Inverse Problems In Engineering Mechanics Iv International Symposium On Inverse Problems In Engineering Mechanics 2003 Isip 2003 Nagano Japan Masa Tanaka Eds

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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: +81-26-269-5124, Tel: +81-26-269-5120
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
L=I
II
I C3-
q {x)=a+bx-^cx^+dx^
location
of sensors
T=0
II
I ^
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

R.A. Bialecki et al.
NUMERICAL EXAMPLE 2. FINITE ELEMENTS USED TO SOLVE THE FOR
WARD PROBLEM
Reference forward problem
As in the previous example, the reference problem is used to simulate measurements.
Here is a detailed description of the problem.
Rectangle with two cooling holes is considered. The reference forward problem is de
fined as a 2D steady state temperature field with no internal heat generation and all
boundary conditions known. The heat conductivity is k = 22W/'mK. The geometry and
the boundary conditions are depicted in Fig. 4.
q=a-\-bx+cx^-^dK+ex^ inverse problem
forward problem
oo
II
fy
0.3
0.4
0.3
location of sensors
L5
q=620[T-713]
0.5
0.2
0.3
0.3
q=h(x)[TA4l^]
2.0
oo
1.0
Figure 4: Geometry and boundary conditions. Example 2
Robin's conditions are assumed along the external perimeter of the domain and within
the cooling holes. While the heat transfer coeflftcient in the holes is assumed to be constant,
known distribution of this coeflficient is assumed along the external boundary. The variation
of the heat transfer coeflBcient is shown in Fig 5.
The forward problem has been solved using FIDAP [12], a FEM commercial pack
age. The mesh consisted of 11072 isoparametris quadratic quadrangle serendipity elements
(Quads) and 33935 nodes.
The results of the forward problem were the temperatures at all nodal points and normal
heat fluxes at nodes located on the boundary. The resulting distribution of the normal
heat flux along the AD side of the domain is shown in Fig 6
Inverse problem
Inverse analysis has been conducted in the described above domain. The boundary
conditions except along the AD edge (cf fig 4) are the same as in the forward problem.
The aim of the inverse analysis was to retrieve the normal heat fluxes along the boundary

Application of the proper orthogonal decomposition
A B
Figure 5: Variation of the heat transfer coefficient along the external boundary
A D
0,5 1,5
Xf m
Figure 6: Resulting variation of the heat flux along the portion of the external boundary
between points A and D.
AD.
This distribution has been postulated as
(8)

10 R.A. Bialecki et al.
The snapshots have been generated by solving a sequence of forward boundary value prob
lems taking as the boundary condition distribution (8). The values of the unknown pa
rameters a, 6, c, d, e have been taken at three levels: -30, 0 and 30.Solution of a sequence
of forward problems using all combination of these values generated a set of snapshots.
The total number of snapshots amounted to 3^ = 243. The POD basis has been generated
assuming that l.E-10 of the energy of the field is neglected. This assumption produced
a POD basis consisting of 6 vectors. The subsequent eigenvalues associated with these
vectors were 0.431E+12, 0.180E+10, 0.436E+8, 0.608E-h7, 0.288E-h6, 0.516E-h4. The re
maining eigenvalues were below E-3. As in the previous example, the rapid decay of the
eigenvalues suggests a strong mutual correlation of the snapshots. A fact in agreement
with the common sense.
Each snapshot consisted of 33935 nodal values of the temperature and 320 nodal values
of the heat fluxes at nodes located along the AD hue.
The measurements have been simulated by taking the values of the temperatures being
a solution of the forward problem. The location of the sensors is shown in Fig 4. The 30
simulated temperature measurements have been biased with a 7, 50 and 100 K amphtude
error of uniform distribution. This error level corresponds to 1, 7 and 14% error related to
the maximum temperature range arising in the problem.
The plot of the error at subsequent sensors is shown in Fig 7.
60
-60
-o- error amplitude 7K
-^ error amplitude 50K
-*- error amplitude lOOK
20 25 30
number of sensor
Figure 7: Plot of random error in measured temperatures for subsequent sensor locations
and different error amplitudes
The Levenberg Marquardt algorithm [11] has been used to obtain the coefficients of the
POD expansion. The retrieved heat flux for different levels of error amplitude are shown
in Fig 8.
It should be noted that even in the presence of unrealistically high measurement error
reaching lOOK, the retrieved distribution of the heat flux is accurate both quantitatively

