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LIBRARY
Michigan State
University

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This is to certify that the

dissertation entitled

Time Discretization of Transition Layer Dynamics
in Viscoelastic Systems

presented by

Hyeona Lim

has been accepted towards fulfillment
of the requirements for

Ph . D. degree in Mathematics

 

 

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Date April 241 2001

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6’01 c:/CIFiC/DateDue.p65-p.1s

 

Time Discretization of Transition Layer Dynamics in

Viscoelastic Systems

By

Hyeona Lim

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Mathematics

2001

ABSTRACT

Time Discretization of 'Iransition Layer Dynamics in Viscoelastic

Systems

By

Hyeona Lim

We investigate how evolution occurs as the strain Du of a viscoelastic system
an = Div(a(Du) + Dut) — it goes towards a state of equilibrium. The physical
description of the system is an elastic material with a nonconvex double-well energy
density and a viscous stress placed on a rigid elastic foundation subject to a zero
displacement boundary condition. The time limit of Du eventually exhibits a finite
number of discontinuous interfaces if the strain starts from the continuous initial data
whose transition layers are steep enough and the initial energy is sufficiently small.
The system conserves the number of phases and the transition layers stay within
the initial interfaces. We first consider the one-dimensional case of the problem by
using the implicit time discretization method and the Andrews-Pego transformed
equations. Numerical computations are conducted and the results are extended to

the two-dimensional system.

To my parents.

iii

ACKNOWLEDGMENTS

I would like to deeply express my heartfelt thanks to my thesis advisor, Professor
Zhengfang Zhou for his warm and patient help, guidance and encouragement. Thanks
to him, the years at Michigan State University became more valuable and memorable.

I would also like to appreciate my committee members, Professor Gang Bao,
Professor Dennis Dunninger, Professor Thomas Pence and Professor Baisheng Yan
for their kind advice.

Special gratitude goes to Professor Seongjai Kim, of the University of Kentucky
for his valuable suggestions on the numerical parts of my dissertation.

I would like to dedicate this work to my family in Korea, especially my parents,
Youngsik Lim and J aebun Kim to express my gratefulnsss for their constant love,
prayer and support.

Finally, I would like to thank God for blessing me to accomplish all these things.

iv

TABLE OF CONTENTS

Introduction 1

1 Time Discretization of Transition Layer Dynamics in one-

dimensional Viscoelastic Systems 8
1.1 The initial-boundary value problem and hypotheses .......... 8
1.2 The time discrete scheme for the solution ................ 10
1.3 Main results ................................ 12
1.4 Energy decay and a-priori estimates ................... 13
1.5 Equilibrium state of the time limit of the solution ........... 21
1.6 Dynamical behavior of the transition layers ............... 34

2 Convergence Analysis of Numerical Solutions in One-dimensional

Systems 42
2.1 Derivation of the matrix equation using the Finite Difference Methods 43
2.2 Existence of the Finite Difference solution ............... 46
2.3 Average approximation of 0(ug‘rj )3 ................... 54
2.4 The Alternating Direction Implicit (ADI) Method ........... 58
2.5 Derivation of the matrix equation using the Finite Element Methods . 60
2.6 Examples ................................. 69

3 Convergence Analysis of Numerical Solutions in Two-dimensional

Systems 72
3.1 Derivation of the matrix equation using the Finite Difierence Methods 73
3.2 Existence of the Finite Difference solution ............... 78
3.3 Average approximation of Div(a(Du""j)) ................ 82
3.4 The ADI method in two-dimensions ................... 83
3.5 Derivation of the matrix equation using the Finite Element Methods . 85
BIBLIOGRAPHY 90

Introduction

There are various results on the phase transitions of microstructured elastic crystals
[1, 3, 6, 12, 13, 16, 23, 25, 26, 28, 29]. Nonconvex double-well free energy induces
hysteretic behavior of the fine microstructures of the material. The usual approach
involves the minimization of the elastic energy. Due to the lack of convexity in the
free energy functional, every minimizing sequence fails to attain the minimizer. In
this situation, a minimizing sequence will undergo finer and finer oscillations [6, 7, 26].
However, the energy dissipation prevents such behavior and the sequence converges
to the minimizer of the energy [4, 15, 25].

This dissertation focuses on the Viscoelastic system
at, = Div(0(Du) + Dut) — u, (0.1a)

where u is a mapping from Q x (0, 00) C IR" x IR to IR” for some open bounded domain

52 satisfying the following boundary and initial conditions

u = O on 69 x [0, oo), (0.1b)

u = uo, m = 120 in Q x {O} - (0.1c)

and 0(X) = «6%,? for some stored energy function W : M N x" -> R.
The system describes a time dependent elastic material with a nonconvex energy

W and a viscous stress Au, with zero displacement boundary conditions. The material

1

interacts with an elastic foundation u. In other words, the material is placed on a
system of linearly elastic springs [28].

Many global existence results for the solutions of similar systems are available
[2, 4, 5, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 20, 21, 22, 24, 25, 27]. The existence of the
weak solution for the Viscoelastic type materials was developed for the cases without
assuming the ellipticity of the free energy W [25], the convexity of W or the Lipschitz
continuity of a [15]. In all three cases, the viscous dissipation term plays a significant
role in the strong convergence of the minimizing sequences. In the higher dimensional
case, G. Friesecke and G. Dolzmann [15] approached the result by an approximation,
called the time discretization method, on each sufficiently small time interval.

The dynamics of the transition layers on the Viscoelastic system (0.1) is the main
topic in this dissertation. Transition layers are defined by the part of the graph of the
strain Du where the norm of Du is sufficiently small and the graph changes the sign,
that is, the small neighborhoods of the solution u where it has local extrema. For the
dynamics of layers in our system, the continuous initial strain must have transition
layers which are steep enough, that is the norm of Div(Duo) should be sufficiently
large. The time limit of the strain Du usually experiences a discontinuity at a finite
number of points. More precisely, the finitely many layers of the strain Du get steeper
as time increases and eventually become discontinuous at the time limit. Away from
these finitely many points, the solution remains continuous. The number of transition
layers and the number of zeros of Du remain the same. The layers of the solution
are always within the intervals of initial layers, which is a comparable result to the
stick-slip motion of layers in a system with nonzero time-dependent displacement
boundary conditions [29]. In [29], it was proven that the layers do not stay in the
initial intervals and will move both forward and backward. G. Friesecke and J. B.
McLeod [16] proved this jump discontinuity of the transition layers at the time limit

using the weak solution of the system. In this dissertation, we use the time discretized

solutions discussed in [15] and the Andrews-Pego transformed equations which were
introduced in [2, 23] to show the phenomenon described above. This approach has
some advantages over the method in [16]. It was proven that the time discretized
solutions aid in the proof of existence of the limit of the minimizing sequences to the
energy functional [15] and several estimates which are essential for the proof of the
results are more easily verified.

The interaction of the material with an elastic foundation u induces a finely lay-
ered microstructure [5]. It has also been shown using the bifurcation analysis that
the elastic foundation induces oscillations in the one-dimensional case of the static
problem [28]. Nevertheless, under the assumption of low initial energy, the results
still hold without the elastic foundation u and only minor change is needed in the
proof. In fact, it can be easily proven that without the u term, the absolute value of
the solution approaches 1 as time goes infinity except for the finitely many isolated
points where the discontinuity occurs, while with the u term, there is a neighborhood
that the time limit of the strain is not 1.

The finite difference methods (FDM) and the finite element methods (FEM) are
used for the numerical observation of the dynamics of transition layers in one and two
dimensional cases. The methods will be described in Chapters 2 and 3 along with
the discussion of efficiency.

In Chapter 1, we use the method of time discretization [15] to prove that the
solution approaches the equilibrium state as time goes to infinity and to describe the

transition layer dynamics in the one—dimensional case of the system

u“ -— (0(u3) + um); + u = 0, (0.2)

Where u maps from Q x (0,00) C IR x IR to IR and Q = (0, 1) with the same boundary

and initial conditions as (0.1b) and (0.1a).

Let m > 0 be fixed and sufficiently small. Let j E N. For each time interval

(( j —1)m, jm], we consider the minimizer M” of the functional defined inductively
by
. 1 1 . . 1 . 1

(0.3)

1 := uo — mun. The minimizer M” is known

assuming u""0 := uo, 12m") := uo, um"
as the time discretized solution of system (0.2) since it can be proven to be the weak
solution of the time approximated equation of the system

1 m, '—1 m, '—2 1 m, '—1
fiW—Zu J +u J )_(U(u$))$—R(u$—u31 )$+u=0

in each time interval ((j - 1)m, jm], j E N. In [15], it was shown that if m —-> 0, a
subset of M” converges to a weak solution of (0.1). Note that um'j is only a function
of a: and is in the Sobolev space Wol’p ((2, IR), where p is the coercivity exponent of
W which is greater than or equal to 2. If W is a convex function or a is Lipschitz
continuous, a standard argument from partial differential equations easily proves the
existence of a minimizer of ftmctional (0.3). However, without such hypotheses, the
third term of the right hand side of (0.3), which is physically interpreted as a viscous
stress, allows the proof of the existence of the functional minimizer. Therefore, in
this problem, we do not have to assume the hypotheses given above.

Like in previous works [16, 25], the decay of the may functional

E(um’j,um’j) 2 f0 [%(u""j(:z:))2 + W(u;"’(x)) + %(um'j(a:))2 d2:

is the crucial point of the proof. An important assumption here is that the initial

energy E(uo, v0) should be sufficiently small. We prove that the transition layers

approach a jump discontinuity as time goes to infinity (j —> 00) by showing that
a finite number of intervals where the time discretized strain 112” is steep enough
are decreasing to a finite number of isolated points as 3' goes to infinity. Unfortu-
nately, the intervals in Q where the norm of 113” is sufficiently small, denote the
intervals as I (uZ‘J ), do not decrease monotonically as j —-> oo in general, that is,
I (ugw‘tl) ¢ I (uZ‘J ) Instead, we introduce the time discretized version of Andrews-

Pego transformed equations
m j l a: m j m j—l 1 1 z m j m j—l
p ’ (x) := - Iu ’ (y) - u ’ (y)]dy — - [u ’ (y) - u ’ (y)]dydz,
m 0 m o o
(1” (:13) == U?” (a?) - pm” (96)

and consider the finite number of intervals in Q where the norm of qm'j is sufficiently
small, denote them as I (qm'j ) We show that the I (qm’j) decrease monotonically and
exponentially in a nested fashion (I (qm’j+1) C I (qm'j )) to the isolated points as j —> co
and the intervals I (ug‘J ) are contained in the I (qm’j ) for each j E N. The solution
approaches the jump discontinuities as j —> 00 because of the decrease of the I (qm'j)
and the fact that the I (ug‘J) are contained in the I (qm’j). We also prove the existence
and the equilibrium state of the time limit of the discretized solution in order to show
the continuity of the time limit of the strain except for the finitely many points which
prove to be the zeros of the time limit of ug‘J .

In Chapter 2, we show the convergence analysis for the numerical results of the
one-dimensional system after obtaining the matrix equation using the FDM. Next
we derive the matrix equation from the FEM and compare the convergence rates
and the accuracy to the FDM. We introduce several finite elements such as linear,
quadratic and Hermite cubic elements and also discuss the convergence rates and
efficiency of these elements. Two difficulties arise in deriving a numerical algorithm
for the Viscoelastic system. First, in the FDM, the central difference approximation

of the nonlinear term 0(uz)z produces a significant tnmcation error as time increases

5

due to the lack of smoothness of u near the position of transition layers. The matrix
derived from the central difference approximation becomes nonsymmetric because of
the term 0(uz)x. However, in the FEM case, numerical integration gives the averaging
effect of the solution and the error is reduced. After modifying the FDM algorithm
by averaging the nonlinear term, the matrix becomes symmetric and the accuracy is
improved. Second, the direct iteration method for the nonlinear term for both meth—
ods is computationally expensive. The alternating direction implicit (ADI) Method
[19], one of the locally one-dimensional (LOD) methods, for the system is introduced.
The computation cost is reduced since the ADI method is a non-iterative method.

The a—priori estimate on the strain ugw‘ given in Chapter 1 confirms the bound-
edness of the strain. The advantage of the method of discretizing the time interval
not only eases the analytical proof but it is also effective in the numerical simula-
tion. The numerical results show a strong agreement with the theoretical predictions.
Here, we discuss the importance of small initial energy in the formation of the jump
discontinuity at the time limit. Examples of large initial energy do not exhibit this
type of behavior. Usually, ugh]. does not obey the results of the layer dynamics given
in Chapter 1 under the initial energy whose norm exceeds some critical point. How-
ever, ugw' decreases quickly and then follows the theory of the layer dynamics even
though it does not preserve the number of transition layers of the initial data. Fur-
thermore, we discuss more examples such as using the Neumann boundary conditions
instead of the Dirichlet boundary conditions or without assuming the steepness of
the initial transition layers. Surprisingly, the results obtained from the numerical
simulations show that the main results of the transition layer dynamics hold while
not theoretically proven.

In Chapter 3, We extend the numerical results of the discontinuity of layers at the
time limit to the two-dimensional case which is a more physically appropriate way to

model experiments. The matrix equations of the system using the FDM and FEM

are derived and the convergence analysis of the numerical solution is discussed for the
matrix equation from the FDM. We also examine the convergence rates by comparing
the results from the numerical computations using both methods. As in the case of
the one-dimensional system, the accuracy is better in the FEM than before averaging
the nonlinear term in the FDM. After the averaging, the FDM error is reduced and
nearly the same as the FEM. The computation cost is also discussed by comparing

the performance of the codes using the direct iteration method and the ADI method.

CHAPTER 1

Time Discretization of Transition
Layer Dynamics in one-dimensional

Viscoelastic Systems

Transition layers, the portion of the strain us where the norm is sufficiently small
and the graph changes the sign, become steeper and eventually discontinuous as time
goes to infinity. The number of transition layers of the strain is preserved. Moreover,
transition layers stay in the intervals where the initial layers occur. Important asp
sumptions are that the initial energy is low, and the transition layers are sufficiently
steep. Time discretized solutions of the Viscoelastic system are introduced in section

1.2 and are used for the proof of the dynamical behavior of the transition layers.

1.1 The initial-boundary value problem and hy-
potheses

In this section, we introduce the initial-boundary value problem in the one dimensional

case and list the hypotheses which will be assumed throughout this Chapter.

Consider the initial-boundary value problem

an - (0(uz) + Um): + u = 0,
u|x=0 = ulz=1 = 0 (t E [0: 00)): (11)

Ult=o = no, ut|t=o = ’00 (27 E [0, 1]),

where u is a function from (0,1) x (0,00) C IR x IR to IR, 0 = WI and W is a stored

energy function from IR to IR satisfying the following conditions

(H1) w e 02(IR),

(H2) There exist c > 0, C > 0, and p _>_ 2 such that
chIP — C s W(z) s C(lzl" +1), Ia(z)l s C(Izl""1 + 1),

(H3) W(zi) = 0 for some 2 = zi and W(z) > 0 elsewhere. 0(2) = z - 5(2) for
some 5 6 (FOR). There exist 21,2, where z- < zl < 0 < 22 < 2+ such

that W”[(zl,22) < 0 and W”|R\[zl,22] > 0.

