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THESIS

Willi "

7599

ll

 

 

 

 

 

 

“limiting;

 

 

1 ms is to certify that the

dissertation entitled

Characterization of the Interior of an Inhomogeneous
Body Using Surface Measurements

presented by

Mahmud Khodadadi-Saryazdi

has been accepted towards fulfillment
of the requirements for

Ph.D . degree in Mechanics

 

 

naming h Ohm—no

Major /P{°f ssor

 

Date August 2, 1990

 

MSU is an Affirmative Action/Equal Opportunity Institution 0-12771

 

 

 

 

LIBRARY
Michigan State
University

 

 

 

PLACE IN RETURN BOX to remove this checpom from your record.
TO AVOID FINES return on or before date due.

DATE DUE DATE DUE DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

l

MSU is An Affirmative Action/Equal Opportunity Institution
snowman-pr

 

 

 

 

 

 

CHARACTERIZATION OF THE INTERIOR
OF AN INHOMOGENEOUS BODY
USING SURFACE MEASUREMENTS

By
Mahmud Khodadadi-Saryazdi
A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Metallurgy, Mechanics and Materials Science

1990

 

 

ABSTRACT

CHARACTERIZATION OF THE INTERIOR
OF AN INHOMOGENEOUS BODY
USING SURFACE MEASUREMENTS

By

Mahmud Khodadadi-Saryazdi

In this study, the feasibility of characterizing the internal structure of an
inhomogeneous body using a discrete number of surface measurements is explored.
The inverse problem is formulated for steady state heat transfer and linear elasticity.
In the heat transfer portion of this study, the problem of determining the location,
size and thermal conductivity of an inclusion in a body of arbitrary shape using a
discrete number of surface temperature and/or heat flux measurements is examined.
In the elasticity portion, the estimation of the location, size, Poisson’s ratio and shear
modulus of the same inclusion using a discrete number of surface displacement
and/or traction measurements is investigated. The boundary element method is
adapted for application to this parameter estimation problem.

Several questions arise in this inverse application of the boundary element
method which have not been adequately addressed in previous investigations.
Among these are (1) does the "initial guess" for the unknown parameters have an
effect on the convergence to the correct parameters and, if so, what is this effect, (2)
how many surface measurements and what combination of measurements are required
to simultaneously estimate the unknown parameters, (3) how should the locations of
these surface measurement points be selected, (4) what effect will the inevitable
errors in experimental measurements have on the ability to estimate the sought
parameters, and (5) what effect does the size of the inclusion have on the estimation
of the unknown parameters? These questions will be systematically addressed using
the boundary element method coupled with the method of parameter estimation.

 

 

 

To my father and mother, and my Wife.

iii

 

Acknowledgments

I would like to express my appreciation to my advisor, Dr. Nicholas J.
Altiero, for his support and valuable assistance throughout this work. I also would
like to thank Dr. James V. Beck and the members of my guidance committee, Dr.
Gary Cloud, Dr. Dahsin Liu, and Dr. David Yen, for their support and assistance. A
final note of appreciation must go to my wife, Mahdokht, for her patience and
encouragement.

Funding of this research by the State of Michigan Research Excellence Fund
through Michigan State University’s Center for Composite Materials and Structures is
most gratefully acknowledged.

 

 

fl'M‘LZLH _ _. . .. . 1’

Table of Contents

 

List of Tables ................................................................................................ vi
List of Figures .............................................................................................. iix
Chapter 1 Introduction and Background .............................................. 1
Chapter 2 Boundary Element Method ................................................... 7
2.1 Heat Transfer ................................................................... 7
2.2 Elasticity .......................................................................... 21
Chapter 3 Inverse Problem and Parameter Estimation ...................... 40
3.1 Heat Transfer ................................................................... 40
3.2 Elasticity .......................................................................... 48
3.3 Sensitivity Coefficients .................................................... 50

3.4 Coupling of the Boundary Element Method
and the Parameter Estimation Method ............................. 52
Chapter 4 Results and Discussion ........................................................... 54
4.1 Heat Transfer ................................................................... 54
4.2 Elasticity .......................................................................... 80
Chapter 5 Conclusions and Recommendations ..................................... 102
References ..................................................................................................... 104
Appendix A: Computer Program .' Heat Transfer ............................ 106
Appendix B: Computer Program .' Elasticity ..................................... 118

v

List of Tables

Table (2.1) - Nodal temperature and heat flux values for heat transfer example

problem.

Table (2.2) - Nodal displacement and traction values for elasticity example problem
#1.

Table (2.3) - Nodal displacement and traction values for elasticity example problem
#2.

Table (2.4) - Nodal displacement and traction values for elasticity example problem
#3.

Table (4.1) - Estimating k2 using one temperature measurement.
Table (4.2) .- Estimating k2 using one heat flux measurement.
Table (4.3) '. Influence of the initial guesses on convergence (heat transfer).

Table (4.4) - Estimating the unknown parameters using four temperature or four heat

flux measurements.

Table (4.5) - Estimating the unknown parameters using five temperature or five heat

flux measurements.
Table (4.6) - Estimating the unknown parameters using six temperature measurements-

Table (4.7) - Estimating the unknown parameters using two temperature and two heat

flux measurements.
Table (4.8) - Influence of experimental errors on the estimations (heat transfer).
Table (4.9) - Influence of the inclusion size on the estimation (heat transfer).

Table (4.10) - Influence of the initial guesses on convergence using displacement and

traction measurements (elasticity example problem #1).

Table (4.11) - Influence of the initial guesses on convergence using displacement

measurements (elasticity example problem #1).

Table (4.12) - Influence of the initial guesses on convergence (elasticity example
problem #2).

vi

 

Table (4.13) -Estimating the unknown parameters using 10 measured displacements in
the x1 direction and 10 measured displacements in the x2 direction

(elasticity example problem #2).

Table (4.14) - Estimating 62 and v2 using displacement measurements (elasticity

example problem #2).

Table (4.15) - Influence of experimental errors on the estimation (elasticity example
problem #2).

vii

 

 

Figure (2.1)
Figure (2.2)
Figure (2.3)
Figure (2.4)
Figure (2.5)
Figure (2.6)
Figure (2.7)
Figure (3.1)
Figure (4.1)
Figure (4.2)
Figure (4.3)
Figure (4.4)
Figure (4.5)
Figure (4.6)
Figure (4.7)
Figure (4.8)
Figure (4.9)

List of Figures

- Plane inhomogeneous body.

- Integration path around singular point.

- Heat transfer example problem.

- Boundary element model.

- Elasticity example problem #1.

- Elasticity example problem #2.

- Elasticity example problem #3.

- Schematic representation of the estimation process.

- Sensitivity of boundary temperatures with respect to k2 .
- Sensitivity of boundary heat fluxes with respect to k2 .

- Sensitivity of boundary temperatures with respect to xc .
- Sensitivity of boundary heat fluxes with respect to xC .

- Sensitivity of boundary temperatures with respect to yc .
- Sensitivity of boundary heat fluxes with respect to yc .

- Sensitivity of boundary temperatures with respect to Rc .
- Sensitivity of boundary heat fluxes with respect to Rc .

- Temperature sensitivity coefficients associated with initial guess of k2
= 100, (xc , yo) = (3.5,3.5) and Rc = 1.8 .

Figure (4.10) - Heat flux sensitivity coefficients associated with initial guess of k2 =

100, (xc , yo) = (35,35) and Rc = 1.8 .

Figure (4.11) - Temperature sensitivity coefficients corresponding to Rc = 0.5 .

Figure (4.12) - Heat flux sensitivity coefficients corresponding to Rc = 0.5 .

Figure (4.13) - Sensitivity of boundary t2 values with respect to Gz and v2 (elasticity

example problem #1).

Figure (4.14) - Sensitivity of boundary t2 values with respect to xC and yC (elasticity

example problem #1).

iix

Figure (4.15) - Sensitivity of boundary ul values with respect to G2 .
Figure (4.16) - Sensitivity of boundary ug values with respect to G2 .
Figure (4.17) - Sensitivity of boundary ul values with respect to v2 .
Figure (4.18) - Sensitivity of boundary u2 values with respect to v2 .
Figure (4.19) - Sensitivity of boundary ul values with respect to xc .
Figure (4.20) - Sensitivity of boundary 112 values with respect to xc .
Figure (4.21) - Sensitivity of boundary ul values with respect to yc .
Figure (4.22) - Sensitivity of boundary uz values with respect to yc .
Figure (4.23) - Sensitivity of boundary ul values with respect to Rc .
Figure (4.24) - Sensitivity of boundary uz values with respect to Rc .

Chapter 1

Introduction and Background

Material characterization is one of the fundamental tasks of engineering and
science. It involves the study of inverse problems which usually imply identification of
inputs from outputs. In this study, the feasibility of characterizing the internal structure
of an inhomogeneous body using a discrete number of surface measurements is
explored. In particular, we examine the problem of determining the location, size and
material properties of an inclusion in a body of arbitrary shape using a discrete
number of surface temperature and/or heat flux measurements and a discrete number
of surface displacement and/or traction measurements. The boundary element method

[1] is adapted for application to this parameter estimation [2] problem.

Other numerical techniques, most notably finite differences and finite elements ,
have been used to investigate various types of inverse problems. However , for the
nonlinear inverse problem investigated here, an iterative scheme is required . Thus the
boundary element method, which requires only the discretization of the boundary,

must be employed in order to avoid the costly and unnecessary task of grid generation

 

2

for the entire multiply connected domain at each iteration.

Application of the boundary element method to the inverse problem of material
characterization is not entirely new. Murai and Kagawa [3] investigated the estimation
of the shape of an inclusion in a two-dimensional region using impedance
measurements on the domain surface. They considered the problem simply as the
interface boundary determination between two domains of different (but known)
conductivities, governed by Laplace’s equation. The boundary element method was
used in conjunction with a simple linearized estimation scheme. The authors did not
address any of the questions regarding convergence, measurement errors, or inclusion

size.

Ohnaka and Uosaki [4] utilized surface temperature measurements to
simultaneously estimate the diffusion constant of a homogeneous body as well as
discrete unknown internal heat sources. The mathematical formulation of the
identification problem was presented using the weighted residual expression and the
boundary element partition. The unknowns were identified using noisy or noise-free
state observations taken at points on the boundary by minimizing a certain criterion
function which is the sum of the squares of the relative errors. The maximum number
of temperature measurements used was 16. Two cases were considered : ( 1) the
internal source location is close to the boundary and (2) the source location is close to
the middle of the body. One set of noisy observation data with a standard deviation of
0.1 was considered. It was revealed that the accuracy of the identification results is
lower when the actual heat source is located close to the boundary. The influence of
the "initial guess" of the unknown parameters was not investigated, nor was the

question of measurement point selection.

Dulikravich [5] investigated the optimal sizes and locations of coolant flow
passages for a user-specified steady distribution of surface temperatures and heat

fluxes. The Laplace equation for steady conduction was treated using the boundary

 

element method. If the temperatures on the boundary were given as the boundary
condition, then an error function based on a normalized least squares formulation for
the difference between the desired and computed surface heat fluxes was formed and if
the heat fluxes on the boundary were given as the boundary condition, an error
function based on a normalized least squares formulation for the difference between
the desired and computed surface temperatures was formed. This function was then
used in a constrained optimization routine to determine the new updated sizes and
locations of the coolant flow passages so that the difference between the desired and
the computed surface heat fluxes and/or temperatures was minimized. As in the other

investigations, questions regarding the convergence process were not addressed.

Kishimoto, et. a1. [6] investigated inverse problems in galvanic corrosion. A
boundary element procedure was developed to estimate the densities across the anods
using potential values which were assumed to be known at several points in the
electrolyte. An approach associated with the single value decomposition of the
coefficient matrix and a criterion for determining its effective rank was presented. It
was concluded that the effective rank should be determined so that both the coefficient

matrix’s condition number and the square sum of the residuals

Q = Z ( ¢inappl - ¢incomp )2

took small values where ¢inappl represent applied potential values and ¢incomp the
values computed using the boundary element method. The authors assumed that the
potential values used as extra information were given in advance. Therefore they did
not address the errors involved if the potentials were to be measured experimentally.
The effect of the initial guesses of the densities on the estimation process and also the
question of how many potential values are needed and the potential corresponding to

what location points in the electrolyte are better to use, were not addressed.

 

 

4

Tanaka and Yamagiwa [7] estimated the shape of an internal flaw using
elastodynamics with given eigenfrequencies. First the eigenfrequencies corresponding
to the assumed defect shape were computed using a boundary-domain element method,
i.e. discretization of the domain was also necessary. Then the boundary integral
equations were solved using the boundary element method to find the displacements
and tractions corresponding to the assumed defect shape. This procedure was repeated

until

8w=(mE—mA) ——>O

where (0E and (0A were the eigenfrequencies corresponding to the exact and assumed
defect shape, respectively. It was revealed that the greater the number of additional
data (eigenfrequencies), the closer and faster the exact defect shape would be obtained.
The authors did not address the effect of the initial shape and size of the assumed flaw
on the ability to estimate it. Although the boundary element was used in this
investigation, the additional information, i.e. the eigenfrequencies, were not measured
on the surface boundary, but were computed using a boundary—domain element
method, at the interface. Therefore, the authors did not address the measurement errors
if the eigenfrequencies were measured experimentally. Also the question of how many
eigenfrequencies are needed, and what are the optimum locations at which to compute

them was not addressed.

Gao and Mura [8] utilized surface displacement data to evaluate the residual
stress field in the vicinity of a damaged region caused by a series of unknown
loadings. A relation between the residual surface displacements and plastic strains was
found. The residual surface displacements were relative and were defined as the
difference between before and after loading. Plastic strains were determined using the

measured residual surface displacement data and then the stress field on the boundary

5

and outside the damaged area were computed. It was shown that the equivalent plastic
strains, though different from the actual ones, induced the actual stresses outside of the

equivalent damage domain.

Das and Mitra [9] found the location and size of a flaw using measured surface
temperatures. An algorithm for the detection of the flaw was employed in solving
several exmple problems. In all the calculations, the boundary element solution for the
real flaw was used as the experimental data. It was observed that for a satisfactory
detection of the flaw, the error in the measured temperatures should be less than

0.25% .

In the heat transfer portion of this study, a body of some arbitrary but given
shape is subjected to a steady thermal state, i.e. a specified temperature is applied to
one portion of the surface and a specified heat flux is applied to the remainder of the
surface. The body is assumed to contain an inclusion of circular shape but unknown
location, size and thermal conductivity. The simultaneous estimation of these
parameters is to be accomplished by measuring temperatures at surface locations
where the flux has been specified and/or measuring fluxes at selected surface locations

where temperature has been specified.

In the elasticity portion, a body of some arbitrary but given shape is subjected to
uniaxial tension. The body is assumed to contain an inclusion of circular shape but
unknown location, size, Poisson’s ratio and shear modulus. The simultaneous
estimation of these parameters is to be accomplished by measuring displacements at
surface locations where the traction has been specified and/or measuring tractions at

selected surface locations where displacement has been specified.

Several questions arise in this inverse application of the boundary element
method which have not been adequately addressed in previous investigations. Among
these are ( 1) does the "initial guess" for the unknown parameters have an effect on the

convergence to the correct parameters and, if so, what is this effect, (2) how many

surface measurements and what combination of measurements are required to
simultaneously estimate the unknown parameters, (3) how should the locations of these
surface measurement points be selected, (4) what effect will the inevitable errors in
experimental measurements have on the ability to estimate the sought parameters, and

(5) what effect does the size of the inclusion have on the estimation process?

The present investigation assumes a two-dimensional body containing a single
inclusion, but the method of analysis is not limited in principle. Extension to three-
dimensional problems and more complex internal structure is straightforward but the
additional question of how many parameters one can hope to determine simultaneously
requires further study. Nonetheless, the success of the technique demonstrated here

shows great promise for application in the area of non-destructive evaluation.

