular surface. These condi-
tions are to be rigorously enforced at each intersecting point of the molecular surface

and the Cartesian mesh lines.

119

5.2 Results and Discussion

In this section, we examine the accuracy and test the convergence order of the present
MIB II, or MIBPB-II when electrostatic potentials are computed in the presence of
geometric singularities. The speed of convergence in terms of the number of iterations
NBiCG is also studied. Comparison is made to the earlier MIB I technique [168], and
the standard second order finite difference (FD) scheme. However, specific MIB I
results presented in this work are generated from a new MIB I code. Moreover, to test
the matrix properties of the MIB I and the MIB II, a common linear algebraic solver,
the BiConjugate Gradient (BiCG), was used, and the number of iterations NBiCG
required by both methods are compared. Electrostatic potential was solved with the
MIBPB-II, and for comparison also with MIBPB-I, PBEQ [78], a representative finite
difference PB solver from CHARMM [22], and APBS [8, 67, 68], a multigrid finite
element and finite difference PB solver recently developed primarily for massively
parallel computing. The finite difference function of the APBS is utilized in our
calculations. Molecular surfaces are generated by using the MSMS program [129]
at density 10. In all test cases, the dielectric constant is taken as e“ = 1 and
(.+ = 80. The probe radius is set to 1.4 A unless specified. From the electrostatic
potentials in vacuum, dime, and in the presence of the dielectric environment, wdielccv
the electrostatic free energy of solvation, AGsolv,elec is calculated from the explicit

charges q, at positions 1‘,- as:
1 ,
AGsolvclec : 72‘ Z (12? [lili’idielec(ri) _ El'vacvifl- (5-4)
2'

To compare the computational performance, we use three error measurements,

the maximum absolute error Loo, the surface maximum absolute error E1, and the

120

surface maximum percentage error E2

L00 2 n1Sz21x|u(.r,y, z) - ucx(;r, y, 2)] (5.5)

E1 : nlf‘LX [11(I, y, 2) _ UCX(I: y‘ 3)]

21(.):, y, z) — vex-(1?, y, .2)

“ex (51:1 1’}: Z)

 

7

E2 = 100 X mlax

where 'u and Hex are numerical and exact solutions, respectively. Here E1 and E2 are
computed over all irregular points near the interface P where the modified difference
schemes are used. The tolerance of BiCG iterations is set to be 1.0 x 10‘6 in all
the cases throughout the chapter. The order of convergence is calculated for the LOC

error.

5.2.1 Validation
Dielectric sphere with a unit charge

To establish the validity and performance of the present MIBPB-II method, we con-
sider a unit charge at the center of a dielectric sphere of 2A diameter, for which the
PBE admits analytical solutions due to Kirkwood [85]. This case has been used to ex-
amine the MIBPB-I in the previous studies [168]. In terms of accuracy, the MIBPB-I
and MIBPB—II give essentially identical results since there is no geometric singularity.
Table 5.1 lists the solvation energies and errors in the electrostatic potential of the
MIBPB-II in a comparison with those of the PBEQ and the APBS. It is observed
that in terms of the solvation energy and E2 error, the MIBPB-II results at a coarse
mesh of 0.5 A are more accurate than those of other two methods at a fine mesh of
0.05 A. Due to the explicit enforcement of flux jump conditions at the interface, the
MIBPB-II is of 2nd order convergence, which is indicated by about fourfold decrease

in all three errors as the grid spacing is halved. In contrast, convergence orders of

PBEQ and APBS are about 0.3.

121

Table 5.1: Electrostatic solvation energy AG in kcal/mol and the error in the surface
potential for a sphere with a centered unit charge. The exact solvation energy is -81.98
kcal/mol.
Mesh Errors in surface potential
AG MIBPB-II PBEQ APBS
(A) MIBPB-II PBEQ APBS E1 52 E1 32 E1 E2
0.50 -81.99 -85.78 -85.85 2.60 6.68 17.05 84.26 17.06 84.26
0.20 -81.97 -82.84 -82.58 0.30 1.57 7.51 74.44 7.50 74.43
0.10 -81.98 -82.49 -82.27 0.04 0.38 3.84 62.30 3.83 62.30
0.05 -81.98 -82.20 -82.03 0.01 0.09 1.94 46.95 1.89 46.18

 

 

 

 

 

Molecular surface singularities of few-atom systems

Having established the MIB II method for solving PBE with the molecular surface
interface, we test its performance on molecular surface singularities. In the diatomic
system, cusps occur in the molecular surface when the atomic distance is enlarged as
shown in Fig. 5.1(a). Both cusps and self-intersecting surfaces occur in a four-atom
system, see Fig. 5.1(b). These singularities pose challenges to numerical methods and
break down the designed 2nd order convergence of the previous MIB I. Note that for
the geometries shown in Fig. 5.1, there is no analytical solution that satisfies PBE
(5.1) with the interface conditions (5.2) and (5.3). In order to test the convergence of
the present MIB II method, we consider the Poisson equation given by Eq. (5.1) with
a = 0. A standard method to construct an analytically solvable system of the Poisson
equation is to allow the source term to vary with the prescribed exact solution. Thus,

we set the exact solution to be

'u_ = 10cosrrcosycos z + 20, 21+ = 10(1‘. + y + 2:) +1. (5.6)

The source term f (2: y) and interface conditions [11], [run] can be easily derived from
the exact solution. Note that. the solutions admit ajump at the interface. To illustrate
the performance of the optimized matrix in MIB II, we include a spherical geometry,
Case I, for which both MIB I and MIB II have similar accuracy, but. different speeds
of convergence.

The numerical errors are computed at three mesh sizes, h = 0.5, 0.25 and 0.125A,

122

 

Figure 5.1: Molecular surface singularities. Radius of carbon atoms is 1.5 A in
both cases. Left: Centers of atoms are (—3.62, 0, 0), (3.62, 0, 0), and the probe radius
is 5.1 A; Right: Centers of atoms are (0,4.2,0), (0, —4.2,0), (5,0,0) and (—5,0,0),
and the probe radius is 4.9 A.

by using the MIB I, MIB II and the FD method. It can be seen from Table 5.2 that
the second-order convergence is achieved by MIB II in all three cases, while MIB I
has second-order convergence only for Case I. MIB I converges slowly and irregularly
in Case 11 and Case III due to molecular surface singularities. Furthermore, the
numerical errors of MIB II are very similar in all cases, which indicates that the
geometric singularities have little effect on the accuracy and convergence order of the
present MIB II scheme. In contrast, the accuracy of MIB I strongly depends on the
location and shape of molecular surface singularities. In Case I, MIB I and MIB
II produce almost identical results because both methods have the same interface
treatment for this case. In Case 11 and Case III, Loo and E1 are always the same in
MIB I, which indicates the maximum errors are originated from the irregular points
near the interface, due to the lack of the treatment of geometric singularities. In
contrast, Loo and E1 errors are quite different in MIB II in all cases because the
largest errors of MIB II occur at the largest value of the solution. At a fine grid, the
MIB I errors are about three orders larger than those of the MIB II. The standard
FD scheme does not converge at all due to the discontinuous nature of the solution
at the interface. Nevertheless, its LDC, errors and E1 errors are identical and its errors
in Case II and Case III are larger than its errors in Case 1. Indeed, the interface and

geometric singularities are the main source of errors in conventional Poisson equation

123

 

(d)

Figure 5.2: The surface projections of numerical errors in MIB I and MIB II. (a)
Numerical errors obtained by MIB II for the diatomic system; (b) Numerical errors
obtained by MIB I for the diatomic system; (0) Numerical errors obtained by MIB II
for the four-atom system; (d) Numerical errors obtained by MIB I for the four-atom
system.
solvers.

The number of iterations. NBiCGs increases in all three methods both as the mesh
is refined and as interfaces get more complex. Case III requires the largest number of
iterations because it has the largest. computational domain among these three cases.
When the grid spacing is repeatedly halved. which enlarges the matrix size by a factor
of 8 each time. NBiCG of MIB II increases slightly from 130 to 153. and 224, while
NBiCG of MIB I increases from 598 to 1125, and 6421. The CPU time required for
a single iteration of BiCG scheme is roughly the same for the matrices generated
by MIB 1. MIB II and FD methods in a given case. It is seen that the NBiCG and
thus the CPU time are significantly reduced in the MIB II comparing to those of
the MIB I attributing to the new matrix optimization procedure. The FD method
requires slightly larger «‘VBiCG than the MIB II method. partially because of the bad

conditional number of the FD scheme for the discontinuous solution and because of

124

optimally symmetrical matrix of the MIB II method. It is worth mentioning that
a common BiCG linear equation solver is used here to test the matrix properties
of three methods. In the original MIB I method, the incomplete LU decomposition
scheme is utilized to accelerate the speed of solving the linear system [168], which
gives competitive CPU time for matrices of size up to 106 by 106, but is infeasible for
larger matrices generated by either a larger computational domain or a smaller mesh
spacing for a given domain.

Fig. 5.2 plots the difference between the exact solutions and the numerical ones
computed by MIB I and MIB II methods at h = 0.5 A. As expected, the MIB I
errors are distributed mainly around the geometric singularities. Whereas, the MIB
II errors are distributed around the local maximum of the solution. The different
scales in the plot indicate the different error. magnitudes in two methods. In fact,
even larger differences can be observed when the mesh is refined as shown in Table

5.2.

Molecular surface singularities of proteins

The geometric singularities of few-atom systems illustrated in the last. subsection are
relative simple. The ultimate computational challenges come from large proteins,
which exhibit. sophisticated topologies and geometric variations. Their molecular
surfaces possess a variety of unnamed geometric singularities, particularly in low
resolution protein structures. Therefore, it is important to examine the convergent
properties of the present MIB II method for solving the Poisson equation with the
molecular surfaces of proteins. To this end, we employ a set of twenty four proteins
used in previous studies [52, 168]. For all structures, extra water molecules are ex-
cluded and hydrogen atoms are added to obtain full all-atom models. Their molecular
surfaces are computed by using the MSMS program [129], and used as the dielectric
interface. We again set the exact solution to that prescribed by Eq. (5.6). The source
terms and the jump conditions of the Poisson equation are derived accordingly for

each molecular surface of the twenty four proteins.

Table 5.3: Accuracy and convergent tests on the molecular surfaces of twenty four

 

 

 

 

 

proteins.

PDB h = 65.4 h = 0.2511

ID Loo E1 E2 NBiCG 1400(Order) E1 E2 NBiCG

1ajj 2.4a — 1 1.1e — 1 9.3a — 1 278 6.08 — 2(198) 1.7a — 2 1.66 — 1 486
2.56 -1 2.56 - 1 1.46 + 0 3809 8.66 — 2(1.53) 8.66 — 2 3.36 — 1

2pde 2.3a — 1 1.26 — 1 8.96 — 1 306 5.9a —- 2(197) 1.9g — 2 2.3g — 1 498
4.76 + 0 4.76 + 0 2.36 +1 4444 1. 76 + 0(1. 47) 1.76 + 0 9.46 + 0

lvii 2.4a — 1 14¢: —- 1 1.4a + 0 273 6. 0e — (21.99) 1.9e — 2 2.0a — 1 488
3.26 + 0 3.26 + 0 6.36 +1 4560 1. 56 + 0(1. 14) 1.56 + 0 7.46 + O

2erl 2.36 — 1 1.16 —- 1 4.26 + 0 265 5. 86 - 2(1. 98) 1.86 — 2 2.06 — 1 491
2.36 — 1 1.16 — 1 4.26 + 0 1888 6. 26 — 2(1. 89) 6.26 — 2 2.36 — 1

lcbn 2.36 — 1 1.46 — 1 9.06 - 1 295 6. 06 — 2.(195) 1.86 — 2 1.56 — 1 497
2.36 -- 1 1.46 — 1 . 9.06 —- 1 5682 8.66 -—- 2.(1 43) 8.66 - 2 3.76 — 1

1bor 2.36 — 1 1.36 — 1 6.36 + 0 319 5. 96 — 2(1. 98) 1.96 — 2 1.86 — 1 538
2.36 -1 2.26 — 1 6.36 + 0 17307 1.16 + 0(—2.27) 1.16 + 0 4.16 + 0

1be 2.4a — 1 1.1.2 — 1 9.3a — 1 322 6.06 — 2(201) 1.86 — 2 2.4g — 1 492
3.46 + 0 3.46 + 0 8.16 +1 17307 2. 36 —- 1(3. 89) 2.36 — 1 1.26 + 0

1fca 2.46 — 1 1.26 - 1 1.26 + 0 249 6.06 — 2.(199) 2.26 — 2 2.06 — 1 423
2.46 —1 1.26 —l 1.16 + 0 4145 6.26 — 2(1. 96) 4.16 — 2 2.16 — 1

luxc 2.46 -—1 1.36 -1 1.56 + 0 281 6.06 — 2(2 00) 1.96 — 2 1.76 —1 516
2.46 — 1 1.36 — 1 1.56 + 0 4031 6.06 —- 2(2 00) 1.96 — 2 1.76 — 1

