Inverse Problems in Engineering Mechanics IV. International Symposium on Inverse Problems in Engineering Mechanics 2003 (ISIP 2003) Nagano, Japan

500 Pages·11.338 MB·English

by  Masa Tanaka (Eds.)

Preface
Inverse Problems can be found in many areas of engineering mechanics. There are
numerous successful applications in the fields of inverse problems. For example,
non-destructive testing and characterization of material properties by ultrasonic or X-
ray techniques, thermography etc. Generally speaking, the inverse problems are
concerned with estimating the unknown input and/or the characteristics of a system
given certain aspects of its output. Mathematically, such problems are in general ill-
posed and have to be overcome through development of new computational schemes,
regularization techniques, objective functionals, and experimental procedures.
Following the first lUTAM Symposium on these topics held in May 1992 in Tokyo,
another in November 1994 in Paris, the ISIP '98 in March 1998, and also the
ISIP2000 in March 2000 in Nagano as well as the ISIP2001, we concluded that it
would be fruitful to gather regularly with researchers and engineers for an exchange
of the newest research ideas and related techniques. The proceedings of these
symposia were published and are recognized as standard references in the field of
inverse problems. The most recent Symposium of this series "International
Symposium on Inverse Problems in Engineering Mechanics (ISIP2003)" was held in
February of 2003 in Nagano, Japan, where recent developments in inverse problems
in engineering mechanics and related topics were discussed.
The following general areas in inverse problems in engineering mechanics were the
subjects of the ISIP2003: mathematical and computational aspects of 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. A number of papers from Asia, Europe, and America were
presented at ISIP2003 in Nagano, Japan. The detailed data of the ISIP2003 is
available on the Internet (http://homer.shinshu-u.ac.jp/ISIP2003/). The final versions
of the manuscripts of fifty-five papers selected from these presentations are contained
in this volume of the ISIP2003 proceedings. These papers can provide a state-of-the-
art review of the research on inverse problems in engineering mechanics. As the
editor of the topical book, I hope that some breakthrough in the research on inverse
problems can be made and that technology transfer will be stimulated and accelerated
resulting from its publication.
As the chairperson of the ISIP2003 Symposium, I 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) as well as the Nagano Prefecture is gratefully
acknowledged. Co-organizership by The University of Texas at Arlington, U.S.A.,
Ecole Polytechnique, France, University of Central Florida, USA, and Technical
University of Silesia, Poland is heartily appreciated. Also, co-sponsorship by the
Japanese Society for Computational Methods in Engineering (JASCOME) and helpful
support by the staff of Shinshu University in managing the financial support from
Monbusho are gratefully acknowledged.
July 2003
Masataka TANAKA, Shinshu University, Japan
Symposium Chair
Prof. Masataka TANAKA
Department of Mechanical Systems Engineering
Faculty of Engineering
Shinshu University
4-17-1 Wakasato, Nagano 380-8553, Japan
Fax: [phone], Tel: [phone]
E-mail: [email protected] jp
International Organizing Committee
Prof Masa. Tanaka(Chair), Shinshu University (Japan)
Prof G.S. Dulikravich, The University of Texas at Arlington (USA)
Prof A.J. Kassab, University of Central Florida (USA)
Prof S. Kubo, Osaka University (Japan)
Prof A.J. Nowak, Technical University of Silesia (Poland)
International Scientific Committee
Prof Masa. Tanaka (Chair), Shinshu University (Japan)
Prof. C.J.S. Alves, Technical University of Lisbon (Portugal)
Prof S. Aoki, Tokyo Institute of Technology (Japan)
Prof T. Burczynski, Silesian Technical University of Gliwice (Poland)
Prof. R. Contro, Politecnico di Milano (Italy)