Application of the proper orthogonal decomposition
B
20
10
0
-10
-20
-30
-40
-50
— exact distribution
-»«- POD no error
-^ POD error amplitude 7K
-^^ POD error amplitude 5 OK
-o- POD error amplitude lOOK
1
1
0.5 1.5
Figure 8: Retrieved heat flux distribution for different values of measurement error
and qualitatively.
CONCLUSIONS
A novel technique of solving inverse problem has been proposed. The first numeri
cal examples indicate that the technique is robust and exhibits favorable regularization
properties. An interesting characteristic of the proposed technique is its insensitivity to
measurements error. Even in the presence of very large discrepancies reaching 20% the
procedure produces stable results. That smoothing properties of POD applied to inverse
problems can be attributed to two features
• POD basis is able to reproduce the important features of the solution using minimal
number of degrees of freedom, thus filtering out the high frequency error
• presence of the values of the retrieved function in the basis introduces additional
regularization.
The extension of this technique to other types of inverse analysis is a topic of ongoing
research.
ACKNOWLEDGEMENT
The work has been partially supported by a NASA grant NAS 3.269L

12 R.A. Bialecki et al.
REFERENCES
PEARSON K., On lines planes of closes fit to system of points in space. The London,
Edinburgh and Dublin Philosophical Magazine and Journal of Science 2, pp 559-572
(1901)
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[9;
[lo:
[11
[12
HOTELLING H., Analysis of complex of statistical variables into principal components,
Journal of Educational Psychology,24: pp 417-441, (1933).
KARHUNEN K. Ueber lineare Methoden flier Wahrscheiniogkeitsrechnung. Annales of
Academic Science Fennicae Series Al Mathematical Physics, 37, pp 3-79, (1946).
LOEVE M.M. Probability Theory, Princeton NY, Van Nostrand. 1955
LY H.V and TRAN H.T Modeling and control of physical processes using proper
orthogonal decomposition. Mathematical and Computer Modelling 33, pp 223-236,
(2001).
ATWELL J.A. and KING B.B Proper orthogonal decomposition for reduced basis
feedback controllers for parabohc equations. Mathematical and Computer Modelling,
33 pp 1-19. (2001)
AZEEZ M.F. and VAKAKIS A.F. Proper orthogonal decomposition (POD) of a class of
vibroimpact oscillations Journal of sound and vibrations. 240(5), pp 859-889, (2001).
BALL K.S. SIROVICH L. and KEEFE L.R. Dynamical eigenfunction decomposition
of turbulent channel flow, International Journal for Numerical Methods in Fluids, 12,
pp 585-604, (1991).
AUBRY N. LlAN W.Y. anb TiTi E.S. Preserving symmetries in the proper orthogonal
decomposition. SIAM Journal of Scientific Computations, 14(2), pp 483-505, (1993)
BERKOOZ G., HOLMES P. and LUMLEY J.L. The proper orthogonal decomposition
in the analysis of turbulent flows Annual Review of Fluids Mechanics 25(N5), pp
539-575, (1993).
6zi§lK M.N. and ORLANDE H.R.B. Inverse Heat Transfer, Taylor and Francis: New
York, 2000.
FIDAP 8.7.0, Fluent Inc., www.fluent.com

