Inverse Problems in Engineering Mechanics. International Symposium on Inverse Problems in Engineering Mechanics 1998 (ISIP '98) Nagano, Japan

583 Pages·1998·10.666 MB·English

by  M. Tanaka and G.S. Dulikravich (Eds.)

Preface
Inverse Problems can be found in many topics of engineering mechanics. There are many
successful applications in the fields of the inverse problems (non-destructive testing and
characterization of material properties by ultrasonic or X-ray techniques, thermography,
etc.). Generally speaking, the inverse problems are concerned with the determination
of the input and the characteristics of a mechanical system from some of the output
from the system. Mathematically, such problems are ill-posed and have to be overcome
through development of new computational schemes, regularization techniques, objective
functionals, and experimental procedures.
Following the first IUTAM Symposium on these topics held in May 1992 in Tokyo,
and another in November 1994 in Paris, we thought it would be very fruitful to
gather researchers and engineers again for an exchange of the newest research ideas.
Consequently, at the International Symposium on Inverse Problems in Mechanics
(ISIP '98) held in March 1998 in Nagano, recent developments in the inverse problems in
engineering mechanics and related topics were discussed. The following general areas
were the subjects of this symposium: mathematical and computational aspects of the
inverse problems, parameter or system identification, shape determination, sensitivity
analysis, optimization, material property characterization, ultrasonic non-destructive
testing, elastodynamic inverse problems, thermal inverse problems, and other engineering
applications. Seventy-two papers from Asia, Europe, and North America were presented
at ISIP '98 in Nagano, Japan. The final version of the manuscripts of sixty-five papers
from these presentations are contained in this volume of the ISIP '98 proceedings.
These papers can provide a state-of the-art review of the research on inverse problems
in engineering mechanics. As the editors of the topical book, we hope that some
breakthrough in the research on inverse problems can be made and that technology
transfer will be stimulated and accelerated through the publication of this book.
As the chairmen of the ISIP '98 Symposium we wish to express our cordial thanks
to all the members of the International Scientific Committee and the Organizing
Committee. Financial support from the Japanese Ministry of Education, Science,
Sports and Culture (Monbusho) is gratefully acknowledged. Co-organizership by the
Pennsylvania State University (USA) and Ecole Polytechnique (France) is heartily
appreciated. Also, co-sponsorship by the Japan Society for Computational Methods in
Engineering (JASCOME) and helpful support by the staff of Shinshu University for the
management of financial support from Monbusho are gratefully acknowledged.
June 1998
Masataka TANAKA, Shinshu University / Japan
George .S DULIKRAVICH, The Pennsylvania State University / U.S.A.
vii
Symposium Chairmen
Prof. Masataka TANAKA
Department of Mechanical Systems Engineering
Faculty of Engineering
Shinshu University
005 Wakasato, Nagano 380-8553, Japan
:leT [phone], Fax: [phone]
E-Mail: [email protected]
Prof. George .S Dulikravich
Department of Aerospace Engineering
The Pennsylvania State University
University Park, Pennsylvania 16802, USA
:leT + 1-814-863-0134, Fax: + 2907-568-418-1
E-Mail: [email protected]
International Scientific Committee
Prof. Masa. Tanaka, Shinshu University (Japan), Chair
Prof. .G .S Dulikravich, The Pennsylvania State University (USA), Co-Chair
Prof. .O .M Alifanov, Moscow Aviation Institute (Russia)
Prof. .S Andrieux, Electricit~ de France (France)
Prof. .S Aoki, Tokyo Institute of Technology (Japan)
Prof. .J Beck, Michigan State Universitry (USA)
Prof. .M Bonnet, Ecole Polytechnique (France)
Prof. H.D. Bui, Ecole Polytechnique (France)
Prof. .H Engl, Johannes-Kepler-University (Austria)
Prof. G.M.L. Gladwell, University of Waterloo (Canada)