Assumption (H 3) indicates that W is a double-well nonconvex function. It is usually

a fourth order polynomial and the most common example is W(z) = fizz —1)2, where

_ ._ _. 1 = _1_
2i — :lzl, 21 — W and 22 «5' Moreover, assume

(A1) no 6 02, 220 e We”, ”menu...” + [Ivollwm s M,

(A2) E(uo,uo) < 6, where
1
E(u, v) :=/ 1u2 + W(ux) + ~1—u2 dm,
0 2 2

(A3) 2,.(0) == {x e [0.11 : I<uo>.<x>l s p} c (0.1),

(A4) |(UO)u(w)| 2 K in 5/40)

for some M, e, p, K > 0. Here, 6, p are sufficiently small numbers and K is a
suficiently large number.

Let the connected components of £,,(0) be denoted by [(ao),, (120),], z' = 1, - . -, N
(0 < (ao)1 < (b0)1 < - - - < (ao)N < (bo)1v < 1). Note that by the assumption (A4),
there exists only one zero of (120),, in each interval [(ao),, (b0),-]. Let the zeros of (no);
he (30),, (2:0), 6 [(ao),, (b0),], 2' = 1, - n ,N.

Before proceeding to the main results, we introduce the time discretized version

of the solution of (1.1) in the next section.

1.2 The time discrete scheme for the solution

Let m > 0 be fixed. m < 1. The m will be the time step size of our problem. Let
u""0 := uo, 21”” := v0, u"""1 := no — mm. For j E N, define the following functional

inductively
' 1 1 ' 1 ’ 22 1 ' 1 2 1 2
J'"”[u] :=/0 [WIu—2um’3" +um’7' | +W(u,,.)+ film” —u;""" I + 2'14 dz:

on the Sobolev space WJ’p(Q,IR), where Q = (0,1) and p is the coercivity exponent
of W in (H 2). It was shown that f” attains a minimum M” if W satisfies the
hypotheses (H 1), (H 2) and (H 3) since the first and forth integrands are convex and
the nonconvex term W(ux) combined with the energy dissipation term 5%; qu. —u;"’j ’1 |2
provides the sequentially weakly lower semi-continuity [15]. It can be easily verified
that for each j E N, M” (m), which is only the function of 3:, satisfies the following

equation

1 m, '—1 m, '—2 1 m, "—1
Ei(u_2u J +u J )“(U(um))z—R(ux"‘u J )z+u=0, (1.2)

I

10

which is the time approximated equation of (1.1). The um'j , j E N are thus called the
time discretized solutions of problem (1.1). We next define the linear interpolation

fimction uj (z, t) of um’j(a:) as follows

 

mat) == (T’mi) um,,--.(,) + (t ‘ "if " 1’) awe). t e «j —1)m, m]

for all j E N. Since uj (m, t) is the piecewise linear function of the time t, it enables us
to differentiate the time discretized solutions with respect to t on each subinterval. It
is now important to define the functions which are called the time discretized version
of Andrews-Pego transformed equations. The equations will play a crucial role for

the proof of main results. Define

190(50):: Axum/)6]!!— fol foz vo(y)dydz,

(10(3) 3: (210),,(53) — 190(33),

>n=mleumj(y)— umj-1(y>]dy———//[um(y)— m 1< )ldydz,

q"”($) = u.',."’()-1D""(-'IB)

 

for all j E N. Note that pZ‘J(x) = "m'j($)—;m'j_l($) . Denote it by vm'j(:r). For all j E N

and (j — 1)m < t S jm, define the interpolation functions of p7(:r,t), qJ (as, t) and

vj(:r,t) of pm'j(:r), qm'j (2:) and um’j(a:) in the same way

was, t) == (inf—t) pm,.-—1(,.) + (t ’ m” ‘ 1)) pram),

 

 

 

 

m m
flat) == (WT—5) «I'M-lo) + (t ‘ ":53 ‘1))q"w'<z),
vj(:z:,t) := (m3; t) uni-1(3) + (t ' mg ' 1)) um’j(a:).

Next, we state the main results on this chapter.

11

1.3 Main results

The following theorem is the main results on Chapter 1 and this describes the dy-

namical behavior of the transition layers.

Theorem 1.1 Suppose the stoned energy function W and the initial data (uo, v0) 6
W01’°° x L2 are assumed to satisfy (H1)-(H3), (A1 )-(A4 ) Then the following holds

(P1) ( Conservation of number of zeros ) The number of zeros of u?” which is denoted
by N (j) is finite, positive for all j E {0} UN, and it is independent of j. Let the

zeros be denoted by 0 < 113;"(3') < - - - < 13530) < 1 Vj E N.

(P2) (Conservation of number of intervals of transition layers). The number of con-
nected components of £g(j) := {:z: E (0,1) : lu;(:r,t)| S g} is finite, positive for
all j E N, and it is independent of j, and in each connected component, u; (at, t)
is strictly monotone and has exactly one zero. Let the connected components of
tee) be denoted by [a?(j),b£"(j)], 2': 1,. - -,N w e N <0 < am) < bra) <

'°°< 0530’) < 533(1) < 1)-

(P3) (Lock-in and exponential steepening of transition layers). For all j E N, i =
1,2,- ~ -,N and for some Co > 0, C0 z 1,

[ain(j)vb:n(j)l C “(1.0),, (bolilv in particular $310.) 6 [(a0)i1(b0)il1

N
lu” (e t)l > K—“em Veece(')=U[a'-"('>b'-"<')1
2:1: 1 _ 2 2 .7 3 J 7 1 .7 1

i=1

 

. . 2p ~
b'." _ T" < “or"
|.(J) a.(J)|_ K008 ,

where 00 := min Io'l > 0.
{-pvp]

(P4) (Convergence of phases). lim mflj) =: (3),)? exists for alli = 1,2,-
J—’°°

-,N and (an)? E [(ag),, (b0),] (in particular, 0 < (:14)? < . - - < (:z:,,)",{,l < 1).

12

(P5) (Jump discontinuity of the limit state). Iim uf’j 2: (um; (which exists as an
J—voo
Lp limit) is continuous on (0,1)\{(x,,)’1",- - o, (ram) but discontinuous at every

(ea)? w = 1,2,. - -,N.

1.4 Energy decay and a-priori estimates

We first prove the decay of the energy functional

E(t) = E(uj,vj) =/(; [$(u’(x,t))2 + W(u_?,,(x,t)) + %(vj(x,t))2] dx

for t E (( j - 1)m, jm], j E N. It is difficult to show the proof of decay of E(t) since the
time derivatives of u’, u; and vi are constant with respect to time. The functional
becomes the combination of the time discretized solutions and their interpolation

fimctions. Thus, one must be careful in the calculation of the following lemma.

Lemma 1.1 E(t) is non-increasing, bounded by the initial data on ((j — 1)m, jm]

for ollj E N.

PROOF. Recall that um'j satisfies (1.2). That is, the following equation

vi - o(u;"’j),, — vg’j + M” = 0 (1.3)

13

is satisfied for all j E N. Then the following estimate holds for (j — 1)m < t g jm

1 . . . , .
52E“) = / (vi - e: + ow.) e1. + u] made
0

1
= / [vm’j - v,’ + o(u;"’j) - ui, + um’j “a: + (vj — vm'j)v{
o

(1.4)
+ (mi) — cum) - at. + (u, — um”) - aide
1 ‘ . c a . .
= / [ewe — our). + W) + (e — vm”)v{
0 (1.5)
+ (out) — «are» «3;. + (u, — W) - unde-
l _ - . .
= / [vm’j - vgj + —-—(t im) - |vm’J — vm’J—IP
0 m
+(e(u;) — are )) u... + 5—1.1) In -u"‘""1|2] de-
=—/1|vmj|2dx+(_—T—n22m—)/ Ivmj— vmj 1|2dx
(1.6)

+ / (cos—em. '))-u;.de+<t—jm) / lemme.

The first three terms of the integrand of (1.4) are the same as the first term of
the integrand of (1.5) since um'j satisfy the same boundary conditions as (1.1) and
u{ = vm’j . By using the Mean Value Theorem and by the fact that the function 0’ is
bounded below by a negative number, that is for all y C IR, o’(y) Z —L for some

L > 0, the integrand of the third term of (1.6) is estimated in the following way

14

< (ui)—a( urn)- u... = e'(e;"J)<u;—umJ)-uj

 

 

 

 

< I I It
_ m 133$ 0(y) m2
my _ m,j-—1 2
S mL(uz 1:3 )
m
= mL(vZ"j)2 (17)

for some 6an between u; and ug‘Jj . Moreover, both the second and the forth term
of (1.6) are negative since t — jm S 0. Since m is sufficiently small, the following

inequalities on the energy E(t) are derived

1
313m 3 (—1+mL)/ |v;"’j|2dx
dt 0

1 .
s —,-nv;JJJnie

for all t E ((j — 1)m,jm] and this proves Lemma 1.1.
Note that by taking an integral from (j — 1)m to jm on both sides of the above

inequality, we get

. . 3"" d 1 m.
E(Jm) - E((J —1)m)= / Emmet s -§mne. Jute.

(j-l)m

After taking a summation from j = 1 to j = S, the following estimate is established

E(Sm) — E(0)=

"Me

(U -1)m)l < -— m2 llvm”||L2

15

Therefore,
m S .
—2— Z Mani. g E(O) — E(Sm) < E(O) < e (1.8)
j=l
for all 3 E N. Next, we show the several estimates on the functions which will play

a significant role for the proof of Theorem 1.1.

Lemma 1.2 The following a-priori estimates hold

(a) SUP SUP llpi(°,t)l|L°° S 7?)
JEN (j-1)m<t$jm

 

 

val/0x My, t)dy)

 

 

(b) sup sup
JEN (j-1)m<t_<_jm

(C) sup sup Ilqj(-,t)||1.oo 5 K0,

jEN (j—l)m<t5jm

1
whe 7rd := — ,
LOOSUOU, re (f) f [of

(d) sup sup “Uj(‘,t)”Loo _<_ K0,
JEN (j—1)m<t$jm

(e) sup sup
JEN (j-l)m<t$jm

g 00”,

[01 o(uf,(x, t))dx

 

 

(f) SUP sup ”vj(‘)t)llL°° =Sup sup l|Pi(J,t)llL°o S K0,
jEN (j—l)m<t$jm jEN (j—l)m<t$jm

for some constants K0, 7] > 0, where 17 is a sufficiently small constant.

PROOF. Since f01p7(x,t)dx = 0, pl(x’,t) = 0 for some x’ in (0,1). Hence, the
following holds

1: 1 1 %
[windy]: / Ip;(y,t)ldys(/ Ipi(y,t)|2dy) .

Since Ilpifetlllm = llvj(°,t)||L2 S VII-3(0)“) S x/E(UO,vo) S 1/3, (a) is accom-
plished by choosing n > 0 such that n > max{\/E, 2\/E/oo, Cs (fl/oo}, where C5 will

lp’lz, t)I =

 

be chosen later.

16

Similarly,

 

 

 

/ uJ(y. t)dy
0

 

L... = [z/oy(pJ(e,t)+qJ(e.t))dzdy L

 

 

 

 

 

 

s fz(pJ(y,t)+qJ(y,t))dy
0 L2

= IIuJ(-,t)um

s f- <U‘J—JJ

which proves (b).

By using equation (1.3),

 

q: = qm'J($)-n:1’"""l(x)

= )0. /./

= Ln’J($)-0(um’ W())+0( ’J(0)Z‘)-’U J($)+'UL"’J(0)

+7ra ”(fogc umJ-J) +/01 o(u;"”) — 0(u;"JJ(0)) +/01 v?” — vZ‘JNO)

 

= -—e.(e(u;"J')) +e. (/ uJJJ') (1.9)
o
= —e(pmJ + W) + eTJ, (1.10)
where em =foo( Z"? (x) )dx + 71’“ (fou umJj( (y)dy). From the hypotheses (H 2) and

(H3), 0(2) 5 W(z) + CI for some C1 > 0, z E IR.

This and the estimate (b) imply

'63le +

/.‘ . (1‘)
(01(|W(u;"'J)l+ 01) + am

< e+C1+oon<C2

 

 

 

 

|/\

17

for some 02 > 0. Here, the energy estimate fol |W(u';"j)|da: g E(t) g E(O) < e was

used. Therefore,
lle’in’JllLoo S 02 (1.11)

for allj E N. Since ||p7||Loo < 17, from (1.10) and (1.11), q: < 0 when qj 2 K1 and
q{ > 0 when qj 3 —K1 for sufficiently large K1 > 0. Let K0 > max{77 + K1,K2},
where K 2 will be chosen later. Hence, qj is bounded by K0 which completes the proof
of (c). Note

IIUillLoo S. Ilflllm + lqullLoo S 17 + K1 < K0. (1-12)
Now, (d) clearly follows from (a) and (0) since
. 1 . l o 1 -
lu’l sf Iuil s/ Ip’l+/ Iq’l stun <Ko.
o o 0
Note that for all j E N,
10013:” S 03 (1-13)

for some C3 > 0 since ||u§||Loo is uniformly bounded for all j 6 N by (1.12). Since qf
satisfies (1.10), (1.13) combined with (1.11) implies that q: is uniformly bounded by
02 + Cg for all j 6 N in L°° norm. Also, (1.12) implies that

|0'(uf,)| 5 C4 (1.14)

for allj E N and for some 04 > 0.

18

From the conditions (H2), (H3) on W, a and by (1.12) and (1.13),

|0(z)|
05 := sup ——
zel—Ko.Kol\{z—,z+} x/W(Z)

is well defined and

S ||0(ui)||z,2

fol o(o;(o, mm;

 

 

1 i
s 05(/ |W(ui)|) «Meson,
0

which proves (6).
It will be shown next that H p31,,“ L2 is uniformly bounded for all j 6 N in order to

prove (f).

Since

where

3

J,”- pz-1(m)+ (“mg " ”)de),

)
J ‘ J) (ii-1e) + (t ‘ ”‘7‘: " 1J) do),

m

 

 

rJ(:2:,t) :=

(
sJ(:1:,t) := (

3

 

 

and ”q: H Leo is uniformly bounded for all j E N, one would only need to show that
“13%|le is uniformly bounded for all j E N. p{ satisfies the following equations

17: = “it —q{
= pZZJ+1ra [0(pm’J +q""J) - f / (pm'J +qm’J)] - (1-15)
0 0

19

The last equality follows from (1.9) and the identity 11;, = 1225].. Let
f (pm’J ) := #13 = 1n, [0(pm’J + (1”) - f: [:me + qm'J)] -
Note that for all j E N,
||f(Pm’J)||L°° < M. (1-16)
where M1 = C2 + C3 since qf is uniformly bounded. From (1.15),

pm — pm’J“ = mApm’J + mf (pm’J ),

 

 

 

 

which implies
(1 - mA)p’"'J = pm"l + mf(10""J)-
Therefore,
"‘0' = pm’j—l + m f( m’J)
1’ (l—mA) (l-mA) 1’
1 pm3j-2 m m,j— 1 m m,j
= (l—mA) ((1—mA) + (1—mA)f(p 1)) + (1—mA)f(p )
= pm’J’J + m ( few-1) f(p""J) J
(1 — mA)2 (1 — mA)2 (1 — mA)
_ po f(p'"'1) f (pm'J )
‘ (1—mA)J' +m [(1—mA)2' +"'+ (1—mA)]'
Thus,
- _ pm” -p'"’j‘1 _ Apo H Af(10""") f (pm'J )
JJJ ‘ m ‘ (1-mA)J' +m;(1—mA)j+1“’° + (1—mA)'

2O

W

Note that the second term on the right hand side is the same as

j—l

1 A m.k
m: (1 — mA)J'-k ' (1 — mA) -f(p )

k=l

 

By incorporating the inequality “(l—_an—mll [,2 S l and (1.16), the following inequalities

 

 

 

 

 

 

 

 

 

 

 

OCCUI
' j-l 1 A mk mj
”pint: s “Apollm + m2 (1 _ mm, - (1_ mm -f(p ' > + um) ' )un
k=1 L2

1‘1 1 .