Prof L. Elden, Linkoping University (Sweden)
Prof. A. Ales Gottvald, Institute of Scientific Instruments (Czech)
Prof. S.I. Kabanikhin, Sobolev Institute of Mathematics (Russia)
Prof. G. Kanevce, St. Kliment Ohridski University (Maccedonia)
Prof M. Kitahara, Tohoku University (Japan)
Prof. A.S. Kobayashi, University of Washington (USA)
Prof F. Kojima, Kobe University (Japan)
Prof P. Ladeveze, ENS de Cachan (France)
Dr. K.J. Langenberg, University of Kassel (Germany)
Prof G. Maier, Politecnico di Milano (Italy)
Prof S. Migorski, Jagiellonian University (Poland)
Prof N. Nishimura, Kyoto University (Japan)
Prof K. Onishi, Ibaraki University (Japan)
Prof. H.R.B. Orlande, Federal University of Rio de Janeiro (Brazil)
Prof M. Reynier, ENS de Cachan/CNRS/Universite Paris VI (France)
Prof R. Rikards, Riga Technical University (Latvia)
Prof H. Sobieczky, DLR German Aerospace Research Center (Germany)
Dr. B. Soemarwoto, National Aerospace Laboratory (The Netherlands)
Prof V.V. Toropov, University of Bradford (UK)
Prof I. Trendafilova, The University of Strathclyde (UK)
Prof L.C. Wrobel, Brunei University (United Kingdom)
Prof A. Yagola, Moscow State University (Russia)
Prof Z. Yao, Tsinghua University (China)
Symposium Chair
Prof. Masataka TANAKA
Department of Mechanical Systems Engineering
Faculty of Engineering
Shinshu University
4-17-1 Wakasato, Nagano 380-8553, Japan
Fax: [phone], Tel: [phone]
E-mail: [email protected] jp
International Organizing Committee
Prof Masa. Tanaka(Chair), Shinshu University (Japan)
Prof G.S. Dulikravich, The University of Texas at Arlington (USA)
Prof A.J. Kassab, University of Central Florida (USA)
Prof S. Kubo, Osaka University (Japan)
Prof A.J. Nowak, Technical University of Silesia (Poland)
International Scientific Committee
Prof Masa. Tanaka (Chair), Shinshu University (Japan)
Prof. C.J.S. Alves, Technical University of Lisbon (Portugal)
Prof S. Aoki, Tokyo Institute of Technology (Japan)
Prof T. Burczynski, Silesian Technical University of Gliwice (Poland)
Prof. R. Contro, Politecnico di Milano (Italy)
Prof L. Elden, Linkoping University (Sweden)
Prof. A. Ales Gottvald, Institute of Scientific Instruments (Czech)
Prof. S.I. Kabanikhin, Sobolev Institute of Mathematics (Russia)
Prof. G. Kanevce, St. Kliment Ohridski University (Maccedonia)
Prof M. Kitahara, Tohoku University (Japan)
Prof. A.S. Kobayashi, University of Washington (USA)
Prof F. Kojima, Kobe University (Japan)
Prof P. Ladeveze, ENS de Cachan (France)
Dr. K.J. Langenberg, University of Kassel (Germany)
Prof G. Maier, Politecnico di Milano (Italy)
Prof S. Migorski, Jagiellonian University (Poland)
Prof N. Nishimura, Kyoto University (Japan)
Prof K. Onishi, Ibaraki University (Japan)
Prof. H.R.B. Orlande, Federal University of Rio de Janeiro (Brazil)
Prof M. Reynier, ENS de Cachan/CNRS/Universite Paris VI (France)
Prof R. Rikards, Riga Technical University (Latvia)
Prof H. Sobieczky, DLR German Aerospace Research Center (Germany)
Dr. B. Soemarwoto, National Aerospace Laboratory (The Netherlands)
Prof V.V. Toropov, University of Bradford (UK)
Prof I. Trendafilova, The University of Strathclyde (UK)
Prof L.C. Wrobel, Brunei University (United Kingdom)
Prof A. Yagola, Moscow State University (Russia)
Prof Z. Yao, Tsinghua University (China)
Symposium Chair
Prof. Masataka TANAKA
Department of Mechanical Systems Engineering
Faculty of Engineering
Shinshu University
4-17-1 Wakasato, Nagano 380-8553, Japan
Fax: [phone], Tel: [phone]
E-mail: [email protected] jp
International Organizing Committee
Prof Masa. Tanaka(Chair), Shinshu University (Japan)
Prof G.S. Dulikravich, The University of Texas at Arlington (USA)
Prof A.J. Kassab, University of Central Florida (USA)
Prof S. Kubo, Osaka University (Japan)
Prof A.J. Nowak, Technical University of Silesia (Poland)
International Scientific Committee
Prof Masa. Tanaka (Chair), Shinshu University (Japan)
Prof. C.J.S. Alves, Technical University of Lisbon (Portugal)
Prof S. Aoki, Tokyo Institute of Technology (Japan)