Prof. .D Ingham, University of Leeds (UK)
Prof. .A .J Kassab, University of Central Florida (USA)
Prof. .M Kitahara, Tohoku University (Japan)
Prof. .S Kobayashi, Kyoto University (Japan)
Prof. .S Kubo, Osaka University (Japan)
Prof. .P Ladev~ze, LMT, ENS Cachan (France)
Prof. G.-L. Liu, Shanghai University (China)
Prof. A. Louis, University of Saarland (Germany)
Prof. .G Maier, Politecnico id Milano (Italy)
Prof. .V Modi, Columbia University (USA)
.rD .C W.J. Oomens, Eindhoven University of Technology (Netherlands)
Prof. .H Sol, Vrije Universiteit Brussel (Belgium)
.rD .H Sobieczky, Deutsche Lufl- und Raumfahrt (Germany)
Prof. .N Tosaka, Nihon University (Japan)
Prof. .R van den Braembussche, yon Karman Institute of Fluid Dynamics (Belgium)
Prof. K.A. Woodbury, University of Alabama (USA)
Prof. .Z Yao, Tsinghua University (China)
Prof. .N Zabaras, Cornell University (USA)
vii
Symposium Chairmen
Prof. Masataka TANAKA
Department of Mechanical Systems Engineering
Faculty of Engineering
Shinshu University
005 Wakasato, Nagano 380-8553, Japan
:leT [phone], Fax: [phone]
E-Mail: [email protected]
Prof. George .S Dulikravich
Department of Aerospace Engineering
The Pennsylvania State University
University Park, Pennsylvania 16802, USA
:leT + 1-814-863-0134, Fax: + 2907-568-418-1
E-Mail: [email protected]
International Scientific Committee
Prof. Masa. Tanaka, Shinshu University (Japan), Chair
Prof. .G .S Dulikravich, The Pennsylvania State University (USA), Co-Chair
Prof. .O .M Alifanov, Moscow Aviation Institute (Russia)
Prof. .S Andrieux, Electricit~ de France (France)
Prof. .S Aoki, Tokyo Institute of Technology (Japan)
Prof. .J Beck, Michigan State Universitry (USA)
Prof. .M Bonnet, Ecole Polytechnique (France)
Prof. H.D. Bui, Ecole Polytechnique (France)
Prof. .H Engl, Johannes-Kepler-University (Austria)
Prof. G.M.L. Gladwell, University of Waterloo (Canada)
Prof. .D Ingham, University of Leeds (UK)
Prof. .A .J Kassab, University of Central Florida (USA)
Prof. .M Kitahara, Tohoku University (Japan)
Prof. .S Kobayashi, Kyoto University (Japan)
Prof. .S Kubo, Osaka University (Japan)
Prof. .P Ladev~ze, LMT, ENS Cachan (France)
Prof. G.-L. Liu, Shanghai University (China)
Prof. A. Louis, University of Saarland (Germany)
Prof. .G Maier, Politecnico id Milano (Italy)
Prof. .V Modi, Columbia University (USA)
.rD .C W.J. Oomens, Eindhoven University of Technology (Netherlands)
Prof. .H Sol, Vrije Universiteit Brussel (Belgium)
.rD .H Sobieczky, Deutsche Lufl- und Raumfahrt (Germany)
Prof. .N Tosaka, Nihon University (Japan)
Prof. .R van den Braembussche, yon Karman Institute of Fluid Dynamics (Belgium)
Prof. K.A. Woodbury, University of Alabama (USA)
Prof. .Z Yao, Tsinghua University (China)
Prof. .N Zabaras, Cornell University (USA)
viii
Organizing Committee
Prof. Masa. Tanaka, Shinshu University (Japan), Chair
Prof. .G .S Dulikravich, The Pennsylvania State University (USA), Co-Chair
Prof. .T Matsumoto, Shinshu University (Japan), Secretary
Prof. .T Adachi, Tokyo Institute of Technology (Japan)
Prof. .T Aizawa, YtisrevinU of Tokyo (Japan)
Prof. .S Aoki, Tokyo Institute of Technology (Japan)
Prof. H.D. Bui, Ecole Polytechnique (France)
Prof. .T Fukui, Fukui University (Japan)
Prof. .K Hayami, University of Tokyo (Japan)
Prof. .S Hirose, Okayama University (Japan)
Prof. .T Honma, Hokkaido University (Japan)
Prof. E Imado, Shinshu University (Japan)
Prof. .Y Iso, Kyoto University (Japan)
Prof. .K Kagawa, Okayama University (Japan)
Prof. .N Kamiya, Nagoya University (Japan)
Prof. .K Kishimoto, Tokyo Institute of Technology (Japan)
Prof. .E Kita, Nagoya University (Japan)
Prof. .M Kitahara, Tohoku University (Japan)
Prof. .S Kobayashi, Kyoto University (Japan)
Prof. .S Kubo, Osaka University (Japan)
Prof. .A Murakami, Kyoto University (Japan)
Prof. .M Nakamura, Shinshu University (Japan)
Prof. .N Nishimura, Kyoto University (Japan)
Prof. .K Onishi, Ibaraki University (Japan)
Prof. .N Tosaka, Nihon University (Japan)
Prof. .M Yamamoto, University of Tokyo (Japan)
INVERSE PROBLEMS IN ENGINEERING SCINAHCEM
.M Tanaka, G.S. Dulikravich (Eds.)