18111 2.46 — 1 1.26 - 1 9.06 — 1 289 6.06 — 2(2. 00) 1.76 — 2 1.56 — 1 525
3.66 + 0 3.66 + 0 1.16 + 2 10366 3. 96 + 0(— —0.13) 3.96 + 0 2.56 +1

1mbg 2.36 -— 1 1.16 — 1 1.86 + 0 290 5.86 — 2(1. 97) 1.76 — 2 1.76 — 1 487
6.76 + 0 6.76 + 0 2.36 + 0 4604 7.76 — 1(3.12) 7.76 — 1 3.36 + 0

1qu 2.4e—1 1.2e—1 1.1e+0 289 5.9e—2(2. 00) 1.86—2 1762—1 525

Continued on the next page

126

Table 5.3 - Continued

 

 

 

PDB h = 0511 h = 02511

ID Loo E1 E2 NBiCG Loo(Order) E1 E2 NBiCG
2.46 —1 1.26 — 1 1.16 + 0 6291 6.96 — 2.(177) 6.96 — 2 3.76 —1

lvjw 2.36 — 1 1.26 —1 1.16 + 0 283 5.96 - 2(1. 95) 1.96 - 2 1.86 —1 443
2.36 — 1 1.26 --1 1.16 + 0 4490 5.96 — 2(1. 95) 3.26 — 2 1.86 —1

lfxd 2.36 —1 1.26 —1 1.76 + O 286 5.96 — 2(1. 99) 1.96 — 2 1.76 — 1 510
2.36 — 1 2.36 — 1 1.76 + 0 3852 1.56—1(0. 63) 1.56 — 1 9.16 — 1

1169 2.36 — 1 1.26 — 1 2.96 + 0 303 5.96 — 2(1. 99) 1.96 — 2 1.86 —1 514
2.36 — 1 1.26 — 1 2.96 + 0 4912 5.96 — 2(1. 99) 2.86 — 2 2.26 - 1

llipt 2.46— 1 1.16— 1 1.16+0 292 6.16—2(1.97) 1.96—2 166—1 502
2.36 — 1 1.76. — 1 1.16 + 0 11346 1.06 + 0(—5.47) 1.06 + 0 4.46 + 0

lbpi 2.46 —1 1.36 —1 1.16 + 0 302 6.16 — 2.(197) 1.96 - 2 1.66 — 1 517
6.56 + 0 6.56 + 0 3.06 +1 10396 4.16 + 0(0. 66) 4.16 + 0 2.16 +1

4516 2.46 —1 1.56 —1 6.06 + 0 342 6. 06 — 2(2 .20 ) 1.96 — 2 2.36 —1 563
6.66 + 0 6.66 + 0 3.36 +1 10049 3. 96 + 0(0. 75) 3.96 + 0 2.16 +1

16128 2.36 —1 1.36 -1 2.46 + 0 379 5. 86 — 2(1. 98) 1.96 — 2 4.06 — 1 651
2.36— 1 2.16—1 2.56-+0 41290 2.76—1(—0.25) 2.76—1 1.66+0

1frd 2.46 — 1 1.26 - 1 3.16 + 0 298 6. O(— 2.(2 03) 1.96 — 2 6.66 - 1 503
1.66+0 1.66+0 4.46+0 17263 1.56—1(345) 1.56— 1 896—1

lsvr 2.46 — 1 1.26 —1 1.56. + 0 391 5. 96 — 2(2. 01) 2.06 — 2 2.06 — 1 678
2.7a + 0 2.76 + 0 9.4e + 0 77080 1.26 + 0(1. 12) 1.22 + 0 5.4a + 0

1neq 2.46 — 1 1.36 — 1 1.26. + 0 369 5. 96 — 2(2. 00) 1.96 — 2 1.66 — 1 688
1.46+0 1.46+0 3.2€+0 17894 1. 56+0(- -0. 13) 1.56+0 6.46+0

18.63 2.46 — 1 1.36 — 1 2.26 + 0 434 6. 06 — 2(1. 99) 2.06 — 2 2.46 —1 665
4.66 + 0 4.66 + 0 4.06 +1 83477 1.16 — 1(5 33) 1.16 —1 5.96 -—I

1a7m 2.56 —1 1.36 — 1 4.06 + 0 419 6. 86 — 2(1. 90) 2.26 — 2 6.36 — 1 726
8.46 + 0 8.46 + 0 4.16 + 2 414908 5. 06. + 0(0. 73) 5.06 + 0 2.36 + 2

 

 

 

 

127

The numerical errors and the number of iterations NBiCG at h = 05.4 and
h = 025.4 are listed in Table 5.3 for both the MIB I and MIB II methods. Although
proteins are ordered according to their gyration radii, their numbers of atoms, Na,
are also listed in the table. In order to compare the performance of these methods,
the numerical results of MIB I at h = 0.25A are needed. However, the original
MIB I method does not converge at such a grid resolution [168]. We therefore have
applied the matrix optimization procedure to the MIB 1, and refer to such a scheme
as ‘accelerated MIB 1’. The difference between the MIB II and the accelerated MIB I
is that the latter does not have the present treatment of geometric singularities. The
MIB II results show great consistency on all twenty four molecular surfaces. With
the same mesh size, the numerical errors are almost the same, which indicates the
robustness of MIB II. Comparing the Loo and the E1 in the table, one may notice that
the Loo error is always greater than the E1 error for MIB II in all cases, indicating
that the molecular surface singularities are not the source of the maximal error. In
contrast, the Loo error is the same as the E1 error for MIB I in most cases. At the
grid resolution h = 0.5A, the MIB I has the largest. E2 error for 1svr, indicating
some special geometric singularity in the molecular surface of lsvr. Furthermore, the
NBiCG required by the original MIB I at h = 0.5A is also provided in Table 5.3 for
a comparison of convergence speed. As it can be seen, the NBiCG of MIB I can be
orders of magnitude larger than that of MIB II. As the number of atoms Na increases,
the NBiCG of MIB 11 increases slightly. For example, the number of iterations NBiCG
only increases less than twice when matrix size is 8 times larger due to halving the
mesh. This confirms the excellent matrix properties of the present MIB II method.

Fig. 5.3 shows the convergence patterns of MIB I and MIB II over molecular
surfaces of twenty four proteins, which are sorted in the same order as that in Table
5.3. The second order convergence of the MIB II on the molecular surfaces of twenty
four proteins is depicted in Fig. 5.3(a). The numerical errors of the MIB 11 method

obtained at h = 0.25A and h = 0.5A are denoted by dots and stars, respectively.

128

 

 

 

 

 

 

 

 

1o 10 .
' 0 go.
. . ‘ H. .*
f o
*
10° 10° * * . .*‘
t
2 K’
o - 9 g .
LE cocoooooooooooooo0.0ooo° Ln 0 900*00 0.... 3
, _ t g,
10'1 7 _ 1 10",. ,, . *
*itfivti-tfiitttttii‘kitttti'k * *‘k ** i
. h=0.5 ’ . h=0.5‘
_2 * h=0.25. _2 * h=0.25
10 ‘ ‘ 10
(a) (b)

Figure 5.3: Accuracy test on molecular surface of twenty four proteins. (3.) L00
errors obtained by MIB II; (b) Loo errors obtained by the accelerated MIB I.

The figure indicates the uniform second order convergence of MIB II. Fig. 5.3(b)
shows the numerical errors of the MIB I obtained at h = 0.5A and accelerated MIB
I obtained at h : 0.25A . The convergence pattern of the MIB I is irregular, and
depends on the occurrence of molecular surface singularities It is worth noting that
at the coarse grid, h = 0.5A, the MIB I has similar accuracy as MIB II for 12 protein
molecular surfaces. When the mesh is refined to h = 0.25A, the accelerated MIB I
has similar accuracy as MIB II has only in 5 cases (1fca, 1r69, luxr, lvjw, and 2erl).
It implies that there are very few molecular surface singularities in the molecular
surfaces of these 5 proteins. However, geometric singularities commonly occur in the

molecular surfaces of most proteins.

5.2.2 Applications
Solvation free energy

After the confirmation of the second order convergence of the proposed MIB II scheme,
we now explore the impact of this new algorithm to the numerical of solution of the
Poisson-Boltzmann (PB) equation for the electrostatic analysis of proteins. The MIB
II based PB solver is denoted as MIBPB-II and its results are compared with those

of the earlier MIBPB-I [168], the PBEQ [78] and the APBS [8, 67, 68}. The same

129

set of proteins [52, 168] used in the last subsection is employed for the present study.
For all structures hydrogen atoms were added to obtain full all-atom models. Partial
charges at atomic sites and atomic van der Waals radii defining the dielectric boundary
were taken from the CHARMM22 force field [82]. The solvation free energies are
calculated by MIBPB-II at the grid spacings of 0.5.3. and 0.25/t, and by PBEQ at
0.5A, 0.25A and 0.15A. The APBS results are reported at a coarse grid spacing of
about 0.4A and a fine grid spacing of about 0.2A, however, it allows grid adjustments
from 0.33A to 0.48A for the coarse grid and from 0.19A to 0.21A for the fine grid.
The results of MIBPB-I were calculated at the grid spacing of 0.5A [168]. Due to its
unfavorable matrix, no result at a refined grid could be produced. These results are
listed in Table 5.4 in the order of gyration radii for proteins. The same order is used
in all figures in the following analysis.

Fig. 5.4(a) gives a bird’s-eye view of the performance of these methods. It is seen
that at a coarse scale of thousands kcal/mol, the results of three methods are in good
agreement with each other. Noted that results from finer grids are selected for APBS
and PBEQ to achieve this effect.

From Table 5.4, it is seen that MIBPB-II results at a coarse grid, h = 0.5A, are
very similar to those at the refined grid h = 0.25A. Indeed, this is true as shown in
Fig. 5.4(b), where differences of these two types of solutions are plotted. Except for
one case, 1be, the MIBPI—II differences between two grids are all within 5 kcal/mol,
disregarding the size of and the radius of gyration of the protein. This means the
MIBPB-II is well converged at a coarse grid of h. = 0.5A, However, this is not the
case for APBS and PBEQ as shown in Fig. 5.4(b). Their differences between two
grids vary dramatically, in the magnitude of 10 - 80 kcal/mol, implying that their
results at the coarse grid are not converged yet. In particular, their differences are
apparently larger for proteins of larger radii of gyration, indicating their convergence
and reliability depend on the radius of gyration, or loosely, on the size of the system.
From Table 5.4, it is seen that APBS and PBEQ results are generally more negative

than those of MIBPB-II. An interesting observation is that, as the grid is refined, the

130

 

 

 

 

 

 

W.
. V -10
—1000 f‘l ~20 /
V: N . «H n ’l 8‘
_ — -30 q . '
o -1500 \ O \/ J J x 0
s l, , g-.. 2.8..8,
§ -2000 1 x _ l h
l,’ 50 l/ l
-2500 l ‘60 3 h
+ MIBPB—ll \ _70L
-3000 + MIBPB-l + MIBPB—II
APBS —80 4- APBS
J- PBEQ O PBEQ
— ‘ —9
35000 5 10 15 20 25 0 5 10 15 20 25

 

 

4“) ‘ff‘ haw h ’

—2o \8 ,/ c I . V
_ , §
,8. w .. MAN

-35 + MIBPB-II

 

 

 

 

 

 

i
+ MIBPB—l '\ f
—40 APBS M 20 + MIBPB-ll
+ PBEQ APBS
“’0 5 1o 15 20 25 oo 5 1o 15 20
(C) (d)

Figure 5.4: Comparison of solvation free energies of proteins, which are listed
in the order of gyration radii as shown in Table 5.4. (a) Solvation free energies
of AGMIBPB—Il(h = 0.5.31), AGMIBPB—I(h = 0.5/X), AGAsz(h ~ 0.2A) and
AGPBEQUL = 0.25A); (b) Differences of solvation free energies between coarse
mesh and fine mesh, i.e., A(AGMIBPB—Il(h = 0-5A)‘AGMIBPB—Il(h = 0.25A»,
A(AGpng(h = O.5A)-AGPBEQ(h = 0.25A)), and A(AGAPBS(h ~ 0.4A)-
AGAsz(h ~ 0.2A)); (c) Difference of solvation free energies between MIBPB II
and other methods, i.e., A(AGMIBPB—Il(h = O-SA)'AGMIBPB—Il(h = 02513.»,
A(AGMIBPB—-I(h = O-SA)'AGMIBPB—II(h = 0.25/h», A(AGPBEQ(h = 0.25/3x)—
AGMIBPB—IIUI = 025A». and A(AGAPBS(h ~ O-ZA)'AGMIBPB—Il(h = 025A»;
(d) CPU time used by APBS at about 0.2A and MIBPB-II at 0.5A for 18 proteins.

131

results of both APBS and PBEQ converge toward those of the MIBPB-II.