S “APOHL2 + li ' Z (1 _ mA)j_k + ”f(pm’J)HL2
k=l L2
j-1 1

s ”Ant. + m - Z (1 _ "no.-. + M
k=l

 

M1 1
< A —— - . —1 M
_ H POHL2 + A1 [(1—mA1)J—1 ]+ 1
l
S ”APOHL’ + (“A—1 + 1) ' M1
S M2

for some M2 > 0. Here, A1 < 0 is the largest eigenvalue of A. Let K2 = M1 + M2.
Then “palm 5 ”pinup 3 K2 5 K0. Therefore, ||p§||L2 is uniformly bounded for all
j E N and this proves the estimate (f). Proof of Lemma 1.2 is now completed.

The equilibrium state of the time limit of the discretized solution is proven in the

next section. This result is sufficient to prove the last part of the main results.

1.5 Equilibrium state of the time limit of the so-
lution

We now introduce the following fimction cp, which is called the phase function. This

function will play an important role for proving the equilibrium state of the solution

21

at the time limit. Fix 7‘ > 0 such that for A E [—r, r], the equation 0(z) = A has

three different solutions zl(/\) < 220‘) < z3(/\). Define

i, zE U z,()\), i=1,2,3,
(p(z) ;= AEl—r, r]

00, elsewhere.

The next proposition states that the discretized solution um'j converges in W01"D

to an equilibrium state as time goes to infinity.

Proposition 1.1 Suppose (H1 )-(H3), (A1 )-(A4 ) hold. Then the solution (umij,vm’j)
of (1.3) converges strongly in W34” x L2 (1 S p < co) to some equilibrium state

(uT,0) E W01’°° x L2 as j -—+ 00.

PROOF. The proof consists of several Lemmas. The following lemma states that
under some appropriate conditions on the elastic stress 0(u;"'j (5:3)) — f0”c M” and the
phase function (,0, the strain uZ‘J converges to an equilibrium state. We must be
careful when choosing the pointwise representatives of u'xn’j since in the measure zero
sets of (0, 1), we never know the behavior of the strain uZ‘J . It is important to choose
the good representatives so that the limit state is continuous except for the finitely

many points which are the zeros of the limit state.

Lemma 1.3 Assume there exists a full measure subset S2 E (0,1)(Measure of (2 is

1. ) and pointwise representatives u?” of uZ‘J such that
(Bl) 0(u7m'j(:1:)) —/ M” =: Ag"(:t) —» Am asj —> 00 for some X" E (—r,r) and all
0
:c 6 Q,

(B2) lim cp(u‘1m'J(a:)) exists and is finite for all z E Q.

.i-+oo

Then _lim 117"“ (x) =2 iDm(ac) assists for all a: E (2. Moreover, the equivalence class 213'"
J—’00

22

«\u

l’l'
‘0

0f 13'" satisfies
.7:
llwmlle S K2, and 243(3) :=/ 111'" is in W01’°°
o
and satisfies

o((u1")x(a=))‘ [qu 2x“ warm» = lim <p(u;"’J($))

J—*°°

and

um’J—vuf‘ in W3“) (1£p<oo) asj—>oo.

PROOF. Recall that SUPHU';“LOO < K2 by (1.12). We first show that fl;c M” is
jEN
convergent in C ([0, 1]). Define

1, acE (”2 and <,0(u7m’j(a:)) =i€ {1,2,3},

0, otherwise,

1, 2: E Q and <,0(1Dm'j(a:)) = 00,

0, otherwise.

Since IDm’j($) = 2,-(fox umJ + Ag"(:z:)) and x3343) = 0 if 90(2Dm’j(a:)) = i, i = 1,2,3

for :1: E Q, the following equation holds in Q

23

w}

 

 

3

 

mea‘ iDmkx = ("Jr-1 at m'j J-"a: — m’kx-z1 tum’k "‘1:
(>- () 2h. ()z.(/ou +A,()) x. () (/0 +1.())]
+X3’J (It) ' wm’J (33) - Xg’kcr) ° wm’k($)-
(1.17)
Note that since 1— — Jlg—(A'J‘) - 7,-(0 o(z,-(/\"‘))) = o’(z,~()\m)) 12,503"),
Iz-(a)—z-(b)l< sup lz’(=v)|-|a—bl< sup ———1-—-|a-bl<-1-|a-b|
t 1 — zE —r,r 1 _ :cE[—r,r] IU’(Zi($))| _ M ,
(1.18)
where M := 6 man ])|o'(z)|. Let 5",,(23) = fox |um1j — um'kl. Then the following
holds
0 d m _ m,j m,k
_. 11—15110”) — (u (<1)—< (1)1
= f (1111-..: k)
-- If i2:x:"1’°( <'>{w"”<< }+2j{x :"(1')}<<m11‘<<')
3’;(;) 171m”(33 )- Xoo’ k$( ')'w m’kiU )ldx'll- (1119)

Note that by (1.12),

/<:;:<1

a:
/ (<21 11
0

—mX

)""J}_ < 2KM?) 6 (0 1) 90(w "“J(x)) 7E r(w'"”°(x))}|

WM) S K2l{a:E (0,1) =<.0(iD'”'J($)) #¢(wm'k($))}|

+ Home e (o, 1) 1 name» = worm» = co}:-

24

Therefore, the last three terms in (1.19) are dominated by

3K2|{x 6 (0,1) : (POI—”J (1‘)) # 90(wm’k($))}|

+2K2|{$€(0 1) <.0('w""J(:II))=<.0("”J""=(11=)) 00}|=

By the assumption (B2), for fixed a: E (O, 1), if j, k are sufficiently large, (p(’le’j (23)) =
<p(u’1m1’°(a:)) = i(.v) for some i(:1:) = 1,2,3. Therefore, $1 —> 0 as min{j, k} ——> 00.

For each i E {1,2, 3},

/ xz"1"<1')(om11a')—<m1k<1'))d<' s (wrux'i—wr’co'ndw’
0 ’1

+ lwm’J($') - u7"""(-’1=')|011?'<
J2

where

J1 == {93' 6 (0,23) = XI" ’J (31") = XI""°($’) = 1},
J2 == {31"6 (0,00) = XT’JCC’) = 1, x?’k($') = 0 or XT’J($') = 0, XI’“’°($’) = 1}-

In the set J1,

Iu‘JmJ(x’) -— iDm’k(a:’)| '21 (for
— U:

l m , m I m I
= fi[§j’k(x)+l)\j ($)—/\k (17)”

I

um’j(s)ds + A3"($')) —— z,- (foz um’k(s)ds + AL"(:I:’)) l

lum’J (8) - um’k (3)1113 + IKE-"($9 - AL"($')|]

 

I

|/\

25

Note that J2 C {2: E (0,1) : <,0(1Dm’j(a:)) 75 <p(iD'"”°(:r))}. Hence,

d m m 1 1 m I m(
8311(2)) S 21 Nat-E]; l’\j($)-chn Mldx'l'fl'jl‘j j.(k$

m 1 m m
2. ,k + fiHA,- — A. no + gains)

l/\

l
S 637k + E $k($)i

where 62”; := 2 1 6;): + if“)? —- Menu“. Since ”Ag" — AZ‘HLi ——> 0 as min{j, k} —> 00 by

the assumption (B 1), 637;, -—> O as min{ j, k} —-> 00. By Gronwall’s Inequality,

$11493 173)S €711M-(exp(%x) —1)—>O as min{j,k}—>oo,

Therefore,

S 61(3) _’ 0

 

:1: ' :1:
/ umg _ / um,k
0 O

as min{ j, k} —-> 00. By combining this with the assumption (Bl), we get

onum’j + W”) - ([u’"* + A?(<))| a 0 an.

as min{j, k} —-> 00. By the assumption (82), for fixed a: E (O, 1) and for some i(a:) =

 

1,2, 3, x?” (2:) = in’k(x) = 1 for sufi‘iciently large j, k and this implies

z.(/ <m1j+1r(1))=z.(/ <m1k+Ar(<>) and xa1j(<)=xe1'°(<>=o.
0 0

Therefore, the right hand side of (1.17) converges to zero and thus wm1j(z)—1Dm1k(a:) —>

0 for all a: E (1 as min{j, k} —-> 00. Hence,

1!

lim M” =: Um exists for all a: E [0,1],
1400 0

lim u‘im'J =: 217'" exists for all a: E (2.

j-*°°

26

(—“
u

(1.)

(I?

 

This implies that uZ‘J converges to the equivalence class 122'" of 112m in L1 as j ——» 00.

Let uf‘(a:) := fox 122'". Then since

1 l
<r(1)=/ <1“: 11m 11‘1“: .lim (<m11'(1)-<m1j(o»=o,

J—’°° 0 :1: J—’°°

u? satisfies the Dirichlet boundary conditions. Moreover, since
. z . x
111(1):] 1131—, f 11'” =ur."(1) in C([0,1])asj —1 oo,
0 o
Um = f: uf‘. Thus
um’j ——> v.1" in Wl’p textas j —+ 00, 1 S p < 00.
Since um'j, 11311 are uniformly bounded, ul" E W01’°°. Thus

um'j —> (um); boundedly a.e.,

Z *

Since 0(u;"1j) — f0:B M“ —> o((uf,")x) — fox uf‘ boundedly a.e., by the assumption (Bl),

X" = o((u1")z) — fox u? a.e. (1.20)

Since a((u§")z) lies in one of the three intervals U 2,1(Am) i E {1,2,3} a.e., we
AE[—r, r]
can choose the nice pointwise representatives 117"” of 1113111 such that (1.20) holds for

the set (0, 1) except for the finitely many points which are the limits (2),)? of finitely
many zeros 2:1”(3'), i = 1,2,1 1 -,N of M” in (P4). Hence, we can conclude that
(wind, vmfi') converges to an equilibrium state (uT, 0) strongly in WOI’P x L2. This
proves Lemma 1.3.

In the next lemmas, we will show that under the low initial energy, the assumptions

27

(BI) and (82) are satisfied. The following Lemma shows that the convergence of
mean elastic stress f01(a(u;"'j) -— f0m um’j)d:c implies the convergence of elastic stress

0(ugnd) _ f: “mi

Lemma 1.4 Let if”, j E N be a solution of (1.2). Assume that

l :1:
lim (0(u;"’j) — / um’j) d3: =: X" emists.
o o

J'“*°°

Then

I
0(u;"”) —/ um” —-> X" a.e. as j —+ 00.
o

PROOF. Recall that

q:’ = -7ra(0(u;"’j)) + wa</x(um'j>)

from (1.9). Thus, the sufficient condition to our conclusion is when qf goes to zero

a.e., as j -+ 00. Define the following modification of the energy fimctional E(t)
~ 1 . 1 . . .
E(t) := / [W(u§,(a:,t)) + §(u’(ar:,t))2 +p’(x,t)w’(a:,t)] dx,
0

where wj(:z:,t) = (mini) q{_1(a:) + (F—figil) q{(x). Note that E(t) is uniformly
bounded and moreover sufficiently small since the first two terms are the part of
energy functional E(t) and the third term is small since p7(:v,t) is small enough by

the estimate (a) of Lemma 1.2 and wj (:13, t) are the interpolation functions of uniformly

28

bounded fimctions q'z Vj E N. By equation (1.3),

|/\

l/\

filo<u;).uz..+uj-uz+pz-wjwinch:

0

1
fmug.)-u;.+uj-uz+pz-qz+p:<wjflaw-wads:
/o[(0(u’) -U( Z"))-Uit+(uj-um’j)'u{

u. M). -:u +um1uz+us qi- ((23)?
+p’ (m((m-t)9{_1 + ( m(jr;1)_m)c13) +fw31dx
anvz‘Juia + (t — mmwniz — / 1(qz)2dx (1.21)
+/. [<2 —v:'> m: Mimi. was
+z:'(“m m3) (qg_qz-1)+p2wg]dx
1
anvz‘jniz— / (<1:de

 

 

 

+/1 [—u;tvmj — u’ vi + uxtuit + utpit (1.22)
+173 - (t — mj) ~wi' +p’w21da:
1 . 1 . - .
lelv;"”||i2 — / (anew / wmt—mj) -p{+p’ld=v

m _ 2 1 . 2 1 , . pm,j _pm,j-1
anv. any — / (qz) dx + f w: [a - my). ( )
o o m
‘ mj — t m,,-_1 t — m0 - 1) m-
+ ( m )p + ( m p (19:

1 1 1
mLIlvf’jlliz —/ (q{)2da: + 2 / wf -pidz -/ wf ~pm'jdx.
o o o

 

 

 

The first term of (1.21) follows from (1.7). (1.22) vanishes because of the identities

29

113,, 2 v3”, ,, = v: and the boundary conditions of system (1.1). Since

1
/ w? -p’"’jdrr
O

S ||19"""||I.2 ' llwi ||L2

 

 

S ”1922me - llwillm

. ._1
“7,ngle . L??—
L2

m
IlvaHLz. 7r (“(u?’j) — “Hind-1)) __ ([5 Um” - um’j‘l)
a: a
m 0 m
. mij __ m’j_1 mij ._ m’j—l
“1):?th2 . ( 7!", (01(6) . U1; U1: ) U uz

3
m
S M3 ° llviwllip,

 

 

 

 

 

 

 

 

L2

.>

|/\

 

 

+
L2

 

 

 

 

 

 

 

 

m

S Hp” llz.2 - Ila)? ”1.2
S ”103.sz - Ilwf “1.2

= ll'villz.2 - llwi'llz.2

1
fwf-pida:
0

 

 

l/\

M4-(llv;"’j‘1||L2 + Ilvin'jllu) - llvén’jllm

S M4 ' (“flu—1”” ° “Ugmllm + -||v;"’j||i2)
and
llvln’j‘llli2 ' IIULWIIL2 S M5-(||v;,"'j"1||i2 + ll?’3"jlli2)
for some M3, M4 and M5 > 0, the following estimate on git-EU) holds
52%) s lelvr'jlliz - [him + M6°(||U;n’j_1”i.2+ IIvZ‘Jlliz) (1.23)

for some M6 > 0, By taking an integral from ( j — 1)m to jm on both sides of inequality

30

(1.23), we obtain the following estimate

J'm d~
—Et
[MM 0

l
—m f (a)? + (mL + Me)mllv;"”lli2 + mMsuer-lniz.
0

E(jm) — E((j — 1)m)

|/\

By taking the summationj = 1, - -- ,S, we get

~ ~ S 1 . S .
E(Sm) - 157(0) S -mZ/O (<11)2 + (mL + M7) Zmllv’xn’JIIiz + mMelKUoMliz
j=1

i=1

for some M7 > 0. By inequality (1.8),

S 1 . ~ ~
m2 [0 (qu g E(O) — E(Sm) +2(mL+ M7)e+ Msmll(vo)x||%2

|E(0)| + |E(sm)| + 2(mL + 147).: + e,

|/\

S 6

0° 1

for some 61,6 << 1. Therefore, m: / (q{)2 _<_ 6 and this implies qf ——> 0 a.e. as
i=1 0

j —> co and this completes the proof of Lemma 1.4.