INVERSE PROBLEMS IN ENGINEERING MECHANICS IV
M. Tanaka (Editor)
© 2003 Elsevier Ltd. All rights reserved. 13
ESTIMATION OF THERMOPHYSICAL PROPERTIES
OF A DRYING BODY AT HIGH MASS TRANSFER BIOT NUMBER
G. H. KANEVCE
Macedonian Academy of Sciences and Arts, Skopje, Macedonia
E-mail: [email protected]
L. P. KANEVCE
Faculty of Technical Sciences, St. Kliment Ohridski University, Bitola, Macedonia
E-mail: [email protected]
G. S. DULIKRAVICH
Multidisciplinary Analysis, Inverse Design and Optimization(MAIDO) Institute
Department of Mechanical and Aerospace Engineering, UTA Box 19018
The University of Texas at Arlington, Arlington, Texas 76019, U.S.A.
E-mail: dulikra@mae. uta. edu
ABSTRACT
The inverse. problem of simultaneously estimating the moisture content and temperature-
dependent moisture diffusivity together with the heat and mass transfer coefficients by using
only temperature measurements is analysed in this paper. In the convective drying practice,
usually the mass transfer Biot number is very high and the heat transfer Biot number is very
small. This leads to a very small temperature sensitivity coefficient with respect to the mass
transfer coefficient relative to the temperature sensitivity coefficient with respect to the heat
transfer coefficient. Under these conditions the relative error of the estimated mass transfer
coefficient is high. To overcome this problem, in this paper the mass transfer coefficient is
related to the heat transfer coefficient through the analogy between the heat and mass transfer
processes in the boundary layer.
KEYWORDS
Inverse approach, drying, thermophysical properties, heat and mass transfer coefficients
INTRODUCTION
Inverse approach to parameter estimation in last few decades has become widely used in
various scientific disciplines. This paper deals with the application of the inverse approaches in
drying.

14 G.H. Kanevce et al.
Drying is a complex process of simultaneous heat and moisture transport within material and
from its surface to the surroundings caused by a number of mechanisms. There are several
different methods of describing the drying process. In the approach proposed by Luikov [1]
from the concepts of irreversible thermodynamics the moisture and temperature fields in the
dried body are described by a system of two coupled partial differential equations. The system
of equations incorporates coefficients, which are functions of temperature and moisture
content, and must be determined experimentally. For practical calculations the influence of the
temperature and moisture content on all the transport coefficients except for the moisture
diffusivity is small and can be neglected. The moisture diffusivity dependence on moisture and
temperature exerts a strong influence on the drying process calculation. This effect cannot be
ignored for the most of practical cases. All the coefficients except for the moisture diffusivity
can be relatively easily determined by experiments. The main problem in the moisture
diffusivity determination by classical or inverse methods is the difficulty of local moisture
content measurements within the drying body.
Kanevce, Kanevce and Dulikravich [2, 3, 4, 5] and Dantas, Orlande and Cotta [6, 7] recently
analysed application of inverse approaches to estimation of drying body parameters. The main
idea of the applied method is to take advantage of the interrelation between the heat and mass
(moisture) transport processes within the drying body and from its surface to the surrounding
media. Then, the drying body parameters' estimation can be performed on the basis of accurate
and easy-to-perform thermocouple temperature measurements by using an inverse approach.
We analysed this idea of the drying body parameters' estimation by using temperature response
of a body exposed to convective drying. An analysis of the influence of the drying air
parameters and the drying body dimensions was conducted. In order to perform this analysis,
the sensitivity coefficients and the sensitivity matrix determinant were calculated.
In the convective drying practice, usually the mass transfer Biot number is very high and the
heat transfer Biot number is very small due to the low moisture diffusivity value relative to the
thermal conductivity for most of the moist materials. This leads to a very small temperature
sensitivity coefficient with respect to the mass transfer coefficient relative to the temperature
sensitivity coefficient with respect to the heat transfer coefficient. This indicates that in these
cases the mass transfer coefficient cannot be estimated simultaneously with the heat transfer
coefficient with sufficient accuracy. To overcome this problem, in this paper the mass transfer
coefficient is related to the heat transfer coefficient through the analogy between the heat and
mass transfer processes in the boundary layer.
The objective of this paper is an analysis of the possibility of simultaneous estimation of the
thermophysical properties of a drying body and the heat and mass transfer coefficients at high
mass transfer Biot number by using only temperature measurements.
MATHEMATICAL MODEL OF DRYING
In the case of an infinite flat plate of thickness 21, if the shrinkage of the material can be
neglected (ps = const), the unsteady temperature field, T{x, t\ and moisture content field, X(x,
0, in the drying body are expressed by the following system of coupled nonlinear partial
differential equations
CO, — = — k— +sp_A//— (1)
' dt dx[ dx ' dt