 

Chapter 2

Boundary Element Method

2.1) Heat Transfer

The differential equation representing steady state heat transfer through an

isotropic, homogeneous, two-dimensional region, Q , is

kV2T=0 in Q (2.1.1)

where T(x1 , x2) is the temperature, k is the thermal conductivity of the material, and

 

 

82 82

2..

V_8x2 8x2.
1 2

The boundary conditions can be written as

T = T0 (8) on ra (2.1.2)
at
q = k a_n = qo (s) on I], (2.1.3)

where T0(s) and q0(s) are specified functions and the boundary of region £2 is given as
F = l‘a+ Pb, 11 is the outward normal direction to I‘, q is the heat flux in the n

7

 

8

direction and s is a coordinate along the boundary as shown in Figure (2.1). Consider
now a weighting function W( x1 , x2 ) which is assumed to be sufficiently
differentiable. Multiplying equation (2.1.1) by W( x1 , x2) and integrating by parts

twice,yields
]TkVZdeldx2+]Wk—ds-jk—Tds=0. (2.1.4)
Q I‘

In order to convert equation (2.1.4) into a boundary integral equation, a weighting
function which satisfies the Laplace equation and represents the field generated by a
unit point source acting at point (x1i , xzi) is used. The governing equation

representing this field is written as
kV2T*=—5(x1—x1i,x2-x2i) (2.1.5)

where 5 (x1— xli , x2 — xzi) is the Dirac delta function. The solution to equation

(2.1.5) is called the fundamental solution and is given by

T— =Et-1-(r( ln—l- ) (2.1.6)

where
1/2
- 2 - 2
r = [ (xl-xll) +(x2-x2‘) ]

Let us choose W=T* ( x1 , x2 , xli , xzi ) and define,

 

 

Figure (2.1) - Plane Inhomogeneous Body .

10

 

* = kaT (2.1.7)
an
Then eq (2.1.4) at a point (x1i , xzi) in (2 becomes
T(X1i , xzi) = j q T" ds - i T q* ds (2.1.8)
I‘ I“
where
q' = ( plnl + 92112 ) , (2.1.9)

_ 21tk

ml and n2 are the components of the outward directed unit normal to I‘, and

i
- Xl—Xl

Pr-

 

I'

Xz‘x2l

P2-

 

1'

Inserting equations (2.1.6) and (2.1.9) into equation (2.1.8), for (in , xzi) in Q yields

. . 11+ 11
T(x1‘,x2‘)=—-1— iT 911922

1 1
d+— l—d . 2.1.10
2” r r s k[camp s < >

Equation (2.1.10) is also applicable as (x1i , xzi) goes to the boundary F , but as the

 

11
integration variables (x1 , x2) go to (x1i , xzi), i.e. r goes to zero, the integrands of

equation (2.1.10) are singular. This is not a problem for the integral containing 1n(%)

since this function is integrable but it is a problem for the integral containing % . To

avoid this singularity, we integrate around singular point (x1i , xzi) as shown in Figure

(2.2). Thus

where

11+ n . . .
ling[Tag—idsqzn—mwnxlule) (2.1.12)
E—) re

and equation (2.1.11) at (x1i , xzi) on 1‘ becomes

911114132112

11+ 11 . .
ITELIT‘D—z-lds=(2n—w’)T1+JT——r——ds (21.13)
F

F

where the integral on the right hand side is interpreted in the Cauchy principle value
sense and (Di = to ( xli , xzi ) is equal to 1: if the boundary is smooth at ( xli , xzi ) .

Inserting equation (2.1.13) into (2.1.10), yields

 

12

 

 

 

T

Figure (2.2) Integration path around singular point.

 

13

. 11+ 11
‘T‘—J‘T 911922

1 1
=— 1 — d 2.1.14
r ds kiln n(r) s ( )

for (x1i , xzi) on r and Ti = T(x1i, xzi) . Equations (2.1.14) are the "boundary-integral
equations" [1]. Since either T or q is specified at each point on I‘, i.e. T is specified on
portion I‘a and q is specified on portion I’b, equations (2.1.14) are employed to

determine T on Pb and q on I’a .

Subdividing the boundary I’ into N segments, equations (2.1.14) become

T —— = —1- £1 [q1n(i) ds . (2.1.15)
. k P r
5:1

|—o

To employ linear isoparametric elements, let :

s = 431 30-1) + 4,2 30')
= ‘91 X104) + 92 X16)
x2 = 4)] x20-1>+ 4:2 x20) (2.1.16)
T = 91 T0-1)+ $2 '16)

on element j, where

 

14

¢1=(1-€)/2 (2.1.17)
¢2=(1+§)/2
are linear shape functions and -1 S E S 1.

Then
ds= __1_ S(j-1)+i 3(1) (1g: hag (2.1.18)
2 2 2

where lj is the length of the element j. Note that the temperature is assumed to be
piecewise linear and continuous whereas heat flux is assumed to be piecewise linear

and discontinuous. Equations (2.1.15) can now be written in the form

1

2 q<2i—2+k> ] gkiidt (2.1.19)
k=1 —1

M2

1
O O N 2 I I.
w‘ T‘ — 2 2 TM) 1 hk‘1d§=
—1

.1.
j=1 k=1 k '

he
ll
3...;

where F0) is taken to be 100 and hkij and gkij are known functions. Equations (2.1.19)

are written in matrix form as :

[Hm 14]} (2......)

where [ H ] has dimension NxN, and [ G ] has dimension Nx2N. The column matrix

{T} contains the values of temperature at the N boundary nodes and column matrix

 

15

{q} contains two values of flux at each boundary node, i.e. the value of flux "before"

the node and the value of flux "after" the node.

For an inhomogeneous body, the domain 9 is divided into two subdomains
S21 and 92 , each having its own thermal conductivity and equations (2.1.20) are

applied to each subdomain. For two subdomains :

OI'

[G11 [0]

V: [01 [612%
T2

Equations (2.1.21) are then reduced by imposition of interface conditions on F1 , the

[H11 [0]

[O] [H] 2 > (2.1.21)

 

 

 

 

r . A H. m
BY—J

 

 

 

 

interface of the two subdomains. At a point i on F1 :

T1i = T; (2.1.22)

ql(21-1) = Char) = _ q2(Zi—1) = _ q2(2i)

where continuity of heat flux at the interface nodes is assumed. Imposition of (2.1.22)

gives us:

16

Wt}

* (2.1.23)
(1} I

r H. H. \
H_J g?)
ll
r—*—fi

[H‘l1

 

 

 

 

where the subscript " I " denotes the interface boundary and the subscript " 1-I "
denotes the outer boundary. The matrix [I-F] is (N1+ 2N2) x(N1+ N2) and
[0*] is (2N1 + 4N2 ) x ( 2N1 + N2 ), where N1 is the number of outer boundary
nodes and N2 is the number of interface boundary nodes. Equations (2.1.23) can be

written as :

[H"]t

FLW

> = [0"] {q} H (2.1.24)

 

 

*4!

where the matrix [H*"'] is (N1+2N2)x(N1+2N2) and [G 1 is
(2N1+4N2)x(2N1). Equations (2.1.24) are reordered based on known outer
boundary conditions i.e. all the known outer boundary temperatures in {T} 1-1 are

taken to the right hand side and all unknown outer boundary heat fluxes in {q}1_I to

the left hand side. Then

[A] H -[B] a...)

 

17

which is solved for { X }, which contains the values of unknown temperatures and
heat fluxes at the outer boundary nodes as well as all temperatures and heat fluxes at

the interface boundary nodes .

Let us consider the example shown in Figure (2.3). A rectangular body with a
circular inclusion of radius 2 located at (3,3) is subjected to the boundary conditions
shown. The coefficient of thermal conductivity of the matrix material is 164 and that
of the inclusion is 73. The outer boundary and the interface boundary are each
divided into 24 linear elements of equal length as shown in Figure (2.4).The unknown

nodal temperatures and heat fluxes are computed and are presented in Table (2.1).

 

 

\\\\\\\\\\\\\\\

 

 

r————.——a

Figure (2.3) - Heat Transfer Example Problem.

 

24 23 22 21 20

 

 

 

 

 

Figure (2.4) - Boundary Element Model .

l3

20

Table (2.1) Nodal Temperature and Heat Flux Values for Heat Transfer
Example Problem.

 

 

 

 

node # x-coord. y-coord. Temperature Flux "before" Flux "after"
1 0.00 6.00 0.00 -88.64 30.00
2 0.00 5.00 0.33 30.00 30.00
3 0.00 4.00 0.52 30.00 30.00
4 0.00 3.00 0.60 30.00 30.00
5 0.00 2.00 0.52 30.00 30.00
6 0.00 1.00 0.33 30.00 30.00
7 0.00 0.00 0.00 30.00 -88.64
8 1.00 0.00 0.00 -25.51 25.51
9 2.00 0.00 0.00 -11.56 -11.56
10 3.00 0.00 0.00 2.41 2.41
11 4.00 0.00 0.00 -0.14 -O.14
12 0.00 5.00 0.00 -6.76 -6.76
13 0.00 6.00 0.00 -9.06 0.00
14 6.00 1.00 0.06 0.00 0.00
15 6.00 2.00 0.12 0.00 0.00
16 6.00 3.00 0.16 0.00 0.00
17 6.00 4.00 0.12 0.00 0.00
18 6.00 5 .00 0.06 0.00 0.00
19 6.00 6.00 0.00 0.00 -9.06
20 5 .00 6.00 0.00 -6.76 -6.76
21 4.00 6.00 0.00 -0. 14 -0. 14
22 3.00 6.00 0.00 2.41 2.41
23 2.00 6.00 0.00 -11.56 -11.56
24 1.00 6.00 0.00 -25.51 -25.51
T25 3.00 5 .00 -0.03 - 10.81 -10.81
26 3.52 4.93 -0.03 - 10.31 -10.31
27 4.00 4.73 0.00 -9.02 -9.02
28 4.41 4.41 0.05 -4.96 -4.96
29 4.73 4.00 0.13 1.67 1.67
30 4.93 3.52 0.19 8.17 8.17
31 5 .00 3.00 0.22 1 0.99 10.99
32 4.93 2.48 0.19 8.17 8.17
33 4.73 2.00 0.13 1.67 1.67
34 4.41 1.59 0.05 -4.96 -4.96
35 4.00 1.27 0.00 -9.02 -9.02
36 3.52 1.07 -0.03 -10.31 -10.31
37 3.00 1.00 -0.03 -10.81 -10.81
38 2.48 1.07 0.00 -11.45 -11.45
39 2.00 1.27 0.08 -9.64 -9.64
40 1.59 1.59 0.02 -2.30 -2.30
41 1.27 2.00 0.34 9.64 9.64
42 1.07 2.48 0.47 20.80 20.80
43 1.00 3.00 0.52 25.42 25 .42
44 1.07 3 .52 0.47 20.80 20.80
45 1.27 4.00 0.34 9.64 9.64
46 1.59 4.41 0.20 -2.30 -2.30
47 2.00 4.73 0.08 -9.64 -9.64
48 2.48 4.93 0.00 -11.45 -11.45

 

 

21

2.2) Elasticity

A linear elastic solid of uniform thickness h, and loaded in some manner, is

considered. The equations of equilibrium are :

 

 

 

 

8011 8012

8x1 3X2 — 0 (2.2.1)
8012 8022 _ 0

8X1 8X2 _

where on, 0'22, 0'12 are the in-plane components of stress and it is assumed there are

no body forces. The boundary conditions are :

ui = f, (s) on 1"a (2.2.2)

ti = Uij 11] h = g (S) 011 Pb (2.2.3)

for i=1,2 and j=1,2, where ui is the component of displacement in the 1 direction, ti is
the component of traction in the i direction, ml and n2 are the components of the
outward-directed unit normal to the boundary I“, s is a coordinate along the boundary
as shown in Figure (2.1), and fi (5) and gi (s) are prescribed functions. To express
equation (2.2.1) and equation (2.2.3) in terms of displacements, Hooke’s law is used,

i.e.

Bul an,

—— — . .4
8x1 + C12 8x2 (2 2 )

011: C11

 

22

8111 61.12
022 = C125; + C223);

-C aul 8112
012- sad-3245;?)

where C11, C22, C12, and C33 are material constants. Substituting equations (2.2.4) into

equations (2.2.1) , the equations of equilibrium are obtained in terms of displacements,

a aul 81.12 a 8112 aul _
(C33 8x1 + C33 8x2 ) — 0 (2.25)

a., (Cue—x1 + 0125; > t a:

a 81.12 aul a aul 8112

a—x1(033‘5;1‘ + 0333;; ) + $50125; + 0225;; ) = 0

and substituting equations (2.2.4) into equations (2.2.3), the tractions in terms of

displacements are obtained,

Bul auz auz aul
t1 = (C115;- + C125“; )nl + (C33a—x1' + (335;; )112 (2.26)
8112 aul Bu allz

l
{=(C —+C—)n+(C —+C —)n
2 33 8X1 33 8X2 1 12 8X1 22 8X2 2

where cij is equal to Cijh . Consider now the weighting functions W1( x1 , x2) and
W2( x1 , x2 ) which are assumed to be sufficiently differentiable. Multiplying the first
of equations (2.2.5) by W1( x1 , x2 ) and the second by W2( x1 , x2 ), and integrating

by parts twice yields

 

 

23

 

5;(Cna—xl+ciza—x2‘)+ —(°3sa——1-x +°33ax12) 1 2 (-~)

 

8X2
3W2 8W1 a 6W1 a 2
+ J; 114—371— (Cm—ax + C33‘“. 8x2 ) ax2(012 a 1 + c22 3x2 ) dxl dxz

 

 

"lg—‘-

aw. aw2 aw2 aw1
111(C11 ax1+ C12_ 8X2 )n1 '1' (C33a—X1_ “1" C33 3X2 )n2 (18

 

 

 

aw2 aw1 awl aw2
112(C33 ax1+ C33—8X2 )n1+(c12 ax1+ C22 3X2 )1’12 (18

+JW1t1ds+IW2t2ds=O
r 1'

where the two expressions have been added. For an isotropic material:

2Gh
011 = 022 = W (2.28)
_ 2v’Gh
C12- —__—7—'( 1 )
C33 = (111
where G is the shear modulus and
, v plane stress
v = v
plane strain

24
where v is Poisson’s ratio. For simplicity let us denote xi = ( xli , xzi ) as the
coordinates of the field point "i" and x = ( x1 , x2) as the coordinates of the source
point.
In order to convert equation (2.2.7) into a boundary integral equation, W1( x ) is

set equal to U1( x , Xi ) and W2( x ) is set equal to U2( x , xi ) such that

“67:5 ( x1 _ x1 , x2 — x21 ) e1( x1 ) (2.2.9)

+
4
N
.9
ll

 

 

 

 

where e1( xi ) and e2( xi) are unit vectors in the x1 and x2 directions,

5 ( x1 - xli , x2 — x2i ) is the Dirac delta function, and

 

 

 

 

 

2 2
V2 = a 2 + a 2
3X1 8X2
In addition
2 aUl Zv’ aU2 aU2 BUI
T =Gh — + —— + + — 2.2.10
1 1—vax1nl 1—v8x2n1 ax,nz ax,r12 ( )
8U2 BU1 Zv’ aUl 2 BUZ
T = Gh — + —-- + — + —
2 8X1 n1 8X2 n1 1 - V 8X1 n2 1 “ V 8X2 n2

 

 

are defined. Inserting equations (2.2.9) and (2.2.10) into equations (2.2.7), yields

25

ul( xi )e1( xi ) + u2( xi )e2( xi ) = —jT1(x,xi)u1(x)ds (2.2.11)
1‘

 

—IT2(x,xi)u2(x)ds+IU1(x,xi)tl(x)ds+'fU2(x,xi)t2(x)ds
I' I" F

for ( xi ) in Q . Now we can write

U1( x , xi ) = U11( x , xi )e1( xi ) + U12( x , xi )e/2( xi) (2.2.12)
U2(x,xi)=U21(x,xi)e1(xi)+U22(x,xi)e2(xi)
T1(x,xi)=T11(x,xi)e1(xi)+T12(x,xi)e2(xi)

T2(x,xi)=T21(x,xi)e1(xi)+T22(x,xi)e2(xi)

where U11 and U21 are the displacements and T11 and T21 are the tractions at a point x
in the infinite plane caused by a unit force in the x1 direction applied at xi . Similarly
U12 and U22 and T12 and T22 are the displacements and tractions at a point x due to a

unit force in the x2 direction applied at x1 . Inserting equations (2.2.12) into equations

(2.2.11) and equating coefficients of e1( xi ) and e2( xi ) , respectively, yields
uj ( xi ) = — j Tkj ( x , xi ) uk(x) ds(x) + ] Ukj ( x , xi ) tk(x) ds(x) (2.2.13)
I“ I‘

where j=l,2 , k=l,2 and summation on k is implied. The Galerkin function, <D( x , xi ),

is introduced such that

26

1 + v’ 32(1’1 82(1’2
2 6x12 axlaX2

 

 

U1=V2¢1—

1 + v’ az(1’1 32%
+
2 axlaX2 8x22

 

U2=V2<I>2-

which is inserted into equations (2.2.9) to obtain

GhV2(V2CDj)=-5(XI—Kliexz-Xzi)ej(xi)

The solution to equations (2.2.15) is

 

where
1/2
' 2 ' 2
r = [ (xl—xl‘) +(x2—x2‘) ] .