Prof T. Burczynski, Silesian Technical University of Gliwice (Poland)
Prof. R. Contro, Politecnico di Milano (Italy)
Prof L. Elden, Linkoping University (Sweden)
Prof. A. Ales Gottvald, Institute of Scientific Instruments (Czech)
Prof. S.I. Kabanikhin, Sobolev Institute of Mathematics (Russia)
Prof. G. Kanevce, St. Kliment Ohridski University (Maccedonia)
Prof M. Kitahara, Tohoku University (Japan)
Prof. A.S. Kobayashi, University of Washington (USA)
Prof F. Kojima, Kobe University (Japan)
Prof P. Ladeveze, ENS de Cachan (France)
Dr. K.J. Langenberg, University of Kassel (Germany)
Prof G. Maier, Politecnico di Milano (Italy)
Prof S. Migorski, Jagiellonian University (Poland)
Prof N. Nishimura, Kyoto University (Japan)
Prof K. Onishi, Ibaraki University (Japan)
Prof. H.R.B. Orlande, Federal University of Rio de Janeiro (Brazil)
Prof M. Reynier, ENS de Cachan/CNRS/Universite Paris VI (France)
Prof R. Rikards, Riga Technical University (Latvia)
Prof H. Sobieczky, DLR German Aerospace Research Center (Germany)
Dr. B. Soemarwoto, National Aerospace Laboratory (The Netherlands)
Prof V.V. Toropov, University of Bradford (UK)
Prof I. Trendafilova, The University of Strathclyde (UK)
Prof L.C. Wrobel, Brunei University (United Kingdom)
Prof A. Yagola, Moscow State University (Russia)
Prof Z. Yao, Tsinghua University (China)
Local Organizing Committee
Prof. Masa. Tanaka (Chair), Shinshu University (Japan)
Prof. T. Matsumoto (Secretary), Shinshu University (Japan)
Prof. K. Amaya, Tokyo Institute of Technology (Japan)
Prof. M. Arai, Shinshu University (Japan)
Prof. H. Azegami, Toyohasi University of Technology (Japan)
Prof. T. Fukui, Fukui University (Japan)
Prof. K. Hayami, The Natioal Institute of Informatics (Japan)
Prof. S. Hirose, Tokyo Institute of Technology (Japan)
Prof. T. Honma, Hokkaido University (Japan)
Prof. M. Hori, University of Tokyo (Japan)
Prof. H. Igarashi, Hokkaido University (Japan)
Prof. F. Imado, Shinshu University (Japan)
Prof. Y. Inoue, Tokyo Institute of Technology (Japan)
Prof. Y. Iso, Kyoto University (Japan)
Prof. K. Kagawa, Okayama University (Japan)
Prof. J. Kihara, Himeji Institute of Technology (Japan)
Prof. K. Kishimoto, Tokyo Institute of Technology (Japan)
Prof. E. Kita, Nagoya University (Japan)
Prof. M. Kitahara, Tohoku University (Japan)
Prof. H. Koguchi, Nagaoka University of Technology (Japan)
Prof. A. Murakami, Kyoto University (Japan)
Prof. M. Nakamura, Shinshu University (Japan)
Prof. N. Tosaka, Nihon University (Japan)
Prof. M. Yamamoto, University of Tokyo (Japan)
INVERSE PROBLEMS IN ENGINEERING MECHANICS IV
M. Tanaka (Editor)
© 2003 Elsevier Ltd. All rights reserved.
APPLICATION OF THE PROPER ORTHOGONAL DECOMPOSITION
IN STEADY STATE INVERSE PROBLEMS
Ryszard A. BlALECKi^ Alain J. KASSAB^
and Ziemowit OsTROWSKl\
Institute of Thermal Technology, Silesian University of Technology, Gliwice, Poland
^ Department of Mechanical Material and Aerospace Engineering
University of Central Florida, Orlando, USA
e-mail: bialeckiOitc. ise.polsl. gliwice. pi, [email protected] . edu,
ostryQitc.ise.polsl.gliwice.pi
ABSTRACT
A novel inverse analysis technique for retrieving unknown boundary conditions has been
developed. The first step of the approach is to solve a sequence of forward problems made
unique by defining the missing boundary condition as a function of some unknown param
eters. Taking several combinations of values of these parameters produces a sequence of
solutions (snapshots) which are then sampled at a predefined set of points. Proper Orthog
onal Decomposition (POD) is used to produce a truncated sequence of orthogonal basis
functions, being appropriately chosen linear combinations of the snapshots. The solution
of the forward problem is then written as a linear combination of the basis vectors. The
unknown coefficients of this combination are evaluated by minimizing the discrepancy be
tween the measurements and the POD approximation of the field. Two numerical examples
show the robustness and numerical stability of the proposed scheme.