The next Lemma shows the convergence of the phase function under the assump—

tions of the convergence of mean elastic stress.

Lemma 1.5 Let um’j be the solution of (1.2). Assume

/0-1(0'(u;”’j)— fun...) do:

exists. Then the assumption (82) in Lemma 1.3 holds.

lim =: X"
j-+°°

 

 

PROOF. By the estimates (b) and (e), mean elastic stress fol (o(u;"'j) — fox um’j) do:
is sufficiently small. Then by Lemma 1.4, lim sup

:1:
0(U;n’j) _ / umo'
190° 0

Small a.e. Note that these inequalities also hold for the interpolation functions uj (:13, t)

is sufficiently

 

 

31

and their derivatives with respect to z for all j E N. Combining this and the estimate
2
(b) of Lemma 1. 2, lim sup |o(uJ) l 3 ~37; a.e. This implies that for almost every so,

J"’°°

there exists J (z) E N such that

{u;(x,t)=12J(z)}§a( (l—r r1) =U Uz

i=1 AE[—r, 1‘]

Since {uZ,(z,t) : j 2 J(z)} is connected, for allj 2 J(z), uf,(z,t) E U z,(/\)
AE[—-r,r]
for some i(z) = 1,2, or 3 and this implies that _lim <p(ufc(z,t)) exists and finite a.e.
J—'°°

Consequently, lim go(u;"’j (z)) exists and finite a.e.
J—’°°
The next lemma shows the convergence of mean elastic stress and this completes

the proof of Proposition 1.1.

Lemma 1.6 Let um'j be the solution of (1.2). Then,

1 :1:
lim [/ (0(u;"’J) — / um’J) dz] ezists.
J"°°k o o J

f

=: co)

PROOF. Suppose this fails. Then there exists a subsequence jk —> 00 such that
C(jk) —> X" and another subsequence j, —> 00 such that c(j,) —> 3"" for some
Am, 5"" E #331,231 and N" < 51'". Then by Lemma 1.4, 0(u;"'j'=) — fox uka —> X"
a.e. as jk —> co and o‘(u;"'j') — f0z um’j' —+ 5"" a.e. as j, —> 00. Also by Lemma 1.5,
lim <p(u;" ’J"), jlim oocp(u "‘7') exist and finite, respectively. Thus, these satisfy the

J): "’00

assumptions (Bl), (32) of Lemma 1.3 and therefore, there exist um, um E W11°°

such that
(7(uZJ),c — um E X" a.e., OWL"); — am E 5"" a.e.,
w(u;")($)=1im 90( 11"“"(3C )) as, <P(fi;")($) = .lim <P(u;"’J’($)) a-e-

jhfim J. —voo

Note that cp(u;”)(z) = <p(um)(z) =: 9000(z) since the limit of the phase function is

32

independent of X" and 5"".
Consider the case where (poo(z) E {1,3} a.e. That is, the measure of the set
(2,, := {z E (0,1) : (poo(z) = 2} is zero. Now we introduce the following principle

whose proof was done in [16].

Comparison Principle for weak solutions of the ordinary differential
equation ”(“3)2: = u.

Assume that u, 17. E WJ’°° satisfy
flux); — u E A a.e., 001$); — a E /\ a.e.,

A < :\, u(O) = 17(0) = O, 0(ux) and dam) E [—'r,r] a.e., 90(ux) = (Mum) a.e.,
90(uz) E {1,3} a.e. Then u(z) < u(z) for all z E (0,1].

Since N" < 33", um, um satisfies the above comparison principle, therefore,
um(l) < am(1). This contradicts to the boundary conditions of (1.1). In the case
when the measure of flu is not zero, contradiction arises from the following modified

principle which was also proven in [16].

Refined Comparison Principle for weak solutions of the ordinary
differential equation (flux);c = u. J

Under the same assumptions as the Comparison Principle but the condition
<p(uz) E {1,3} a.e. is replaced by f01W(u$) < 61 and IQul # 0, the following
inequality

u(l) < 11(1)

holds.

Now the Proposition 1.1 is finally completed.

33

In the next step, we will show that the set of z where the transition layers are

steep, is decreasing to the finitely many isolated points.

1.6 Dynamical behavior of the transition layers

If the set

5510) =12 e (0.1) : lui(:v,t)l s

Nl'b

}

is monotonically decreasing to the finitely many isolated points as j goes to infinity,
we obtain the desired conclusion since this is equivalent to the fact that the layers
are getting steeper and steeper as j —> 00 and eventually become discontinuous.
But unfortunately, the set £5 ( j ) is not always decreasing as j ——> 00. We define the
following set Z( j ) instead and show our set L; (j) is contained in the newly defined
set E( j ) We will show then the set Z( j ) is decreasing to the finitely many isolated

points. Let 17 E (O, 3). Set p0 := p — 17. Define

5(1) := {x e (0.1) : lqj(x,t)| 5 p0}.

The following Lemma states that the set of transition layers are always in the set
5(3) and furthermore in the set of initial transition layers £p(0). This Lemma plays

an important role for showing the preservation of the number of transition layers.

Lemma 1.7

PROOF. If a: E [35(3), then |u;(z,t)| g ’5. Therefore, by the estimate (a) of Lemma

1.2,

- - - p p p p
qu($,t)|=IUi-p’ls§+n<§+1=p-Z<p-n=po

34

which implies

I(uo).l s Ip9'(:c,0)l + Iqj(x,0)l s n + P0 = p.

This proves Lemma 1.7.
We show next the set E(j) is exponentially decreasing to the finitely many isolated

points.

Lemma 1.8 For allj E N and for some Co > 0, CO z 1,

(i) Iai(:v,t)l Z CoeJmJ°|(Qo)x| 2'f In E Z(j) (exponential growth),

(ii) E(j + 1) g E(j) {monotonicity).

PROOF. We will show (i) by induction. Fix 3' E N and fix z E E(j). Then a: E £,,(0)
by Lemma 1.7. By the hypothesis (A4), |(uo)u($)| 2 K. Suppose (uo)m(z) 2 K.
Since (po)x(z) is bounded by the estimate (f) of Lemma 1.2, (qo),,.(z) = (uo)m(z) —
(p0)x(z) > 0. By differentiating both sides of equation (1.9) with respect to z for
j = l, and by using the estimates (d), (f) of Lemma 1.2 and equation (1.14), we get

the following estimate

(121(3) - 4;"‘°($) {-[0(u;"’1($))lz + u""J(-’I=)}m
{-0'(UZ"J(-’B))(PZ"J($) + qZ"J($)) + u"“J(93)}m

—o’(u;"’1(z))q;"'1(z)m — C4K3m — K27".

IV ||

IV

-0’(UZ"J($))qL"’J($)m - Cam

for some 06 > 0. Thus,

(1 + 0'(u;"’1($))m)qg"l($) Z (12" ’°($) - Cam.

35

Since m is sufficiently small and qr,“ = (go); > 0, q;"'1(z) is also positive. Therefore,

the following inequality holds
(1 — 00m)q;"’1($) 2 (1+ 0'(U?’1($))m)q;"’l($) Z q;"’0($) — C6m-
Recall that do = [min] Io’l. By induction, suppose qgn’i-l > 0. Then,
(1 + 0'(u;"’1($))m)q;n’J($) Z Qin’J"l($) - 06m
which implies that qghj > 0 and the following inequality holds

(1 — 00m)q;"’J(z) Z qr’J_J(z) — Cam.

By iterating this, we get the following inequality

 

 

 

 

. 1 . 1
mg > __ . m,J—1 _
q: _ 1 — 00m qz 06m 1 — 00m
1 l 1 1
Z qu’J 2 — 06m — Csm
l—oom 1—oom l-oo l—oom
l . l 1
= mg 2 _ C 1-
(1 — 007702 q: 6m L1 — 00m (1 — 00m)2]
l 1 1
= a: — C
(1 " 00mlJ ((10) 6m l1 " 00m + (1 — 00mlJl
1 F__1_ ._ (__1T+_1.
= . . z _ C m l—oom l-oom J
(1 _ 00171)] ((10) 6 - 1:29;; :I
1 1" (17%
= __. e z o C
(1 - (70m)J ((10) + 00 6
1 Ce Cs

Similarly,

(13"J_1 Z 60'1”“ - ((10)::-

Since m is sufficiently small, e‘m‘J0 z 1. Therefore, we can establish the exponential
growth of qi, that is

l9i($,t)| 2 CoeJm"o - |(QO):cl

for some Co z 1. Similarly, for the case (uo)w(z) S —K, qgn'j < O for allj E {0} UN

and the following inequalities hold

(1 - dom)QL"’J(x) S ain'J‘JCv) + Cam,

61.2"” S eJm° '(QO):1:1 (AW—1 5 80“)”o - ((10)::-

Hence, we get the same conclusion

Iané(x,t)| Z 006””0 - |(40)z|

and this proves (i) of Lemma 1.8.
Note that for K > 4K0,

|01($,t)| Z CoeJmJ°°|(qo)x|
Z Coejma°(l(u0)xxl - K0)
2 Z—KCoejM. (1.24)
Also, Notice that
IUiI=|pJ+qJ| Sn+po=p (1.25)

37

when IQJI 3100- Iqu =Po, thenu; =PJ+QJ =PJ+P0 Z -fl+P0 > Oandiqu = —P0,

then u; = p7 — p0 S 17 — p0 < 0, which implies sign(ui) = sign(qj) at |qj| = p0. By

using this and equation (1.9) and also by using the estimates (b), (e) of Lemma 1.2,

we have at Iqj | = p0 and for some can”. between 0 and uZJ'j ,

d .

IV

IV

IV IV IV

IV

 

meat». [111190 311%]

am (a?) - qm’j’1($)]

 

sign<u1<x,t>> - I

signage, t» - [0(0) — 0(uZ""(x))

+ f care] + «.(f use]

- . . l . x .
—a'(c;"")-u;"1-sz'gn<u;>—I] oars 7r. (/ um)
0 0

—a’(c;J’J) - ui - sign(ui.) -— o'(C;n’J) - (U210 " U1) ' 33.974141)
1 I
/ 0W) m] W)
0 0
0‘0 - Inil - 0765'”) - sign(Ui) -
no - (1qu ~— In" I) - owes") - signal) m («4:1 — W“) — 2001

m - , - m — t
on - (p0 — n) — 20017 — out. '1) ~szgn<u1> . J

m
m—t

 

 

 

 

 

LC”

 

 

 

 

 

 

Loo
'm—t

 

(a? - 2121-1) — 20077

jm—t

 

(21:11 — raw-1)

(uZJ’J -- mind—J). (1.26)

 

ao - (p — 4n) — axes”) - signmi) - J

Note that Hail < 1. By the estimate (a) of Lemma 1.2, lpm’j —pm1j‘1| S 217 << 1 and

Iqm’J — qm'j‘ll = mqul 3 li << 1 by (1.16) and m < 1. Hence, |u;"*j — uZ‘J‘lI g

IPm’J - find—1| + Iqm’J — qm’j‘ll << 1 and this enables the second term of equation

(1.26) to be small. Now we can say

(1 .
— J t >

which implies Iqj(z,t)| g |qj+1(z,t)| for allj E N when |qj(z,t)| = p0. By (i), qj is

38

strictly increasing or decreasing on Z( j ) which implies

Z(j+1) g: 5(1).

Now (ii) is proved and this completes the proof of Lemma 1.8.

From the part (i) of Lemma 1.8, the estimate (f) of Lemma 1.2 and the hypothesis

(A4), if z e 5122(7) and K > max{4Ko, 4&9},

Iui.(x,t)|

IV IV IV IV IV

IV

I‘li($,t)| — IPflxatH
00‘9""I0 ' I((10)a:l — K0

Coejmao ' “(“0)le - I(p0)zll - K0

. K
come - (I(uo)..l - K0 — 3°
0
mo K K
Coe’ (K 4 4 )
éKCoejm‘m. (1.27)

From Lemma 1.7, inequality (1.27) and the fact that “um’jllcz < oo Vj E N, [3122(7)

has a finite number of components [a{"(j),b}"(j)], 0 < a'1"(j) < b'I"(j) < <

a'§(j)(j) < b’fimfi) < 1, and in each of which, u;(z, t) is strictly monotone and has

exactly one zero z?(j). Also, N(j) _>_ 1 since uj(0, t) = uj(1,t) = 0 Vj E N.

Lemma 1.9 N(j) E const. Vj E N.

PROOF. For all j E N, define

gJ(cc.t) := u;(a:,t), (j — 1)m < t g jm.

Since 9’} i E C((011)><((J'-1)m,jml) andat each zero (xo(j).to(j)) ofg’} Igi($1t)l 2

5299 > O by inequality (1.27), for each to(j), {gj(zo(j),to(j))|(z0(j),t0(j)) is a zero

39

of 91} does not contain a critical value of gj (-, t(j)). By Implicit Emotion Theorem,
the number of zeros of gj(-,t) is independent of t for ( j — 1)m < t g jm. Since this
holds for all j E N, number of zeros of gm’j(z) := u;"’j(z) is independent of j which

implies N (j ) E const. This proves Lemma 1.9. Similarly, by defining
gJ(z,t) := u;(z,t) — g, gJ(z,t) := u;(z,t) + g, (j — 1)m < t S jm,

the number of connected components of £1; ( j ) is independent of 3'.
Now, the proof of (P1), (P2) is completed.
From Lemma 1.7, [af‘(j),b§"(j)] Q [(ao),, (b0),], i = 1,2, - -- ,N. Moreover,

p = IUi(bI"(j).t)-Ui(aI"(J'),t)|

binU) _
= / Inside
“i"(j)

1 . , m .
2 §KCoeJmJ° - IbI"(J) -a.- (J)|,

which implies

 

2 .
(bro) — arm) 5 KEG-12‘3“” Vt: 1,2,--.,1v

for fixed j. This proves the last part of (P3). The rest of (P3) was already proved.
From (1.24) and from similar analysis as in the case £122 ( j ), Z( j ) has a finite number
of components [01"(j)153"(j)], 0 < ainO') < WU) < '" < ail/‘0') < WU) < 1- By
Lemma 17’ a31"‘(J'I E I02"(J'),bi"(j)l E [02"(3')153"(J')I Q [a?.b?]. By (ii) of Lemma 18,
[summon g [ai"(j)fi:"(j)]- Therefore. the set of momma} forms a nested

40

family of intervals. Thus

2p>2po = lqj(fi?‘(j),t)-qj(ai"(j),t)l
are) _
= / lqildrv

I"(J')

3 .
Exams) — are» - em

IV

which finishes the proof of (P4).
(P3) and (P4) automatically imply that (“L"): is discontinuous at every (22*)? It
remains to show that (211"), is continuous on (0,1)\{(:c*)§",- - -, (23.)?) Since 11.1" is

an equilibrium state, it satisfies the following equation

e<<ur>.(e)> = [724") + A”

for some constant X" > 0. We know that the first term on the right hand side
of the above equation is small by the estimate (b) of Lemma 1.2. Furthermore,
X" is sufficiently small on (0,1)\{(:c*)'1",- - -, (3*)73} from the proof of Lemma 1.5.
Therefore, (u?),,, the inverse image of a is continuous on those intervals which proves

(P5) and Theorem 1.1 is completed.