Thermophysical properties of a drying body 15
^A-_^o'JL,D,'L^ (2)
dt dx y dx dx)
Here, t, x, c, k, AH, s, 8, D, ps are time, distance from the mid-plane of the plate, heat capacity,
thermal conductivity, latent heat of vaporization, ratio of water evaporation rate to the
reduction rate of the moisture content, thermo-gradient coefficient, moisture diffusivity, and
density of the dry plate material, respectively.
Fig.l. Scheme of the drying experiment.
As initial conditions, uniform temperature and moisture content profiles are assumed
^ = 0 T{X,0) = TQ, X(X,0) = XQ (3)
In the convective drying experiment (Fig. 1) the surfaces of the drying body are in contact with
the drying air thus resulting in a convective boundary conditions for both the temperature and
the moisture content
The convective heat flux, 7^(0, and mass fiux, jm(t), on these surfaces are
J,=h{T,-T,.,) ^5^
where h is the convection heat transfer coefficient and ho is the mass transfer coefficient, while
Ta is the drying air bulk temperature.
The convection heat and mass transfer coefficients can be expressed by the Nesterenko's
relations [1] for the heat and mass Nusselt numbers in drying conditions
Nu = 0.0210Pr'''Re''Gu'''' (6)
Nu^=0.024SSc'''Re''Gu'^'' (7)
where Pr, Sc, Re, Gu are Prandtl, Schmidt, Reynolds, and Guhman number, respectively.
The water vapor concentration in the drying air, C«, is calculated by
c =—JlM^A^ (8)
" 461.9(7^+273)
whcTQps is the saturation pressure. The water vapor concentration of the air in equilibrium with
the surface of the body exposed to convection is calculated by

16 G.H. Kanevce et al.
"^^ 461.9(7^^^+273)
The water activity, a, or the equilibrium relative humidity of the air in contact with the
convection surface at temperature TX=L and moisture content JG=z, is calculated from
experimental water sorption isotherms.
The problem is symmetrical, and boundary conditions on the mid-plane of the plate (x = 0) are
^1 =0, W =0 (10)
In order to approximate the solution of Eqs. (1, 2), an explicit procedure has been used [8].
ESTIMATION OF PARAMETERS
The estimation methodology used is based on minimization of the ordinary least square norm
E(P) = [ Y - T(P)]^ [ Y - T(P)] (11)
Here, Y^ = [>^i,>2, ••• ,^imax] is the vector of measured temperatures, T^ = [^i(P), ^2(P), •••
^imax(P)] is the vcctor of estimated temperatures at time /j (i = 1,2, ..., imax), P^= [P\,P2, ••
PN] is the vector of unknown parameters, imax is the total number of measurements, and N is
the total number of unknown parameters (imax > N).
A version of Levenberg-Marquardt method was applied for the solution of the presented
parameter estimation problem [9, 10]. This method is quite stable, powerful, and
straightforward and has been applied to a variety of inverse problems. It belongs to a general
class of damped least square methods [11], The solution for vector P is achieved using the
following iterative procedure
P^^'=P^+[(J^)''J^+^^I]-'(J0^[Y-T(P0], (12)
where r is the number of iterations, I is identity matrix, |i is the damping parameter, and J is the
sensitivity matrix defined as
dT, dT,
(13)
The presented iterative procedure stops if the norm of gradient of E (P) is sufficiently small, if
the ratio of the norm of gradient of E (P) to the E (P) is small enough, or if the changes in the
vector of parameters are very small [12].
RESULTS AND DISCUSSION
The proposed method of the moisture difflisivity estimation by temperature response of a
drying body was tested for a model material which was a mixture of bentonite and quartz sand
with known thermophysical properties [8]. From the experimental and numerical examinations