KEYWORDS
inverse problems, Proper Orthogonal Decomposition, steady state conduction
INTRODUCTION
Reduction of the number of the degrees of freedom in an inverse problem is a well known
technique of filtering out the higher frequency error. The Proper Orthogonal Decomposi
tion (POD) offers an elegant way to cut down the number of unknowns without loosing
the accuracy. The method has been developed about 100 years ago as a tool of processing
statistical data [1, 2, 3, 4]. It has been also used in signal processing, pattern recognition,
control theory, fiuid flow and dynamics [5, 6, 7]. Another important area of apphcation
is in the turbulence where the technique, know also as Karhunen-Loeve method, has been
4 R-A. Bialecki et al.
used to detect the spatial large scale organized motions [8]. Some more recent description
of POD theory can be found in [9, 10].
The present paper presents a technique of applying POD in inverse analysis and, to the
best of the knowledge of the authors, it is the first attempt to use POD in this context. The
proposed technique uses the Proper Orthogonal Decomposition (POD) as a regularization
technique. The idea is to solve a sequence of forward problems within the body under
consideration. The solution of each problem is sampled at a predefined set of points. Each
sampled field corresponds to a certain set of assumed values of the parameters defining
the distribution of the function to be retrieved. POD detects the correlation between the
discretized fields leading to a significant reduction of the degrees of freedom necessary to
describe the field with a high accuracy.
BASICS OF POD
The fundamental notion of POD is the snapshot being a collection of A^ sampled values
of the field under consideration. The snapshot is stored in a vector U\2 = 1,2,... ,M.
A collection of all snapshots is a rectangular N by M matrix U. The snapshots are
generated by changing the values of some parameter(s) upon which the field depends on.
Time, parameters entering the boundary conditions, internal heat generation or material
properties are examples of such parameters. The snapshots may be obtained either from
a mathematical model of the phenomenon or from experiments. The aim of POD is to
construct a set of vectors (basis) ^^ resembhng the original matrix U. The basis is stored
in another rectangular matrix ^ of the same dimensionality as U. The elements of the
basis are defined as
$ = U-V (1)
where V is a modal matrix defined as a nontrivial solution of a problem
C • V - AV (2)
In the above A is a diagonal matrix storing the eigenvalues A^ of the positive definite
covariance matrix C. The entries of the latter are defined as
N
C.j = J2^lUi (3)
the eigenvalues are real, positive and distinct and should be sorted in an descending order.
The basis vectors associated with the eigenvalues are orthogonal ie
^^ ^ = A. (4)
It can be shown [10], that the jth eigenvalue is a measure of the kinetic energy transferred
within jth basis mode ^^ (strictly speaking this is only true, when the field under consid
eration is the velocity field). Typically, this energy decreases rapidly with the increased
number of the mode, which permits discarding the majority of modes. This can be done
by deciding which fraction of the energy may be neglected in further calculations. The
resulting POD basis # consists oi K < N elements.
This basis captures, in an optimal way, the spatial variation of the field. To have a full
picture of the field, dependence on additional parameters used in generating the snapshots
Application of the proper orthogonal decomposition 5
needs to be built into the approximation formula. This is accomphshed by expressing the
field represented by an arbitrary snapshot as
K
U^Y^a^^' (5)
with the unknown scalars a^ depending on the parameters. The a-^'s are found by an
appropriate procedure, say the least square fit or weighted residuals.
INVERSE PROBLEM
If the points at which the snapshots were evaluated coincide with the location of the
sensors, the values of the unknown parameters a^ in eq. (5) can be found by least square
curve fitting. Note that only few terms would typically be retained in eq. (5), thereby
significantly reducing the number of unknowns, which is a very desirable feature in any
inverse analysis algorithm.
To make the discussion more concrete, assume that the aim of inverse problem is to
retrieve a distribution of a heat flux on a portion of the boundary. Steady state heat
conduction with other boundary conditions known is considered. Both heat conductivity
and distribution of the internal heat generation are known. The functional form of the
unknown heat flux is postulated, say as a polynomial of a given degree. The coefficients of
the polynomial (parameters of heat flux distribution) are unknown. Additional information
is produced by temperature sensors located at some points of the domain.
To generate the POD basis, a sequence of direct problems is solved using any analytical
or numerical technique. For each problem, other combination of values of the unknown
parameters is taken. All solutions are sampled at the same set of points. When a numerical
technique is used, a natural snapshot is the set of all nodal temperatures. Adding normal
heat fluxes at nodes located on the surface where the heat flux distribution is to be retrieved,
significantly improves the stability of the algorithm. In the numerical examples discussed
hereafter, snapshots gathering both temperatures and heat fluxes have been used.