Since the results of APBS and PBEQ converge to those of MIBPB-II, which are
well converged at h = 0.25A, we choose the solvation free energies of the MIBPB-II
as references to show the convergence property of other methods. In Fig. 5.4(c), the
differences in the solvation free energies are plotted between PBEQ and MIBPB-II,
APBS and MIBPB—II, and MIBPB-I and MIBPB-II. The results of MIBPB-I calcu-
lated from a coarse grid of h = 0.5A are essentially converged, confirming the earlier
claim [168]. The PBEQ method at the grid resolution of h = 0.25A produces the
largest differences in solvation free energies as shown in Fig. 5.4(c). The magnitudes
of such differences range from a few kcal/mol to 43 kcal/mol, and exhibit a growing
trend in the increase of the radius of gyration. This indicates the lack of reliability of
the PBEQ method at the grid resolution of h = 0.25A in practical applications. Only
when the grid is refined to h = 0.15A could PBEQ further reduce its differences to
less than 20kcal/mol as indicated in Table 5.4. However, such a grid resolution may
not be very practical in most electrostatic analysis due to large computational costs.
The results of APBS obtained at h ~ 0.2A generally show much smaller differences
and a weak dependence on the radius of gyration, partly due to its ability of adap-
tation to the interface. However, MIBPB-II is more accurate at the grid resolution
h = 0.5A than the APBS at finer grid of h N 0.2A.

It is interesting to compare the CPU cost of the MIBPB-II with that of APBS at a
similar level of accuracy for computing the solvation free energy. Fig. 5.4(d) depicts
the CPU time of MIBPB-II and APBS at the grid resolutions of h = 0.5A and
h ~ 0.2A, respectively. A dedicated PC of 2.8GHz Pentium D CPU with 2GB RAM
was used for the CPU time comparison. However the CPU time of the APBS for the
last six largest proteins, Ineq, 1a2s, lsvr, 1frd, 1a63 and 1a7m, is inaccessible because
their memory requirements are either extremely close to or exceed the 2GB limit of
the dedicated Win32 system. The computation of these six proteins are therefore
completed in a larger computer where the precise CPU time cannot be recorded

because of disturbance from other users. It is to be noted that at the same grid

132

resolution, the present MIBPB-II solver with a BiConjugate Gradient linear solver
requires more CPU time than the APBS solver. However, at a given accuracy as

shown in Fig. 5.4(c), the MIBPB-II requires less CPU time than APBS.

Electrostatic potentials

Finally, we consider the surface electrostatic potential of a heme-binding protein,
Fe(II) cytochrome C551 from the organism Pseudomonas aeruginosa (PDB ID: 451C).
The electrostatics distribution near the heme-bonding site and inside the binding
cavity is very important for the electrostatic steering effect during the protein-heme
docking process. The potential is computed with MIBPB-II, MIBPB-I and PBEQ
at the grid resolution of h. = 0.5A. Fig. 5.5 (a) illustrates the surface electrostatic
potential obtained with MIBPB-II, and for a comparison, the differences between the
surface electrostatic potentials of MIBPB-II and MIBPB-I, and between MIBPB-II
and and PBEQ. It is seen that the inner surface of the cavity is mostly positively
charged except of middle section there the Fe(II) ion would locate, which is consis-
tent with the fact that the heme surface is mostly negatively charged because it has
a positively charged Fe(II) core. From the difference plots, we found that MIBPB-I
potential differs from that MIBPB-I only for a few isolated spots due to the molecular
surface singularities. The light color in these spots indicates small differences. How-
ever, the discrepancies between MlBPB—II and PBEQ are more intensive, and mostly
distributed around convex surfaces where there are more untreated irregular grid
points. We expect that these discrepancies of about 5kcal/ mol/ e would have a signif-
icant consequence in a quantitative analysis of the interaction between cytochrome C

and heme.

5.3 Conclusion

This work reports a new generation of interface based Poisson-Boltzmann (PB) equa-
tion solvers that take explicit care of geometric singularities, such as cusps and self-

intersecting surfaces in the molecular surface definition [89]. The previous matched

133

' S-M
1.01
, 4.51

| -.-..m

   

(a)

Figure 5.5: Comparison of surface electrostatic potentials of cytochrome 0551 at h =
05A- (a) AGMIBPB—II; (b) A(AGMIBPB—II - AGMIBPB—I); (C) A(AGMIBPB—n -
AGPBEQ)-

interface and boundary (MIB) method based PB solver, denoted as MIBPBI, was
the first PB solver to explicitly enforce the. flux jump conditions at solvent-solute in-
terfaces and could consequently provide highly accurate biomolecular electrostatics in
continuum dielectric environments [168]. However, the MIBPB-I cannot maintain its
designed second order convergence whenever there are geometric singularities More-
over, the MIBPB-I matrix is not optimally symmetrical and diagonally dominant,
resulting in a severe convergence problem. The present method, denoted as MIBPB-
II, is designed to overcome the aforementioned difficulties. A new MIB scheme is
proposed to rigorously treat arbitrarily complex interfaces in the Cartesian represen—
tation [160, 161]. The MIB matrix is made optimally symmetrical and diagonally
dominant. The proposed MIBPB-II is extensive validated by the molecular surfaces
of few—atom systems and a set of twenty four proteins. Uniform second order conver-
gence of MIBPB-II is confirmed for singular molecular surface interfaces of proteins,
disregarding of the radius of gyration. The MIBPB—II matrix is optimally symmetri-
cal, which leads to a significant reduction in the number of iterations required by a
linear equation solver.

The MIBPB—ll is applied to electrostatic potential calculations of a dielectric
sphere with a unit charge, which admits an analytical solution [85], and a set of twenty
four proteins. As a comparison, the finite difference based PBEQ and APBS are
employed for the same electrostatic calculations. For the. analytical case. the MIBPB—
II at a. coarse grid of h : 0.5A was found to be more accurate than the PBEQ and

APBS at fine grid of h = 0.05A in terms of both surface electrostatic potential and

134

solvation energy. For twenty four proteins, we found that the solvation free energies
computed by the MIBPB-II has essentially converged at the grid resolution of h =
0.5A. The results of APBS and PBEQ converge toward those of the MIBPB-II when
their grids are refined. The discrepancies between converged MIBPB-II solvation free
energies, which are obtained at the grid of h = 0.5A, and those of PBEQ obtained
at h = 0.15A are are nearly 20 kcal/mol for proteins with a large radius of gyration.
Slightly smaller discrepancies are found between converged MIBPB-II solvation free
energies and those of APBS obtained at about h ~ 0.2A. The accuracy of both PBEQ
and APBS shows a dependence on the radius of gyration. The present MIBPB-II
requires more CPU time than APBS and PBEQ at given grid resolution, but less
CPU time at a given accuracy.

As the interface flux jump conditions are rigorously enforced at complex solvent-
solute interfaces and convergence order is systematically restored at geometric singu-
larities, the proposed MIBPB-II solver ought to deliver more reliable surface electro—
static potentials. Indeed, converged surface electrostatic potentials are obtained at
the resolution of h = 0.5A. The discrepancies between converged MIBPB-II surface
electrostatic potentials and those of PBEQ at the same grid resolution are about
5kcal/mol/e. Such discrepancies would induce a consequence in a quantitative anal-

ysis.

135

 

 

 

 

 

 

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137

Chapter 6

MIB method for the nonlinear

Poisson-Boltzmann equation

In this chapter, we consider the numerical solution of the nonlinear Poisson-Boltzmann

equation (NLPBE). In the case of a 1 : 1 electrolyte, this equation can be written as
-—v . (C(r)Vu(r)) + n2(r)sinhu(r) = m), (6.1)

where e, u, w, fi(l‘), and f (r) are defined the same as in Chapter 5. There have
been many numerical techniques employed to solve the NLPBE, such as nonlinear
Gauss-Seidel methods [2], nonlinear SOR method [117], nonlinear conjugate gradient
methods [109], nonlinear multigrid methods [69, 11.9], and inexact-Newton—multilevel
methods [69, 70].

The nonlinear relaxation methods, such as nonlinear Gauss-Seidel methods and
nonlinear SOR methods, can be viewed as a fixed-point iteration: an“ = C(u").
The implementations of these methods require the solution of a sequence of one-
dimensional nonlinear problems by Newton’s method. Therefore, these methods are
also referred to as Gauss-Seidel-Newton and SOB-Newton methods. The convergence
properties of these types of methods are discussed in details by Ortega and Rheinboldt
[120]. As the size of the problem is increased, these methods converge more and more

slowly. As a result, we will consider alternative methods. Luty et. al [109] applied a

138

nonlinear conjugate gradient method to the N LPBE. It is remarked by in the paper
that solution time for the nonlinear conjugate gradient method for the full nonlinear
problem is five times greater than that for the linear method applied to the linearized
problem. Nonlinear multigrid method has been considered by Oberoi and Allewell
[119], and Holst [69] for the nonlinear Poisson-Boltzmann equation. It has been shown
that this method is efficient for simple Poisson-Boltzmann equation problems. But
it is remarked in [70] that even a quite sophisticated implementation of nonlinear
multigrid may diverge for difficult problems such as the nonlinear Poisson-Boltzmann

equation with complex, large, or highly charged molecules.

6.1 Inexact-Newton methods

Given the nonlinear operator F, a generalization of the classical one-dimensional

Newton’s method for solving the problem F (u) is as follows:

F’(un)vn = —F('u.n) (6.2)

u"+1 = II," + v" (6.3)
where F,('ltn) is the Jacobian matrix of partial derivatives:

aWW] , (6.4)

, I

F (u) = [$717-—
Inexact-Newton methods refers to several variations of the standard Newton iteration.
A quasi-Newton method refers to a method which uses an approximation to the true
Jacobian matrix for solving the Newton equation. A truncated-Newton method uses
the true Jacobian matrix, but solves the system only approximately using an iterative
linear solver in which the iteration is stopped early or truncated.

The inexact-Newton, especially the truncated-Newton method, could be an effi-

cient. method for the nonlinear Poisson-Boltzmann equation since the diagonal prop-

139

erties of the nonlinear term. The nonlinear operator of NLPBE could be written

as

[7(a) = An + N0!) —— f = 0 (6.5)

and

F’(--) = A + N’(u). (6.6)

Therefore, F’ (u") could be calculated at a low cost. We hope that the compu-
tational cost could be dramatically reduced by solving the Jacobian systems only
approximately even though a few more Newton iterations may be required due to the

inexactness of the Newton steps.

6.1.1 Inexactness and superlinear convergence

Dembo et a1. [41] establish conditions on the accuracy of the inexact J acobian solves
at each Newton iteration which will ensure superlinear convergence. Their main result

is the following theorem:

Theorem 6.1.1. [41] Assume there exists a unique u such that F (u) = 0, that
the inexact-Newton iterates {un} converge to u, and that both F () and F ' () are

sufficiently smooth. Then:
0 The convergence is superlinear if: limnHoc nn = 0

o The convergence is at least Q-order(1 + p) if F’(u) is Holder continuous with

ezponent p, and

7),, = ()(||F(u"‘)|[p), as n —> 00 (6.7)

140

where fin is the forcing sequence 77;; < 1 for all n and

.77.
F, n ,,Tl = _F Tl 7t [[7 H < 68
(u )L (u )+ r , wllFtunhl _ 7772 ( )
As a result of this theorem,
[[rnll S C [F('u.")|]p+1, (hp > 0 (6.9)

 

is suggested by Holst. [70] to guarantees local Q-order(1 + p) convergence.

6.1.2 Inexactness and global convergence

If the initial approximation is close enough to the solution, then superlinear or Q-
order(p) convergence occurs in N ewton-like methods. It is well known that N ewton’s
method will converge slowly or fail to converge at all if the initial approximation is
not good enough.

Holst et a1. [70] establish conditions to improve the robustness of a Newton
iteration without loosing the favorable convergence properties close to the solution.

The global convergence of Newton’s method is forced by requiring that
||F(U"+1)|| < llFfunhl (6-10)

Their main result is the follow:

Algorithm (Damped-Inexact-Newton Method)

71-
F’(U-n)’Un : —F(un) + Tn, ]]T [I l S Tin, (6.11)

[[F('u.")[

un+l = u" + Ant?"