Substitution of equations (2.2.16) into equations (2.2.14) yields

 

1
U:—
1 81tGh{

 

j=1,2.

(2.2.14)

(2.2.15)

(2.2.16)

(3 — v’) 1n % +( l + v’)p12]e1+( 1+ v’ )p1 p2 e2} (2.2.17)

27

 

(3-v’)ln%+(1+v’)pz2 62+(1+v’)pl p261}

 

1
U2 " 87tGh {

where

x1 ‘ Xli

 

P1

Xz—Xz

 

P2 =
and substitution of equations (2.2.17) into equations (2.2.10) yields

1 , I /
T1 = 21:; {[2( 1+V )(n2 p23-n1 p13 )_( l-V )n1 p1_( 3+V )n2 p2]el

+[2(1+v’)(n2p13+n1p23)-(3+v')n2p1—(l+3v’)n1p2:|e2}

 

(2.2.18)

1 , , ,
T2=ZE{[2(1+V )(n2p13+n1p23)-(1+3v )nzpl-(3+v )n1p2]e1

+[2(1+V')(n1p13-n2p23)—(3+V')n1p1—(l—V')n2p2]62}.

The influence functions are determined by comparing equations (2.2.17) and equations

(2.2.18) to equations (2.2.12), i.e.

 

 

 

 

 

 

28

 

 

 

1 I 1 I 2
=— - 1— 1+

U11 81tGh (3 V)nr+( V)P1]
U = 1 (1+v’)p p
12 81tGh 1 2
U21=_81tGh (1+V')P192

U22: 1 (3-V’)1n—1-+(1+V')p22

81tGh r

 

and

1 , z I
T11=E[2(1+V)(H2P23—H1P13)—(1-V)H1P1—(3+V)n292]

L

T:
12 41tr

[2(1+V’)(n2013+n1923)-(3+V’)n2p1-(1+3V’)nlpz]

1 , , ,
T21=4—m—[2(1+v)(n2p13+n1p23)—(3+v)nzpl—(1+3v)nlpz]

_1_

T22: 4m

Recall that equations (2.2.13) are valid at any point xi in Q. They are also applicable

as xi goes to the boundary F but, as x goes to xi , i.e. r goes to zero, the integrands

are singular. Here Tkj varies as % and Ukj varies as In ( —:-) . The first of these

requires special treatment. As before, we integrate around xi and write

29

llji = —\iji uki "' J. Tkj Uk (18 + I Ukj tk dS (2.2.19)
1‘ I"

where uji = uj ( xi ) and

$3.} = 1130 11 Tkj ds

where e and I‘8 are shown in Figure (2.2). The first integral on the right hand side of
equations (2.2.19) is interpreted in the Cauchy principle value sense. Equations

(2.2.19) can be rewritten as
akji uki + J, Tkj 11k (18 = J Ukj tk dS Xi OI] F (2.220)
F T

where akji = Skj + ‘iji and 519- is the Kronecker delta. Equations (2.2.20) are the

"boundary integral equations" .

Subdividing the boundary I“ into N segments, equations (2.2.20) become

. . N . N .
akj‘ uk1+ Z Juk( x ) Tkj( x , xl ) ds = Z I tk( x ) Ukj( x , x1) ds . (2.2.21)
$1 F, r=l F,

To employ linear elements, we introduce

 

30
S = $1 5&4) + $2 Sm
X1 = 491 X104) + ¢2 X1“)
x2 = ()1 x2(“l) + ()2 x20) (2.2.22)
u1 = 431 1110-1) + $2 “1(r)
112 = $1 112“” + ¢2 112“)
t1 = 451 Hat—1) + 432 Mar)

... 2r-l 2r
‘2-¢1t2( )+¢212( )
on element r, where

¢1=(1—§)/2

¢2=(1+§)/2

are linear shape functions and —l S g S l . Note that the displacements are assumed
to be piecewise linear and continuous whereas tractions are assumed to be piecewise

linear and discontinuous. Equation (2.2.21) can now be written in the form

1 1
. . N 2 . N 2 ,
ak; uk1+ z z mgr-2+?) j hpur dg = z z tk<2r-2+P> j gpudE, (2.2.23)
r=l p=1 —-1 r=l pr=l —l

where ukm) is taken to be ukm and hpiI and gpif are known functions. Equations

(2.2.23) are written in matrix form as :

 

Wu}: [em (2.2.2..

where [ H ] has dimension 2Nx2N, and [ G ] has dimension 2Nx4N. The column
matrix {11} contains the values of displacements in the x1 and x2 directions at the N
boundary nodes and column matrix {t} contains four values of tractions at each
boundary node, i.e. the value of tractions in the x1 and x2 directions "before" the node

and the values of tractions in the x1 and x2 directions "after" the node.

As before, for an inhomogeneous body, the domain (2 is divided into two
subdomains £21 and £22 , each having its own Poisson’s ratio and shear modulus and

equations (2.2.24) are applied to each subdomain. For two subdomains :

OI'

[H11 [0]
[0] [H12

1

 

 

 

 

 

 

 

 

[011 [0]
* t. (2.2.25)
.}.

H = [01 [612%
112

Equations (2.2.25) are then reduced by imposition of interface conditions on F1 , the

interface of the two subdomains. At a point i on I] :

32
uli I1 = u,i I2 (2.2.26)
uzi II = uzi I2
t1(21—1) '1 = t1(2i) II = _ t1(21—1) l2 = _ t1(21) I2

t2(21—1) I1 = t2(2i) l1 : _ t2(21—1) I2 = _ t2(21) l2

where continuity of tractions at the interface nodes is assumed. Imposition of (2.2.26)

gives us:

‘

({u} H {I} H
[11*] 1 r = [6*] 1 * (2.2.27)
111 {l

J 5 J

 

 

 

 

where the subscript " I " denotes the interface boundary and the subscript ” l-I "
denotes the outer boundary. The matrix [ H” ] is ( 2N1 + 4N2 ) x ( 2N1 + 2N2 ) and
[ G,“ ] is ( 4N1 + 8N2 ) x ( 4N1 + 2N2 ), where N1 is the number of outer boundary
nodes and N2 is the number of interface boundary nodes. Equations (2.2.27) can be

written as :

 

_ [if] i{u}1 r = [6”] H14 (2.2.28)

 

 

where the matrix [H"“‘] is (2N1+ 4N2)x (2N1+ 4N2) and [6“] is
(4N1+ 8N2 ) x ( 4N1 ). Equations (2.2.28) are reordered based on known outer
boundary conditions i.e. all the known outer boundary displacements in {u}1_I are
taken to the right hand side and all unknown outer boundary tractions in {t}1_I to the

left hand side so that

[4M 43] (2.2.29.

which is solved for {X}, which contains the values of unknown displacements and
tractions at the outer boundary nodes as well as all displacements and tractions at the

interface boundary nodes .

Let us consider the examples shown in Figures (2.5), (2.6), and (2.7). A
rectangular body with a circular inclusion of radius 2 located at (3,3) is subjected to
the boundary conditions shown in each figure. The shear modulus and Poisson’s ratio
of the matrix material are 3.0 x 106 and 0.3 respectively, and those of the inclusion are
1.0 x 106 and 0.2 . The outer boundary and the interface boundary are each divided
into 24 linear elements of equal length as shown in Figure (2.4). The unknown nodal
displacements and tractions are computed and are presented in Tables (2.2), (2.3), and

(2.4).

34

 

 

 

 

 

$0 a
my

t=0
1

Figure (2.5) - Elasticity Example Problem #1.

35

Table (2.2) Nodal Displacement and Traction Values for Elasticity
Example Problem #1.

 

 

 

 

node # (X,y) 111 112 t1 t2 t1 t2
coord. "before" "before" "after" "after"

1 (0.00.6.00) 0.1 lE-04 0.1lE-02 0.00 1000.00 0.00 0.00

2 (0.00.5 .00) -0.22E-04 0.92E-03 0.00 0.00 0.00 0.00

3 (0.00.4.00) -0.99E-04 0.75E-03 0.00 0.00 0.00 0.00

4 (0.00.3 .00) -0.16E-03 0.52E-03 0.00 0.00 0.00 0.00

5 (0.00.200) -O.1 1E-03 O.29E-03 0.00 0.00 0.00 0.00

6 (0.00.1.00) -0.33E-04 0. 1215—03 0.00 0.00 0.00 0.00

7 (0.00.0.00) -0.73E-06 0.00E+00 0.00 0.00 0.00 -876.69

8 (1 00.000) ~0.35E-04 0.00E+00 0.00 -1142.70 0.00 -1 142.70

9 (2.00.0.00) -O.37Eo04 0.00E+00 0.00 -984.68 0.00 984.68
10 (3 .00.0.00) 0.00E+00 0.00E+00 0.00 -866.64 0.00 -866.64
1 1 (4.00.0.00) 0.37E-04 0.00E+00 0.00 -984.68 0.00 -984.68
12 (0.00.5 .00) 0.35E-04 0.00E+00 0.00 -1 142.70 0.00 -1 142.70
13 (0.00.6.00) 0.73E—06 0.00E+00 0.00 -876.69 0.00 0.00
14 (6.00.1 .00) 0.33E-04 0. 1215-03 0.00 0.00 0.00 0.00
15 (6.00.200) 0.11E—03 0.29E—03 0.00 0.00 0.00 0.00
16 (6003.00) 0. 16E-03 0.52E-03 0.00 0.00 0.00 0.00
17 (6.00.400) 0.99E~04 0.75E-03 0.00 0.00 0.00 0.00
18 (6.00.5.00) 0.22E.04 0.92E-03 0.00 0.00 0.00 0.00
19 (6.00.6.00) -0.1 1E-04 0.1 1E-02 0.00 0.00 0.00 1000.00
20 (5.00.6.00) 0.30E-04 0.10E-02 0.00 1000.00 0.00 1000.00
21 (4.00.6.00) 0.41E-04 0.1 113-02 0.00 1000.00 0.00 1000.00
22 (3 .OO.6.00) -0.30E-09 0.1 1E-02 0.00 1000.00 0.00 1000.00
23 (2.00.6.00) -0.41E~04 0.1 1E-02 0.00 1000.00 0.00 1000.00
24 ( 1 .00,6.00) -0.30E—04 0.10E~02 0.00 1000.00 0.00 1000.00
25 (3.00.5 .00) 08313-10 0.99E-03 0.00 675.65 0.00 675.65
26 (3 52.4.93) 0.35E-04 0.97E-03 56.35 637.87 56.35 637.87
27 (4.00.473) 0.72E—04 0.92E-03 1 18.08 542.15 118.08 54215
28 (4.41441) 0. 1213-03 0.84E-O3 194.65 414.03 194.65 414.03
29 (4.73.4 .00) 0.17E-03 0.74E-03 294.19 269.61 294.19 269.61
30 (4.93 ,3 .52) 0.22E-03 0.64E-03 406.34 124.97 406.34 124.97
31 (5.00.3.00) 0.25E-03 0.52E—03 469.11 - 10.77 469.11 -10.77
32 (4.93 .248) 0.23E-03 0.40E-03 422.61 - 144.76 422.61 -144.76
3 3 (4.73 .200) 0.19E-03 0.30E-03 317.40 -286.99 317.40 -286.99
34 (4.41.159) 0.14E.03 02015-03 227.22 429.93 227.22 429.93
35 (4.00.1.2?) 0.95E-04 0.13E-03 165.04 -543.00 165.04 -543.00
36 (3 .521 .07) 0.50E-04 0.89E-04 96.04 -60236 96.04 -60236
37 (3.00.1 .00) 0. 1215-10 0.74E-04 0.00 -616.90 0.00 -616.90
38 (2.48.1 .07) -0.50E-04 0.89E-04 -96.04 —60236 -96.04 -60236
39 (2.00.1.27) —0.95E-04 0. 1313-03 - 165 .04 -543.00 -165.04 -543.00
40 (1.59.1 .59) -O.14E-03 02015-03 -227.22 429.93 -227.22 429.93
4 1 (1.27.200) -0. l9E-O3 0.30Er03 -317.40 -286.99 -3 17.40 ~286.99
42 (1.07.248) -0.23E-03 0.40E-03 422.61 -144.76 422.61 -144.76
43 (1.00.3.00) -0.25E—03 0.52E-03 469.11 -10.77 469.11 - 10.77
44 (1.07.352) -0.22E-03 0.64E-03 406.34 124.97 406.34 124.97
45 (1.27.4.00) -O.17E-03 0.74E-03 -294. 19 269.61 -294. 19 269.61
46 (1.59.4.41) -0.12E-03 0845—03 -194.65 414.03 -194.65 414.03
47 (2.00.4.73) -0.72E-04 0.92E-03 -l 18.08 542.15 -1 18.08 542.15
48 (2.48.4.93) -O.35E-04 0.97E-03 -56.35 637.87 -56.35 637.87

 

 

-" “n.4, up-

36

 

 

 

 

t =1000
2
t =0
1
11
Q
1
t =0
2
t =0
1
. A .
V l l l J l l V
t =-1000
t =0

Figure (2.6) - Elasticity Example Problem #2.

37

Table (2.3) Nodal Displacement and Traction Values for Elasticity
Example Problem #2.