To solve an inverse problem, some measured values are needed. The simplest way to
approach this problem, is to locate the sensors at a set of points being a subset of points
used to create the snapshots. Otherwise, some additional interpolation would be necessary.
Solution of each direct problems, ie., each combination of the unknown parameters gen
erates one snapshot. From the collection of ah snapshots, the truncated POD bases in eq.
(5) can be constructed, based on the user specified fraction of the energy to be neglected.
The values of the unknown multipliers in expansion (5) are calculated by a least square
fit of the measurements and the model. Once the values of a^ are known, the values of
temperatures at all snapshot points can be determined. As the snapshot definition encom
passes also the sought for heat fluxes, the procedure also yields the values the retrieved
heat flux. Two simple numerical example are used to demonstrate the effectiveness of this
technique.
R.A. Biatecki et al.
NUMERICAL EXAMPLE L ANALYTICAL METHOD USED TO SOLVE THE FOR
WARD PROBLEM
Steady state heat conduction in a unit square ^ < x,y < I with unit conductivity is
considered. Edges x = 0 and x = I are insulated. At ?/ = 0 temperature is zero. The
unknown distribution of the normal heat flux g^ at ?/ = 1 is to be retrieved. It assumed
that this distribution can be described by a cubic polynomial
'^ — a -\- hX -\- ex ^- dX (6)
where a. h. c. d are unknown parameters. The sketch of the problem has been shown in
Fig 1. Every of these parameters has been given a value of -L, -L5, -2, and -2.5. The
y
q {x)=a+bx-^cx^+dx^
L=I
II
II
I C3- location I ^
of sensors
T=0 L=I
Figure 1: Sketch of the geometry, imposed boundary conditions
snapshots have been generated taking every combination of these values leading to a total
of 4^ = 256 combinations. An analytical method (separation of variables) has been applied
to solve the resulting 256 direct problems. The temperature has been sampled at uniformly
spaced set of 121 points corresponding to 0.1 x and y increments. The heat flux at ?/ = 1
has been sampled at x = 0.1, 0.2, ..., 0.9, thus, each snapshot had 130 entries.
Based on this collection of snapshots the covariance matrix in eq. (3) has been generated
and the appropriate eigenproblem of eq.(2) solved. The magnitude of the eigenvalues of
the POD system decreased rapidly. Here are the first 5 eigenvalues: 0.113E-I-6, 0.230E+4,
0.127EH-3, 0.350E+1, 0.035. Remaining were below E-8. The neglected energy fraction
was set to be lower than E-10 resulting in a truncated POD system consisting of 5 modes.
Such distribution of the eigenvalues indicates strong correlation between the snapshots. As
a result, the temperature field can be approximated with good accuracy using only few
POD basis vectors.
Measurements have been simulated by computing the values of the temperatures at 27
nodes located at ?/=0,8, 0.9 and 1.0 at x locations 0.1, 0.2,...0.9. The values of parameters
describing the distribution of the heat flux have been taken as a=-100, 6=-200, c=-300,
d=-400 (note the significant extrapolation with respect to the values used to generate the
snapshots!). The resulting temperatures have been'biased with a lOK amplitude error of
uniform distribution.
The values of the coefficients a^ in eq. (5) have been obtained using the Levenberg
Marquardt algorithm [11] to minimize the objective function
Application of the proper orthogonal decomposition
* = E[2^'-E"'*'i (7)
where i/i denote the error ladened measured yalues of the temperature.$] is the coordinate
of the jth POD base vector associated with the iih sensor location.
Once the coefficients a^ of the POD expansion (5) are known, the values of the tempera
ture at nodes where the snapshots were produced can readily be calculated. The snapshots
contain also the heat fluxes at points where these quantities are to be retrieved. Thus,
the same coefficients resulting from the least square probelm, can be used to retrieve the
values of the heat fluxes.
Fig. 2 presents the comparison of the retrieved and accurate distribution of the heat
flux at y=l. Fig. 3 shows the error in temperatures. Note the small amplitude of the error
in Fig 3 and the decreasing of the error for points closer to the sensor location. Within the
layers where temperatures have been measured, the error, similarly to the measurements
errors, behaves randomly.
Figure 2: Comparison of the retrieved and accurate distributions of the heat flux at y=l
0.1 0.2 0.3 0.4
y coordinate
Figure 3: Comparison of the retrieved and accurate distributions of temperature versus y
coordinate for different values of the x coordinate