Theorem 6.1.2. [70/ [neared-Newton (Algorithm 6.11) yields a descent direction v

141

at u if and only if the residual of the Jacobian system, r = F’(u)v + F(u), satisfies:

(FM-7") < (17(10- F(U)) (6-12)

Corollary 6.1.2.1. [70/ Inezmct-Newton (Algorithm 6.11) yields a decent direction

12 at the point u if the residual of the Jacobian system, r = F'(u)v + F(u), satisfies:

Hrll < IIF('u)l| (6-13)

6.1.3 Damped-inexact-Newton algorithm for PBE

A pseudocode of damped-inexact-Newton algorithm is given as follows:
1: u" = 0 D Initial Value

2; while error > 1.0 x 10-9 do

3: Cond = |]r"|| 3 HF (1I")|| D Global convergence condition
4: repeat

5: Iteratively solve F ’ ('u")v” = —F(u") + r"

6: i.e., iteratively solve (A + N’(u"))v" = — [Aun’ — f(:l:, y, z)] — N(u") + r"

7: until Cond == True

 

 

8: An = 1

9: repeat D Search for damping factor An
10: An = ratio * An D ratio < 1
11: until F(u" + Anv")|] < ||F(u")||

12: u" = u” + An 2 ratio * A”

13: if A == 1 then D no damping needed
14: Cond = ||r"]] g C|]F(u")|]1+p and ”7"” g ||F(u")|] D Both superlinear

convergence condition and global convergence condition must be satisfied
15: end if

16: end while

142

The existence of the damping factor An is guaranteed because the global convergence
condition must be satisfied in the previous step. The calculation of F (an + Ann")

could be simplified as follows:

F(u" + Ant-'71) = A(un + Ana") + N(un + Anon) — f(:r, y, 2) (6.14)

= [Ann — f(:r, y. 3)] + An [Ann] + N(u.n + Anon)

where [Ann —— f (:r, y, 3)] has been calculated in the previous step, and [AW] only need

be calculated once for each Newton direction v".

Damped-inexact-Newton—SOR(DIN-SOR) method

The Jacobian system could be solved iteratively by the successive overrelaxation
(SOR) method. Relaxation methods split the sparse matrix that arises from finite
differencing, and then iterate until a solution is found. It can be proved [140, 157]
that overrelaxation with 1 < a) < 2 can give faster convergence than the Gauss-Seidel
method under certain mathematical restrictions. A pseudocode of the SOR alogrithm
for a linear system AX = B is given as follows:

1: X1 = 0

2: r1 = B

3: repeat

4: Xn‘l'1 = X" + w * rn/D D D is the diagonal part of A

5: r"+1 = B — AX"+1

6: until Cond == True

The major advantage of SOR is that it is very easy to program. Its main disad-

vantage is that it is still very inefficient for large problems.

Damped-inexact-Newton—BICG(DIN-BICG) method

Preconditioned biconjugate gradient(BICG) method has been proved to be an ef-

ficient method for solving linear PBE in Chapters 4 and 5. A pseudocode of the

143

preconditioned BICG for a linear system AX = B is given as follows:
1: X1 = 0
2: r1 = 7"1 = B
3: :51: 31: Tl/D
4:repeat

7771+1 . z‘n'l'l

T
51 f)” = ’7
1- l , zn

6: if n == 1 then

7 p" 2 [371 = 2n
8 else
9: p" = z" _+_ flrzpn—l
10: 'p‘" = 511 + Jul—)n—l
11: end if
12: Q" = 12:.

[7" - A - p"
13: r"+1 = 1'" — (tnA '1)"
14: F"+1 = F" — (WA .17”

15: X"+1 = X" + onpn
16: zn+1 = Tn+l/D
17: 2"“ : 7""+1/l)
18: until Cond == True

The NLPBE could therefore be solved by the nonlinear MIBPB solver in the
following steps. The linear term Au in the nonlinear operator F (u) is generated by
the MIBPB If in the same way which is used to solve the linear PBE. The discretized
NLPBE could then be solved by the DIN-SOR method or the DIN-BICG method.

The numerical tests and results are discussed in the next section.

6.2 Results and Discussion

To test the performance of dan11.)ed-inexact-N ewton method, we consider the Poisson
equation given by Eq. (6.1). A standard method to construct an analytically solvable

system of the Poisson—Boltzmann equation is to allow the source term to vary with

144

the prescribed exact solution. Thus, we set the exact solution to be
u— = cosxcosycosz. 21+ = ———1—. (6.15)
with
r = 1, (.+ = 80, (6.16)

where n— represents the solution inside the interface and 11+ represents the solution
outside the interface. The source term f (:r, y) and interface conditions [11], [can] can
be easily derived from the exact solution. Note that the solutions admit a jump at

the interface.

6.2.1 Case I: Spherical interface

Consider a case with a spherical interface of radius r = 2.5151, the convergence of
the solution and the corresponding CPU times when h: = 1.0 and K. = 100.0 solved
by the DIN-BICG method are reported in Table 6.1. The computational domain is
{—5, 5] x {—5, 5] x [—5, 5]. This case has been also solved by the DIN-SOR method.
But it only converges for h = 0.5, which takes 159 inexact-Newton iteration and
47 seconds. Comparing with the DIN-BiCG method, it is not an efficient method
to solve the nonlinear Poisson-Boltzmann Equation. It is seen in Table 6.1 that the
present method has second-order accuracy. It converges fast for relative small matrix.
For strong nonlinear case with r: = 100.0, the computational time doubles comparing

to the case with K = 1.0.

6.2.2 Case II: van de Waals surface of two atoms

The interface is van de \N’aals surface of two atoms of radius 1.5 with centers ( —1.52, 0, 0)
and (135,0,0). The computational domain is [—5,4] x [—3,3] x {—3, 3]. The conver—
gence of the solution and the corresponding CPU times when Ii. = 1.0 and K = 100.0

are reported in Table 6.2. This interface has been tested in Chapter 3, shown in Fig.

145

Ill

 

3.9. The intersection of two spheres create a sharp edge, which is treated by the
MIB method for geometric singularities to achieve the second order accuracy. Table
6.2 shows that the numerical solutions converge in the second-order for the nonlinear

case with geometric singularities as well.

6.2.3 Case III: Molecular surface of protein lajj

The interface is the molecular surface of protein 1aj j. The computational domain
is {—6.25} x [—10, 23] x {—1621}. The numerical solutions and the corresponding
CPU times when r; = 1.0 and K. = 5.0 are reported in Table 6.3. It is seen that for
large matrix system, the present method converges slowly and the number of inexact-
Newton iterations is also large. In the strong nonlinear case when a: = 5, it converges

very slowly

6.3 Conclusion

The MIB method is implemented to solve the nonlinear Poisson-Boltzman equation.
The equation is discretized by using fictitious values obtained by the MIB method.
The nonlinear system is solved by the damped inexact-Newton method. The precon-
ditioned biconjugate gradient method is used as the inexact solver. Three test cases
are studied. It is shown that the present nonlinear MIB-PB solver has the second-
order accuracy even in the presence of geometric singularities. It converges fast for
both weak nonlinearity and strong nonlinearity cases when the matrix is relatively
small. But it converges slowly for a large system, such as protein laj j. Therefore,

further research is required to accelerate the convergence rate of the nonlinear solver.

146

 

 

 

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Table 6.3: Numerical errors for the nonlinear Poisson-Boltzmann equation when
h = 0.5. Case III: Protein 1ajj;

 

 

K.(A) Loo E1 CPU time(s) NDIN
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5.0 2.976 — 2 2.016 - 2 1818 55

 

 

 

149

Chapter 7

MIB Method for the vibration

analysis of plates

Plates, beams, and shells are basic elements in engineering structures and are of great
practical significance to civil, mechanical, and aerospace engineerings. For instance,
bridge slabs, floor systems, window glasses and airplane wings can be modeled as
plates or shells with various internal and boundary supports. A pioneer study in
this field is due to Chladni [29] who analyzed the nodal patterns on square plates
at their resonant frequencies. Apart from a few analytically solvable cases that are
insignificant for real world applications, there is no exact solution for the practical
analysis of structures. Since analytical methods often fail or become too cumbersome
to use, numerical simulation is one of major approaches in structural and mechanic
engineering. Computational methods which determine the performance of numeri-
cal simulations have a crucial impact to the accuracy, robustness and reliability of
structural analysis.

In the past three decades the vibration analysis of rectangular plates with uniform
and nonuniform edge supports has attracted considerable attention. A variety of com-
putational methods have been successfully employed for such analysis. Earlier success
in the vibration analysis of plates includes methods of finite strip [26, 27], spline fi-
nite strip [49], Ritz variational methods [91, 92], Rayleigh methods [146], Galerkin

approaches [28], and series expansions [116]. Recently, the least squares technique

150

[126], meshless methods [43], Rayleigh-Ritz methods [102, 156] and finite element
methods [93, 171] have been introduced to the plate vibration analysis. Differential
quadrature (DQ) methods [13, 14, 79] have found many successful applications in the
vibration analysis. This method is based on the idea that the partial derivative of a
function with respect to a spatial variable at a given discrete point can be expressed
as a weighted linear sum of the function values at all the discrete points in the com-
putational domain. Shu and Richards [134] introduced the generalized differential
quadrature (GDQ) [44, 133, 135] to simplify the calculation of the weight coefficients.
Their method has been successfully applied to the vibration analysis of plates with
free edge supports [133] and [135].

All the above mentioned methods are either local or global, with distinct advan-
tages and disadvantages. Global methods are very accuracy but are not flexible for
irregular geometry and complex boundary conditions. In contrast, local methods
are suitable for irregular geometry and complex boundary conditions, but are less
accurate than global ones. More recently, the discrete singular convolution (DSC)
algorithm [149, 152] has emerged as a local spectral method to combine the accuracy
of global methods with the flexibility of local methods. The mathematical founda-
tion of the DSC algorithm is the theory of distributions and the theory of wavelets.
The DSC algorithm has been realized in both collocation and Galerkin formulations
[74, 101, 152]. The utility of the DSC algorithm for the vibration analysis has been
extensively explored [150—152, 155, 166]. This method provides some of the best
predictions for high frequency problems [153, 165]. The superb performance of the
DSC algorithm for the vibration analysis has been independently verified by Civalek
[30—36]. By extending the computational domain according to the boundary condi-
tions, the DSC algorithm works well for simply supported, clamped and transversely
supported edges. Nonetheless, the earlier DSC algorithm has its difficulty in imple-
menting the free edge conditions [152].

Most'recently, the matched interface and boundary (MIB) method has been de-

veloped for solving partial differential equations with discontinuous coefficients and

151

singular sources [161, 163, 169, 170]. The MIB method is capable of dealing with ar-
bitrarily complex interfaces and geometric singularities in three dimensions [158, 160].
The essential idea of the MIB method is to smoothly extend computational domains
near the interface or the boundary so that. the standard central finite difference (FD)
discretization can be conveniently carried out. The interface or boundary conditions
are strictly enforced in the domain extension. Although domain extensions were com-
monly used in the DSC algorithm [152], where each boundary condition is used only
once. In contrast, the MIB method found a way to repeatedly use the lowest order
interface or boundary conditions, and thus is able to rigorously enforce complex con-
ditions over a wider extended domain and achieve desirable higher order accuracy
[163]. This approach works because more regular grid function values are used in
the enforcement of the interface or the boundary conditions. An interpolation for-
mulation of the MIB method, in which one does not need to repeatedly enforce the
interface or the boundary conditions, has also been proposed [169]. It becomes clear
in this new formulation that the number of unknowns is consistent with the sum of
the number of the interface (or the boundary) conditions and the number of function
values on regular grid points. In a preliminary study, the MIB method has been
utilized to overcome the difficulty of implementing free boundary conditions of beams
in the DSC algorithm [164].

The objective of the present work is to introduce the MIB method for the vibration
analysis of plates with arbitrary boundary supports and their arbitrary combinations.
Since the governing equation of plate vibration involves fourth order derivatives, a
normal stencil that would support a fourth order discretization for an elliptic equation
can only provide second order accuracy in dealing with fourth order equations. There-
fore, wider extended domains are required to maintain high order accuracy. The most
challenging issue in the plate vibration analysis is the presence of cross derivatives in
the boundary conditions of free edges and free corners. The present work overcomes
these difficulties by developing a set of new MIB schemes. Specifically. the domain

extension cannot be pursued along a given meshline at a time as in the previous MIB

152

schemes.

The rest of this chapter is organized as follows. In Section 7.1, we briefly review
the governing equations of the plate vibration and three types of boundary condi-
tions. In Section 7.2, theory and algorithm of the new MIB method is developed
for plate vibration analysis. In Section 7.3, we validate the proposed MIB method
by convergence studies. We select a few typical combinations of simply supported,
clamped and free edges to illustrate and validate the proposed method. Comparison

is given to other established methods. Conclusion remarks are given in Section 7.4.

7. 1 Governing Equations

To establish notation and simplify our discussion, we briefly review the Kirchhoff
theory of plates. The non-dimensional governing equation for a thin rectangular
plate is given as [91]

8411. 4 r9411

‘94" + 2A2 + A o2 (71)
———"— -"—— I 'll. .
0X4 ('IX2UY2 0Y4 ’

 

where u is the dimensionless mode shape function, X = 117/11: and Y = 1// /y are
dimensionless coordinates with [I and [y being the lengths of the plate edges. Here
A 2 l1 / ly is the aspect ratio, Q = we? p/ D is the dimensionless frequency with to
being the dimensional circular frequency, and D = Eh3[12(1 — 112)] being the flexural
rigidity. Here E is Young’s modulus, I/ is Poisson’s ratio, p is the density of the plate
material and h is the plate thickness. Let us consider three typical types of boundary

conditions, namely, simply supported, clamped and free edges.