 

 

 

 

node # (x,y) 111 112 t1 t2 t1 t2
coord. "before" "before" "after" "after"

1" (0.00.6.00) 0.94E-05 0.11E-02 0.00 1000.00 0.00 0.00

2 (0.00.5 .00) -0.29E-04 0.95E-03 0.00 0.00 0.00 0.00

3 (0.00.4.00) -0.11E-03 0.79E-03 0.00 0.00 0.00 0.00

4 (0.00.3 .00) -0.17E-03 05513-03 0.00 0.00 0.00 0.00

5 (0.00.2.00) -0. 11E~03 0.32E-03 0.00 0.00 0.00 0.00

6 (0.00.1.00) -O.29E-04 0.15E-03 0.00 0.00 0.00 0.00

7 (0.00.0.00) 09413.05 0.27E-04 0.00 0.00 0.00 -1000.00

8 (1.00.0.00) —0.31E-04 0.46E-04 0.00 -1000.00 0.00 - 1000.00

9 (2.00.0.00) -0.41E-04 0.23E-04 0.00 - 1000.00 0.00 -1000.00
10 (3 00.0.00) 0.00E+OO 0.00E+OO 0.00 -1000.00 0.00 -1000.00_
1 1 (4 00.0.00) 0.41E-04 0.23E-04 0.00 -1000.00 0.00 -1000.00
12 (0.00.5 .00) 0.31E-04 0.46E-04 0.00 -1000.00 0.00 -1000.00
13 (0.00.6.00) —O.94E-OS 0.27E—04 0.00 -1000.00 0.00 0.00
14 (6.00.1 .00) 0.29E-04 0.16E-03 0.00 0.00 0.00 0.00
15 (6.00.2.00) 0.11E-03 0325-03 0.00 0.00 0.00 0.00
16 (6.00.3.00) 0. 1713-03 0.55E-03 0.00 0.00 0.00 0.00
17 (6.00.400) 0.11E-03 0.79E-03 0.00 0.00 0.00 0.00
18 (6.00.5 .00) 0.29E-04 0.95E-03 0.00 0.00 0.00 0.00
19 (6.00.6.00) -0.94E-05 0.11E-02 0.00 0.00 0.00 1000.00
20 (5.00.6.00) 0.31E-04 0.11E-02 0.00 1000.00 0.00 1000.00
21 (4.00.6.00) 0.41E-04 0.1 1E-02 0.00 1000.00 0.00 1000.00
22 (3 00.6.00) 0.00E+00 0.1 1E-02 0.00 1000.00 0.00 1000.00
23 (2.00.6.00) -0.41E-04 0.11E-02 0.00 1000.00 0.00 1000.00
24 (1.00.6.00) —0.31E-04 0.1 1E-02 0.00 1000.00 0.00 1000.00
25 (3 00.5.00) 030E08 0.10E-02 0.00 669.62 0.00 669.62
26 (3 524.93) 0.38E-04 0.10E-02 60.68 632.64 60.68 632.64
27 (4.00.4.73) 0.79E-04 0.94E-03 126.79 538.56 126.79 538.56
28 (4414.41) 0. 13E-03 0.87E—03 208.39 412.49 208.39 412.49
29 (4.73.4.00) 0.18E-03 0.77E-03 313.39 271.23 313.39 271.23
30 (4.93.3.52) 0.23E-03 0.67E-03 427.85 13 1.19 427.85 13 1. 19
31 (5.00.3.00) 0.26E-03 0.55E-03 485.14 0.00 485.14 0.00
32 (4.93 .248) 0.23E-03 0.44E-03 427.85 -131.19 427.85 -131.19
33 (4.73 .200) 0. 1813-03 03315-03 313.39 -271.23 313.39 -271.23
34 (4.41.1.59) 0.13E-03 0.25E-03 208.39 412.49 208.39 412.49
35 (4.00.1.27) 0.79Eo04 0.161303 126.79 -538.56 126.79 -S38.56
36 (3.52.1.07) 0.38E-04 0.1 1Er03 60.68 632.64 60.68 -632.64
37 (3.00.1 .00) 0.30E-08 0.86E-04 0.00 669.62 0.00 669.62
38 (2.48.1 .07) 038E—04 0.1 1E-03 -60.68 -632.64 -60.68 -63264
39 (2.00.1 .27) -0.79E-04 0.16E-03 -126.79 -538.56 -126.79 -538.56
40 (1.59.1.59) —0.13E-03 0.24E-03 -208.39 412.49 -208.39 412.49
41 (1.27.200) -0.18E03 0.33E—03 -3 13.39 -271.23 -313.39 -271.23
42 (1.07.2.48) -0.23E-03 0.441303 427 .85 -131.19 427.85 -131.19
43 (1.00.3 .00) -0.26E-03 0.55E-03 485.14 0.00 485. 14 0.00
44 (1.07.3.52) —O.23E-03 0.67E-03 427.85 131.19 427.85 131.19
45 (1.27.4.00) -0. l8E-03 0.77E-03 313.39 271.23 -313.39 271.23
46 (1.59.4.41) -0.13E-03 0.871303 208.39 412.49 -208.39 412.49
47 (2.00.4.73) -0.79E-04 0.94E-03 -126.79 538.56 -126.79 538.56
48 (2.48 .493) -O.38E-04 0.10E-02 -60.68 632.64 -60.68 632.64

 

 

"“"-IIWQ-u

38

 

 

 

 

u =0
11 =0.00l
Q
1
1 =0
2
t =0
1
u =0
u =0
2

Figure (2.7) - Elasticity Example Problem #3.

 

39

Table (2.4) Nodal Displacement and Traction Values for Elasticity
Example Problem #3.

 

 

 

 

node # (x,y) “1 u2 tl t2 t1 12
coord. "before" "before" "after" "after"

1 (0.00.6.00) 0.00E+00 0.10E-02 -219.37 658.79 0.00 0.00

2 (0.00.5 .00) -0.36E-04 0.91E-03 0.00 0.00 0.00 0.00

3 (0.00.4.00) -0.16E-03 0.75E-03 0.00 0.00 0.00 0.00

4 (0.00.3 .00) -0.24E-03 0.50E-03 0.00 0.00 0.00 0.00

5 (0.00.200) -0. 1613-03 0.25E—03 0.00 0.00 0.00 0.00

6 (0.00.1 .00) -0.36E-04 0.93E-04 0.00 0.00 0.00 0.00

7 (0.00.0.00) 0.00E+00 0.00E+00 0.00 0.00 -219.37 -658.79

8 (1.00.0.00) 0.00E+00 0.00E+00 172.16 -1081 .03 172.16 - 1081 .03

9 (2.00.0.00) 0.00E+00 0.00E+00 191.67 - 1059.01 191.67 -1059.01
10 (3 00.000) 0.00E+00 0.00E+00 0.00 -961.46 0.00 -961.46 .
1 1 (4.00.0.00) 0.00E+00 0.00E+00 -191.67 -1059.01 - 19 1.67 -1059.01
12 (0.00.5 .00) 0.00E+00 0.00E+00 -172.16 -1081.03 -172.16 - 1081.03
13 (0.00.6.00) 0.00E+00 0.00E+00 219.37 -658.79 0.00 0.00
14 (6.00.1.00) 0.36E-04 0.93E-04 0.00 0.00 0.00 0.00
15 (6.00.200) 0.16E-03 0.25E-03 0.00 0.00 0.00 0.00
16 (6.00.300) 0.24E-03 0.50E-03 0.00 0.00 0.00 0.00
17 (6.00.4 .00) 0.16E-03 0.75E-03 0.00 0.00 0.00 0.00
18 (6.00.5.00) 0.36E-04 0.91E-03 0.00 0.00 0.00 0.00
19 (6.00.6.00) 0.00E+00 0.101502 0.00 0.00 219.37 658.79
20 (5 00.6.00) 0.00E+00 0.10E-02 172.16 -1081.03 172.16 - 1081.00
21 (4.00.6.00) 0.00E+00 0.10E-02 191.67 -1059.01 191.67 ~1059.01
22 (3.00.6.00) 0.00E+00 0.10E-02 0.00 -961.46 0.00 -961.46
23 (2.00.6.00) 0.00E+00 0.10E-02 -191.67 -1059.01 -l91.67 -1059.01
24 (1.00.6.00) 0.00E+00 0.10E-02 -172.16 -1081.03 -172.16 ~108103
25 (3.00.5.00) 0.69E-11 0.921303 0.00 620.83 0.00 620.83
26 (3.52.4.93) 0.45E-04 0.90E-03 69.23 598.37 69.23 598.37
27 (4.00.4.73) 0.96E-04 0.861503 137.63 523.43 137.63 523.43
28 (4.41.4.41) 0.16E-03 0.79E-03 221.16 393.49 221.16 393.49
29 (4.73.4.00) 0.23E-03 0.711303 355.67 240.15 355 .67 240.15
30 (4.93 .352) 0.31E-03 0.61E-03 523.44 106.42 523 .44 106.42
3 1 (5.00.3.00) 0.34E-03 0.50E-03 609.45 0.00 609.45 0.00
32 (4.93 .248) 0.31E-03 0.40E-03 523 .44 -106.42 523 .44 -106.42
33 (4.73 .200) 0.23E-03 0.30E-03 355 .67 -240. 15 355.67 -240.15
34 (4.41.1.59) 0.16E-03 0.21E~03 221.16 -393.49 221.16 -393.49
35 (4.00.1 .27) 0.96E-04 0.14E-03 137.63 -523.43 137.63 -523 .43
36 (3.52.1.07) 0.45304 0.98E—04 6923 -598.37 69.23 -598.37
37 (3.00.1.00) 0.37E-11 0.84E-04 0.00 -620.83 0.00 -620.83
38 (2.48.1 .07) -0.45E-04 0.98E-04 -69.23 -598.37 -69.23 -598.37
39 (2.00.1 .27) 0.961504 0. 1413-03 -137.63 -523.43 -137.63 -523 .43
4O (1.59.1.59) —0.16E-03 0211303 -221. 16 -393.49 -221.16 -393.49
4 1 (1.27.200) 0.23E-03 0.30E—03 -355.67 -240. 15 -355.67 -240.15
42 (1.07.2.48) ~0.31E-03 0.40E-03 -523 .44 -106.42 -523.44 -106.42
43 (1.00.3.00) -0.34E-03 0.50E-03 -609.45 0.00 -609.45 0.00
44 (107.3 .52) -0.31E-03 0.61E-03 -523.44 106.42 -523 .44 106.42
45 (1.27.400) -0.23E-03 0.71E-03 -355.67 240.15 -355.67 240.15
46 (1.59.441) -O.l6E-O3 0.79E-03 -221.16 393.49 -221.16 393.49
47 (2.00.4.73) -0.96E-04 0.86E-03 -137.63 523.43 -137.63 523 .43
48 (2.48.493) -0.45E-04 0.90E-03 -69.23 598.37 -69.23 598.37

 

 

 

 

Chapter 3

Inverse Problem and Parameter Estimation

In the previous chapter the use of the boundary element method to solve the
direct steady state heat transfer and linear elasticity problems for homogeneous and
inhomogeneous bodies was demonstrated. In the direct problems, the governing
equation, geometry. material pr0perties and the boundary conditions are given and the
unknnown boundary data is computed. In the inverse problem some of the geometric
features and/or material properties are unknown. but some of the unknown boundary
data can be measured and used as additional information necessary to estimate the

unknown input parameters.

3.1) Heat Transfer

Let us investigate the simplest case of estimating one parameter, i.e. the thermal
conductivity of the circular inclusion. 92, using temperatures measured on the outer
boundary I‘b. Let Yi denote the measured boundary temperatures, Ti denote the
boundary temperatures corresponding to a guess of the unknown parameter. computed
using the boundary element method, and k2 the sought thermal conductivity of the

inclusion. The sum of the squared differences between Yi and Ti is

z: i (vi—T.)2 (3.1.1)

1:1

40

 

41

where n is the number of boundary temperature measurements used to estimate the
unknown parameter. In order to find the best estimate of k2. Z is minimized by setting

its derivative with respect to k2 equal to zero. Note that

dZ _ n dTi( k2)
m-—2i=ZI[—akg_][Yi—Ti(k2)] . (3-1-2)
Let us define
dT- k
Xi(k2)= ——‘( 2) (3.1.3)

dkz

where Xi are called the sensitivity coefficients. Since temperature is a nonlinear
function of k2, Xi will be nonlinear functions of k2. Inserting equations (3.1.3) into

equations (3.1.2) and setting it equal to zero, yields
i=1

which gives the value of k2 at which Z is minimized. Note that equation (3.1.4) is
nonlinear in k2. Suppose T has a continuous first derivative in k2. Then using the first

two terms of a Taylor Series for T (k2) about 152 which is an estimate of k2. i.e.

- dT(lE2) -
T(k2)=T(k2)+T(k2—k2) (3-1-5)

2

and approximating X. as function of 122, i.e.

 

42

.. d'ri (122 )
X- k = ——...——— ,
1( 2 ) dkz
equation (3.1.4) becomes
n ..
in|:Yi-Ti —Xi(k2‘k2)]=0 (316)

i=1

where Xi and Ti are functions of E2. Note that now equations (3.1.6) are linear in k2 .

To indicate an iterative procedure let

an

k2 = R2 (M) , k2 = R2 (M+1) , T = T(M) ’ X = X(M)

so that equations (3.1.6) become

11
.2 Xi<M> [Y1 _ Ti (1%)]
12‘1““): 12‘“ > + F1 (3.1.7)

[xi wt

where iteration on M is required. It is possible to estimate thermal conductivity by

 

'M=’

l

y-a
ll

only one measurement of the temperature on the boundary. For this case. equation

(3.1.7) simplifies to

[Y—fim]
x(M)

 

12‘1““) = 12‘“) + (3.1.8)

To estimate k2 using heat flux measurements, the same procedure is used to obtain :

 

 

43

mam

 

122““): 12““ > + i=1 n 2 (3.1.9)
2 [Xi (1%)]
i=1
where
~ dCIi ( lE2 )
Xi( k2 ) = ———:—
dkz

and Qi denote the measured boundary heat fluxes and qi denote the boundary heat
fluxes corresponding to a guess of the unknown parameter. computed using the
boundary element method. For the case that only one boundary measured heat flux is

used to estimate k2. equation (3.1.9) simplifies to

(w)

1(2(M+1) = R20“) '1’" XM)

 

(3.1.10)

Next let us discuss the estimation of thermal conductivity, location, and size of
the inclusion shown in Figure (2.1) using temperatures measured on 1], and/or heat

fluxes measured on Fa .
Let us introduce the following column matrices :
{T}: = a column matrix containing m measured boundary temperatures,

{TL = a column matrix containing the same m boundary temperatures.

computed using the boundary element method,

 

 

44

{(116 = a column matrix containing n measured boundary heat fluxes.

{q}. = a column matrix containing the same It boundary heat fluxes.

computed using the boundary element method.

and

{B} =1 ’

 

 

which contains the thermal conductivity of the inclusion. k2. the coordinates of the

center of the circular inclusion, (xc , y.) . and the radius of the inclusion, R. .

The sum of the squared differences between measured and computed

temperatures and heat fluxes is :

Z={{T}.—{T}.}T{{T}.—(T}.}+ kl}.— (q}.}T {{q}. -{q}.} (3.1.11)

where the superscript T indicates the transpose of the mauix, and {T}c and {q} are
functions of {B} . In order to find the best estimate of the unknown parameters,
function Z is minimized by setting its derivative with respect to {0} equal to zero.

The matrix derivative of Z with respect to { B} is :

{8.}z=—2 2[{8 8.3}{1‘ TH{T}. —{T}.}—2[{a}3}{q). W -{q}.} (3.1.12)

 

 

45

where {83} is the matrix derivative operator,

 

 

 

 

 

Let us define

[ xT ] =[{BB}{T}. T] T (3.1.13)
1 Xq 1 =[{as}{q}. T] T

where [ XT ], which is mx4. and [ Xq] .which is nx4. are both functions of {[3}, and
are called sensitivity matrices. The "ij" components of the sensitivity matrices are
called the sensitivity coefficients. [ XT ] contains the sensitivities of temperatures with
respect to {B} and [ Xq ] contains the sensitivities of heat fluxes with respect to {[3}.
Since the temperatures and heat fluxes are nonlinear functions of {B} , the sensitivity
coefficients are also nonlinear functions of {B}. Inserting equations (3.1.13) into

equations (3.1.12) and setting {8.} z = 0, yields

4..

._ and ‘

 

 

46

[XT1T{{T1e‘iT}c} +[Xq]T{{q}e—{q}c} =0

(3.1.14)

which gives the value of {[3} at which z is minimized. Note that equations (3.1.14)

are nonlinear in {B}. Therefore the first two terms of a Taylor series in matrix form

for {Tic and {q}c about {B} are used. where {B} is an estimate of {B}, and

approximate [ XT ] and [ Xq ] as functions of {B}. i.e.