Remark. The result of transition layer dynamics works for the Viscoelastic

system without the elastic foundation term 11., that is, for the system
u“ — (0014-) + 111;); = 0.

The proof is similar to the proof of the original system. Only the minor change of
the proof of energy decay (Lemma 1.1), the estimate of (c) of Lemma 1.2, Section 1.5

and the estimate of filqj (:12, t)| in Section 1.6 is needed.

41

CHAPTER 2

Convergence Analysis of Numerical
Solutions in One-dimensional

Systems

We obtained the theoretical results on the dynamics of the strain um in Chapter 1.
Next we will derive the numerical results on the behavior of at by using the finite
difference methods (FDM) and the finite element methods (FEM) using the linear,
quadratic and Hermite cubic elements.

The nonlinear term 0(Uzlz is treated by the direct iteration method for both
schemes. However, two types of problems arise. First, the steepness of transition
layers affected by the term 0(u3)m leads to the truncation error as time approaches
infinity. Second, iteration in each time step is computationally expensive. The first
case is treated in Section 2.3 by averaging the term ”(712): and the error is greatly re-
duced. In the second case, the alternating direction implicit (ADI) method combined
with the explicit method for the system is derived in Section 2.4 and it is shown that
the computational cost is reduced.

Even though the computational cost is high, using a higher order element in the

FEM can reduce the order of error in the space dimension. In our problem, due to the

42

time derivative of a energy dissipation term amt, the error for the quadratic element
is of order 1 and is of order 3 for the Hermite cubic elements in the space direction.
However, the fact that the time limit of the solution u which is only in 00(0) and
its derivative has a singularity prohibits the improvement of the error for the higher
elements. The numerical- results presented in Section 2.5 shows that there is not much
difference between the three types of elements.

In the first two sections of this chapter, we derive the matrix equation for the
one-dimensional systems which is obtained from the FDM using the standard second
order central difference approximation for 0(ux), and prove the convergence of the

solution.

2.1 Derivation of the matrix equation using the

Finite Difference Methods

Let m, j be fixed. Recall that the governing equation for the discretized solution is

1 m,' m, '-—1 m,'—2 m,' 1 m,' m, '-1 m,'
mh‘ 3—2u 3 +u J )=(o(ux 3))x+T—n-(uz’—u33 )z—u 7. (2.1)

Note that W, a = W’, 0’ are all bounded for small a?” . Divide the interval (0, 1) into
n subintervals (xk_1,:rk) with length i, k = 1,--- ,n, O = 2:0 < 2:1 < < xn_1 <

11:" = 1. By using the central difference scheme, the following approximation will be

43

used

um’j ($k+1) — um’j(1‘k—1)

 

 

 

 

m,j z
m . 11"” :1: ' — 211"” a: + um’j a: _
MW z < a.) h; I.) ( k I),
0(U?’j($k))z = 0'04"" (3%)) ' UL?" (931:)
~ 0, ume(ek+1)— Wm-» ume'(a=k+1)— 2am“) + Wet—1)
"’ 2h h2

(2.2)

l
for k = 1, - - ~ , n —- 1, where h = R By substituting these equations for the terms in

equation (2.1) and multiplying by m2 on both sides, we get the following equation

(1+ m2) - tweet) — £3. - (tweet) — zumtwek) + awash—1))

T: , z (um’j ($k+1) - um’j($k—1)

h2 2h ) ' (um’j ($k+1) — Zum’j (min) + um’j (mk_1))
m

= ‘77; - (wee-1e...» — zest-1(a) + amt-lea» + amt—Wet) — amt—2m)-

 

After arranging the above equation in terms of if” (mk_1), umtj (wk) and 21"” (n+1),

we get

Mum” ($0) ° um’j($k—1) + 3(um’j (9%)) ° um’j (xk) + A(U'"”j (5%)) - um’j (n+1)

= F (um’j "l($k)), (2.3)

44

fork=l,-~ ,n—lwhere

 

 

 

m - 7" m2 um’j («751ml — “WNW—1)
A<u 0(a)) == W - 7,; - e'( 2, ). (2.4)
- 2m 2m2 um'j ($1: ) —— um'j(:z: _ )
m, ._ 2 +1 I: 1
B(u ’(xk)).—l+m +-h—2+7co'( 2h ), (2.5)
F(u’"”“(zt)) == 7,; - (“m’1_1($k—1)+ um”_1(l‘k+1))
2m ‘ . (2.6)
+ (2 + 71—2“) ° Um’J—l($k) — um”_2(xk)
for k=2,-~- ,n—2. Note that for k: 1,n— 1,
m . m m2 um'j(:1:2)
AW ”(31” = “71—5 — 717 ' 0' (T)’
m ' m m2 I _um,j(xn_2)
A“ mm» = —fi _ I? ' a ( 2h )
since um'j(:ro) = um’j(:rn) = 0 for all j E N. The similarities hold for

B (amt-7 (2:1)), B (um’j (zn-1)), F(u""""1 (31)) and F(um’j'1(:cn_1)). Note also that the
definitions of A(u'"'j (xk)), B (wind (2%)) indicate they depend on if” (xk_1), um'j(:rk+1)
and F(um’j‘1(zk)) depends on um'j’1(:rk_1), um’j‘1(mk), um’j‘1(zk+1) and «NJ-2(a).

Equation (2.3) now becomes the following matrix equation

C(um‘j ){um’j } = {Mum—1)},

45

where C(um'j) = {01,3 (um’j (xk))}(n_1)x(n_1) is the tri-diagonal matrix

 

 

/ Home») Ammo.» o o )
A(um’j($2)) 3(um’j($2)) A(um’j($2)) 0 0

0 Mum” ($3)) 301"” (133)) Mum” (563)) - ~ - 0

0 A(um’j($n—3)) B(U"‘”($n—3)) A(um’j(xn—3)) 0

o o A(um'j(a:n_2)) B(um'j(xn_2)) A(um'j(xn_2))
\ o o A(um’j(xn_1)) B(um'j(:cn_1)) )
and

{um’j} = {um’j($1)i”°:um’j(xn-1)}Ti

{E(um’j—1)} = {F(um,j_l($1)" ' ° ,F(um’j-l(xn-1)}T'

Note that the matrix C(um’j) is nonsymmetric.
The matrix C(um'j) involves the nonlinear term 01 which is the function of the
unknown present time solution 11"”. Thus in each time step j E N, we assume that

C(u?’j) = C(ufli), 216"” = MN"1 for 2' e N which allows the use of an iterative

solution method to compute um’j . We stop when the norm of the difference of urn”

and ufli is sufficiently small. This method is called the direct iteration method.

2.2 Existence of the Finite Difference solution

We now prove the existence of the solution of the matrix equation which we obtained

from the previous section.

Theorem 2.1 The solution of the matrix equation

C(um’j ){um’j } = {11“(um’j "l)},

46

ezm'sts.

PROOF. Let j E N be fixed. Consider the following iteration
C(uiJi ){ 11;"3}: {Nu m’ 1)} VieN,

where 113"]. = um’j‘l. By subtracting the equation C(u:{){u:"j } = {E(um’j '1)} from
C( um”){uz+’1} = {E(um’j—1)}1 we get

C(U?’){U?‘.fi1-U'—"’j} = (C(UITi) -C(UI"’j)){uI-"’j}- (2-7)

The kt" element of the right hand side of the above equation is

:{Ckflu s "301710) Ck,s(uin’j ($10)} ° ”PW-Ts)-

By the Mean Value Theorem, it becomes

n—1 n—1

:Z[Ck3(u “3““) ),°]Xt (ui— 1($1)— u:n,j(xl)).u:n,j(xs),

8:11: 1

where the value uzf’j($k) is in between ufl’flxk) and uzn’j(a:k) and [Ck,,(u;:"j(xk))]x,
represents the partial derivative of Ck,,(u;:"j (1:0) with respect to uzf’j (3,). Now, the

right hand side of equation (2.7) becomes

n—1 11—1

2 gum...) - I01..(u::‘"'<wk)>],x. - (ammo — we.»

(=1 8=1

E(ulf”){UI'i’i - HIM},

47

where E(UK’j ) is the following matrix

 

 

( 2W (x. ),(u.[C1. ($1))l.x1 ~ iWe.»[01..<u::‘”(x1)>1,x._. )
Euimj ($3)[C2s(u u“ m’j($2))l, X1 '- i:ninjas)[C2,s(u:-:"j(a:2))],xn_l
3:1 3:1
i:u?”(a=.)[Cn_2,.(u;ffd(xn_2))],xl -- i:u?’j(ms)[Cn_2,s(uzf’j(xn_2))],xn_l
K :u?‘j(x8)[0 “113(u$,j($n-1))],X1 " :umj($s)[0 —1.:Ts(uj—($n 1))]X,,_1)

Since 01 ,Us(- m’j(-’131)), s = l, - -- ,n —- 1 only contains 113%332),
_ n—l
DUMT'J) = Zum'j.($s)[01 A“: ’jfl3( 1))l,x1 = 0
3:1

ifl#2.W'henl=2,

01.2(u3’j) = :um1(.(1‘$s)[01am.fj($1))lx2

= u:’j($1)[01.1(u.'. d.($1))lxz+um’j($2)[01.14?2( J.($1))]X2

m2 “‘2' -7 m . 777,2 u, (13 m -
= F . an (_*_2l(l_1)) 41,- ”(1131) - 2‘}; - U" (:21?!) u, ”(532)

s h (5)210": - (lu?”(w1)l+%lu?"j(x2)l) . (2.8)
Let r = 7?}. Since m and h2 are both small, we can make 1' bounded by letting m
be sufficiently small. The first equality comes from the equations (2.4), (2.5) which
are the components of the matrix C(ufi’j). Here, it is sufficiently small and if” is
bounded from the a-priori estimate ((1) of Lemma 1.2. Moreover, a” is bounded and
these enable (2.8) to be bounded by a small number. Therefore, all the elements

in the first row of the matrix IlD(u '7‘ J ) are zero except for the second element and

48

the second element is sufficiently small. Similarly, Cn_1,s(u:"j (zn_1)) only contains

771,]
3*

u (xn_2), s = 1, - -- ,n — 1, which implies

Dn_1,¢(uz"j) = 0
iflaén—2. Whenl=n—2,

Dn—1,n—2(u:’j) = “T’j($n—2)[C -1,n-2(u:’j(xn—l))l,Xn—2
+“In'j($n—l)[0 —1.n-1(u$’j($n-1))l.Xn—2

"7’2 II ‘uln'j (En- m '
. 0' ( t2; 2)) . ui ”(mu—2)

W4 .2; 2>)....a(xn_.)
1

mm (11.3%.-.» + —|u?"j(rvn—2)I)

61

 

 

|/\
to

|/\

for some 61 << 1. Thus all the elements in the (n — 1)“ row of the matrix E(uzf’j ) are
zero except for the (n — 2)“ element and the (n — 2)“ element is bounded by a small
number. I
Since only uz’j (ask_1) and 11:“. (n+1) are in the Ck,,(uf:"j(zk)), k 74 1,n — 1, s =
1, . .. ,n _ 1,

Dk,,(u:f'j) = 0

49

iflyék—l,k+1.

Dk,k—1(u:’j)

l/\

|/\

Whenl=k—l,

“I” (inc—1)[Ck.k-1(uif’j($k))lxk-1 + u?” (17k)[Ck,k(u$’j($k))l.xk_1

+11?“ (3H1)[Ck,k+1(u3’j($k))l.xk_1

m2 .0” (uzj ($k+1) " “Edwin-1)

 

2h

m2 . 0,, (uE’j($k+1) - “Z’j($k—l)

 

2h

1 m . m . 1 m .
hr"’|0”l (5m.- "<xk_1>| + In. ”(anal + gnu.- ’J($k+1)|)

62

for some 62 << 1. Similarly,

Dk,k+1(u:j)

|/\

 

m2 . gl’ (usjwkn) — ux’j(mk_1)

) ' (“EM-(331:4) + “In’j($k+1))

 

2h3 2h
2 ("J __ (mi .
+%.gn (U1. ($k+1)2h“z. (WC-1)) -uI"’J(a:k)
62.

Therefore, the matrix Mar”. ) is of the following form

f o

 

50

D1304”) 0
D2,1(u:':’j) o D2,3(u:-':’j) 0 0
0 D3,2(u:':’j) 0 ... 0
0 0 Dn—3,n—2(u?:’j) 0
o 0 Dn_2,n_3(u:':’j) 0 D.._2,.._1(u?,"")
K o o D._1,._2(u::"j) 0 )

 

where the nonzero elements are sufficiently small. Thus, the norm of the matrix
may), which is defined by sup ||Ill>(u:"j)m||L2, is small and bounded.
mElR""1
Now, it remains to show that C(uzn”) is invertible and bounded away from zero.

Consider the matrix C(uzn’j) as the sum of three matrices .11, K and ]L(u:"’j ), where

.11, K are the following matrices respectively

 

 

 

 

{1+m2 0 0 \ {2r —7' 0 0\
O 1+m2 0 0 —7‘ 27‘ —7‘ O O
0 0 1+m2 0 0 0 —r 21' —r

K 0 0 Hm?) \0 0 —r 2r)

and 1L(u:"’j ) is the following tri-diagonal matrix

 

 

( 2L($1) —L($1) 0 ' ' ° 0 \
—L($2) 2L($2) —L(.’£2) 0 ' ' ' O
0 ° ' ° 0 —L($n_2) 2L($n_2) —L(£Bn_2)
\ 0 ... 0 —L(a:n_1) 2L(a:,,_1) /

51

 

if” x — If” as _
Where 11(3):) 2: 0J( ( k+1)2h ( k 1)) .

The matrix K is positive definite since eigenvalues of the matrix are positive by the
following inequality
lAi — kiil S E Ikijl)

j?“
where 19,-.- = 27' and Z Ikijl 2 2r.
1'?“
The matrix C(u?” ) is positive definite and bounded away from zero since
€T-C(u:"'") -E = (1+m") - I£I2+£T-K-£+£T-L(u2""')-£
2 (1 +m2) - |€|2 — 3m . 1' - maxla'l - [g]2
= (1+m2 — 3m - r - ma:z:|a’|)|§|2
1 2
> §|€|

>0.