(cid:14)9 1998 Elsevier Science .V.B llA rights reserved.
SPECTRAL AND WAVELET METHODS FOR SOLVING AN INVERSE HEAT
CONDUCTION PROBLEM
LARS ELDt~N and FREDR/K BERNTSSON
Department of Mathematics, Linkiiping University
185-S 38 ,gnip'bkniL Sweden
ABSTRACT
We consider an inverse heat conduction problem, the Sideways Heat Equation, which is a model
of a problem, where one wants to determine the temperature on both sides of a wall, but where
one side is inaccessible to measurements. Mathematically it can be formulated as a Cauchy
problem for the heat equation, with data given along the line x - 1, where one wants the solution
to the left of that line. The problem is ill-posed, in the sense that the solution (if it exists) does not
depend continuously on the data. The problem can be stabilized by replacing the time derivative
in the heat equation by a wavelet-based approximation or a spectral-based approximation. The
resulting problem is an initial value problem for an ordinary differential equation, which can be
solved by standard numerical methods, e.g. a Runge-Kutta method. As test problems we take
model equations as well as problems from industrial applications.
KEYWORDS
Inverse heat conduction problem, ill-posed, temperature measurement, wavelet, Fourier analysis,
sideways heat equation
NOITCUDORTNI
In many industrial applications one wishes to determine the temperature on the surface of a body,
where the surface itself is inaccessible for measurements 1. It may also be the case that locating
a measurement device (e.g. a thermocouple) on the surface would disturb the measurements so
that an incorrect temperature is recorded. In such cases one is restricted to internal measure-
ments, and from these one wants to compute the surface temperature. The measuring situation is
illustrated in Figure .1
4 L. Eld~n, E Berntsson
Thick wall
L\'/,
Hot gas
or
l "qr/ / / / / 1/
liquid
I
0 1
Fig. "1 Interior temperature measurements.
In a one-dimensional setting, this situation can be modeled as the following ill-posed problem
for the heat equation: Determine the temperature u(z, t) for 0 _< z < 1 from temperature and
heat flux measurements, g,,,(') := u(1, .) and hm(.) := u~(1, .), when u(z, t) satisfies
I (a(:c)u~)~ = ut, O < z < ,1 t _> ,O
u(z, )O = ,O 0 _< z _< ,1 (1)
~(1, t) = g~(t), t _> 0,
u~(1, t) = hm (t), t __> 0,
The coefficient a(x) that represents the heat conducting properties of the medium is assumed to
be positive, a(x) _> ao > .0 We refer to (1) as the sideways heat equation .1 It is an ill-posed
Cauchy problem for a parabolic equation, see e. g. ,1 2, 3.
In principle, the problem (1) might be reformulated as f2 k(t - r)f(r)dr = gin(t), which is
an integral equation of the first kind, and standard techniques for ill-posed problems, such as
Tikhonov regularization, could be used. However, the kernel function k(t) is explicitly known
only in the case a(z) - const. 4. Therefore, for (1) we propose to keep the problem in the
differential equation form and solve it essentially as an initial value problem in the space variable
("space-marching", ,1 5, 6, 7, 8, 9, 10, 111).
Space-marching methods can be implemented efficiently, if the time-derivative is approximated
by a bounded operator 7, 8. In this way we obtain a well-posed initial-boundary value problem
for an ordinary differential equation (ODE). The initial value problem can be solved numerically
by a standard ODE routine.