0 Simply supported edge (S)

u = 0, _' : 0 (7.2)

153

at X = constant. and

 

 

8211.
1!. - 0. (if—'2 = 0 (7.3)
at Y = constant.
0 Clamped edge (C)
0
u = 0, 70—}, = 0 (7.4)
at X = constant, and
Ba
= — = 7.
u 0, BY 0 ( 5)
at Y = constant.
0 Free edge (F)
0211. 9 0211. (7311. 9 0311,
‘—.+ /\"_—=O, —',—. 2— AH—Tf—=O 7.6
axZ V av? aw H V) ex 011/2 ( )
at. X = constant, and
82a 0211 9 03a 8311
A2— .—=0, A“ .+2— —=0 7.7
8Y2 + ”ax? aw ( ”)ax2ay ( )
at Y = constant,and
8211
= 0 7.
(9XOY ( 8)

at the corner of two adjacent free edges. In most numerical approaches, it is the
free edge boundary condition that. requires special treatments. Certainly, the

method developed in this work is able to handle other boundary conditions.

154

7 .2 Theory and Algorithm

New .\IIB schemes are developed in this section for the eigenvalue problem of plate
vibration analysis. We first present a second order scheme which illustrates the algo-
rithm of the MIB method. In the process of enforcing the given boundary conditions,
all the fictitious values on the extended domain are resolved in terms of function
values on the interior grid points. Therefore, the final discretization matrix for the
governing equation is expressed solely on the information of interior grid points. Af-
ter the illustration, we discuss the general MIB schemes for achieving arbitrarily high

order accuracy in the plate vibration analysis.

7 .2.1 Second order MIB scheme for plate vibration analysis
Domain extension

Let NI and Ny be the total numbers of grid points in the .r— and y-directions, respec-
tively. To maintain a second order discretization, we need to extend the computa-
tional domain by two layers of fictitious grid points at. each edge, see Fig. 7.1. One
of this two layers of extended grip points is on the boundary. There are a total of
4( N I — 2) + 4(Ny —— 2) + 4 grid points on the extended domains near four edges. For
each grid point on the boundary. there are two boundary conditions. There is one
boundary condition for each corner point. Therefore, the number of available bound-
ary conditions is the same as the number of fictitious grid points (4N; + 4N1, — 12).

The discretized boundary conditions for a grid point (I j) on an edge can be

written as

0 Simply supported edge

“111' = 0 (7.9)
fVJj+1 (9)

Z (”..rjifiuk-J : 0’ (7.10)
1:20

l:1,f\rx,j:2,...,1~ry—1

 

 

 

 

I
? ? v v
o o ------------- i ------------- —0— 3
O o ———————— 'r---AI--—-w'--—»I———4I --¢
1 1 I I I
0 o--—---- ----- ; ————— .——--o —-¢
l ‘ l I
o onu————l~—-~--~1—--~4_—-—4Ino
l I I I
o o---«'—-—-'r-——-.—---J ———————— 4) 0'
4'. J. A J. A
V V V V v
l l l I I l
I | I U l l
----- u... o- -o- o--—--o-—-—-

Figure 7.1: The distribution of first two layers of fictitious grid points (N3 = Ny =
7, M = 2). The solid line ‘—’ indicates the boundary of the plate.

at X = constant, and

um- = 0 (7.11)
Ny+1

2
Z 0353,“th = 0, (7.12)

k=0
2:2,”..Nx—1;j=1,fvy

at Y = constant. Here c are FD coefficients and can be evaluated as described

by Fornberg [55].

o Clamped edge

WJ=0 (in)
Nm-f'l (1)

Z cx,i,kuk.j=01 (7.14)
k=0

i=1,NI; j=2,...,N,,—1

at X = constant, and

um- : 0 (7.15)

Ny+1

2: cm u- = 0 (716)
yij‘k' 2,53 ’ .

k=0

1:2,...,1Vx-1;J'=1INII

156

at Y = constant.

0 Free edge

Nay-1'1
kg: Crzkth+VA2Z aijcuf-k: (7.17)
Alf+1 (3) ‘N‘I'l'l Ivy +1

2
Z Cx,i.kuk~j+(2_u))‘ :5 6:17-11: E byIIIUkF (7-18)
k=0 k=0

i=1,1\=},; j=2,...,Ny—1

at. X = constant, and

Ny +1
A2 2c; Iku7h+VZ kaukJ 0 (7.19)
ry‘l‘ ( Ny+1(
3
A2 Z Icy-lkttiak + (2 — V) Z('( ('ylj 1: N21 1133. k (111 k— — 0, (7.20)
k—O k=0

i=2,...,Nx—1;j=1,Ny

at Y = constant,

where i and j are I— and y— indexes of the grid point where the boundary conditions

are discretized, respectively. k and l are the summation indexes of differentiation.
9 8211
Here aflfifik = 1,2, . . .,Nx) are the FD coeffiricnts [55] for— .19—X2 at (X— -- i, Y- —

1) and (X = i,Y = Ny), respectively. Let vector w( 2.2)(k1: k2) denotes the FD
coefficients on the klth through the kgth grid points for the mth order derivative at

the ith point in x direction. And let vector 1115;) (11 : 12) denotes the FD coefficients

on the 11th through the 12th grid points for the mth order derivative at the jth point
(2)

in y direction. Then the matrix representation of a k can be given as

 

 

 

 

(2) (2) ,(2)
(1117.2,1 . l . al‘.2.f\r13 u).17,2(1 1V1 )
(2) _ . . . __ .
(antic) — 3 " 3 _‘ 3 (7'21)
(2) <2) ,(2) ,
_ aJ'JVr—lgl . ' . al'.1Nr_l;—1,J\fx ‘ _ “(L‘JVI—1(1 ' [\‘E) _ (AYI__2)XATJ.

157

r)
Here (1ng A: I are the FD coefficients for

sentation can be written as

 

82a . . . . .
X2 at. grld pomt (1,17). Thelr matrix repre—

 

 

 

 

 

2
0 O b13102
(‘3) 2
0 bx,i,1.l br.i.1,2
(2) (2) (2)
:r,i,2,0 br,i,2,1 br,i,2,2
(bx,i,k.l) = ' ' E (7.22)
bf?) bf?) (2)
I.i,Ng;-1,0 LLNx—IJ I.i,Nx—1,2
(2) ('2)
O b$,i.N1:._1 $,i.f\rx,2
0 0 (1(2)
L I,1,N$+1,2
(2) -
bI.'f,0.1VJj—1 0 0
(2) (2) 0
I.i,1,f\rx—l 17.2.1JV1;
(2) (2) ('2)
banana—1 b.17,i,2,NI bx,i,‘2.Nx+1
be) (2) be)
3.1.NI—1JVI—1 $.1..[\rI—1,1\TI 1,7,1NrI—1,NI+1
(2) ('3) 0
JT,‘II,"VI,I\[I—1 17,7,1N'xjvx
('2)
b$,i.1Vr.r+1.IVx—1 0 0 d
0 0 qu,l(2 : N, — 1) 0 0
0 “182% I : NI) 0
111532 (0 : N3; + 1)
11:23“) : N, + 1)
0 .l ”:22 ( 1 : 1V1: ) 0
,(21 2. _
_ 0 0 11.I.,l-( . AI; 1) 0 0 _ (IN’y+2)x(1\’I:+2)

(7") , d 2"”) ( —1 2 3 k—O 1 v +1)-~ a PD I-ffi ' Harm-fl
C113,],k (n Lil'INJNk In _ , ,., .— , ,...,I J: are 1e C)C C19 . 8X7"
at. (1, j) and (N17, j), respectively. Their matrix representation can be written as

1"") _ (m) (m) (m) _ [ m) . ]
((1131.11‘) _ [ 6.510 6331.1 Cm,1,N;r+1 — [UL] (0 - IN.” + 1) 1><(N,,~+2) (7.23)

158

(m) ) _ (m) (m) (711)
(CI,1V;L'.k — C$.N_1:.0 CQTII’VJHI ‘ ' ' CIJVIJVI'i'l (7.24)

(m) .
lwiCeNIm ° N1: + 1)11X(N;1;+2)

Similarly, (a(2) )’ (bf?)

y j k ijc l) and (C(m) ) are the FD coefficients for the partial

yokk
derivatives with respect to Y. Their matrix representations can easily be obtained
by changing subscription 1' into y in Eqs. (7.21)-(7.24).

At the corner of two free edges, the discretized boundary conditions can be written

38
NF (1) 7y“ (1)
Z ex,i,k Z qyJJcJukJ = 0 i: 11N$1 J = 1, Ny (7.25)

(1)

1',i,k

(1)

where the matrix representations of e and qy j k l are

(e212,) = [wggu : N,)]1xN$ (7.26)

 

 

F 0 111133-(1 : NI) 0 -
Inf/3(1): NI + 1)
(75)-M) = s (7.27)
wf‘llj)-(0 : NI +1)
_ 0 “511)“ I N‘”) 0 _ Nxx(Ng,-+2)

In order to provide a matrix representation of the fictitious values, we need to
establish some notations. Referring to Fig. 7.1, we denote the fictitious values by a
vector F M, where M = 2 is a stencil parameter. The 1.7-component. of F2, Fg‘k, is

denoted by

F21 = “251‘ (7.28)

159

where the index k is restricted to

f

7—1 V2§iSNI-1,j=0
Nz—2+i Vlging,j=1
N—2+N +4'—2 +1+i vogztgi,2_<_'gN—1
k=( :1: :1: (J l J 31 (7.29)

.Nx—2+l\r$+4(1My—‘2)+z VlsiSNz,j=]\ry

 

.Nx—2+.’)Vx+4(i’Vy—-2)+NI+Z—1 V2S’SANI_1,j=Ny+1

Here, the label 13 counts only all the fictitious points.

To resolve the elements of F2, we consider a matrix representation of the form

F2 = IQU (7.30)

where vector U denote all the function values on the interior grid points. The com-

ponents of U are given by Uk = “iJ with

Here, the label k counts only the all the interior points. The matrix C2 can be
determined as the follows. First, let us cast the discretized boundary conditions

(7.9)-(7.20) and (7.25) in the matrix form

Abp2 - F2 + AbU - U = 0, (7.32)
where Ang and AW respectively represent the coefficient matrix of fictitious values
F2 and interior grid point values. By using Eq. (7.30), the above equation can be

rewritten

’4ng - CQU + AbU ' U = 0. (7.33)

160

Therefore, we can determine the matrix C
C2 = —A;F19 . Aw. (7.34)

It is noted that the dimensions of matrices Abp2 and AbU are (4N; + 4Ny — 12) by

(4N; + 4Ny — 12) and (41%; + 4Ny —— 12) by (Na; — 2)(Ny — 2), respectively.

Discretization matrix

The above section provides a representation of fictitious values in terms of the function
values on the interior grid points by using the boundary conditions. This enables us
to construct a matrix representation of the governing equation solely in terms of
function values on the interior grid points. A second order discretization (M = 2) of

the governing equation can be given

2 < > 2 >Ny+1 ( >
Z Smueka ”A ' 2 335,7. 2 ty.i+k.j,lui+k-1 (7-35)
k=—2 k=—2 1:0
2 <4)
2
+ Z Sy.k"i..j+k : Q '"iJ
k=~2

with 2 S 2' S N; — 1, 2 S j S Ny — 1. Here 33.7,? and 5;")? are central FD coefficients,

which can be written as

. , ( ,
(.533) = ”(757—2 ; 2), (.1173) = “(337.2 . 2). (7.36)
(2) . . 02a . . ,. . ,
Here (y. are the FD eoefhments for — at (z + A33). Vt hen 2 _<_ z+k S A; — 1,

i+k.j.l ayQ
.. . . 2
they are standard central FD coefficients. The matrlx representation of l( )

‘y.i+k.j,l can

161

be written as

 

wile = 4) 0 0 °
0 u’flu I 5) 0 0
2 0 0 0
(ti/,2‘)+k.j,l) = 0 0 0 111:”) (j — 2 3 j + 2) (7.37)
0 0 0 0
0 0 i
0 0
0 0
0 0
0
0 'll’f}\.~'y_1(Ny _ 3 5 N9 +1) - (Ny—2)><( ’y+2l

 

(2

i 2 2 . . 2 .
Note that 'wy‘2)(0 : 4) = w( )(1 : 5) = 11);)(3 — 2 : j + 2) = IUZS’Ary_1(./\’y — 3 :

373 J

Ny + 1) since all of them are standard central finite difference schemes with the
same bandwidth. Near the corners, i.e., when i + k = 0, 1, NE, NI + 1, the standard
central FD schemes cannot be used in the cross derivatives because there are not
enough grid points on the side where grid point (i + k. j) is near the upper or lower

side of the plate. Therefore, asymmetric (i.e., one-sided) FD schemes are used in

these cases. Parameter L _>_ 2 is used to describe the bandwidth of the derivatives.