{T}. ((13)) ={T}. ({Bb + 1 XT] ({B} —{B}>
{q}.<{B}>={q}.<{B}>+1xq1((B}—(B}>.

Substituting equations (3.1.15) and (3.1.16) into equation (3.1.14), yields

(x. 1T

 

1T1. —iT}. —[XT1{{ B}—{B}}} +

 

1xq 1T

Note that now equations (3.1.17) are linear in {B} .

To indicate an iterative procedure we let :

131:113}(M) {B} ={ B}(M+1)

{T}e ={T}e(M) [XT1=[XT1(M)

{q}e -{q}c -[Xq]{{B}—{”}}} =0 .

(3.1.15)

(3.1.16)

(3.1.17)

 

 

47

{q}c ={q}.‘M’ 1X.1 1 =1Xq1<M>

and obtain four equations for the four parameters, i.e.

 

 

 

 

' ) (M+1) ' 1 (M)
k2 k2
XC x0

i * =1 * + (3.1.18)
ye ye
Re Re

 

where

1P1={1[XrlTleHMH[Xq]T[Xq]](M>}

Iterations stop when the following criteria are satisfied :

" M1 " M
[[311 + )_Bj( )l

lBj(M)l+ 81

 

< 5 forj=1.. . ,4 (3.1.19)

where 5 and 51 are small numbers. Here. following [2]. 8 is set equal to 10“, and

51 = 10_10 .

 

48

3.2) Elasticity

In this section. a body that is known to contain a circular inclusion but its
location, size and mechanical properties are unknown is considered. We wish to
determine these unknown parameters by using the measurements of displacements on
the portion of the boundary where tractions are prescribed and/or tractions on the

portion of the boundary where displacements are prescribed.

Let us introduce the following column matrices :

{‘1}c = a column matrix containing m measured boundary displacements.

{u}.

a column matrix containing the same m boundary displacements.

computed using the boundary element method.

{t}

C

{t}

C

a column matrix containing n measured boundary tractions,

a column matrix containing the same 11 boundary tractions,

computed using the boundary element method.

and

 

 

where G2 is the shear modulus of the inclusion. v2 is the Poisson’s ratio of the
inclusion, (x. . y.) are the coordinates of the center of the circular inclusion, and R. is

the radius of the inclusion.

 

49

Following the same procedure as in section 3.1. an equation is obtained to
estimate the five unknown parameters in {B} using the displacements measured on I],

and/or tractions measured on I}, i.e.

' ‘<M+1) ' ‘(M)
G2 Gz
V2 V2
1 Xe 1 =1 x. > + (3.2.1)
Ye yC
RC RC
1 J L J

 

 

 

 

[P]-1 [Xu ](M){ {u}: _{U}C(M)} +[ x[ ]<Ml{{t}e -{t}.<M)}}

 

where
1P1={[[XulT[Xu]](M)+[[x.]T[xt]1am}

Iterations stop when the criteria indicated in equations (3.1.19) are satisfied. Note that
estimation of the parameters using equations (3.1.18) and (3.2.1) involves computing

[ P ]‘1 which requires [ P ] to be non-singular.

 

50

3.3) Sensitivity Coefficients

Sensitivity coefficients are very important in the parameter estimation method.
They indicate the magnitude of change of the responses such as temperature, heat flux.
displacement. or traction due to a small change in the values of the unknown
parameters. There are some cases for which it is not possible to uniquely estimate all
parameters from the measurements, but it is possible to estimate certain functions or
ratios of the parameters. The sensitivity coefficients can provide insight into the cases
for which parameters can or cannot be estimated. Parameters can be estimated if the
sensitivity coeflicients over the range of the observations are not linearly

dependent [2]. Linear dependence occurs when

8T 8T 8T 8T-

c1———ak +c2— ax +c3— 8y +c4 8—.—Rl= -0 (3.3.1)

is true for all i measurements, and not all Ci ’8 equal zero. One way to check linear
dependence is to find the determinant of the matrix [ P ]. and see how close it is to
zero. In order to uniquely estimate all parameters in the parameter vector {B} , it is

necessary that the determinant of the matrix [ P ] not be equal to zero.

51

Finite differences are used to approximate the sensitivity coefficients, i.e.

 

8Ti = Ti ( 131 MB]. + 8 B.,..,B4) — Ti ( [’31 ,..,B-,..,B4)

 

 

 

X .. = _
( T )1] BBJ- 5 Bj
(X ).. = 8i = qi ( B1 ""Bj + 8 B'v'9B4) - qi ( B1 ,..,B',..,B4)
q 1] 8‘13.- 5 “Bj
(3.3.2)
( X ).. = 31: : ui ( 131 ,..,Bj + 5 Bj,...B5) - “i ( B1 ,n,“}'3jm'}'35)
u 1] 313]- 8 B].
(X. M = 8h ___ M131 ,..,t3.- + 5 131...,135)- t. ( Bi ,,_,ij,}35)

313: 531'

which are the sensitivities of the iLh temperature, ith heat flux, iLh displacement, and ith
traction with respect to the j‘h parameter. respectively. Note that equations (3.3.2) lead
to difficulty if the initial value of Bj is chosen to be zero or if it approaches zero.
There are two kinds of errors which contribute to inaccuracies in computing the
sensitivity coefficients. The first kind is the rounding error which is found when two
close values of temperature or heat flux are subtracted and the second kind is the
truncation error which is due to the inexact nature of the finite difference method. In
order to reduce the truncation error the difference 5 in equation (3.3.2) is made small.
However, if it is too small, then the rounding error becomes important. The central
difference method which is more accurate could also be used, but it requires twice as

many values of temperature to be calculated as does forward difference.

52

3.4) Coupling of the Boundary Element Method and Parameter Estimation Method

A computer program has been developed which employs the boundary element
method as a subroutine. A first guess of the unknown parameters is made and the
boundary element subroutine is called to compute the boundary data corresponding to
the guessed parameters. Then, using the parameter estimation technique, the computed
data at selected locations are compared with their measured values taken at the same
locations. and "corrected" values of the parameters are found. Using equations (3.1.18)
for the heat transfer problems or equations (3.2.1) for the elasticity problems the
unknown parameters are estimated. The iterations continue until the criteria indicated
in equations (3.1.19) are satisfied. Figure (3.1) shows the schematic representation of
the estimation method. Complete listings of the computer programs are given in

Appendix A and Appendix B.

53
Step 1.

 

Make the first guess of the unknown parameters.

 

 

 

 

A $1 Step 2.

 

Compute the temperatures and heat fluxes or displacements and
tractions corresponding to the guessed values of the parameters,
and calculate the sensitivity coefficients.

 

 

 

1 Step 3.

 

Find the correction in the values of the guessed parameters and
compute the new values of the parameters.

 

 

 

Step 4.

 

 

Check for the three groups. i.e. Yes
groups (1). (2), (3). which ___.) STOP

 

 

 

 

 

 

cause divergence. *
i No

 

 

Check if the criterion for Yes
convergence is satisfied. ___). STOP

 

 

 

 

 

 

{No

Go back to step 2, and
< use the new values of the
parameters.

 

 

 

 

 

 

* These three groups will be discussed in the next chapter.

Figure (3.1) Schematic Representation of The Estimation Process.

Chapter 4

Results and Discussion

4.1) Heat transfer

The first problem investigated is the estimation of only one parameter. i.e. the
thermal conductivity of the inclusion shown in Figure (2.3). by using one temperature
measured at a location on the portion of the outer boundary where heat flux has been
specified or one heat flux measured at a location on the portion of the outer boundary
where temperature has been specified. The exact value of the inclusion’s thermal

conductivity is equal to 73.

Table (4.1) shows the results when one temperature measured at different
locations on 1‘. is used and Table (4.2) shows the results when one heat flux measured

at different locations on F. is used to estimate k2 .

It is observed that it is possible to estimate the thermal conductivity of the
inclusion using only one measurement of temperature or heat flux. It is also observed
that the first guess of the parameter could be very far from the exact value and still
convergence is possible. Initial guesses of 1 and 100 are considered. With the initial
guess of k2 = 100, all cases converged. However for the initial guess of k2 = l, the
cases involving nodal locations 2. 3. 4. 5. 6, 8. 11, 21 and 24 diverged due to the fact
that the estimated thermal conductivity became negative during the estimation process.
Since the inclusion is assumed to be located at the center of the body. the heat fluxes
are symmetric about the x1 axis, and thus the heat fluxes measured at the nodal
locations 8. 9, 10. 11, and 12 give the same results as the nodal locations 20. 21, 22,
23. and 24. Figure (4.1) shows the sensitivity of the temperatures with respect to k2
and Figure (4.2) shows the sensitivity of heat fluxes with respect to k2. Note that

sensitivity coefficients are nonlinear functions of the unknown parameters and the plot

54

55

Table (4.1) Estimating k2 Using One Temperature Measurement

 

 

 

 

 

Case # Nodal Location Numbers Converged Diverged
Iteration # Due to
First Guess of k2 = 1.0

1 2 Unrealistic Value of k2
2 3 Unrealistic Value of kg
3 4 Unrealistic Value of k2
4 5 Unrealistic Value of k2
5 6 Unrealistic Value of k2
6 14 3

7 15 3

8 16 3

9 17 3

10 18 3

First Guess of k2 = 100.0

1 1 2 4

12 3 4

13 4 4

14 5 4

15 6 4

l6 l4 3

17 15 3

18 16 3

19 17 3
20 18 3

 

 

 

56

Table (4.2) Estimating k2 Using One Heat Flux Measurement.

 

 

 

 

 

 

Case # Nodal Location Numbers Converged Diverged
Iteration # Due to
First Guess of k2 = 1.0

1 8 Unrealistic Value of k2
2 9 4

3 10 6

4 11 Unrealistic Value of k2
5 12 2

6 20 2

7 21 Unrealistic Value of k2
8 22 6

9 23 4

10 24 Unrealistic Value of k2

First Guess of k2 = 100.0

1 1 8 4

12 9 3

13 10 4

14 l 1 4

15 12 3

16 20 3

17 21 4 .

18 22 4

19 23 3
20 24 4

 

 

57

of sensitivity coefficients presented here. Figure (4.1) through Figure (4.8).

correspond to the exact values of the parameters.

Again consider the problem shown in Figure (2.3). This time the location. the
radius. and the thermal conductivity of the circular inclusion are not known a priori
and the estimation of these four parameters simultaneously is desired by measuring (a)
at least four temperatures on the portion of the boundary where heat fluxes are
specified. (b) at least four heat fluxes on the portion of the boundary where
temperatures are specified, or (c) some combination of measured temperatures and heat

fluxes

The "experiment" is simulated as follows. The body is analyzed by the boundary
element method using the exact values of the four parameters and the unknown
boundary temperatures and heat fluxes are computed. These computed values are then

used as the "experimental" results.

The first question addressed is the influence of the initial guesses on convergence.
A total of 32 sets of initial guesses were examined and "experimental values" were
used for all boundary nodal points. Convergence was defined in accordance with
equations (3.1.19). Results for the 32 cases are given in Table (4.3). Note that 82.0%
of the cases converged to the correct values of the parameters whereas 18.0% of the
cases diverged. The reason for divergence becomes apparent after one or two iterations
and can be classified into one of three groups: (1) the determinant of [ P ] for an
iteration is zero; (2) the estimated values of (XO , y.) and Rc for an iteration are
unrealistic. i.e. the estimated inclusion does not lie entirely within the matrix domain;
(3) the estimated value of the inclusion thermal conductivity for an iteration is greater
than (less than) the thermal conductivity of the matrix material but the actual thermal
conductivity of the inclusion is less than (greater than) that of the matrix. All the
above three groups can be identified after one or two iterations and the program will

stop if any of the above situations occurs. Table (4.3) shows that 3.0% of the cases

58

diverged due to (1). 9.0% of the cases diverged due to (2). and 6.0% of the cases
diverged due to (3). Figure (4.1) through Figure (4.8) show the sensitivities of
temperatures and heat fluxes with respect to the four parameters. Figure (4.1) and
Figure (4.7) show that the sensitivities of temperatures with respect to k2 and R. are
very similar in shape which indicates linear dependence of the columns corresponding
to the above parameters in the [P] matrix. Similarly Figure (4.2) and Figure
(4.7) reveal the same observation about the sensitivities of the heat fluxes with respect
to k2 and R.. This is the reason that most of the cases in Table (4.3) which diverged
were because k2 or R. did not converge and went in the wrong direction until the
condition indicated by groups (2) or (3) were observed and the program stopped.
Figures (4.3). (4.4), (4.5), (4.6) show that the sensitivities of temperatures and heat
fluxes with respect to the location of the inclusion (x. , y.) are not similar in shape but
at some boundary nodal locations are very small or zero. and thus the measurement of
temperature or heat flux at those locations is not good. Since the initial guess given by
case #24 resulted in the most rapid convergence. this guess is used in addressing all of

the following questions.

The second question addressed is the number and combination of surface
measurements required to simultaneously estimate the unknown parameters. The
minimum number of measurements needed is four. We considered 20 different
combinations of four temperature locations and 20 different combinations of four heat
flux locations on the boundary. The results are presented in Table (4.4). These results
show that convergence can be achieved using four measurements. When four
temperature measurements were used. 20% of the cases considered converged, 5.0% of
the cases diverged due to (1), 25.0% of the cases diverged due to (2), 10.0% of the
cases diverged due to (3). and 40.0% of the cases diverged due to (2) and (3). The
number of iterations ranged from 4 to 7. When four heat flux measurements were

used, 10.0% of the cases considered converged, 20.0% of the cases diverged due to

59

Table (4.3) Influence of the Initial Guesses on Convergence (Heat Trasfer).

 

 

 

 

Initial guesses -
Case # k2 (X. . y.) R. Converged Diverged
Iteration # Group #
1 50.0 (2.5.2.5) 1.6 7
2 60.0 (2.5.2.5) 1.6 7
3 85.0 (2.5.2.5) _ 1.6 6
4 100.0 (2.5.2.5) 1.6 7
5 50.0 (2.5.2.5) 1.8 5
6 60.0 (2.5.2.5) 1.8 5
7 85.0 (2.5.2.5) 1.8 6
8 100.0 (2.5.2.5) 1.8 6
9 50.0 (2.5.2.5) 2.2 6
10 60.0 (2.5.2.5) 2.2 6
1 1 85 .0 (2.5 .2.5) 2.2 8
12 100.0 (2.5.2.5) 2.2 (2)
13 50.0 (2.5.2.5) 2.4 7
14 60.0 (2.5.2.5) 2.4 8
15 85.0 (2.5.2.5) 2.4 (2)
16 100.0 (2.5.2.5) 2.4 9
17 50.0 (3.5.3.5) 1.6 (l)
18 60.0 (3.5.3.5) 1.6 (3)
19 85.0 (3.5.3.5) 1.6 (3)
20 100.0 (3.5.3.5) 1.6 6
21 50.0 (3.5.3.5) 1.8 9
22 60.0 (3.5.3.5) 1.8 7
23 85.0 (3.5.3.5) 1.8 5
24 100.0 (3.5.3.5) 1.8 4
25 50.0 (3.5.3.5) 2.2 6
26 60.0 (3.5.3.5) 2.2 6
27 85.0 (3.5.3.5) 2.2 6
28 100.0 (3.5.3.5) 2.2 7
29 50.0 (3.5.3.5) 2.4 7
30 60.0 (3.5.3.5) 2.4 7
31 85.0 (3.5.3.5) 2.4 (2)
32 100.0 (3.5.3.5) 2.4 9

 

 

6O

 

 

 

 

 

 

1.0

4.1

CI ‘.