In this paper we mainly treat the sideways heat equation formulated as in (1), and we discuss
the stabilization and numerical solution of this equation. Theoretical results for the quarter plane
1 Also the Inverse Heat Conduction Problem (IHCP) .1
lartcepS dna televaw sdohtem for gnivlos na esrevni taeh noitcudnoc melborp 5
problem with constant coefficients are given in 12, 7, 8, 13, 14. We discuss two different
methods for approximating the time derivative and their numerical implementation. Numerical
examples are given.
APPROXIMATING THE TIME DERIVATIVE
We rewrite the sideways heat 0( equation (1) as a first order initial value problem in the space
variable
1/a(x)) (u)
0 v ' 0<x<l, t>__0, (2)
v = O/Ot
x
with initial values
(u(1,t)) (gin(t))
v(1,t) = hm(t) '
and the extra condition u(x, 0) = 0, for 0 < x < .1 The ill-posedness of this problem can be
seen by taking the Fourier transform with respect to time. For simplicity, in this discussion we
temporarily assume that the coefficient is constant, ooL a(x) = a. Let
1 e-iOu(x t)dt
denote the Fourier transform of u(x, t), where we have assumed that u(x, t) is identically zero
for t < 0. Then, in frequency space the initial value problem becomes
0
0 "6 ' 0<x<l, -oo<~<co, (3)
which is a family of initial value problems parameterized by .~ The solution in frequency space
is
(~(x, ~)) exp(B(x _ 1)) (~'m(~))
~(~,r = ~(r
,
and we can write the solution in the time domain formally sa
( )
)t,x(v)t,x(u = 1~- oc e iOexp(B(x - 111 hm(r (4)
The eigenvalues of the matrix B are
A=+~=+ 1 + i sign(~) v/l~i/a.
v~
Therefore, since there are eigenvalues of unbounded magnitude with positive real part, the solu-
tion operator defined by (4) is unbounded; perturbations in high frequencies will be magnified
and will destroy any numerical solution unless some regularization is done.
One obvious way of removing the ill-posedness and stabilizing the problem is to replace the
unbounded differentiation operator 0/at by a bounded approximation D. We will consider two
6 .L ,n.~dlE E nosstnreB
different methods for doing this. In the following we assume that the data functions are sampled
in n discrete points 0 < t 1 < ... ~,t < .1 Then the unknown functions are taken to be
u(~) = (cid:12)9 , v(~) = (cid:12)9 ,
u(~,t~) v(~,t~)
and the approximate differentiation operator D is a matrix.Thus we want to solve numerically
the initial value problem
(U) (0 1/a(x)I) (U)
(U(1)) (Gin)
V = D 0 Y ' 0<x<l, V(1) = Hm " (5)
and we will see that this can be done using a standard ODE solver.
lartcepS noitamixorppA
The conceptually simplest way of approximating the time differentiation operator to make it
bounded is to cut off all frequencies above a certain threshold. This can be done by putting
D _= ~D F ,FLAH
(6)
=
where cA is a diagonal matrix corresponding to differentiation of the trigonometric interpolant,
but where frequency components with ~ greater than a cut off level ~ are explicitly set equal to
zero 14, 15, Section 1.4. This will filter the data and numerical solution by removing high
frequency components. The matrix F is the Fourier matrix 16, and the multiplication of F by
n(O
a vector can be performed in log n) operations using the fast Fourier transform (FVr). Thus,
in the ODE solver, multiplication by ~D is carried out as a FFT, followed by a multiplication
n(O
by a diagonal matrix, and finally an inverse FFT, giving a total operation count of log n)
operations.