162

K)
(1‘51“?)+ k j l)( , T could be represented as the following:

 

103.3(2 2 + L) 0 0 0
(2) .
wy’2+L/2(2 . 2 + L) O 0 0
, ,(2) .
0 uy.3+L/2(3.3+L) 0 0
0 0 0
0 o 0 u§12J)(j— L/2 :j + L/2) (7-38)
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0
0 10(2) (N-2-L-v—2) 0
yJVy—2—L/2 A y ' 1 y
0 0 “(2) (v — 1 — I - N — 1)
y.i’\’y—1—L/2 ‘ y 1' y
(2) V, . r
O 0 'II’ZIJVry—1(f\y — 1 _ l1 . 4N3] — 1) -

 

The FD weights in the first and the last L / 2 rows are different from each other since
one—sided FD schemes are used. The FD weights in the middle rows are the same

since they are standard central FD weights with the same bandwidth. Finally, the

163

discretized governing equation can be written in the following matrix form
2
AgF2°F2+AgU-U=Q -U (7.39)

where 4491.1) and AgU represent the coefficient matrix of fictitious values and function
values on interior points obtained from the discretized governing equation in Eq.

(7.35). Substituting Eq. (7.34) into Eq. (7.39). one obtains
(Agp2 . Ab‘é aw + AgU) - U = 92.11. (7.40)

It is noted that the dimensions of matrices 149172 and AgU are (N1, -- 2) - (Ny - 2) by
(4N; + 4Ny — 12) and (NI - 2) - (Ny — 2) by (NI — 2) - (Ny —— 2), respectively. Finally

Eq. (7.40) is to be solved for the eigenvector U and eigenvalues.

7 .2.2 High order MIB schemes

To construct high order MIB schemes, we first create more fictitious values outside
the boundary. For example, three layers of fictitious values are needed for a fourth
order MIB scheme. Let M denotes the total number of layers of fictitious points. In
general, M layers of fictitious values can be used to construct (2M — 2)th order MIB
scheme. Unfortunately, the total number of the boundary conditions is fixed. One
therefore encounters a situation that the number of unknowns (i.e., fictitious values) is
greater than the number of equations (i.e., boundary conditions). This difficulty has
been resolved in the MIB method by repeatedly using the boundary conditions [163]
and appropriate interior function values. This idea and its iImplementation details

were discussed in the original work of the MIB method [163].

Domain extension

To obtain a matrix representation of the fictitious values, we provide an iterative

scheme. Assume that the first p — 1 layers of fictitious points have been found as

164

 

 

 

 

 

 

 

 

 

. . . J J v v v v v

r I L r v5 vlflvl- v-v-,--, ‘ ........ 4.5.6 .-—6-- 1
. 9+ -< +4 . , ... -s
L I I~ %¢ ¢ v v .1 : e e c - _‘-_'
~—v ... o i I —«I~v o-v—-— v—»Q~o~o ------ ‘ eeeeeeeeeeee o AQAAQLV
‘ vA-o—o ,1,__--_‘_.v__,__..V-,j voo o ------------------- 0~vo1
«A. 0 v .,‘- 0 .7 ,__-' who’re .0 eeeeeeeeeee , 777777 I) erory
yo 9- I; A A Louvre-~- v~-6‘~+Aro~»1 ------ L ----- 0 »ov»o~v
_-+_-._ o-~--- —A-——.-——‘.--0~~o--—v‘~- v~-O» 9 -0---, ------ ‘-r-‘~—,A--¢ 4779 1
s g 3 a4; a. ¢ ¢ .A ¢ ¢ .4 :..
......... 0.... ————— I0 9009
, , , My , , +- V ,_ J- 1, - ........... as b one»;
-_-r--._--.-l-..---,.i_-. ........ J--_ - J r 1 'm‘?‘

Figure 7.2: The distribution of fictitious points: (a) p = 3, (b) p = 4; ‘V’ indicates
the next layer of fictitious points to be extended, ‘0’ indicates the fictitious points
have already been found by previous steps.

shown in Fig. 7.2 and can be written as
F1, _1 = Clo-1U. (7.41)

We look for the expression of PP in terms of C -1 and U.

There are 2N1, + 2Ny — 8 fictitious points on each layer except the first one. To
obtain the matrix representation of the pth layer of fictitious values, 2N; + 2Ny - 8
unknowns can be solved from 4Nx + 4Ny —- 8 boundary conditions on 2N],- + 2Ny — 4
edge points. Therefore, only one boundary condition on each non-corner edge point,
i.e., Eqs. (7.10), (7.12), (7.14), (7.16). (7.17) and (7.19), is required. In fact, each
boundary condition has to be enforced at least once. After the first round enforcement
of all boundary conditions, we normally use only the easy-to-implement boundary
condition. In order to involve all the 7) layers of fictitious points in these boundary

("1) (m) ("1)

conditions the FD coefficients (:11 k’ c LNz k (1,11 k an nd “$113.1; k in these equations

should have the following matrix representations

((5:90: “($99 P- L+ 1) (“:33”) =“ i‘IIfiJ (.Jv ‘ Li“'1+7”1l(7'42)

(cg/"1016) = 11);?)(2 - P : L +1), (C(IIRIk) = u';,1\?yy(1\’y — L : Ny + p — 1),

165

(2) 02.1

for

 

where L is the bandwidth of the FD coefficients. The FD coefficients (1

:r,i,k 0x2 ’
. . . . (2) 0221 . .
at pomts (1., 1) and (I. Ny), and the FD coefficwnts 0y]. k for W at POlnt-S (1.)) and
a 7 ( .—

(NI, j) are also defined with bandwidth L + 1 to achieve better stability. The matrix

(2)

representation of (a1, ,- k)(V 9) (N )
” ‘- ‘ I $—~ X I

is given as

 

(113(1 L + 1) 0 0 0
u1;%i+L/2(1:L+1) 0 0 0
0 ui2g+L/2(2z L + 2) 0 0
0 0 0
0 0 0 1182(13 — L/2 : i + L/2) (7-43)
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 I
0 0 0
0 0 0
0 0 0
0 0 0 ,
0 0
0 wfjvr2_L/2(1\IJ — 1 — L: NJ — 1) 0
0 0 (1,§i?VI_L/2(NJ — L— : NJ)
0 0 (“EILVMNJ — L ; NJ) _

 

166

and (am

as

. ,- ) , r is given as
yjk (.NIy—2)X(i\y)

 

IUEE§(12L +1) 0 0
u'SI)+L/2(1 : L + 1) 0 0
111;?2)+L/2(2:L+2) 0
0 0
0 0 ((5)0 — L/2 = j + L/2)
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0
0 u1£?}vy_1_L/2(Ny — 1 — L: Ny — 1) 0
0 0 Jug/'23),” __ 1. /2(1vy — L: NJ)
0 0 "“(fivfy—1( (’y — I. : Ny)

 

_l

C)

OOOOOOOO

(7.44)

The matrix representation of 2N; + 2Ny — 8 boundary conditions can be written

ACFC - Fe + IACFP—

1

' Fp—l + AeU ' U = 01

167

(7.45)

where vector Fe denotes the pth layer of fictitious values. The dimensions of matrices
f1€F€,Aer_—1 and fleU are (QJIVIx + QI‘NIy ‘_ 8) l3} (21V'x + 21Vy — 8). (ZINE; "l" 21V}, _ 8) by

respectively. Combining Eq. (7.41) with Eq. (7.45), one obtains
’4ch - Fe + A6Fp—l - Cp_1 - U + ACU ' U = 0. (7.46)
Here, one can resolve Fe
Fe = —A;Fle - (.zi,,.~p_1 -C,,_1+ ACU) - U = Ce - U. (7.47)

Therefore, the 1) layers of fictitious values can be represented in terms of interior

points

By repeating the above procedure until 1) = M, we will have the matrix represen-

tation of M layers of fictitious values in terms of interior values
FM = CMU. (749}

where the vector of interior values having its components U k = aid as defined earlier.

Here FM’S k-component, FMJJ, is defined by

FALL- = Ur.) (7.50)

168

where the index k is restricted to

f

 

(.41—2+j)(NJ—2)+i—1 V2gz'g1vJ—1,
—M+2gng
(M—1)(NJ—2)+( ' V1gz’gNJ,j=1
(M—1)(NJ—2)+NJ V—M+2gigi,
k=( +2.\AI(j—2)+M—1+z' nggNy—l (7.51)
(M—1)(.-’\('J—2)+NJD v —M+2§ig 1,
+2111(j—2)+1111+1+z'—NJ 2_<_jgNy—1
(M—1)(NJ—2)+NJ+2A.I(Ny—2)+z' VigigNJ, j=Ny
(M—1)(NI—2)+NJ;+2M(Ny—2)+Nx V2_<_ 2' g NJ—i,
(+(j—(Ny+1))(1vJ—2)+-1—1 Ny+1gjgNy+M+1

Here, we count only all the fictitious points.

7.2.3 Discretization matrix

The governing equation can now be discretized by 2(M — 1)th order MIB schemes as

M (1) M (2) Ny+1 (2)
‘ 2
Z s,,u.,.,,,-+2,\ ' Z 8.1. Z tJ..-+k.J-,zuz:+k,z (7.52)
.M (1)
‘ 2
+ E '“y,],-”'i.j+k:n utm-
[CZ—AI

with 2 _<_ i g NI — 1, 2 S j 3 Nu - 1. 5;")? and SOIL) are (llth order central FD
. ., y, .

coefficients, which can be written as

(In) _. (m) ., . , _(m) _. (m) . ,.
("ml _ mm, (—M . M), (..M) _ my“ (—(\~I . .11). (7.53)

169

,(2)

. . . can be written as

The matrix representation of (

 

.
(113‘,ng — M : 2 + M) 0 0 0
0 (113,33 — M : 3 + M) 0 0
0 0
9 (7.54)
0 0 11153.0 — M : j + M)
0 0 0 0
t 0 0 0 0
0 0
0 0
0 0
0 0
0
0 (“(2) (iv — 1 — M - N — 1+ M)
yJVy—l ‘ y 1 ° ‘ y _

 

when 2 S 11 + k g NI — 1. Similarly to Eq.(7.37) in the second order scheme, all the
nonzero weights in each row are the same. (ti/21) +k,j.l) is given by Eq. (7.38) when
2—2Mgz'+k_<_lorNJ._<_z'+k§ Nz+2M—1.

The governing equation can be written in the following matrix form

.4 FM + AgU . U = e? . U. (7.55)

I
9F M

Substituting Eq. (7.41) into Eq. (7.55), one obtains

(.4

'IgFM -CM + AQU) - U = e? - U. (7.56)

Y
1 x
, ,

, . ‘. ‘_ ‘ .H‘ I ‘ I a (7 _ _ r H .
The dimensions of matrices AgFM and AgU are (Am 2) - (Ag 2) by (2.111% +

21111N’y — 8M + 4) and (...-"\v";1C — 2) - (Ny — 2) by (NI — 2) - (My — 2). respectively.

170

7 .3 Results and Discussion

In this section, we demonstrate the utility, test convergence and examine the accuracy
of proposed new MIB method for the vibration analysis of plates. We consider three
boundary conditions, the simply supported, clamped and free edges. The performance
of proposed method is also explored on a few typical combinations of these three edges.
Special attention is paid to free edges and their combinations with other two types
of edges. Since the convergence can be improved by using adaptive grids, we also
constructed the present MIB method in nonuniform Chebyshev grids. Comparison is
made to the results of the GDQ method [133]. As an established global method, the
GDQ method is known for its accuracy in the vibration analysis of rectangular plates.
A standard eigen solver is used to obtain eigenvectors and eigenvalues. To evaluate

the performance, we compute the relative L2 error by using the first five eigenvalues,

5 2
1 lfli - “fl
L 3 Z < 9f > , (7.57)

i=1

 

to
II

where the ‘cxact’ solution S26 are the frequency data of Leissa [91].

7.3.1 Uniform meshes

Let us consider the computational domain of [0, 1] x [0, 1]. The coordinates of uniform

grid points are chosen as

 

'—1

X,=——I,———, 2L—2gigNJ+2L—1 (7.58)
“—1

37:.)I/J—1’ 2L—2gjgNJ,+2L—1. (7.59)

The convergence analysis of plates with different edge supports is presented in Tables
7.1-7.10. In these tables, N = N I = Ny is the number of grid points used for the
discretization in each dimension. Thus, N : 7 represents a 7 x 7 mesh. The half
bandwidth of the discretization is given by M , which also represents the number of

layers of fictitious values extended along each boundary. As 1W increases, the conver-

171

gence order of the MIB schemes increases. We use L to represents the stencil width
used for calculating fictitious values. In the present work, N, M, and L together
determine the accuracy of the numerical results. Once L is given, M could be in-
creased to certain level to obtain stable and highly accurate results. However, the
accuracy is eventually constrained by L. Therefore, in Tables 7.1—7.5, the value of M
is associated with the value of L. That is, M = 2 when L = 6, M = 3 when L = 8
and M = 4 when L = 11. Choosing an M that is relatively larger than a fixed L
will not increase the order of accuracy, but it may make the numerical solutions more
stable, which is a crucial feature for plates with at least one free corner.