.9 0.8-

.9. .

3: 0.6-

o .

C)

o 0.4-

3‘ 1 °

...>.. 0.2“ o o o

1; ‘ o ° o o o o

7;,- 00

c ..

fl —o.2-

8 —o.4—

.E ‘

'6 —0.6-

E ..

5 -o.8—

7— _1 o ‘ Fo k2 cit/816]
2 3 4 5 6 14 15 16 17 18

Nodal Location Number

Figure (4.1) - Sensitivity of boundary temperatures with respect to k2 .

61

50.0

 

 

 

 

 

 

 

4...:

C:

.22 40.0-

.9

04...

*8 30.0-

0

o 20.0—

>N

.1: 10.0— ° °

"2— 0.0 O o

()3 --10.0— ° °

'8 -20.0-

.13

'6 -30.0-

E

:5 -40.0-

2-50.0 e..fi-.,[.0k,2_és£§£2:l_
23 23 24

8 9 10 11 12 20 21

Nodal Location Number

Figure (4.2) - Sensitivity of boundary heat fluxes with respect to k2 .

 

Normalized Sensitivity Coefficient

-0.2-
-0.4-
—O.6-
—O.8-
—1.0

1.0

62

 

0.8-
0.6-1
0.4-
0.2-
0.0

 

 

LA xc dT/dxé}

 

 

Figure (4.3)

2 3 4 5 6 14 15 16 17 18

Nodal Location Number

- Sensitivity of boundary temperatures with respect to x. .

 

 

Normalized Sensitivity Coefficient

50.0

63

 

400‘
30.0-
20.04
100*

 

0.0
-10.0-
-20.0-
-30.0-
-40.0-

 

 

 

-50.0

 

r I A xc dgzdxcl .

10 ll 12 20 21 22 23 24
Nodal Location Number

Figure (4.4) - Sensitivity of boundary heat fluxes with respect to x. .

Normalized Sensitivity Coefficient

-1.0

64

 

1.0
0.8-
0.6-
0.4-
0.2-‘

i

 

0.0

-0.2-
-0.4-
-0.6-
—0.8-

 

 

 

.1

[0 yc <17de]
4 5 6 14 15 16 17 18

Nodal Location Number

Figure (4.5) - Sensitivity of boundary temperatures with respect to y. .

 

65

 

 

 

 

50.0
4.:
C:
.9.) 40.0- o
.9
‘45 30.0-
o
O 20.0“ 6 o
>\
.t’ 10.0-
._>_ o
*4 o
'5 0.0
C o o
()3 -10.0—
8 -20.0- . .
.fl
‘6 —30.0~
E
*5 -4o.0- .
2

-50.0 . r . g , . r Lg) trade/gala

 

 

8 9 10 11 12 20 21 22 23 24
Nodal Location Number

Figure (4.6) - Sensitivity of boundary heat fluxes with respect to y. .

Normalized Sensitivity Coefficient

66

 

1.0
0.8=
0.6-
0.4—
0.2-

 

0.0
—0.2-
-0.4-
-0.6-
—0.8-

 

-—1.0

 

[9K Rc dT/dRV}

 

F—‘ T r I'— l'

4 5 6 14 15

16 17 18

Nodal Location Number

Figure (4.7) - Sensitivity of boundary temperatures with respect to R. .

 

67

 

 

 

 

 

50.0

+2

CI

.2 40.0- . .

.9

‘4: 30.0-

Q)

C)

o 20.0- . I. 1‘
a)? 10 04

.2 °

4..)

"as 0.0

C:

$ ~100-

8 —20.0- “ "

.u e
B —30.0-

E

5 --40.0-

2 —50.0 . , . . - (_x Redo/3128

10 ll 12 20 21 22 23 24
Nodal Location Number

Figure (4.8) - Sensitivity of boundary heat fluxes with respect to R. .

 

68

Table (4.4) Estimating the Unknown Parameters Using Four Temperature or Four
Heat Flux Measurements.

 

Case # Nodal Location Numbers Converged Diverged
Iteration # Group #

 

Four Temperature Measurements

 

 

 

 

1 2 , 3 .14 , 15

2 2 . 3 , 15 . 16 (2)
3 2,3,16,17 (2).(3)
4 2 . 3 .17 . 18 (2)
5 3 ,4 , 14 , 15 (1)
6 3.4.15.16 (2).(3)
7 3 .4 .16 .17

8 3 , 4 .17 .18 (2)
9 4,5,14,15 (2).(3)
10 4.5.15.16 (3)
11 4.5.16.17 (2).(3)
12 4 . 5 .17 .18

13 5 .6 , 14 , 15 (2), (3)
14 5.6.15.16 (2)
15 5 ,6 , 16 , 17

16 5.6.17.18 (2)
17 2.3.4.5 (2).(3)
18 3.4.5.6 (2).(3)
19 14 .15 . 16, 17 (3)
20 15 .16 . 17 , 18 (2), (3)

Four Heat Flux Measurements

21 8,9,20,21 (1)
22 8,9,21,22 (3)
23 8.9.22.23 (2)
24 8 ,9 , 23 .24 (2). (3)
25 9 .10 . 20 . 21 (1)
26 9 .10 .21 .22 (2), (3)
27 9 .10 .22 . 23 (3)
28 9 .10 , 23 .24 (2), (3)
29 10 .11 .20 , 21 (2)
30 10 .11 .21 .22 (3)
31 10 .11 .22 . 23 (2), (3)
32 10 .11 . 23 . 24 (2). (3)
33 11 .12 , 20 . 21 (2)
34 11 .12 . 21 . 22 (3)
35 11 .12 . 22 . 23 (1)
36 11.12.23.24
37 8.9.10.11 (1)
38 9 .10 .11 , 12 (2)
39 20 .21 .22 , 23 (2)
4O 21 .22 . 23 .24

 

 

69

(1). 25.0% of the cases diverged due to (2). 20.0% diverged due to (3), and 25.0%
diverged due to (2) and (3). Table (4.5) shows the results when five temperature
measurements or five heat flux measurements were used. When five temperature
measurements were used. 65.0% of the cases considered converged. 5.0% of the cases
diverged due to (1), 20.0% diverged due to (3). and 10.0% diverged due to (2) and
(3). The number of iterations ranged from 5 to 8. When five heat flux measurements
were used. 41.0% of the cases considered converged, 4.5% of the cases diverged due
to (2), 41.0% of the cases diverged due to (3). and 13.5% of the cases diverged due to
(2) and (3). The number of iterations ranged from 5 to 6. For the case of five
temperature measurements. it is observed that the locations on the top are better than
those on the bottom. and for the case of five heat flux measurements, it is observed
that the locations on the right side are better than those on the left side. This is
because the first guess of the inclusion location is close to the top-right corner and the
sensitivity coefficients of the top and right side nodal locations are thus higher than
those of the bottom and left side as shown in Figure (4.9) and Figure (4.10). Table
(4.6) shows the results when six temperature measurements were used. It is observed
that 70.0% of the cases considered converged. 20.0% of the cases diverged due to (3),
and 10.0% of the cases diverged due to (2) and (3). The number of iterations ranged

from 5 to 7.

Table (4.7) shows the results when two temperatures and two heat fluxes were
used. It is observed that when this combination of temperatures and heat fluxes was
used. 40.0% of the cases converged. 26.0% of the cases diverged due to (2), 22.0% of
the cases diverged due to (3), and 12.0% of the cases diverged due to (2) and (3). The

number of iterations ranged from 5 to 10.

The next question addressed is the effect that the inevitable errors in experimental
measurements will have on the ability to estimate the sought parameters. For this case.

the "experiment" is simulated as follows. The body is first analyzed by the boundary

70

Table (4.5) Estimating the Unknown Parameters Using Five Temperature

or Five Heat Flux Measurements.

 

 

 

 

 

 

Case # Nodal Location Numbers Converged Diverged
Iteration # Group #
Five Temperature Measurements

1 2.3.4.5,14 7

2 2.3.4.5.15 (2)
3 2.3.4.5,16 7

4 2.3.4.5,17 (3)
5 2.3.4.5.18 (2).(3)

6 3.4.5.6.14 7

7 3.4.5.6,15 8

8 3.4.5.6,16 (3)
9 3.4.5.6,17 (3)
10 3.4.5.6.18 (3)
11 14.15,16.17,2 5

12 14.15.16.17,3 5

13 14,15,16.17.4 6

14 14.15,16,17.5 6

15 14.15,16,17.6 6

16 15,16.17.18,2 5

17 15.16.17,18.3 5

18 15.16,17.18,4 6

19 15,16.17,18,5 7

20 15,16,17.18.6 7

21 14.15.16.17,18 (2).(3)

Five Heat Flux Measurements

22 8,9,10,11,20 (2).(3)

23 8,9,10,11,21 (3)
24 8,9,10,11,22 (3)
25 8,9,10,11,23 (3)
26 8,9,10,11,24 (3)
27 9,10,11,12,20 (3)
28 9.10.11.12.21 (2)
29 9.10.11.12.22 (3)
30 9.10.11.12.23 (3)
31 9.10.11.12.24 (2).(3)

32 20,21.22.23,24 5

33 20.21,22,23,8 5

34 20.21,22,23.9 (3)
35 20.21,22,23,10 6

36 20.21,22,23,11 5

37 20.21,22,23,12 (3)
38 21.22.23,24,8 5

39 21,22.23,24,9 6

40 21,22,23.24,10 5

41 21,22,23.24,11 6

42 21,22,23.24,12 5

43 8,9,10,11,12 (2).(3)

 

 

71

 

 

 

 

 

 

 

 

1.0
4.4
C .
.92 0.8- u
.9 .. u
4“: 0.6-
0 ll
0 ¢ ¢ 0
O 0.4“- ¢ ¢ . 0
>4 . .
'4: 0.2- Q 0 o O t
._>_ . o 0 ° 0 at o
:1: O O a L O o 1 ‘
g . . . at ,. U U o
6}} -0.2- . . ' "‘ x
8 -0.4-
.E i
B -O.6-‘ a R‘dT/dRc
O ' . O x‘clT/dxc
Z _1 0 I I I I I O ksz/dkz
° 2 3 4 5 6 1'4 is 1'6 ‘17 T‘ 18

Nodal Location Number

Figure (4.9) - Temperature sensitivity coefficients associated with initial guess of k2

= 100, (x. , y.) = (353.5) and R. = 1.8 .

72

 

 

 

 

 

 

 

 

 

.5 50.0
r: O kzc'lq/dk2
.3 40.0- o xglq/dxc
°- 9* ydq/dy ' 2
‘4— _ C c g
0 20.0" 3 ¢ 2 O
3" 10 0- '
:E . g o . ‘ g 0
d:
m 0.0 g . °
E; o I Q 0 ‘
00 "1013‘ 9 . o
8 -20.0- s
.9. .
‘5 —30.0- . . c
E -40 0— -
O .
2: o
—50.0 I I T 1 I I r 1 r r
8 9 10 11 12 20 21 22 23 24

Nodal Location Number

Figure (4.10) - Heat flux sensitivity coefficients associated with initial guess of k2 =

100, (x. , y.) = (35,35) and R. = 1.8 .

 

73

Table (4.6) Estimating the Unknown Parameters Using Six Temperature

 

 

 

 

Measurements.

Case # Nodal Location Numbers Converged Diverged

Iteration # Group #
Six Temperature Measurements

1 2,3,4.5.6.14 7

2 2.3,4.5.6,15 7

3 2,3,4,5,6,16 (3)

4 2,3,4,5.6,17 (3)

5 2,3,4,5,6.18 (2).(3)

6 14.15,16.l7.18,2 5

7 14.15,16.l7.18,3 5

8 14,15.16.17.18,4 6

9 l4,15.l6.17,l8,5 7

10 14,15.16.17.18,6 6

 

 

74

Table (4.7) Estimating the Unknown Parameters Using Two Temperature

and Two Heat Flux Measurements.

 

 

 

 

Case # Nodal Location Numbers Converged Diverged
Iteration # Group #
Two Temperatures.and Two Heat fluxes
1 ‘ 2 . 3 , 8 . 9 (3)
2 2 , 3 . 9 , 10 (3)
3 2 . 3 , 10 , 11 (2)
4 2 , 3 , 11 . 12 7
5 3 , 4 , 8 , 9 (3)
6 3 , 4 . 11 , 10 (2)
7 3 , 4 , 11 . l2 9
8 3 .4 . 12 , 13 7
9 4 . 5 . 8 , 9 10
10 4 . 5 , 9 , 10 (2)
11 4 , 5 , 10 . 11 8
12 4 , 5 , 11 , 12 7
13 5 .6, 8 , 9 (3)
14 5 , 6 , 9 , 10 (2)
15 5.6.10.11 (3)
16 5,6,11,12 (2)
17 14 .15 , 20 . 21 (2). (3)
18 14 .15 . 21 . 22 9
19 14 .15 . 22 . 23 6
20 14 .15 . 23 . 24 (3)
21 15 .16 . 20 , 21 (2), (3)
22 15 .16 , 21 . 22 6
23 15 .16 . 22 , 23 5
24 15 .16 . 23 . 24 (2), (3)
25 16 .17 . 20 . 21 (2). (3)
26 16 .17 , 21 . 22 (2)
27 16 .17 . 22 , 23 (2)
28 16 .17 . 23 , 24 5
29 17 .18 , 20 . 21 (2)
30 17 .18 .21 . 22 6
31 17 .18 . 22 , 23 (3)
32 17.18.23 . 24 6

 

 

 

75

element method using the exact values of the four parameters. Then random errors are
added to the computed boundary temperatures and/or heat fluxes, and these are taken
to be the "measured" data. The statistical assumptions regarding the introduced errors
are that they are additive, non-correlated, normally distributed and have zero mean and
constant variance. These errors are generated following a procedure discussed in [2].
Table (4.8) shows the results when ten temperature measurements with 0.0%, 0.5%,
1.0%, and 2.0% random errors or ten heat flux measurements with 0.0%. 0.5%. 1.0%.
and 2.0% random errors are used. Comparing the results of Table (4.8) shows that
when heat fluxes are used, convergence is much faster and leads to more accurate
values of the unknown parameters. The case of six temperature measurements with
0.0%. 0.5%, 1.0%. and 2.0% random errors to estimate the four unknown parameters
is also considered. It is very interesting to note that better results are obtained here
using six temperature locations than when ten locations were used. When five heat flux
measurements with 0.0%, 0.5%. 1.0%, and 2.0% random errors are used to estimate
the four unknown parameters. the results are not as good as when ten locations were
used. It should be noted that all the results in Table (4.8) are rounded off to one
significant figure. Thus, the results for cases 6, 7 and 8 are not exact but are correct

when rounded to one decimal place.

The final question addressed is the effect of the inclusion size on the
convergence. Table (4.9) shows the results when the size of the actual inclusion
decreases. The sensitivities of temperatures and heat fluxes decrease as the the size of
the inclusion decreases. Figure (4.11) and Figure (4.12) show that the sensitivity of
temperature and heat flux become very small as R. decreases to the value of 0.5, but it
is interesting to note that it is possible to estimate the unknown parameters
corresponding to a very small circular inclusion with R. = 0.1 . The number of

iterations increases as the inclusion size decreases.

76

Table (4.8) Influence of Experimental Errors on the Estimations ( Heat Transfer).