Thermophysical properties of a drying body 17
of the transient moisture and temperature profiles [8] it was concluded that for the calculations
in this study, the influence of the thermal diffusion is small and can be ignored. It was also
concluded that the Luikov's system of two simultaneous partial differential equations could be
used. In this case, the transport coefficients can be treated as constants except for the moisture
diffusivity. The appropriate mean values for the model material are:
Density of the dry solid, ps =1738 kgW,
Heat capacity, c = 1550 J/KTkg db.
Thermal conductivity, k = 2.06 W/m/K,
Latent heat of vaporization, A//= 2.3110^ J/kg,
Phase conversion factor, s = 0.5, and
Thermo-gradient coefficient, 5 = 0.
The following expression can describe the experimentally obtained relationship for the
moisture diffusivity.
-12 ^-2^^ +273V'
D=:9.0\0-''X-'\ (14)
V 303 j
The experimentally obtained desorption isotherms of the model material is presented by the
empirical equation
a = I - exp(-l.5 • 10^(r + 273)-'-'^ X^-'''' (^-273H3.91)^ ^^^^
where the water activity, a, represent the relative humidity of the air in equilibrium with the
drying object at temperature T and moisture content X.
For the direct problem solution, the system of equations Eq. (1) and Eq. (2) with the initial
conditions Eq. (3) and the boundary conditions Eq. (4) and Eq. (10) was solved numerically
with the experimentally determined thermophysical properties.
For the inverse problem investigated in this paper, values of the moisture diffusivity, D, and,
heat and mass transfer coefficients, h and ho, are regarded as unknown. All other quantities
appearing in the direct problem formulation were assumed to be known.
The moisture diffusivity of the model material has been represented by the following function
of temperature and moisture content
2^7 + 273 y^
t 303 J
(16)
where Dx and DT are constants. Thus, the vector of unknown parameters is
F'' =[D^,D^,h,h^] (17)
For the estimation of these unknown parameters, the transient readings of a single temperature
sensor located in the mid-plane of the sample were considered (Fig. 1). The simulated
experimental data were obtained from the numerical solution of the direct problem presented
above, by treating the values and expressions for the material properties as known. In order to
simulate real measurements, a normally distributed error with zero mean and standard
deviation, a of 1.5 *^C was added to the numerical temperature response.
The sensitivity coefficients analysis has been carried out for a plate of thickness 2Z = 4 mm,
with initial moisture content of X(x, 0) = 0.20 kg/kg and initial temperature T(x, 0) = 20^C.

18
G.H. Kanevce et al.
Following the conclusions of the previous works [3, 4, 5] the drying air bulk temperature of Ta
= 80 C, and drying air speed of F^, = 10 m/s, have been chosen. The relative humidity of the
drying air was cp = 0.12.
CO
c
O
!tE
0)
o
u
>%
>
0)
CO
70-
50-
so
lo-
-10-
-30-
/ \ ^^^
1 \^^^
/ \ ^^
/ ^/\ ^^^^T
_ 1 i ~~^p^ \^
\ ho
1 1 1 1
15 20
time [min]
Fig. 2. Relative sensitivity coefficients for the convective drying experiment
Figure 2 shows the relative sensitivity coefficients PmdTi/dPm, i = 1, 2,..., 101, for temperature
with respect to all unknown parameters, Dx, DT, /Z, /ZD (m = 1, 2, 3, 4). It can be seen that the
temperature sensitivity coefficient with respect to the convection mass transfer coefficient ho is
very small relatively to the temperature sensitivity coefficient with respect to the convection
heat transfer coefficient h. The very high mass transfer Biot number and the very small heat
transfer Biot number can explain this. The heat transfer Biot number is 0.08. The mass transfer
Biot number ranged from 200 to
content and temperature change.
MO and changes during the drying with local moisture
Figure 3 presents transient variation of the determinant of the information matrix if four, (Dx,
DT, h, ho) and three {Dx, Dj, h) parameters are simultaneously considered as unknown.
Elements of this sensitivity determinant were defined [10] for a large, but fixed number of
transient temperature measurements (101 in these cases).
Fig. 3. Determinant of the information matrix