The proof of Theorem 2.1 is now complete.
The following pictures are some examples of the dynamical behavior of the solution
u and the strain u,: for the polynomial function u1(:r, 0) = 1011:4 — 212:3 + 13.42:2 — 2.42:

and the sine function u2(x, O) = 61-0 sin(107ra:(a: + 1)).

52

 

(3) ~- time = 0
0.3, ........ : ....... ..... -+-time=500
- ; . : -— time = 1500
.... time = 2500
unit = sec/800

   
  

 

 

 

 

 

 

.2 '—+—time=500 ........ ...... fl

 

 

 

 

 

 

 

 

 

 

 

-°—time=1500
_02 i i i 4 g _3 +fime=2500 . .‘
0 0.2 0.4 0.6 0.8 1 0 "00:5901300 .6 0.8 1
X X
c d
1.2, ................... (.) ....... —time=0 4 ................... (.) ...................
; s 2 --time=soo i z 2 2 z s
1, ........ j ........ :........;....—o—time=1000 ; ; g j -
2 2 2 +time=1soo 2 ......... s ........ ........
0.8 ......... :.. ...... I ..... unitgsecjam 1 _ A i
.' I . . . ' « A
......................................... Ax \‘ A ,A‘ fl M
an 3010+ .. . .. v
_2 _, —time=0‘
—v—time=500

 

 

—- time= 1000

412* . ; . , , _4_+fime=1500 . .- ;

' o 0.2 0.4 0.6 0.8 1 o 1"“52‘21800 1.6 0.8 1
X X

 

 

 

 

 

 

Figure 2.1. Transition Layer Dynamics for (a) u1(a:,0) = 10m4—2l$3+13.4x2—2.4x,
(b) (u1)z(a:,0) (c) u2(a:,0) = ésin(107rz(x + 1)) and (d) (u2)z(x,0) using the
FDM.

We can see the steepness behavior of the transition layers from Figure 2.1. From
(b) and (d) of Figure 2.1., we can also see that on the compliment of transition layers,
the graphs are decreasing or increasing near 1 or —1 until 1500 time steps and this is
because of the energy decay which is proven in Lemma 1.1 in Chapter 1. Note that

the energy functional E(t) has minima near 1 and — 1 since the stored energy function

53

W(u;"'j) has minima at 1 and —1. The other integrands of E (t) are negligible since
they are sufficiently small.

The graph of ug‘J except for the finitely many zeros is not exactly approaching
l or —1 and the reason is the following. Recall that by Proposition 1.1, the time
limit u? of the discretized solution if” is in the equilibrium state and satisfies the
following equation
)2); — ul" = 0 a.e.

0((u.

Thus, this indicates that there is a neighborhood such that (UT); 7é 1. For the
Viscoelastic system without the elastic foundation term if” , the absolute value of
graph is approaching 1 and it is also clear since (211"); satisfies

a((u1")m)x = 0 a.e.

At 2500 time steps in (b), the graph, moves above and below 1 and -—1. This type
of behavior is even worse in the sine function (d). It blows up even after 1000 time
steps. This is because of the truncation error of the numerical solution due to the
effect of the nonlinear term 0(u;"’j)z on the transition layers. We improve the error

in the next section by averaging of the term 0(ur’j)x in the FDM algorithm.

2.3 Average approximation of 0(ugl’j)x

When the time is sufficiently large, the approximation of 0(u;”'j)z (2.2) produces a
significant error since the difference between um'j(a:k+1) and um'j(a:k_1) can be very
large near the position of transition layers. Since 0(u;"'j )3 = (E(ugn’i ) -u;"'j)x, the non-
linear coefficient E(ug‘d ) of the Laplacian ug'j leads to the nonsymmetry of the matrix
C(um’j ) in Section 2.1. Therefore, we modify (2.2) by using the average approximation

instead of the central difference approximation to recover the symmetry of C(um’j).

54

Let mk+§ be the point in the middle of wk and “+1. Similarly, let mhé be the point

 

' ' m.’ ~ u""j(k+1)-u’""'lki m.‘ ~
in the middle of zk_1 and zinc. Then um 3(mk+%) ~ h , u:c J(:1c,c_%) ~

um'j(k)—u""j(k-l)
h

 

and the following average approximation holds

 

 

 

 

0(u;"’j(xk))x = (E(UZ‘jj(wk))-u;’"j(rf=k))z . .
~ Garnet...» - army — anyway) - army
~ ~ . h
z 0(u; lffk—é-D .um'j($k_1)
(auras/st,»+6<u;"'j(x.+,)» M )
— ’12 '11. £13k
0(ux ISM?) . um” ($k+1),

where

 

 

Hence, we obtain

E(um’j ($0) - um’j (mm) + E(Um’j (3%)) mm” (mic) + E(um’j (n+0) - ”um” (mm-1)

= Hum—WED),

where
"’ mj m m2 ~ mj
A(u ’ ($16)) := —713 _ W ' 0(ux ’ (wk—:15»)
.... m , 2m m2 ~ m . ~ m .
B(“ ”(3710) == 1 + m2 + W + 32— ' [0(1‘2 ’J($k—%)) + ”(71; ”(fig)”

55

and F(um’j‘1(xk)) is the same as (2.6) for k = 1,-~ ,n — 1. We now derive the

following modified matrix equation for (2.1)
(E(um”){um’j } = {Mum—1)},

where (E(um'j) = {51,8(um'j (zk))}(n_1)x(n_1) is the following symmetric tri-diagonal

 

 

matrix
{§(um'j(xl)) Emma») o 0 \
Kama/:2» E(Woz» Kama-3)) o o
0 Iowa») E(um'jm» E(Ww) o
o E(umvi(x,,_3)) §(um’j(a:n_3)) E(umd(xn_2)) o
0 . . . 0 E(um'j (Inn-2)) E(‘um’j (zit—2)) E(um’j (Tn—1))
( o 0 E(umd(x,,_1)) Emma.-.» )

We compare the standard central difference scheme to the average approximation

in the next pictures.

56

 

 

 

 

_2... -+—time=500 ........ ........

 

 

 

 

 

 

 

 

 

 

 

      

 

 

 

 

 

 

 

 

 

 

_ : : —°— time=1500 .
_._ time = 2500 ; 3 _._ time = 2500 : :
'30 w. ""“uisemgu o 8 1x '3 ofl “n": sea/80° ‘ 0A8 ix
(0) (d)
4 ................................. . ........ ’ 3 ......................................... ll
: E l 3 l
3 Q 3 3 2 ........................................
2‘, ................ ........... l
A A A A ’ "
A” l l l l ‘
3“0-"'
2 ....................... .
— time=0 : . 4 time=0 i
_4___-°—time=500 i :x _2 -0—time=500 ; :x
o —- time=1000 5 0.8 1 o -— time=1500 0.8 1
—~— time =1500 —0— time = 3000
unit = sec.l800 unit = sec.l800

 

 

 

 

 

 

Figure 2.2. Dynamics using the central diflerence scheme for (a) u1(a:, O) = 10:1:4 -
21:1:3 +13.4:z:2 — 2.42:, (c) u2(a:, 0) = % sin(107r:1:(a: +1)), the average approximation
for (b) u1(x,0) and (d) u2(a:,0).

At 2500 time steps for the polynomial function (a), the central difference approx-
imation produces an error and it starts blowing up. However, the graph (b) for the
same function using the average approximation still converges near 1 and — 1. We
can see from (c) and (d) that the average approximation is more efficient in the sine

ftmction. Unlike (c), the absolute value of the graph converges near 1 in ((1) even

57

after 3000 time steps.
Note that the matrix (E(um'j) still involves the current time step which increases
the computation time. In the next section, we introduce a new noniterative method

which saves the computation cost.

2.4 The Alternating Direction Implicit (ADI)

Method

While it is common to use the explicit methods to the hyperbolic equations, it is

useful to apply the ADI method since it can reduce the order of error and improve

the stability. In [19, Chapter 8], the improved ADI method which is combined with

the explicit procedure was introduced for the following second-order wave equation
0%

w+fiu=fl $Efl,t>0,

1
where [l = 2A,, l is the dimension of :13. Given 2", - -- ,zj‘l, the approximated
i=1

solution 2’ at the time tj is given by the following explicit formula

2’30 — 2zj"l + 2’"2

'm,2 + AZj—l = fj-l (2'9)

 

and the implicit stepping

j,n_ 135-1 _ . . -
z z + (1145(2'7" — 22"1 + 23—2) = 0, "3 = 1’ ° " ’l’

 

m2
zj = z“, (2.10)

58

where a E [.25, .5]. By applying this ADI procedure to our system, we get

umtm) = law-1m)—X(u'"J-l(xk>>warm—1)
_(§(um,,--.(,k) + m2) - uni-1m)

Jew—lee.» «Mi-1m...) + fi<u'"""1(wk)),

(am W 1(a))— —-;,—"Z) ~um'j(mk_1) + (1 + mum-1m» + 12,-,3) «we»
+ (er/Mum 1( (‘$k+1)) ,',—”-.) «Macao
= um’*(zk) + aA(u""j—1(a:k)) - fim'j’1(:1;k_1) + a§(um’j"l(a:k)) -17"’j"1(:ck)

+ozf’l\('um’j—l ($k+1)) ' End—l (mu-1),
where

find—1(a) := 2um'j_1(mk)—um’j—2($k),

flaw-1m» == Foam 1(a2k))— 27'” 1(a),

Emma-1(a)) := -fi-a(u;"d-l(z._%>>,
A . m2 ~ . ~ .
Emma-1(a)) == fi'[0(“Zl’J—1($k—§))+0(u;n’3_1($k+§))l

for k = 1, - . - ,n — 1. We can compare the direct iteration method to the ADI method

from the following table that shows the computation time.

59

 

Methods t = 500 t = 1500 t = 2500 t = 3500
Direct Iteration Method 24.11 71.52 119.03 168.29
ADI Method 18.07 52.21 90.46 127.27

 

 

 

 

 

 

 

 

 

 

 

Table 2.1. CPU time (sec) unit oft— = sec. /800, h- — Tb—o’ m = 87130“

Table 2.1 shows that the computation time in the ADI method is faster than the

Direct iteration Method.

2.5 Derivation of the matrix equation using the

Finite Element Methods

Recall that the discretized solution if” satisfies equation (1.2) in Chapter 1. Since we
consider the Dirichlet boundary conditions um'j(0) = um'j(1) = 0, these become the
essential boundary conditions for constructing the weak formulation for the system.

Multiply equation (1.2) by a test ftmction w E W3’°°(Q), where Q = (O, 1) and in-
tegrate over the finite element (use, me“) with the length h. Then we get the following
expression

3e+l . .
/ (wt),J — wo(u;"”)x + wum —wv£§cj)d:c = 0
3e

and this implies
xe-l-l
/ (wvf + wzo(um 'j) + wumj + wmv’" ”)dx = 0 (2.11)

for all j E N. The boundary terms after integration by parts are zero since the test

function w satisfies essential boundary conditions.

Let u "‘J( :c(): = :um’zbshz), w(a:) := 44(2)), k = 1,--- ,ne. Here, ugh], s =

1, - - - ,ne are undetermined constants and z/J,(:z: ) are the interpolation functions. Then

60

equation (2.11) becomes

2M 5M ()5: )¢,(5)d5~](u mi+um5)+; [fohd d2/2k(i :)d¢;§(ci)di] 112,,-

+ fond,” (MM ”(12:11: “Mg: ——)dr‘i=0, (2-12)

. . umij _ umvj—l .. . umg’ _ 2um,j—1 + um,j—2
where u?” z ‘9 3 , u?” z 3 8 2 ‘9 . Define the (16,3) com-
m m

 

 

 

 

ponents of ne x ne matrices A8, 18" as follows

_fh Cit/11:57 )d¢s($)d-

{A8 }k s .—'—/h i/Jk($)1/Js($)d$ {Be}k3 dff; div

and let the 19‘" component of ne x 1 vector lFe(u'"'j) be the following

{Fe<um’j)}k==/hd$ “’W )U(Zus ”’d—d—w‘f —)d-

O

 

After multiplying the both sides of (2.12) by m2 and moving the previous time solu-

tions to the right hand side, we get the following matrix equation

[(1 + m2)Ae + mlBe]{um’j} = Ae(2{um’j"1} — {MW—2}) + mlBe{um’j_1}

—m2]Fe(um’j) (2.13)

on each finite element (are, $84.1). It is known that for the linear elements, A3 and Re

are the following matrices

A8

ll
Oil 3"
a
«i
ll

61

For the quadratic elements, the following are the matrices A8 and Be

4 2 —1 7 —8 1
h 1

A8 = _ e := — _ _

30 2 16 2 , B 3h 8 16 8

—1 2 4 1 —8 7

For the hermit cubic elements,

 

 

 

 

 

 

/ 1_3h_ _11h2 % 13h2 \ f 6 1 6 1
35 210 70 420 5h 10 5h 10
11h2 h: 13h2 h3 _1_ a i __h_
210 105 420 140 10 15 10 30
A5 = , B8 =
9h 13h2 13h 11h2 _3 _1_ _6_ _1_
70 420 35 210 5h 10 5h 10
1
13h2 h3 11}:2 h3 K _i _l _1_ g};
420 m 210 153 10 30 10 15 /

For n subintervals (xk_1,:ck) of (0, 1) with length h = i, k = 1, - -- ,n, O = 1170 <
2:1 < < 22”-; < 23,, = 1 we get the following global matrix equation of n — 1 by

n — l for our system after assembling the local element equations (2.13).

[(1+m2)A+mlB]{u"‘"} -—— mum-1 _um,._.}+mB{um,,-_1} —m21F(u’""').

(2.14)

Where {MW} := {113“ , - - . ,uIfiJT and the similarity holds for {um’j‘l}, {rim—2}.

62

A, B are the following global stiffness matrices of (n — 1) x (n - 1)

(410 ---0) (2—10 mo)

 

 

 

 

(0... 014) (on. 0—12)

for the linear elements. Define ”31.1 = 112']. = 0. Then 1F(um'j) is the following vector,

{F1(um’j)v ' ' ' 1Fn—l(um‘j)}Ta

 

 

 

 

where
F (um,,) ;= [h d¢2(5) .0 um,- 551(5)+ umdzpd2__(_5) d_
8 0 di‘ 8-1 (if; Us
In — -
+ j W .. (5561—1552 gm ). .
dx
0
maj _ m’j mi] _
= 0(5, hu,_1)_a(u.+,h u, l

for s = 1, « - ,n — 1 since ”if = —%, “if: = %. For the quadratic elements, A, B

63

are the following, respectively.