 

 

 

 

 

 

 

 

 

 

Case # % Error Nodal Location Numbers Iteration #. and Converged Parameter
Ten Temperature Measurements
1 0.0 2.3,4.5,6.14.15.16,17.18 4, k2=730. (x. . y.)=(3.0,3.0). R.=20
2 0.5 2.3,4.5,6.14.15.16,17.18 7, k2=79.9, (x. , y.)=(2.9,3.0), R.=20
3 1.0 2.3,4.5,6.14.15.16,17.18 9, k2=84.7, (x. . y.)=(2.9,2.9), R.=l.9
4 2.0 2.3,4.5,6.14.15.16,17.18 10. k2=89.7. (x. , y.)=(2.8.28), R.=l.9
Ten Heat Flux Measurements
5 0.0 8.9.10.11,12,20,21,22.23.24 4 . k2=73.0, (x. , y.)=(3.0,3.0). R.=20
6 0.5 8.9.10.11.12,20,21.22.23.24 4 . k2=73.0, (x. . y.)=(3.0,3.0). R.=20
7 1.0 8.9.10.11,12,20,21.22.23.24 4 ,k2=73.0. (x. , y.)=(3.0,3.0). R.=20
8 2.0 8.9.10.11.12,20,21.22.23.24 4 , k2=73.0. (x. . y.)=(3.0,3.0). R.=20
Six Temperature Measurements
9 0.0 14.15,16.l7.18,2 5 ,k2=730. (x. , y.)=(3.0,3.0). R.=20
10 0.5 14.15,16.l7.18,2 5 . k2=75.1. (x. , y.)=(3.0,3.1). R.=20
11 1.0 14.15,16.l7.18,2 5 .k2=77.3. (x. , y.)=(3.0,3.1). R.=20
12 2.0 14.15,16.l7.18,2 5 . k2=821, (x. , y.)=(2.9,3.2). R.=20
Five Heat Flux Measurements
13 0.0 20,21.22.23,24 5 ,k2=73.0. (x. , y.)=(3.0,3.0). R.=20
14 0.5 20,21.22.23,24 5 ,k2=74.2, (x. . y.)=(3.0,3.1). R.=20
15 1.0 20,21.22.23,24 5 ,k2=75.4, (x. . y.)=(3.0,3.1). R.=20
16 2.0 20,21.22.23,24 5 ,k2=77.8, (xc , y.)=(3.0,3.2), R.=l.9

 

 

77

Table (4.9) Influence of the Inclusion Size on the Estimation.

 

 

 

 

Initial guesses
Case # k2 (x. , y.) R. Actual Converged
Inclusion size Iteration #
1 73.0 (3.0.3.0) 1.8 2.0 4
2 73.0 (3.0.3.0) 1.3 1.5 4
3 73.0 (3.0.3.0) 0.8 1.0 5
4 73.0 (3.0.3.0) 0.7 0.5 5
5 73.0 (3.0.3.0) 0.5 0.3 6
6 73.0 (3.0.3.0) 0.3 0.1 8

 

 

78

50.0

 

40.0—
30.04
20.04
10.04

0.0 +rg—g RAJ 3 8 a

 

 

Normalized Sensitivity Coefficient

 

 

 

—10.0—

—20.0-

-300- at RC dq/dRc

~490- 1 123223.:

_50... . f . T f f .4 o k2d cit<2_1

8 9 10 11 12 20 21 22 23 24
Nodal Location Number

Figure (4.11) - Temperature sensitivity coefficients corresponding to R. = 0.5 .

 

 

79

 

 

 

 

 

 

1.0
4.;
C .
.3.) 0.8-
.9. J
4“: 0.6-1
g i
o 0.4-1
>‘ J
a: 0.2—
.2 4
i500 what F"‘=*—=‘+' "
c .1
.33 _023
.1
8 —0.4—
.L“. ‘
‘6 -O.6- are Re dT/dRc
E ' 0 ye dT/dyc
O 7'08: A xc dT/dxc
Z _10 o k2dI/dk2
2 3 4 5 6 14 15 16 17 18

Nodal Location Number

Figure (4.12) - Heat flux sensitivity coefficients corresponding to R. = 0.5 .

 

 

80

4.2) Elasticity

Consider the problem shown in Figure (2.5). The location, the radius, the shear
modulus and the Poisson’s ratio of the circular inclusion are to be estimated
simultaneously using the displacements measured on the portion of the boundary
where traction is specified and/or the tractions measured on the portion of the
boundary where displacement is specified. The "experiment" is simulated as described

in the previous section.

The first question addressed is the influence of the initial guesses on convergence.
The first attempt was to estimate all five parameters. but it failed despite many
different choices of the first guess of the unknown parameters and induced boundary
conditions. Next the estimation of four parameters was attempted and it was observed
that estimation of any combination of four of the five parameters is possible. A total of
40 sets of initial guesses were examined and "experimental values" at all the boundary
nodal locations were used. Convergence was defined in accordance with equations
(3.1.19). The first problem investigated is shown in Figure (2.5). Estimation of the
location, shear modulus and Poisson’s ratio of the circular inclusion with known size,
using all measured displacements in the x1 and x2 directions and tractions in the
x1 and x2 directions is attempted and as is shown in Table (4.10). Note that 15.0% of
the cases considered converged whereas 85.0% of the cases diverged. The reason for
divergence becomes apparent after 3 or 4 iterations and can be classified into one of
three groups: (1) the determinant of [ P ] for an iteration is zero; (2) the estimated
values of (x. , y.) and R. for an iteration are unrealistic, i.e. the estimated inclusion
does not lie entirely within the mauix domain; (3) the estimated value of the inclusion
Poisson’s ratio for an iteration is less than zero or greater than 0.5. All the above
three groups are identified during the estimation process and the program will stop if

any of the above situations occurs. For the above problem 40.0% of the cases diverged

81

Table (4.10) Influence of the Initial Guesses on Convergence Using
Displacement and Traction Measurements (Elasticity Example Problem

 

 

 

#1).
Initial guesses
Case # G2 V2 (x. . y.) Converged Diverged
Iteration # Group #

l 2.0E+06 0.30 (2.5.2.5) (l)
2 1.5E+06 0.30 (2.5.2.5) 15

3 1.2E+06 0.30 (2.5 .2.5) (2)
4 0.8E+06 0.30 (2.5.2.5) (2)
5 0.5E+06 0.30 (2.5.2.5) (1)
6 2.0E+06 0.25 (2.5.2.5) (2)
7 1.5E+06 0.25 (2.5.2.5) (3)
8 1.2E+06 0.25 (2.5 .2.5) (3)
9 0.8E+06 0.25 (2.5 .2.5) 8

10 0.5E+06 0.25 (2.5.2.5) (1)
1 1 2.0E+06 0.15 (2.5.2.5) (2)
12 1.5E+06 0.15 (2.5.2.5) 14

13 1.2E+06 0.15 (2.5.2.5) (1)
l4 0.8E+06 0.15 (2.5.2.5) (1)
15 0.5E+06 0.15 (2.5.2.5) (1)
l6 2.0E+06 0.10 (2.5.2.5) (2)
l7 1.5E+06 0.10 (2.5.2.5) (3)
18 1.ZE+06 0. 10 (2.5.2.5) 9

19 0.8E+06 0.10 (2.5.2.5) (1)
20 0.5E+O6 0.10 (2.5.2.5) (l)
21 2.0E+06 0.30 (3.5.3.5) (2)
22 1.5E+06 0.30 (3.5.3.5) 12
23 1.2E+06 0.30 (3.5.3.5) (2)
24 0.8E+06 0.30 (3.5.3.5) (1)
25 0.5E+06 0.30 (3.5.3.5) (3)
26 2.0E+06 0.25 (3.5.3.5) (l)
27 1.5E+06 0.25 (3.5.3.5) 10
28 1.2E+O6 0.25 (3.5 ,3.5) (3)
29 0.8E+06 0.25 (3.5.3.5) (3)
30 0.5E+06 0.25 (3.5.3.5) (3)
31 2.0E+06 0.15 (3.5.3.5) (l)
32 1.5E+06 0.15 (35,35) (1)
33 1.2E+06 0.15 (3.5.3.5) (2)
34 O.8E+O6 0.15 (3.5.3.5) (l)
35 0.5E+06 0.15 (3.5.3.5) (l)
36 2.0E+06 0.10 (3.5.3.5) (2)
37 1.5E+O6 0.10 (3.5.3.5) (2)
38 1.2E+06 0.10 (3.5.3.5) (2)
39 0.8E+06 0.10 (3.5.3.5) (l)
40 0.5E+06 0.10 (3.5.3 .5) (1 )

 

 

 

82

due to (1). 27.5% of the cases diverged due to (2), and 17.5% of the cases diverged
due to (3). The number of iterations ranged from 8 to 15. The same problem was
investigated using only the displacement measurements at all nodal locations where
traction was specified and, as is shown in Table (4.11). 67.5% of the cases converged,
27.5% of the cases diverged due to (l), and 5.0% of the cases diverged due to (2).
The number of iterations ranged from 5 to 20. As the result of this observation.
another problem with fewer displacement boundary conditions, shown in Figure
(2.6), was examined. 18 displacements in the x1 direction and 18 displacements in the
x2 direction were used and. as is shown in Table (4.12), 90.0% of the cases
converged 2.5% of the cases diverged due to (1), and 7.5% of the cases diverged due
to (3). A third problem, shown in Figure (2.7), with an equal number of displacement
boundary conditions and traction boundary conditions, was investigated. All the cases
involving different first guesses of the unknown parameters diverged. It is concluded
that if the body under investigation has more traction boundary conditions, the
estimation of the unknown parameters is more successful. The sensitivity of traction in
the x2 direction with respect to parameters 62 and v2 is shown in Figure (4.13). It is
observed that the sensitivity coefficients of t2 with respect to Poisson’s ratio and shear
modulus are identical in shape and are linearly dependent. This would lead to
difficulty when simultanious estimation of these two parameters is attempted by using
the measurements of t2 as extra information. Figure (4.14) shows the sensitivity of t2
with respect to the location of the circular inclusion and it is observed that the

sensitivity coefficients corresponding to x. and y. are not linearly dependent.

Since the second problem, shown in Figure (2.6), resulted in the best
convergence, it is used to investigate the remaining issues. Figure (4.15) through
Figure (4.24) show the sensitivity of displacements in x1 and x2 directions with
respect to the four parameters being estimated. They correspond to the exact values of

the parameters. Figure (4.15) and Figure (4.17) show that the sensitivities of 111 with

83

Table (4.11) Influence of the Initial Guesses on Convergence Using
Displacement Measurements (Elasticity Example Problem #1).

 

 

 

 

Initial guesses
Case # G2 V2 (x. . y.) Converged Diverged
Iteration # Group #
1 2.0E+06 0.30 (2.5.2.5) 9
2 1.5E+06 0.30 (2.5.2.5) 20
3 1.2E+06 0.30 (2.5.2.5) 6
4 0.8E+06 0.30 (2.5.2.5) 6
5 0.5E+06 0.30 (2.5.2.5) (1)
6 20E+06 0.25 (2.5.2.5) 9
7 1.5E+06 0.25 (2.5.2.5) 11
8 1.2E+06 0.25 (2.5.2.5) 6
9 0.8E+06 0.25 (25.25) 6
10 0.5E+06 0.25 (2.5.2.5) (l)
11 20E+06 0.15 (2.5.2.5) (1)
l2 1.5E+06 0.15 (2.5.2.5) 8
13 1.2E+06 0.15 (2.5.2.5) 6
14 0.8E+O6 0.15 (2.5.2.5) (1)
15 0.5E+06 0.15 (2.5.2.5) (l)
16 2.0E+06 0.10 (2.5.2.5) 7
l7 1.5E+06 0.10 (2.5.2.5) 9
18 1.2E+06 0.10 (2.5.2.5) 5
19 0.8E+06 0.10 (25.25) 7
20 0.5E+06 0.10 (2.5.2.5) (1)
21 2.0E+06 0.30 (3.5.3.5) (2)
22 1.5E+06 0.30 (3.5.3.5) 15
23 1.2E+06 0.30 (3.5.3.5) 11
24 0.8E+06 0.30 (3.5.3.5) 6
25 0.5E+06 0.30 (3.5.3.5) (1)
26 20E+06 0.25 (3.5.3.5) (2)
27 1.5E+06 025 (3.5.3.5) 7
28 1.2E+06 0.25 (3.5.3.5) ll
29 O.8E+06 0.25 (3.5.3.5) 6
30 0.5E+06 0.25 (3.5.3.5) (1)
31 2.0E+06 0.15 (3.5.3.5) 15
32 1.5E+06 0.15 (3.5.3.5) 6
33 1.2E+06 0.15 (3.5.3.5) 6
34 0.8E+06 0.15 (3.5.3.5) 9
35 0.5E+06 0.15 (3.5.3.5) (1)
36 2.0E+06 0.10 (3.5.3.5) (1)
37 1.5E+06 0.10 (3.5.3.5) 6
38 1.2E+06 0.10 (3.5.3.5) 5
39 0.8E+06 0.10 (3.5.3.5) 6
40 0.5E+06 0.10 (3.5.3.5) (1)

 

 

Table (4.12) Influence of the Initial Guesses on Convergence (Elasticity Example

84

 

 

 

 

Problem #2).
Initial guesses
Case # G2 V2 (x. . y.) Converged Diverged
Iteration # Group #

1 20E+06 0.30 (2.5.2.5) 12

2 1.5E+06 0.30 (2.5.2.5) 7

3 1.2E+06 0.30 (2.5.2.5) 5

4 0.8E+06 0.30 (2.5.2.5) 5

5 0.5E+06 0.30 (25.25) 6

6 20E+06 0.25 (2.5.2.5) 7

7 1.5E+06 0.25 (2.5.2.5) 7

8 1.2E+06 0.25 (2.5.2.5) 5

9 0.8E+06 0.25 (2.5.2.5) 5

10 0.5E+06 0.25 (25.25) 7

11 20E+06 0.15 (2.5 .25) (1)
12 1.5E+06 0.15 (2.5.2.5) 9

13 1.2E+06 0.15 (2.5.2.5) 6

14 0.8E+06 0.15 (2.5.2.5) 6

15 0.5E+06 0.15 (2.5.2.5) 8

16 2.0E+O6 0.10 (2.5.2.5) (3)
17 1.5E+06 O. 10 (2.5.2.5) 7

18 1.2E+06 0.10 (2.5.2.5) 6

19 0.8E+06 O. 10 (2.5.2.5) 6

20 0.5E+06 0.10 (2.5 .25) 9

21 2.0E+06 0.30 (3.5.3.5) 18

22 1.5E+06 0.30 (3.5.3.5) 8

23 1.2E+06 0.30 (3.5.3.5) 5

24 0.8E+06 0.30 (3.5.3.5) 6

25 0.5E+06 0.30 (3.5.3.5) 8

26 20E+O6 0.25 (3.5.3.5) (3)
27 1.5E+O6 0.25 (3.5.3.5) 7

28 1.2E+06 0.25 (3.5.3.5) 5

29 0.8E+06 0.25 (3.5.3.5) 6

30 0.5E+06 0.25 (3.5.3.5) 8

31 20E+06 0.15 (3.5.3.5) 11

32 1.5E+06 0.15 (3.5.3.5) 11

33 1.2E+O6 0.15 (3.5.3.5) 5

34 0.8E+06 0.15 (3.5.3.5) 6

35 0.5E+06 0.15 (3.5.3.5) 9

36 20E+06 0.10 (3.5.3.5) (3)
37 1.5E+O6 0.10 (3.5.3.5) 9

38 1.2E+O6 0.10 (3.5.3.5) 6

39 0.8E+06 0.10 (3.5.3.5) 6

40 0.5E+06 0.10 (3.5.3.5) 9

 

 

85

 

 

 

 

 

 

 

 

4. 500 g
c o 02 dtz/dcz o o
{9, A v2d12/dv2
E 400~
Q)
C)
o
>K
:t.’ 300- o
._>.
:1 A A
(D
C:
(3’, 200- ‘
8 ° 0
.5
B 100-1
g ‘ ‘
<3
2
O T 1 1 r f
8 9 10 ll 12

Nodal Location Number

Figure (4.13) - Sensitivity of boundary t2 values with respect to G2 and v2 (elasticity

example problem #1).