Table 7.1: Convergence of the 8883 plate.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

M L N 91 (12 Q3 Q4 Q5 L2

2 6 7 19.510 46.029 46.100 74.362 80.222 9.74E - 2
9 19.611 47.739 47.743 76.882 90.335 4.481? — 2
11 19.657 48.297 48.297 77.617 92.856 3.07E -— 2
13 19.683 48.618 48.618 78.036 94.649 2.13E — 2

3 8 9 10.735 49.063 49.135 78.720 95.198 1.63E - 2
11 19.738 49.281 49.288 78.863 97.963 3.46E - 3
13 19.739 49.313 49.314 78.910 98.162 2.4713 - 3
15 19.739 49.330 49.330 78.933 98.455 1.13E - 3

4 11 11 19.739 49.341 49.346 78.943 98.799 4.79E - 4
13 19.739 49.346 49.346 78.955 98.620 3.4313 — 4
15 19.739 49.347 49.347 78.956 98.679 7.6713 - 5
17 19.739 49.348 49.348 78.956 98.688 3.44E — 5

‘exact’ 19.739 49.348 49.348 78.957 98.696

 

 

Table 7.1 gives the convergence analysis for the 8888 plate, which admits an
analytical solution. This is a relatively easy case. The proposed MIB method shows
an excellent convergence trend as M and N increase.

Results for the CCCC plate is given in Table 7.2. The numerical results are very
close to the ‘exact’ solutions. As the computational grid size N and the order of the
MIB scheme M increase, the numerical solutions converge to the ‘exact’ ones very
fast. Under the same N, the larger M and L usually result. in smaller errors.

We next consider plates with one free edge. Tables 7.3 and 7.4 show the con-

vergence analysis of FSSS and FCCC plates, respectively. The numerical results of

172

Table 7.2: Convergence of the CCCC plate.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

A] L N 01 92 Q3 Q4 Q5 L2
2 6 7 36.229 68.497 68.518 103.539 103.659 1.06E — 1
9 36.151 71.448 71.451 107.281 120.213 4.26E - 2
11 36.100 72.141 72.142 107.780 123.492 2995' — 2
13 36.068 72.544 72.544 107.993 126.085 2.04E — 2
3 8 9 35.885 72.339 72.387 106.946 124.969 2.50E — 2
11 35.935 73.033 73.038 107.618 129.674 7.94E — 3
13 35.946 73.167 73.167 107.718 130.297 5.55E — 3
15 35.964 73.266 73.266 107.904 130.939 3.11E - 3
4 11 11 35.990 73.385 73.386 108.259 131.436 7.34E — 4
13 35.988 73.401 73.401 108.263 131.484 5.41E — 4
15 35.986 73.396 73.396 108.242 131.560 3.37E — 4
17 35.985 73.394 73.394 108.225 131.573 3.46E - 4
‘exact’ 35.992 73.413 73.413 108.270 131.640
Table 7 .3: Convergence of the F SSS plate.
M L N 511 02 Q3 Q4 95 L2
2 6 7 11.714 27.672 34.557 55.264 56.663 8.63E — 2
9 11.676 27.704 40.479 59.123 59.643 1.79E - 2
11 11.682 27.729 40.326 58.739 60.403 1.43E — 2
13 11.684 27.740 40.606 58.857 60.858 9.81E — 3
3 8 9 11.632 14.177 14.177 27.739 42.686 4.58E — 1
11 11.682 27.743 37.346 57.609 61.639 4.33E — 2
13 11.681 27.748 48.345 55.034 55.034 9.69E — 2
15 11.683 27.752 41.265 59.098 61.808 8.70E — 4
4 11 11 11.682 27.743 37.346 57.609 61.639 4.33E — 2
13 11.685 27.756 41.352 59.183 61.859 190E — 3
15 11.685 27.756 41.193 59.064 61.859 5.50E — 5
17 11.684 27.757 41.196 59.066 61.860 3.13E - 5
GDQ 15 11.685 27.756 41.197 59.066 61.861 0.00E — 0
‘exact’ 11.685 27.756 41.197 59.066 61.861

 

 

 

 

 

 

 

 

 

 

173

Table 7.4: Convergence of the FCCC plate.

 

 

 

 

 

 

 

M L N 01 02 Q3 :24 {25 L2
2 6 7 24.257 41.402 56.145 71.334 77.538 6.51E-2
9 24.238 40.949 60.878 75.203 80.984 2.33E—2
11 24.202 40.709 61.807 75.819 80.940 15517—2
13 24.157 40.547 62.517 76.161 81.095 1.01E-2
3 8 9 24.231 40.198 63.146 63.146 66.383 11215-1
11 24.682 40.538 58.197 76.489 78.650 4.13E—2
13 23.738 39.051 62.029 71.671 71.671 6.04E—2
15 24.381 40.014 63.630 76.489 80.351 7.26E-3
4 10 11 24.118 40.353 61.677 76.850 80.019 1.39E—2
13 23.665 40.073 63.766 76.933 81.259 75913—3
15 23.987 40.194 63.325 76.913 80.809 24219—3
17 23.985 40.194 63.454 76.913 80.967 2.50E—3
GDQ 15 24.025 40.147 63.494 76.845 80.901 1.67E—3
‘exact’ 24.020 40.039 63.493 76.761 80.713

 

 

 

 

 

 

 

 

 

 

the GDQ method [133] with the mesh size of 15 x 15 are provided for a comparison.
Both cases involve a free edge which is designed to test the proposed algorithm. The
convergence of the numerical solutions to the ‘exact’ ones is very fast. When the mesh
size is 15 x 15. the relative error L2 for the FSSS plate is 8.7 x 10"4 for M = 3 and
5.5 x 10‘5 for M = 4. With the same mesh size, the relative error L2 for the FCCC
plate is 7.3 x 10‘3 for M = 3 and 2.4 x 10‘3 for M = 4. These differences again
indicate that it is relatively easy to deal with simply supported edges. The numerical
results also show that when N is relatively small, i.e. N < 13, M = 2 is a better
choice than M = 3 and M = 4. The GDQ method [133] produces similar but slight
better accuracy in these two cases. Nevertheless, the proposed MIB method works
well for plates involving one free edge.

We next consider a few plates with one free corner, which is a more challenging
situation. Tables 7.5, 7.7 and 7.7 show the convergence analysis of FF SS, FF SC,
and FFCC plates, respectively. For the GDQ method [133], when a uniform grid is
applied, the relative error L2 is up to 1.18 x 10’1 in a mesh size of 15. For the present
work, with the same mesh size and M = 4, the relative error [.2 is 1.70 x 10‘2 for
the FFSS plate, 3.36 x 10‘2 for the FFSC plate, and 1.63 x 10‘2 for the FFCC plate.

These results indicate that the present method is about 10 times more accurate than

174

Table 7.5: Convergence of the F FSS plate.

 

 

N

91

Q2

Q3

 

7
9
11
13

2.729
3.089
3.238
3.320

15.248
17.391
17.363
17.350

17.792
18.878
19.042
19.151

37.094
38.220
38.320
38.401

48.009
44.457
49.842
50.357

1.12E —1
7.13E - 2
2.29E — 2
1.19E — 2

 

9
11
13
15

14.691
4.013
3.563
3.339

17.633
17.292
17.306
17.311

20.100
22.196
20.067
19.300

34.884
41.877
39.674
38.287

34.884
41.877
40.018
51.012

1.51E -— 0
1.4213 — 1
1.04E — 1
5.66E' - 3

 

11

11
13
15
17

4.069
3.784
3.483
3.394

17.317
17.320
17.321
17.320

22.266
20.500
19.623
19.277

43.893
39.629
38.647
38.257

43.893
51.145
51.089
51.078

1.47E — 1
6.30E - 2
1.70E — 2
4.97E — 3

 

GDQ

15

2.549

17.316

17.662

36.576

51.039

1.18E —1

 

‘exact’

 

 

 

3.369

 

1 7.407

 

19.367

 

38.291

 

51.324

 

 

 

Table 7.6: Convergence of the FFSC plate.

 

 

N

01

Q2

93

115

L2

 

7
9
11
13

4.772
4.921
4.781
4.908

20.235
19.344
19.296
19.455

20.235
24.460
24.801
24.677

41.746
42.767
43.297
43.135

51.279
49.634
52.463
52.747

1.00E — 1
4.73E — 2
4.89E — 2
3.87E — 2

 

9

11
13
15

3.129
4.044
7.710
5.778

17.183
20.18
21.567
19.528

19.274
24.31
21.567
24.428

25.322

43.46
43.271
43.475

43.949
52.593
51.026
53.828

2.95E — 1
1.13E — 01
2.12E — 01
3.68E — 02

 

11
13
15
17

4.686
8.252
5.746
5.419

20.535
21.295
19.467
19.210

24.506
21.295
24.553
24.853

43.898
43.179
43.597
43.559

54.736
52.351
53.701
53.214

6.70E — 2
2.54E - 1
3365’ — 2
6.5013 — 3

 

GDQ

15

5.780

20.703

20.926

40.296

52.255

9.07E — 2

 

‘exact’

 

 

 

5.364

 

19.171

 

24.768

 

43.191

 

53.000

 

 

 

175

Table 7 .7: Convergence of the FF CC plate.

 

 

M

N

01

Q2

Q3

 

L
6

7
9
11
13

6.465
6.386
6.919
7.068

22.119
24.668
24.244
24.210

25.469
25.090
27.969
27.737

48.406
40.059
44.640
46.397

60.285
40.059
61.676
61.981

5.5213 — 2
1.84E - 1
3.80E — 2
2.4813 — 2

 

9
11
13
15

3.727
8.696
18.003
8.154

20.375
23.148
18.003
24.125

24.557
25.602
24.454
26.764

24.557
47.405
44.430
50.101

47.760
61.942
44.430
63.415

3.28E —-1
1.1613 —1
7.35E' — 1
8.11E — 2

 

11
13
15
17

5.252
3.716
7.149
6.542

24.199
23.677
24.323
23.735

33.573
29.657
26.249
25.747

45.287
54.279
47.865
47.547

45.287
63.305
62.719
62.692

2.04E — 1
2.22E — 1
1.63E — 2
3.08E — 2

 

GDQ

15

7.873

23.615

23.873

44.587

62.730

8.23E — 2

 

‘exact’

 

 

 

6.942

 

24.034

 

26.681

 

47.785

 

63.039

 

 

 

the GDQ method for handling a free corner. Overall, the present method works very

well for plates involving a free corner.

Table 7.8: Convergence of the FF FS plate.

 

 

M

L

N

01

Q2

93

 

2

6

7
9
11
13

5.660
6.068
6.306
6.618

12.013
14.981
14.827
14.801

19.158
25.270
25.005
25.337

26.226
25.699
26.048
26.655

47.165
39.212
46.947
47.648

1.58E' — 1
9.5913 — 2
3.00E — 2
1.52E — 2

 

9
11
13
15

11.815
8.474
7.318
6.605

11.815
15.118
14.954
14.894

20.797
30.600
26.850
25.393

25.391
37.322
28.249
26.119

25.684
37.322
49.635
48.601

4.26E — 1
2.66E — 1
6.32E — 2
5.191? — 3

 

11
13
15
17

8.684
6.900
6.866
6.666

15.113
14.979
14.919
14.912

29.525
25.673
25.934
25.436

38.934
27.157
26.347
26.171

38.934
48.596
48.598
48.475

2.83E — 1
2.60E - 2
1.73E — 2
4.32E — 3

 

GDQ

15

5.161

14.725

23.082

24.156

46.296

1.16E-1

 

‘exact’

 

 

 

6.648

 

15.023

 

25.492

 

26.126

 

48.711

 

 

 

We next. consider two plates that. have more than one free corners. These cases are
considerably more challenging. The convergence analysis of FFF S and FF F F plates
is listed in Tables 7.8 and 7.9, respectively. It is seen that the numerical results in

both configurations have a similar level of accuracy comparing to the configurations

176

Table 7 .9: Convergence of the FF FF plate.

 

 

M

N

91

Q2

Q3

 

7
9
11
13

8.632
11.574
12.240
13.080

17.203
19.564
19.585
19.595

20.565
24.175
24.165
24.119

22.502
33.361
34.563
34.547

35.474
34.771
34.867
36.270

2.41E - 1
6.56E - 2
4.09E — 2
2.33E — 2

 

9
11
13
15

0.000
0.002
15.783
13.662

7.418
19.552
19.574
19.584

7.418
23.279
24.841
24.319

14.236
27.607
36.741
34.617

19.468
37.326
38.585
35.491

6.94E — 1
4.58E — 1
9.49E - 2
1.13E - 2

 

11
13
15
17

0.000
16.885
14.571
13.505

19.604
19.601
19.599
19.597

27.209
25.222
24.406
24.335

30.625
37.724
35.488
34.361

30.625
38.772
35.761
34.943

4.57E — 1
1.3113 — 1
3.97E - 2
6.191? - 3

 

GDQ

15

10.303

19.596

22.146

30.026

30.803

1.38E - 1

 

‘exact’

13.468

19.596

24.271

34.801

34.801

 

 

 

 

 

 

 

 

 

 

with only one free corner. For the GDQ method [133], the overall relative error L2 is
larger than 0.1 in these cases with a uniform mesh size of 15 x 15. The relative GDQ
error for a single mode is as large as 0.23 and the absolute error for a single mode is
as large as 4.8. For the MIB method, the overall relative error L2 is 1.73 x 10"2 for
the FFF S plate and 3.97 x 10“2 for the FF FF plate. The relative MIB error for a
single mode is less than 0.1 and the absolute error for a single mode is less than 1.0.
The proposed MIB method works very well for multiple free edges and free corners.
These results indicate the merits of using central finite difference schemes in vibration

analysis.