{16 2 0

 

 

0
2 8 2 -1 0 0
0 2 16 2 0 0
0 —1 2 8 2 —1 0 0
O 0 0 2 16 2 0 0
h
A’s—o
0 O 2 l6 2 0 O 0
0 0 —1 2 8 2 —1 0
0 O 2 16 2 O
0 0 —1 2 8 2
( 0 0 2 16 j
( 16 —8 0 0 )
—8 14 —8 1 O 0
O —8 16 —8 O 0
0 l -—8 14 —8 1 0 O
0 0 O —8 16 —8 0 0
l
B ‘ 55

0 —8 l6 —8 0 0 0
0 1 —8 14 —8 1 0
0 ——8 16 ——8 O
0 1 —8 14 —8

 

 

OOOOO

0 —816)

64

The vector lF(u""j) has the following components

 

 

 

 

 

 

 

 

 

 

 

 

h .. - .. _
- €17,020?) - d¢1($) -d1/J2($) - d¢3($)
Fa m,] 1: . mg _— m,] __ m,] d—
(u ) [0 d5: 0 ““1 d5: +118 d5: +15“ d5 5:
if s=2k—l, k=1,--- H;
h _ _ ._ _
- (ii/’30”) ' d¢1($) ' d1/22(a:) -d¢3($)
F8 771,] := . mi] my] 771,] d-
(U, ) A di 0 H’s—2 (bi + us-l d5: + “'3 d-
h _ _ _ _
(MICE) m 11710105) m,'d¢2($) m,'d1/13(-’13) _
+/o d2 '0 u” J d5 +51%], d5: +1552 d5: d
— 2
if s = 2k, k 1, , n 2 .
For the Hermite cubic elements, let 113;, = u?’j(xk_1), k = 1,--- ,n, U21: =
um’j(a:k), k = 1,--- ,n — 1, 123:,” = ufcn'j(x,,). Then we get the 271 x 211 matrix
equation (2.14) where
113 131:2 h3
( 1% _W —m 0 0 \
13h2 26h 9h 13h2
‘22—0‘ ‘3? 0 7‘0 m 0 0
h3 21:3 13h2 1:3
‘14—0 0 m ”455 "m 0 0
9h 13h2 26h 9h 13h2
0 56 ’72? a 0 56 Tim 0 0
13h2 h3 21:3 131:2 h3
0 W ‘W 0 as ‘m‘ “if: 0 0
A. = ,
9h 13h2 26h 9h 13h2
0 0 To ’725 3—5 0 "To 725 0
3h2 113 2h3 13h2 h3
0 0 1470 .170 0 5m ‘72? ‘555 0
9h 13h2 26h 13h2
0 0 5‘0 “750— 55‘ 0 a)“
3h2 1:3 2113 1.3
0 0 1420 —m 0 1m ”1%
2 3 3
K 0 o — —:’1—. — j

65

 

 

 

 

h h
f is 116 ’56 0 0 )
5 — o 5 —:—. o 0
’55 0 % 1% *3'3 0 0
0 ‘56:; 1% $72. 0 :5. -$ 0 0
0 ’11_0 '5 % 5 -3'2, 0 0
B:
0 0 _56h 1—10 51,725 0 ’52,, —11—0 0
0 0 —5 —— o — s—. -— o
0 0 —56h 1_10 51:72; 0 ‘11—0
h h
0 0 _110 _3—’b' 0 :7, ”'fi
\ 0 0 “110 ”5% '1‘? j
and
h - — _
' 552(5) 552(5) d___¢3(5) 514(5) _
m"? 2: . "‘13— m] m,j d
mu ) [0 die a ”1 d5 "L“? d5: Jr“ d5: 5,
- ”1545) 551(5) - d_¢__2(5 ) 554(5)
F" m"? I: —. m] m,] m,j CC,
2(u ) f0 .15 0(u,,g;——, 25 +5 um .15 M," 55 __)d-

 

 

 

F,(um»1') : f0}. d¢4(i) (um,jd¢1(5) mddzpzfic’)

 

 

 

 

 

d5: “3 d5 3-2 d5
+1111 mg?) + uf‘ 3 dz??? d5
+1121”; ”3? + 11.3 “3?» 5

if 3=2k+1, k:1,...,n_1,

66

d5: "2 d5: ““1 d5:
(17103 (i) m,j d¢4(i) _

+ /" 521(5) _0(um,,-dwl(5) + 5.512(5)
0

d5: 8 d5 us“ d5:

 

F8(um'j) .___ [ohm/’30?) 0(um,jd¢l(i)+ m,jd¢2(i)

+11?”

 

 

 

m'd¢3 5‘) m'd¢(i) _
+11, 4’2 d; + “545—353 due

if s=2k, k=1,--- ,n—l,

 

As in the case of FDM, the difficulty arises for solving the matrix equation (2.14)
since it involves the vector 1F(um'j) containing the nonlinear terms, but again the
numerical solution is obtained using the direct iteration method by treating the term
11"” which is inside the nonlinear term a as the known solution in the previous time
step.

The truncation error is reduced in the FEM case since the numerical integration of
F, (um'j ) has an effect of averaging the nonlinear term 0(u;"'j )3. The following pictures
are the examples using the FEM with the linear, quadratic and Hermite cubic elements

and using the FDM without averaging the nonlinear term, respectively.

67

 

.......................

 
 
 
 

 

 

—time=0 8

+ time=500

—~—time=1500

+ time=3000
unit=se01800 4

 

 

  

 

 

 
 
 

   

.......................

........................

........................

 

 

«_— time=0

: —1—time=500
Q-o—time=1500
f—o— time=3000

 

 

 

unit = sec.l800

 

 

 

3x 1 ..................... 3X 2 ...............
o .......................................... 0 ...........................................
_1 .. ............... _2 ............................. .
_2 . . . ; 3 4 1 .« . i. g
o 0.2 0.4 x 0.6 0.8 1 o 0.2 0.4 x 0.6 0.8 1
(C) , _ (d) — time = 0
— 9m 3 0 -«— time = 500
3 ...................... —o—t!me_500 4. ........_._time=1500
*“me=15°° ;—+—time=2soo
2 ....................... "' "m9 = 3000 3 ------------------------ unit = sec.l800

 

 

 

 

  

unit = sec1800

 

1 ...........
3X
0 _
-1 .. ...............
_2 g ; . is g
0 0.2 0.4 x 0.6 0.8 1

 

 

 

 

 

Figure 2.3. Transition Layer Dynamics using the (a) linear elements , (b) quadratic
elements (c) Hermite cubic elements for the FEM and (d) FDM without averaging
the nonlinear term 0(uz)z.

From Figure 2.3, one can see that the three finite elements have almost the same

behavior. However, the graphs are still convergent at 3000 time steps while in ((1),

using the central difference scheme induces a blow up at 2500 time steps.

indicates that the FEM is still much more accurate than the FDM without averaging

the term 0(uz)x.

68

This

2.6 Examples

Note that if initial energy is large, the dynamics no longer follow such a behavior and

the following pictures show the importance of this assumption.

 

 

 

 

 

 

 

Figure 2.4. Transition Layer Dynamics for the (a) strain when |(u1)x(a:, O)| > 20 for
some a: 6 (0,1), (b) strain when |(u2)z(a:,0)| < 20.

The graph in part (b) of Figure 2.4 decrease fast and still has the same number
of zeros and the transition layers are getting steeper. However, in the graph in
part (a), the shape of the initial value (ul),(a:, 0) is the same but its absolute value
is greater than 20 for some a: E (0, 1), the graph does not preserve the number
of zeros and deve10ps new zeros and transition layers. Thus, in this example, the
critical point is when um (:13, 0) = 20. The equation for the initial data is (111),, (:13, O) =
400:1:3 — 630:1:2 + 268:1: — 24 for (a) and (112),,(92, 0) = 280:1:3 - 4412:2 + 187.62: — 16.8
for (b). i

We next consider the Neumann boundary conditions instead of the Dirichlet

69

boundary conditions. Although we did not prove the dynamics for this type of prob-
lem theoretically, we can see that the dynamics also hold for the Neumann problems in

Figure 2.5. We used the initial data uz(a:, 0) = 500225 — 12501134 + 10802:3 — 370:1:2 + 40:13.

 

 

 

 

 

. .. .3. ‘ “ ____J”'
+time=3000 ; v
unit=sec1800 ; ; ,

_15 1 1 1 1 1 1 1 1 L 1

 

 

Figure 2.5. Transition Layer Dynamics using the Neumann boundary conditions.

70

One more interesting example is when the assumption (A4) is omitted. We used
the polynomial flmction 113(3, 0) = 60:125 — 120:1:4 + 80:1:3 — 21.122:2 + 2.208051: — 0.0640
as an initial data. Note that u, (2:, O) has a local minimum at :1: = 0.2 which is also
a zero of uz(x, 0). At a: = 0.2, the graph in Figure 2.6 pushes down to the negative

values and deve10ps the transition layers.

 

1.5 1 r r 1 1 1 1 1 1

 

+time=0
+time=500
+time=1000
+time=2000
+time=3000
+time=4000
—time=5000
+time=7000
unit=sec1800

 

 

 

 

 

 

 

 

 

 

 

Figure 2.6. Transition Layer Dynamics for the problem for the strain whose initial
data does not obey (A4).

71

CHAPTER 3

Convergence Analysis of Numerical
Solutions in Two-dimensional

Systems

Even though we have not achieved the analytic proof of the transition layer dy-
namics for the multi-dimensional systems, the extended numerical methods to the
two-dimensional systems from the one-dimensional systems also show the transition
layer dynamics. The finite number of portions of the surfaces become steeper and
discontinuous at the time limit. In this chapter, we derive the matrix equations from
both FDM and FEM. We also prove the convergence of the numerical solution for
the two-dimensional systems from the FDM with the standard central difference ap—
proximation. Average approximation for the nonlinear term Div(o(Du’"'j)) and the

ADI method are discussed in Section 3.3 and 3.4.

72

3.1 Derivation of the matrix equation using the

Finite Difference Methods

Consider the two-dimensional Viscoelastic system,

11“ = Div(o(Du) + 012,) — 11,
“Ian = 0 (t 6 [0100»,

141:0 = no: Ut|t=o = ’00 (33 E 9),

(3.1)

where u is a mapping from D x (0, 00) to 1R, (2 = (0, 1) x (0,1). It was proven [15]

that for higher dimensions, the discretized solution 21"” , j E N, of system (3.1) is

the minimizer of the following ftmctional

W111] := 1 In — zumd-l + wad-212 + W(Du) + —1—|Du — Dam-112 + 115112 55.
9 2m 2

2m2

Therefore, 11"” , j E N satisfies the following equation

1 m, ‘—1 m, “—2 - 1 m, '—1 _
WW—Zu 3 +11 3 )—D1v(a(Du))—T—n-:A(u—u 3 )+U-—0.

The second term of the above equation can be rewritten by

2 2 2 D 2 D
Div(o(Du)) ___ a W(Du) 6 W( 11) . + 6 W( 11) .
For simplicity, we use the following stored energy function

2
uy.

NIH

W(Du) = 211113—1)? +

73

(55:12 '“m” (02511011.) “x” (M2 “y”

(3.2)

(3.3)

Hence, the second term of equation (3.3) is zero and

52W(Du) _ 3212 _ 1

62W(Du) _
(5“ch _ I

(any)? “ 1'

As in Chapter 2, divide the interval (0,1) in the a: direction into n subintervals
(344,151,) of uniform length i, k = 1,--- ,n, O = :50 < £1 < < xn_1 < 93,, = 1
and divide the interval (0,1) in the y direction into the same number of subintervals
(yk_1,yk) of uniform length i, k = 1,--- ,n, 0 = yo < yl < < 31,.-1 < ya = 1.
By using the same method as in Chapter 2 (Central difference scheme for the FDM),

the following approximations hold.

. mg. m _ umsj _
ugg,3(xk,y8) z u ( k+l?y8) (mic 1,318),

 

 

 

 

2h
, ma _ ma’ _
u;n,1(xk,ys) R: u (“Will/8+1) u (kays 1),
2h
m ' um,j($k 1,?! ) _ zum’j($k,y ) + um’j($k—lays)
um?) (£13k, ya) z + 8 h2 8 a
m - um'j($k,ya+1) - 2am” (xk, ya) + um” (13k, 313—1)
uyg’cvk, 313) m h? -

Thus, equation (3.2) becomes

(1 + m2)um,g (3k: ya) _ E(umd(mk+11y8) — 2am“? (mic, ya) + um,_7 (:L'k_1, ya»

m ' ' m '
‘mum’uxhz/m) - 2am“, y,) + u ”(whys—1))

2 m,j _ 7nd 2 , .
—% [3 (u (xk+l,y8)2hu (wk—by») — 1] (um'1($k+1,ys)- 2um’3(xk: 318)

 

2

. m - ' '
+Um’J (mic—1) ya) - (uma (13k: ys+l) ‘ 271mg (3k, ys) + um,J(xk? yB—l»

h?
_ ._ m -_ -_
= Zum’J-l($k:y3)-Um"7 2(ka,ys)-7,3[um” l(1731c4r1,1/3)+um” Vick—1,310

+um’j_1($k, ys+1) + um’j_l(1‘k, ys—l) — 4um’j_l($ka yd],

74

for k, s = 1, - - - ,n — 1. After arranging the above equation in terms of

um’j(xk—l:y8)a um’j(xkay8)) um’j(xk+lays)a um’j($kay8-1) and um’j(xk1ys+1)’

we get the following equation

02(um’j(wk, ys))um’j($k—1,ys) + 02(um’j (ark, ys))um’j (n+1, ya)

+Gl(um’j(wk, ys))um‘j (ark, ya) + GEWm’j (wk, ills—1) + 03"“m’j (wk, ya“)

 

 

= H (um’j‘lhmym, (3.5)
where
- 4m 2m2
01(um”($k,ys)) 3: 1+ m2 + 'h—2 + 7‘2—
+3_"_l: . 3 um’j($k+1,ys) - Um’j(-’L‘k—1,ys) 2 _ 1
h2 2h ’
2 m,j mhj 2
ma’ .= _fl __ 1, u ($k+1,ys) - u (wk—1,313) _
02(11. (23k, y8)) ‘ h2 h2 [3 ( 2h, 1
2
m ._ m m
03 -— -p — F’ (3.6)
and
H(um”“(mk,ys)) := Zum’J‘1(xk,ys) - um"‘2(xk,ys) — 713 ' [um’J‘1(-’Ek+1,ys)

+um'j‘1(xk_1,ys) + um’j‘1(xk,ys+1) + um’j”1(mk, 313-1)

-4um’j"1($k, ys)]- (37)

Equation (3.5) becomes the following matrix equation
G(um’j){um’j} = {mum—1)},

75

where C(um'j) is a (n —- 1)2 x (n — 1)2 matrix and can be represented by the matrix

of the (n — l) X (n — 1) submatricw Gz,s(um’j), l,s = 1, - -- ,n — 1. That is,

 

 

K G1,.(um) G1,._.<w> )
G2’1(umij) . . . G2,n_1(um’j)
Gn—2,1(um'j) ° ' ' Gn-2,n—1(um’j)

K Gn—1,1(Um’j) " ‘ Gn—1,n—1(um’j) /

The main diagonal entries G1,z(um'j), l = 1, - - - ,n — 1 and the other diagonal entries
G1,,_1(u'"'j), G¢_1,¢(um’j), l = 2, - - - , n—l are the only nonzero elements. We denote
Gi(um'j(zk(l),ys(¢))), the nonzero elements of G1,;(um'j),Gz,z-1(um'j) or Gz-1,1(um'j),
by (G?’j)k(z),,(z), z' = 1,2. Then, the G1,;(um'j), l = 1,-o- ,n — 1 is the following

(n — 1) x (n — 1) tri-diagonal matrix.