86

 

 

 

 

 

 

 

 

4.: 800' ,
C .1 O xc dtZ/dxfi
.2 o
.9 600‘ jC dt2/dyc
u— .1
“a3
8 400'] . 0 .
g 200- .
.2 .
‘5;- o .
c ‘1
Q)
<0 -200-l °
'0 .1
Q) 0
.L“. -400-i
‘6 l
E — ._. O O
5' 600 .
z .
_800 ‘ I l I— I—
8 9 10 11 12
Nodal Location Number

Figure (4.14) - Sensitivity of boundary t2 values with respect to x. and y. (elasticity

example problem #1).

Normalized Sensitivity Coefficient

87

 

0.001 0

0.0008;
0.0006;
0.0004;

0.0002- ,

 

0.0000
-0.0002;
—0.0004j
-0.0006-‘
-0.0008-:
-0.0010 ‘

 

4 5 6 8 9

 

‘ [e 02 du1/dGZ
11 12 14 15 16 17 18 20 21 23 24

j I I I

Nodal Location Number

Figure (4.15) - Sensitivity of boundary “1 values with respect to G2 .

 

88

 

 

 

 

 

0.0010
*5 .
.9 0.0008-
.9 «
‘4: 0.0006-
8 .. O O
0 0.0004- ° .
>‘ ‘ O O . .
:g 0.0002- . . . .
’5 0.0000 ’ ’
C 1 . o 9
Q) ’ o
m -0.00024
8 -0.0004—
.24. ~
2 -0.0006-
5 -0.0008-1
2 .
-0.001O I ' I t t j I j 1 fl ' L. IGz‘dU'Z/d‘Gz

2 3 4 5 6 8 91112141516171820 212324

Nodal Location Number

Figure (4.16) - Sensitivity of boundary u2 values with respect to G2 .

 

 

89

 

 

 

 

 

 

0.0010 +

4.)

C .

.9. 0.0008-

.9. ~

‘4: 00006-

9 -

C) o

0 0.0004- . .

>x .

a; 0.00024 . ,

35 0.0000 H—r—s 1—-r—*—~—’
c: .

fl -0.0002- ' '

3 -0.0004- _ ' ‘

bl . O

‘5 —0.0005-

E ..

*5 -0.0008~

2 _00010' . . . . . ' j Le vzdutgdvzl

 

2 3 4 5 6 8 9 11 12 14 15 16 17 18 20 21 23 24
Nodal Location Number

Figure (4.17) - Sensitivity of boundary ul values with respect to v2 .

90

 

 

 

 

 

 

0.0010

+5 .

.2 0.0008~

.9 -

it: 0.0006-

g .

0 0.0004-

.2)? 0.0002; ° ° ’ ’ . .
.2: . 9 9 0 O O o
’5, 0.0000 —r-‘ t ’ .__.

C ..

(g -0.0002-1

8 -0.0004-

.5 J

283 -0.0006-

3 -0.0008-

2 -00010' . ' . . Iofivz du2Zdvfl

 

 

 

2 3 4 5 6 8 9 11 12 14 15 16 17 18 20 21 23 24

Nodal Location Number

Figure (4.18) - Sensitivity of boundary uz values with respect to v2 .

91

 

 

 

0.0010 -
4E . o o
.9 0.00081
.9 "‘ O O O O
‘4: 0.0006—
8 .
0 0.0004— . . . .
>. .
3:): 0.0002- . . . .
.— " O O
’5 0.0000 ' '
c .
3}; —0.0002—
8 —0.0004-
.5 -
'6 -0.0006-
ES .
5 -0.0008-
2 I1
"0.0010 I I I I r l r t I v I r rrje rxcrdUIZduxc—l

 

 

 

2 3 4 5 6 8 9 11 12 14 15 16 17 18 20 21 23 24

Nodal Location Number

Figure (4.19) - Sensitivity of boundary ul values with respect to x. .

Normalized Sensitivity Coefficient

92

 

0.0010
0.0008-
0.0006d

0.0004“
1

0.0002‘
.1

 

0.0000
-0.0002—‘
-0.0004-}
-0.00064
-0.0008:

 

-0.0010

0

o

 

[ O xc «241ch

 

6 8 9

11 12 14 15 16 17 18 20 21 23 24

Nodal Location Number

Figure (4.20) - Sensitivity of boundary uz values with respect to x. .

 

 

Normalized Sensitivity Coefficient

' -0.0006-

93

 

0.001 0

0.0008:
0.0006:
0.0004:
0.0002:-

 

0.0000
-0.0002-
-0.0004-J

-0.0008-

 

-0.0010

 

 

2

3

4

T T

Nodal Location Number

Figure (4.21) - Sensitivity of boundary ul values with respect to y. .

 

l 9 ye du1[dyfl
5 6 8 9 11 12 14 15 16 17 18 20 21 23 24

Normalized Sensitivity Coefficient

94

0.0010

 

0.0008-
0.0006-

0.0004-
0.0002-

 

0.0000
—0.0002; ’ ’ ° ’
—0.0004:- . . . .
-0.oooe- ° ° ' o

4.0008;

 

—0.0010 . fl.

2 3 4 5 6 8 9 11 12 14 15 16 l7 18 20 21 23 24

Nodal Location Number

Figure (4.22) - Sensitivity of boundary uz values with respect to y. .

 

Normalized Sensitivity Coefficient

95

 

0.0010

0.0008;
0.0006;
0.0004;
0.0002:

 

0.0000
-0.0002-:
—0.0004:-
«0.0006'1
-0.0008-:

 

-0.0010

 

1 9 Rc duMdRc

 

2

3

456891'11'2141518171820212324

Nodal Location Number

Figure (4.23) - Sensitivity of boundary ul values with respect to R. .

 

96

 

 

 

 

 

0.0020 G v
+2
5 .
.6 0.0016: . .
“40030.001zi..... ..,..
0 0.0008": 9 o
>\
.4: 0.0004-
.2 . . .
"5 0.0000
C ..
()3 -0.0004-
8 -0.0008-
.5 ~
6 -0.00124
E ..
5 -0.0016-

 

2 3 4 5 6 8 9 11 12 14 15 16 17 18 20 21 23 24

Nodal Location Number

Figure (4.24) - Sensitivity of boundary 0) values with respect to R. .

 

97

respect to G2 and v2 are very similar in shape. Similarly Figure (4.16) and Figure
(4.18) reveal the same observation about the sensitivities of uz with respect to G2 and
v2 . Also Figure (4.20) and Figure (4.21) show that the sensitivty of uz with respect
to x. is similar in shape to the sensitivity of 111 with respect to y. . All of these
combinations lead to the problem of linear dependence of the columns corresponding
to the above parameters in the [P] matrix. Comparing the sensitivities of temperature
with respect to the sought parameters in the heat transfer problem and the sensitivities
of displacement with respect to the sought parameters in the elasticity problem shows
that the temperature sensitivities are much larger in magnitude than the displacement
sensitivities. The implication of this is apparent when the number of cases which
diverged because the determinant of [P] becomes zero. group #(1). in the heat transfer
and elasticity problems are compared. It is found that in the elasticity problem. the
determinant of [P] becomes zero more frequently. Since the initial guesses given by
case #3 of Table (4.12) resulted in the most rapid convergence, they are used in

addressing all of the following issues.

The second question addressed is the number and combination of surface
displacements required to simultaneously estimate the four unknown parameters, i.e.
shear modulus, Poisson’s ratio, and location of the inclusion. The minimum number of
measurements needed to estimate the above four parameters is 20. We considered five
different combinations of ten displacements in the x1 direction and ten displacements

in the x2 direction and. as is shown in Table (4.13), only one case converged.

Estimation of only two parameters, i.e. shear modulus and Poisson’s ratio of the
circular inclusion with known location and size is investigated next. Table
(4.14) shows that by selecting the right locations on the surface of the boundary, it is
possible to estimate the above two parameters by measuring only eight displacements.
This requires analyzing the plots of the sensitivity coefficients, Figure (4.15) through

Figure (4.18), and selecting the nodal locations with the highest value of 111 and uz

98

Table (4.13) Estimating the Unknown Parameters Using 10 Measured Displacements
in the x1 Direction and 10 Measured Displacements in the
x2 Direction (Elasticity Example Problem #2).

 

 

 

 

uz at 14.15,16.l7.18,8.9,11.12.4

 

 

Case # Nodal Location Numbers Converged Diverged
Iteration # Group #
1 111 at 2.3,4.5,6.14.15.16,17.18 5
u2 at 2.3,4.5,6.14.15.16,17.18
2 111 at 2.3,4.5,6.8.9.11,12,16 (2)
u2 at 2,3,4,5,6,8,9.11.l2.16
3 111 at 2,3.4.5.6.20.21.23.24,16 (2)
u2 at 2,3.4.5.6.20.21.23.24,16
4 ul at 14.15,16.l7.18,20.21.23.24,4 (2)
Hz at 14.15,16.l7.18,20.21.23.24.4
5 111 at 14,15.16.17.18,8.9.11.12.4 (2)

 

 

99

Table (4.14) Estimating G2 and v2 Using Displacement Measurements

(Elasticity Example Problem #2).

 

 

 

 

112 at 14.15.22

 

 

Case # Nodal Location Numbers Converged Diverged
Iteration # Group #
1 111 at 2,3.4.5.6.20.21.23.24 6
u2 at 2,3.4.5.6.20.21.23.24
2 111 at 2,3.4.5.6.8,9,11,12 (1)
uz at 2,3.4.5.6.8,9,11,12
3 111 at 14.15.16.17.18.20,21,23,24 6
u2 at 14.15.16.17.18.20.21.23.24
4 111 at 14,15.16.17.18,8,9,1l,12 (1)
uz at 14,15.16.17.18,8,9,11,12
5 1.11 at 8.9.11.12.20.21.23,24 (1)
uz at 8,9,11,12,20,21,23,24
6 “1 at 3,4,16,17 6
uz at 14.15.2223
7 111 at 3.45.15.16.17 (1)
1.12 at 22
8 u1 at 3.4.16 (1)

 

 

100

sensitivities with respect to the two parameters. Table (4.14) shows that even with 18
displacement measurements, if the locations are not selected carefully. the estimation
will diverge (cases #2 and #4). Estimation of G2 and v2 using less than eight

measurements was 1101 SUCCCSSflll.

The final question addressed is the effect that the inevitable errors in experimental
measurements will have on the ability to estimate the sought parameters. For this case.
the "experiment" is simulated as follows. The body is first analyzed by the boundary
element method using the exact values of the four parameters. Then random errors are
added to the computed boundary displacements, and these are taken to be the
"measured" data. The statistical assumptions regarding the introduced errors are the
same as described in the previous section. Table (4.15) shows the results when 18
displacements in the x1 direction and 18 displacements in the x2 direction. with
different percent errors were used to estimate the four unknown parameters. It is
observed that as the %error increases, the number of iterations also increases, but it is
possible to estimate the unknown parameters with experimental errors as high as 4.0%.

All the results in Table (4.15) are rounded off to three significant figures.

101

Table (4.15) Influence of Experimental Errors on the Estimation (Elasticity Example

 

 

 

Problem #2).
Nodal Locations where “1 and uz Are Measured
2,3.4.5.6.8,9.11.12.14,15,16,17,18,20.21.23,24

Case # % Error Iteration #. And Converged Values of Parameters
1 0.0 5 . Gz=l.OOOE+06. V2=0.200, (x. , y.)=(3.00.3.00)
2 0.5 8 . G2=1.015E+06. v2=0.195, (x. . y.)=(3.00.3.00)
3 1.0 10 , G2=1029E+06. V2=0.190. (x. , y.)=(3.01.2.99)
4 2.0 13 . G2=l.049E+06, V2=0.183. (x. , y.)=(3.03,2.97)
5 3.0 22 . G2=0.966E+06. V2=0.217, (x. . y.)=(3.03,3.01)
6 4.0 32 , G2=0.967E+06. v2=0.217, (x. , y.)=(3.03,2.99)

 

 

 

 

 

 

 

Chapter 5

Conclusions and Recommendations

A technique has been proposed which couples the boundary element and

parameter estimation methods for the purpose of characterizing the interior of an

inhomogeneous body utilizing surface measurements only. The parameter study

presented here addressed several questions which arise regarding implementation of

this technique in the heat transfer and elasticity problems. Although the technique does

not always converge, the cases which do converge give excellent results. Also, the

method never converges to an incorrect solution. Based on the results of this

investigation, the following conclusions are drawn.

1.

It is possible to estimate the four unknown parameters simultaneously using only

four measurements in the heat transfer problem.

It is better to use two temperature and two heat flux measurements than four

temperature or four heat flux measurements.

As the number of measurements increases. the percentage of cases which

converge increases.

Heat flux measurements with experimental errors give better results than

temperature measurements with experimental errors.
It is possible to estimate parameters corresponding to a very small inclusion.

Better results are obtained for elasticity problems using more displacement

measurements .

Estimating the unknown parameters in the heat transfer problem appears to be

easier than in the elasticity problem.

102

103

8. More success was achieved in estimating only the shear modulus and Poisson’s

ratio of the inclusion in the elasticity problem.

9. It would appear that the best procedure would be to estimate the thermal
conductivity. size. and location of the circular inclusion using the boundary
temperature and/or heat flux measurements and then, taking the location and size
of the inclusion as known, to estimate the mechanical properties of the inclusion

using the boundary-measured displacements.

It should be emphasized that for this nonlinear inverse problem, the sensitivity
coefficients depend on the current guess of the parameters. Thus if one wants to use
the sensitivity coefficients as a predictor of "best" measurement locations, one must
recognize that these best locations will change from one iteration to the next. One
possible approach would be to test various initial guesses and corresponding sensitivity
coefficients a priori and to select the initial guesses based on optimal initial
sensitivities.

Several additional questions need to be addressed. In particular, the effectiveness
of this technique for characterizing more complex situations such as several inclusions
or inclusions of unknown shape needs to be examined. Also extension to anisotropy
and/or three-dimensional problems would be of practical interest. However, based on

the results obtained so far. the method shows considerable promise.

REFERENCES

[1]Brebbia, C.A., and Dominguez J., 1988, Boundary Elements: An
Introductory Course, Computational Mechanics Publications.

[2] Beck, J.V., and Arnold, K.J.. 1977, Parameter Estimation In Engineering
And Science, John Wiley and Sons.

[3] Murai, T., and Kagawa. Y., 1986, "Boundary element iterative techniques
for determining the interface boundary between two Laplace domains: a
basic study of impedance plethysmography as an inverse problem", Intl. J.
Num. Meth. Engrg., Vol. 23, pp. 35—47.

[4] Ohnaka, K., and Uosaki, K., 1987, "Simultaneous identification of the
external input and parameters of diffusion type distributed parameter

systems", Intl. J. Control, Vol. 46. pp. 889-895.

[5] Dulikravich, GS, 1988, "Inverse design and active control concepts in

strong unsteady heat conduction", App]. Mech. Rev., Vol. 41, pp. 270-277.

[6] Tanaka, M., and Yamagiwa, K., 1989, "A boundary element method for
some inverse problems in elastodynamics", Appl. Math. Modelling. Vol.

13) pp. 307‘ 312.

[7]Kishimoto, K., Miyasaka, H., and Aoki, S., 1989, "Boundary element
analysis of an inverse problem in galvanic corrosion", JSME Intl. J. Series

I. Vol. 32, pp. 256-262.

104

105

[8] Gao, Z., and Mura, T., 1989, "On the inversion of residual stresses from
surface displacements", J. Appl. Mech., Vol. 56, pp. 508-513.
[9] Das, S., and Mitra, A. K., 1989, "An algorithm for the solution of inverse

Laplace problems using BEM (abstract)", Developments in Mechanics,

Vol. 15, pp. 525-526.

 

Appendix A

Computer Program : Heat Transfer

106

 

 

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