In 12, 41 error estimates are given and it is shown how to choose the cut off level c~ in the case
of a quarter plane problem. We will demonstrate later that this works also in the present case.
televaW noitamixorppA
Another way of of stabilizing the problem is to solve it in a wavelet space. Let {Vj}jez be a
noituloseritlum sisylana
(MRA) 17, 81 generated by
2J/2r k), j,k e Z,
jV = span{r Cjk(t) =
where r is the scaling function of Meyer wavelets 81 (for a discussion of the application of
Meyer wavelets to the sideways heat equation, see 13). The functions kjC constitute an orthog-
onal basis in Vj, and since the Fourier transform satisfies
^ 4r 2j
Cjk(~) = 0, for 1~I _> -5- =: ~j'
Spectral and wavelet methods for solving an inverse heat conduction problem 7
we can consider the projection Pj onto Vj, defined by
Pjf(t) ,)t(kjpd)kjpd
= Z(f,
kEZ
as a low pass filter. In our context this means that if we expand the solution of the sideways heat
equation in terms of the basis of Vj,
ZE,~
then no frequencies larger than j~ will be present in it. Thus, the choice of the wavelet space
corresponds to the choice of the cut-off level in the spectral approximation above.
In 31 we described a wavelet-Galerkin approach for the sideways heat equation. In this method
the time differentiation operator is approximated by a bounded operator, which is the differenti-
ation operator Dj in .jV We proved almost optimal error estimates in the case of a quarter plane.
Also a recipe for choosing jV was given, depending on the error level of the data.
SOLVING THE ODE PROBLEM
In the numerical solution of the initial value problem (5) it is important to have information about
the behaviour of the analytic solution of the ordinary differential equation. Since the system is
linear, the behaviour is, to a large extent, determined by the eigenvalues of the matrix
0 1/a(x)I)
B(x)= D 0 '
where D is the matrix corresponding to time derivative. In Fig. 2 we illustrate the eigenvalues of
-hB a(z)
for = i and D = De, the spectral approximation of the derivative (6), and h is the step
length in space. The eigenvalues of-hB in the left half plane correspond to solution components
that decrease when z goes from 1 to 0, whereas those in the right half plane correspond to
increasing solution components.
In Fig. 2 we have also plotted the stability region of an explicit fourth order Runge-Kutta method.
This is a standard method for solving numerically initial value problems for ODE's 19. The
significance of the stability region is that the numerical method gives a non-increasing solution
for solution components corresponding to eigenvalues inside the region, and an increasing solu-
tion for components corresponding to eigenvalues outside the stability region. Thus, if h is taken
small enough, then all the eigenvalues in the left half plane will be inside the stability region,
and the numerical solution has the correct properties with respect to increasing and decreasing
solution components.
In 8 it is shown that for realistic problems a standard Runge-Kutta method, with automatic step
length control, is adequate.
8 .L ,n~dlE E Berntsson
2
1 x x x xxxxxxxx~
0
x x x .xxxX ~
-1
-2
%
. . . . . . o'.,
Fig. 2: Eigenvalues in the complex plane of the matrix -hB, for D = De, cut off level c~ = 90
and step length h = 0.2. The closed curve is the stability region of a fourth order Runge-Kutta
method.
NUMERICAL EXAMPLES
Here we present a few numerical experiments conducted using our methods. These tests are not
to be considered as a comprehensive evaluation, but rather they are intended to demonstrate the
usefulness of the approach. In the first numerical experiment we solved the problem (1) using
the coefficient function
1
a(x) = ~(1 + e-Z), x > .0
The test was performed in the following way: We selected a solution u(0, t) = f(t), 0 < t < 1
and computed data functions u(1, t) = g(t) and u~(1, t) = h(t) by solving a well-posed quarter
plane problem for the heat equation using a finite difference scheme. Then we added a normally
distributed perturbation of variance 10 -3 to each data function, giving vectors ~,G and H,~. From
the perturbed data G,~, Hm we reconstructed the surface temperature, and compared the result
with the known solution. The results are displayed in Fig. 3.
In a certain industrial application one is interested in finding the temperature on the surface of a
particle board 14. In this particular example it is difficult to measure the surface temperature
directly, and instead a thermocouple placed inside the board records the temperature history,
u(L, t), at a distance of 2.9 mm from the surface, and at 8192 points in time. In this application
it is natural to assume that the temperature is symmetric with respect to the center of the board.
Therefore we can compute u~(L, t), by solving a well-posed problem for the heat equation. Thus
no heat-flux measurements are required. The computational results are displayed in Fig. 4. Here
we present only the results from the wavelet method, but similar results are obtained using the
spectral method.
ACKNOWLEDGEMENT
The work of Fredrik Berntsson is supported by a grant from the National Graduate School for
Scientific Computing.