7 .3.2 Non-uniform meshes

It is reported [133] that the GDQ method is very sensitive to the grid point distribu-
tion for plates with at least one free corner. Adaptive grids were proposed to stabilize
the GDQ method. In this work, we are interested to know how the adaptive grids

will improve the. results of the proposed MIB method. Let us consider adaptive grids

177

given by the Chebyshev coordinates

NIH NIH lav-I

 

Mt“ MP4 NH

3+cos

1-co~..:(7,L'TI
(‘7‘:—

Ll—cos(1

 

h -(
3+cos (1471—1 '77)

cos( _11’ 77)—
JII

jfi—i

71'

 

)f

, 2L—2SiSO

 

ISiSNI

, 2L-2SjSO

(7.60)

, Nx-l-ISing+2L—1

(7.61)

, NJ+1gigNy+2L—1

Non-uniform central FD weights are generated for this grid mesh by using the stan-

dard Lagrange polynomials [55].

Table 7 .10:

The numerical results obtained with adaptive grids (N = 13,M =

5, L = 5). The relative errors obtained with uniform grids is denoted as L2’uni and
are listed for a comparison.

 

 

 

N 91 Q2 Q3 Q4 05 L2 L2,uni
SSSS 19.739 49.345 49.345 78.946 98.643 2.49E - 4 2.13E - 2
CCCC 35.985 73.386 73.394 108.253 131.424 7.651? — 4 2.04E — 2
FSSS 11.720 27.773 41.486 59.268 61.864 3.76E — 3 9.8113 — 3
FCCC 24.322 40.066 64.639 76.583 81.332 1.04E — 2 1.01E - 2
FFSS 3.412 17.437 19.439 38.522 51.556 6.911? — 3 1.19E — 2
FF CC 6.953 23.786 26.869 47.434 63.384 6.96E - 3 2.4813 — 2
FFSC 5.443 19.155 25.003 43.406 53.079 8.19E — 3 3.87E — 2
FFFS 6.681 15.012 25.611 26.160 49.243 5.82E - 3 1.52E' — 2
FFFF 13.603 19.851 24.269 35.301 35.362 1.21E - 2 2.331? — 2

 

 

 

 

 

 

 

 

 

Table 7.10 shows the numerical results of nine plate configurations under an adap-

tive mesh of 13 x 13. As mentioned before, when N is small, a relatively small L pro-

duces better results than a relatively large L. Therefore, when N = N1 = Ny = 13,

L = 5 is used in this study. It is known that when L = 5, M = 2 has the same order of

accuracy as M = 5. The reason of using M = 5 here is to increase the stability of the

method with the adaptive mesh. Comparing with the relative error L2 obtained with

the uniform mesh when N = 13 and L = 6, it is seen that although all the numerical

results are slightly improved, the improvements on plate configurations involving at.

least one free corner are not. as large as on those without free corners.

178

On uniform grids, the performance of the present MIB method is similar in all the
configurations. In contrast, on adaptive grids, the accuracy decreases as the number
of free edges increases in most cases. A possible reason is that for plates with simply
supported and clamped edges, the boundary stress is a major source of numerical
errors. Therefore, adaptive grids which provide more computational nodes near the
edges increase the overall accuracy. Whereas for plates with at least one free corner,
the errors induced by free motions cannot be simply suppressed by adaptive grids

near the boundary.

7 .4 Conclusion

The matched interface and boundary (MIB) method [161, 163, 169, 170] is general-
ized for the free vibration analysis of rectangular plates. This work is motivated by
the difficulty of implementing free edge boundary conditions in our earlier discrete
singular convolution (DSC) algorithm [149], which has been applied to a variety of
vibration analysis of plates [150—152, 155, 166]. However, the proposed MIB method
is independent of our DSC algorithm. To test the performance the present method,
nine plates of simply supported, clamped and free edges, and some of their important
combinations are considered. To test the accuracy, first five eigenmodes are examined
by relative L2 errors. To test the convergence, mesh sizes of 9 x 9, 11 x 11, 13 x 13,
15 x 15 and 17 x 17 are employed. The number of layers of fictitious values, M, is
selected in association with stencil width L. Different combinations of M, L and N
are studied in the convergence analysis. It is shown that the present method provides
accurate and stable solutions for all nine plate configurations. As M, L and N in-
crease, the numerical solutions converge to the ‘exact’ solutions provided by Leissa
[91]. Comparison is made to the nun‘ierical results by the GDQ method [133]. As a
global method, the GDQ performs better for plates without free corners. However,
the proposed MIB method delivers much better results for plates with one or multi-
ple free corners. A possible reason is that the GDQ method enforces the boundary

conditions through one—sided discretizations while the present method enforces the

179

boundary conditions through symmetric discretizations.

It is well known that the performance of the GDQ method can be improved
by using the so called adaptive grids, which distribute dense grid nodes near the
boundary. Such an approach provides results similar to those obtained by the present
method with uniform grid. We have also implemented the MIB method with adaptive
Chebyshev grids by using non-uniform central finite difference schemes. The present
study shows that the MIB method with adaptive grids reduces the errors of 8888 and
CCCC plates up to 90 times. However, the improvement is not obvious for the rest
configurations which involve free edges or free corners.

The MIB method is a general approach for solving partial differential equations
with material interfaces and/or special boundary conditions. The essential ideas of
the MIB method is to employ simple Cartesian grids to avoid grid generation even
if the interface or the boundary is complex and irregular. The standard (higher-
order) central finite difference (FD) schemes are utilized for the discretization of
the governing equations in the entire domain, including the interface and boundary
region. Since the central FD schemes reach across the interface and / or the boundary,
fictitious values are used to ensure that the numerical derivatives are locally computed
on a “smooth function”. Therefore, fictitious values are the smooth continuation of
the original function across the interface and/ or the boundary. This is the so called
domain extension. We make use of the interface and/or boundary conditions to
determine the fictitious values, which in turn, rigorously enforces the interface and / or
boundary conditions. In order to construct higher-order schemes, the interface and / or
boundary conditions are repeatedly utilized. For elliptic equations, the domain is
extended along one meshline at a time. Consequently, the 2D or 3D interface problem
is locally reduced into 1D-like ones. However, due to the cross derivatives, the 2D
domain extended has to be pursued simultaneously. It is possible to recast the present
MIB scheme in terms of an interpolation formulation which bypasses the fictitious
values or extended domains. It was shown that such an MIB approach is equivalent

to the fictitious domain formulation [169].

180

A motivation of the present work is to study the possibility of generalizing the
MIB method developed for elliptic equations with arbitrarily complex interfaces [59,
158, 160] to the structural analysis with similar geometric complexity. A relevant
finding in the present work is that when M = 2 and N Z 9, the proposed method
yields excellent results for all the plates examined. For arbitrarily complex geometries,
M = 2 is likely the best one can achieve, according to our experience in handling
arbitrarily complex elliptic interface problems [59, 158, 160]. Therefore, the present
results indicate a great promise for this future generalization. This aspect is beyond

the scope of the present thesis and is under our consideration.

181

Chapter 8

Thesis Achievement and Future

Work

8. 1 Thesis Achievement

The main contributions of the present thesis are follows. First, the MIB method is
generalized to solve 2D and 3D elliptic problems with geometric singularities. The
presence of geometric singularities, such as cusps and self-intersecting surfaces, is
a major obstacle to the accuracy, convergence and stability of numerical methods.
Technically, it is much more challenging to construct efficient numerical methods for
this class of problems due to the lack of regularity in the interface. The original
MIB method is designed to solve elliptic problems with smooth interfaces. Fictitious
values are obtained along interfaces and are utilized to solve the governing equation
with standard finite difference schemes on each side of the interface separately. But
in many physical applications where there are geometric singularities on the interface,
it is difficult to find enough fictitious values. The new MIB method developed sev-
eral strategies and successfully resolves the aforementioned challenges. The two main
ideas in the MIB method for geometric singularities are to solve fictitious values iter-
atively around singularities and to use two sets of jump conditions around geometric
singularities simultaneously.

Second, the fourth—order and sixth-order MIB schemes are successfully extended

182

from two dimensions to three dimensions. A systematic algorithm is developed to an-
tomatically simplify jump conditions for any local geometries. This algorithm greatly
reduces the complexity of the formulations of 2D and 3D MIB schemes, which makes
the treatments of complex geometries by high-order MIB schemes possible. Extensive
numerical tests show that the fourth-order accuracy is achieved for complex interfaces
with moderate geometric singularities and the sixth—order accuracy is achieved for
smooth curved interfaces.

Third, a matrix optimization strategy is developed, which not only significantly
reduces the cost of solving linear systems, but also resolve the convergence issue arising
in solving large linear systems, whose size might be as large as 106 by 106. The main
idea of the strategy is to optimize the distribution of auxiliary points which are used to
represent fictitious values. A systematic scheme is adapted to select auxiliary points
surrounding irregular points as symmetrically as possible. The CPU time required
by the optimized MIB scheme could be less than film of that required by the original
MIB scheme. Furthermore, as the mesh is refined, the CPU time required by the
optimized MIB scheme only increases linearly while the original MIB scheme fails to
converge for large linear systems.

Fourth, a new MIB method based Poisson-Boltzmann (PB) equation solver, MIBPB-
II, is developed to calculate the electrostatic potentials and solvation free energies of
proteins. The geometric singularities, such as cusp and sharp edges, are widely found
on the molecular surfaces of proteins. The MIBPB-II is the first PB solver that pro-
vides the second order accuracy for realistic molecular surfaces of proteins. Moreover,
the MIBPB-II converges much faster than the MIBPB-I when the same linear solver
is used in both methods, and it is able to provide convergent results for calculations
of large proteins in a. very fine mesh while the MIBPB-I is not. This improvement
is caused by the matrix optimization strategy which makes the matrix more banded.
The extensive comparison of MIBPB-II, MIBPB-I, APBS and PBEQ shows that
MIBPB-II is the fastest PBE solver under the same level of accuracy.

Fifth, an MIB based nonlinear Poisson-Boltzmann equation solver is developed.

183

Damped-inexact-Newton method is adapted to solve the nonlinear system. In each
step of Newton iteration, the discretized linear system is established by the MIB
method and is solved by preconditioned biconjugate gradient method inexactly. The
criteria of inexactness and damping parameter are carefully chosen according to the-
orems provided in [70]. The numerical results show that the nonlinear MIBPB solver
not only converges superlinearly, but also preserve second-order accuracy in the pres-
ence of geometric singularities, which has never been reported in literatures.

Finally, the MIB method is adapted to solve eigenvalue problems in plate vibra-
tion analysis. The first two layers of fictitious values are obtained by using boundary
conditions instead of interface conditions. More layers of fictitious values are obtained
iteratively by using the same boundary conditions along each side of a rectangular
thin plate. The fourth-order governing equation is then discretized by standard finite
difference schemes. The cross derivative 53%;; is treated carefully based on the avail-
ability of fictitious values. Different combinations of simply supported, clamped and
free boundary conditions are tested. The proposed method works for both uniform
grids and adaptive grids. The numerical results show that by extending the com-
putational domain using fictitious values, the MIB method is much more stable and
accurate than the GDQ in the same grid setting.

In summary, the MIB method for geometric singularities is a robust, highly ac-
curate, fast convergent, and easy to be implemented method for both interface and
boundary value problems in complicated geometric shape. It has been successfully

applied in molecular biology and structure analysis.

8.2 Future work

It would be an interesting topic to extend the present MIB method from solving two-
domain problems to solving multi-domain problems, where different jump conditions
exist between background domain and each subdomains. Moreover, the MIB based
Poisson-Boltzmann solver, the linear and nonlinear MIB—PB II, provides accurate

electrostatic potential of proteins under low computational cost. It. is crucial to the

184

future work of the calculations of pKa values and electrostatic force, which could be
further applied to molecular dynamics simulations. Furthermore, the 2D rectangu-
lar plate vibration analysis by the MIB method opens a large number of new topics
in structural analysis. The vibration analysis of arbitrary shaped plates might be
done by iteratively extending fictitious values outwards using boundary conditions.
The flexibility of the new MIB method allows one to deal with complicated geome-
tries automatically. The simplified formulations and optimized matrix distributions
of the MIB method give it great potential to conduct three dimensional structural
analysis. Finally, the proposed MIB method can be generalized to solve the Navier-
Stokes equation for computational fluid dynamics and the Maxwell’s equations for

electromagnetic waves.

185

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