 

 

( (G?'j)(t—1)n—(z—2),1 GE" 0 0 \
G5" (Gin'j)(l—1)n—(z-3),2 GE," 0 . . . 0
0 " ' 0 0'3" (Gln’j)1(n—1)—1,n—2 GE,"

\ O . l . 0 GS" (Gln‘j)l(n—1),n—1 )

76

The two other diagonal elements G1,z._1(u’"'j), G1_1,z(um'j), l

the following diagonal matrices, respectively

( (Ghn'j)(l—1)n—(z—2),1 0
0 (012" 'j)(z—1)n—(z—3),2 0
0 0 (G?’j)z(n_1)_1,n-2
0 0

 

 

/ (ng'j)(z—2)n—(t—3),1 0

0 (Ghn’j)(z—2)n—(z—4),2 0

0 0 (G?’j)(z—1)(n—1)—1,n-2
\ 0 0

-,n-—1are

0
(Ghn'j)l(n—1),n—1 /

 

0

(ng’j)(l-l)(n—l),n—l )

 

The {wind} is a (n — 1)2 x 1 vector and it is also considered as the (n — 1) x 1 vector

of n — 1 subvectors {up}, 2': 1, - -- ,n — 1. That is,

}}T,

{um’j} = {{UT’j}, - ~ , {111311

where

{uln’j} 3: {um’j($i,y1),' " ,Um’j($iayn—1)}T-

Similarly,

{Wum’j‘lfl = {{HT’j}, - ~° ,{HZ‘le}}T,

77

where

{Hind} :: {Hm’j(ziay1)i ° ' ' ’Hm’j(miiyn-1)}T'

3.2 Existence of the Finite Difference solution

Theorem 3.1 The solution of the matrix equation
G(um’j){um’j} = {H(um’j‘1)},

exists.

PROOF. As in Theorem 2.1, consider the following equation

G(u:"”'){u?:;{ — W} = (Gerri) — Gardner's. (3.8)

By using the same argument as in the one-dimensional case, the right hand side of

the above equation becomes the following matrix equation

1P’(um’j ){UI'l’i - “In” },

i.

where Mug” ) is the following matrix

 

 

( 0 1031‘ 0 --- 0 )
szjj 0 P53] 0 . . . O
0 "' O P(f::71)2—1,(n—1)2—2 0 PfSZIV-lln-l)’
\ 0 W 0 P(’::71)2,(n-1)2—1 0 /

78

Here, denote 131,,(u3’j) by Pf? .

a3_w

P < hrz-constu
12 — Bug

' IuI"”(x1, 311) + ”PR“: 312) + uln’j($2,y1)|

 

 

S 63.

Recall that r = 72% Similarly,

83W
Bug

+ UT’P($n_1,yn_1)|

' lurk]- (3711—2, yn—l) + “In”. (mu—1, yn—2)

 

P(n_1)2,(n_1)2_1 S hT2°CO713t.'

 

 

 

S £4,
for some 63, 64 << 1.
For general H,,+1(u$’j ),
33W m. . .
P 5 ’"2 ' mt" 75.73- . Iu. ”(z-.-.,y.) + u?“ my.) + amass/...»
I
S 65.
Similarly,
IDH-IJ S 661

for some 65, 66 << 1. This proves that the matrix P012” ) is bounded and small.

m3]
1

Now, it remains to show that G(u ) is invertible and bounded away from
zero. Consider the matrix G(u:"’j) as the sum of three matrices 11K and may"),

where 3,112 are the following matrices, respectively.

79

 

Here, denote Pl,,(u:.:"j) by PI'Z’P .

my

' luln'j($1,311) + uln'j($1,y2) + “i (932, 311“

 

 

 

|/\
:9

Recall that r = inf. Similarly,

(33W
0n:

+ U?’j($n—1,yn—1)|

 

' luzn’j (mu—2, yn—l) + ”Ind ($71—13 git—2)

 

P(n—1)2.(n—1)2—1 S hTz ° const. -

|/\

641

for some 63, 64 << 1.

m.)'
For general Pz,z+1(u,~, ),

63W
. 0u3

Z

- |uI"’j (331—1, ya) + U?” (x1, 31.) + ”$711+“ ys)|

 

Pl,l+1 S hrz-const.

 

S 65.
Similarly,
IDH-IJ S 66:

for some 65, 66 << 1. This proves that the matrix Max”) is bounded and small.

Now, it remains to show that C(uf’j) is invertible and bounded away from
zero. Consider the matrix C(uln’j) as the sum of three matrices 11K and E(uznj),

1

where TI, 1K are the following matrices, respectively.

79

{1+m2 0 0 \

 

 

0 1+m2 0 0
0 0 1+m2 0
\ 0 0 mm
(47' —T 0 ... ... 0 _7‘ O ... ... 0\
—r 41" —7‘ 0 0 0 —r 0 0
0 —r O
—'T' 0 O —’I‘ )
0 —r 0
0 0 —r 0 0 0 —r 47' —r
\0 0 _7. 0 0 _7. 47.)

 

 

where in the first row, the second —r appears at nth column. Similarly, in the first
column, the second -r appears at n‘h row.

The matrix 11:01:”) has the similar form. It has non-zero terms in the tri-diagonal
positions and nonzero terms appears at nth column of first row and nth row of first

column. That is,

80

 

 

 

 

 

 

(2f,(.) _Z2(.) 0 _fl(.) 0 0 )
—Z2(-) 250) —Z2(-) 0 —L(-) 0 0
-E1(-)
hr
—E1(-)
0 o —E.(-) o —Z2(-) 211-) 41320)
K 0 0 _fi1(.) 0 _f2(.) 2Z(.))
where
m...) egg;(“<wk+ws>,-,y<xk—w>),
~ 83W 21.23, 8+1 -U£L'k, s—l
......) = MU” ),,< v >),
Z = 31+52.

Note that this matrix is not symmetric since each row depends on the components 22,,
and y,. The matrix IR is positive definite since eigenvalues of the matrix are positive
by the following inequality.

IAi — 75,4 S 2 lzijl:

1'9“

81

where hi,- = 4- r, ZIEJ) Z 4-1:
.1?“ _
Now, the matrix G(u§"” ) is positive definite and bounded away from zero since

éT-G(u:"")-£ = (1+m2)«I€I2+£T-R-s+€T-Hi(u:"")-é
Z (1+m2)-LEI")—5m°r-ma:z:|o'|-|§|2
= (1+m2—5m-r-maa:|o’|)|§|2
> 3&2

>0.

This proves Theorem 3.1.

3.3 Average approximation of Div(o(Dum’j))

As in the one—dimensional case, we modify (3.3) by applying the average approxi-
mation to the term Div(o(Du”"j)). Since 0(Dum’j) = (H2133. ) - offing”), where

E(ULM) = (11?" )2 - 1,

Div<o<Dumo> = mum - um. + (um...

As in Chapter 2,

5W?” (1%-; , 31.)) .
h2 2 .um’J(mk-l:y8)
(E(U?’j($k-;,ys)) + answers») .. .
— h2 .u ,J(mkay8)

a:(um'j (mk+l 1 ya» -
. hz’ wants.)

 

”(“2” (wk, 31.9))2: R5

 

 

+

82

Therefore, we get the modified equation of (3.5)

62(um'j (wk, 31.)) - um'j(rrk_1, y.) + 52(um’j (n+1, 31.)) - um’j(ark+1,y3)
+51(um’j($k, ysl) ' um’j (17k, 313) + GB" ° um’j($k, 318—1) + GS." ° um” (11k, ys+1)

= H(um’j—l($k, 313)),

where

~ ' 4m 2m2
01(um’3($k1 31.9)) 3: 1+ m2 + F + F
2

m ~ - ~ m '
+75 ° l0(ué"”($k—%,ys)) + ”(“x ’J(‘Pk+%’y‘))]’

2
m m~m,.

62(um'j(x.,y.)) := —E,—fi.a(u.1(x._,,y.))

and GE," and H(um'j‘1(zk,y,)) are the same as (3.6) and (3.7), respectively for k, s =
1, - - - , n — 1. The ADI method is more useful in the two dimensional case since this
locally one-dimensional method enables us to solve the two onedimensional (n — 1) x
(n — 1) tri-diagonal matrix equations instead of dealing with the (n — 1)2 x (n — 1)2

banded matrix with 5 nonzero diagonals.

3.4 The ADI method in two-dimensions

We apply the explicit formula and the ADI approximation (2.9), (2.10) to the two-

dimensional system (3.1). Since ft = 2, we get the following 3 equations

83

“WW-Talk) = meow.)-@2(u'"J-l(xk.y.))mm’j-wasshy.)
—02(umd-1<xk.l,y.» - um'j'1(xk+1,ys)
—(c?1(um’j-1(sck. y.» + m2) «ms-let y.) + 6;." - um’j-lm. yr.)

—2c?a“-um'j-1(x.,y.) + 5;" - zed-1m, ya.) + fiw-‘(zt y»),

A m ._ m m“
(cram ’1 lawn—p) -u ' easy.)

A m ._ 2m m“
+(1+aGl(u " Nasal/3H?) 'u ’ (whys)

A m '— m mu
+(aG2(u ’1 1($k+1,ys))—7{2‘)'u ’ ($k+1,ys)
= um'*(zk, 31..) + 052(um’j‘l(zk,y3)) -fim’j‘1($k-1,ys)

+a@1(um’j-l($ka ya» ' find—1(xka ya) + 062(um’j—1($k+lv 318)) ' awn,j—l(zk+1, ya),

. A 2 ,
0‘an “ 2) 'um’3($k, sis—1) + (1 - 2003" + h?) Pumquws)

A m m .
+ ((10? — E5) ' u 'J($kays+1)
= umka, y.) + aég" . rind-1m, y,_1)

-2033" - end-1o... y.) + a6? aroma/3+1),

84

where

m’j—l($k)y8) I: 2um’j_l($k1y3) _ um’j_2($k, ya),
fi(um’j_l($k,ys)) I: H(um’j_1($k,ys))“m’j—l($k,ys),
A ._ m2 ~ -_ ~ m-_
01(um” 1(xkiys» 5: fi-[a(u;n'3 1($k_%,ys)+0'(uz’3 1($k+%vys))l,
" mj—l m2 ~ mj—l
G2(u ’ (“shay-9)) 3: "F'Ucux' ($k_%,ys)),
2
Am ._ m
G3 .— _W.

3.5 Derivation of the matrix equation using the

Finite Element Methods

Multiply equation (3.2) by the test flmction w 6 W01’°°(Q), where (2 = (O, 1) x (0,1)
and integrate over the finite element 523. Then the final weak formulation holds

+ um'j = 0

 

 

m,j _ m,j—1 m,j—2 . - m,j _ m,j—1

/ w- [u 2a 2 + u _ Div(o(Dum")) _ D1v(Du Du )
e m m

for all j E N. Since w satisfies essential boundary conditions, the boundary terms

vanish after the integration by parts of the above equation and we get the following

equation.

 

umaj .... zum’j-l + um’j—2 , . .
w - + 21); - 01(Dum”) + w - 02(Dum”) + wum”
m2 y
C
I

+ wm - ‘P + w,,- y y ) dmdy = 0, (3.9)
m m

 

 

where 01, 02 are x and y components of a, respectively. By (3.4), these are given by

a. = award)? -— 1), 02 = u;"

85

Similarly, as in the one dimensional case, define
u“(26, 31) == ZUT’WAHC, y), w(x, y) == «Mar, 31)-
3:1

Here, ufhj are undetermined constants and 1b,.(x, y) are interpolation functions. Then

equation (3.9) becomes

 

 

 

 

i F 9 «Mac mm y)didy]( uZJJ+uZJJ>
.. ’ 61,026?) 02.,(2) 6¢k(17)6¢s(z7) - - .m-
+ 82;“ 6:7: 6:: + 617 337 )dxdylu'
02/120: ...,6_____¢.(:c) +6224?) "‘ 2.2-WM) - __
+11?“ <2 —> «2 2 ,0..-

where 11;”, it?” have the same meanings as in Chapter 2. For k, s = 1, - - - he, let

{A622 == /‘2 ¢k<i.y>¢.(2.y>d2dy,

e ,_ 6114(5) 31PM) 61PM) 31/4431)
{B }k,s .— L( 0:? 5:7: + (9y 6y )dxdy,

{1FJ(u""J)}k == / (egg—L)“ (Z WW5?)
a k ne mja s _ _
+ Egy- y-) (”(14:22, ”P Ja;))>dxdy.

After multiplying the both sides of (3.10) by m2 and moving the previous time solu-

 

 

 

tions to the right hand side, we get the same type of matrix equation on each finite

element (28, as in the one dimensional case as follows

{(1 + m2)Ae + mBe}{um'j} = Ae(2{um’j‘1} — {uni-2}) + mBe{um'j_l}

—m2{1Fe(um’j)}. (3.11)

86

The assembly of the global stiffness matrix from this finite element equation depends
on the elements. Let (2:1, yl), (x2, yg), ($3, 313) be the three components of the
triangle. Then the following flmctions are interpolation functions for the triangular

elements

1
2A1},

e e e ._
01 = $2173 - $3332, 02 = $3331 - $1133, (13 — $15132 — $2131,

 

$50041) = (05+fi§x+7§y), (8= 1,2,3), where
m=m—m,%=m-m,%=m—m,
7i=$3—$2, ”6:31-33, 7§=$2--’E1,

Are = Area of the triangle.

Choose the right triangle on each equilengthed cubic of length h. Then we can obtain

the local matrices Ac, Be as follows

2—1—1 211
81 eh2
A—-2-—110,B—§4-121

—10 1 112

The following pictures are the layer dynamics of the two-dimensional surface using
FDM and FEM linear triangular elements, respectively. We used the initial data

uo(:z:, y) = —4(5:1r:3 — 7:1:2 + 21:)(1:2 — as).

87

 

(uo)x(x.y)
T

 

 

.............................................

    

...............................................

................

. ‘ .Z- l . Ir...“ .7 . .. .. 1; n W. . r ,' _'.. I]!

n ‘15.;1211. \1 : .‘v'fv- Q‘HT. ,v‘ I?" V‘bfi‘ ."
' " 1fife—29 knu.~.1.7.r.ri.2§' 2.3“ It. ks... uhV-w‘g

...-...... .....

 

Figure 3.1. Transition Layer Dynamics in two-dimensions for (a) initial data (no);
and (b) um after 1000 (sec. / 800) time steps using the FDM.

88

(uo)x(x.y)

 

 

 

 

 

Figure 3.2. Transition Layer Dynamics in two-dimensions for (a) initial data (no),c
and (b) uI after 200 (sec. / 800) time steps using the FEM linear triangular elements

89

BIBLIOGRAPHY

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