Thermophysical properties of a drying body 19
Case
P3
P4a
P4b
Table 1.
Estimated
|s| [%]
Estimated
|s| [%1
Estimated
|s| [%1
Exact values
. Estimated parameters (a
DxlO^^ DT
[m'/s] [-]
8.93 9.95
0.83 0.54
9.87 8.79
9.61 12.08
8.89 10.02
1.27 0.24
9.00 10.0
= 1.5°C)
h
[W/m^/K]
83.03
0.09
83.17
0.08
83.02
0.09
83.1
fe-10^
[m/s]
8.28
10.87
9.28
0.09
9.29
The drying time corresponding to the maximum determinant value was used for the
computation of the unknown parameters. Table 1 shows the computationally obtained results.
For comparison, the exact values of parameters are shown in the bottom row. The relative
errors of the estimated parameters, s, are also shown in the table.
From the obtained results in the case P3, it appears to be possible to estimate simultaneously
the moisture diffusivity parameters, Dx and Dj, and the convection heat transfer coefficient, h,
by a single thermocouple temperature response with the relatively high noise of 1.5 ^C. But,
the accuracy of computing parameters in the case of simultaneous estimation of the moisture
diffusivity parameters, Dx and DT, and the convection heat and mass transfer coefficients, h and
ho (case P4a) is small. The very small values of the relative sensitivity coefficient with respect
to the mass transfer coefficient (Fig. 2) can explain this.
To overcome this problem, in this paper the mass transfer coefficient is related to the heat
transfer coefficient through the Eqs. (6) and (7), obtained from the analogy between the heat
and mass transfer processes in the boundary layer over the drying body. From Eqs. (6) and (7),
with accuracy within 1%, following relationship can be obtained
h^=0.95^h (18)
where Da and ka are moisture diffusivity and thermal conductivity in the air, respectively. The
obtained relation is very close to the well-known Lewis relation. By using the above relation
between the heat and mass transfer coefficients, they can be estimated simultaneously with the
moisture diffusivity parameters with high accuracy (case P4b in table 1). However, local
minimums have been obtained depending on the initial guesses. To overcome this problem, a
hybrid optimization algorithm like OPTRAN [13] will be used in the future work.
CONCLUSIONS
An analysis of the possibility of simultaneous estimation of the thermophysical properties of a
drying body and the heat and mass transfer coefficients at high mass transfer Biot number by
using only temperature measurements was presented. By using an interrelation between the
heat and mass transfer coefficients, they were simultaneously estimated with the two moisture
diffusivity parameters with high accuracy. Depending on the initial guesses, local minimums
have been often obtained during the analysis. To overcome this problem, application of the
hybrid optimization algorithm OPTRAN will be analysed in the future.

2^ G.H. Kanevce et al.
REFERENCES
1. Luikov, A.V. (1972). Teplomassoobmen (in Russian), Moscow, Russia.
2. Kanevce G. H., Kanevce L. P. and Dulikravich G. S. (2000). In: Inverse Problems in
Engineering Mechanics II- ISIP 2000, eds: Tanaka, M. and Dulikravich, G. S., Elsevier,
Amsterdam.
3. Kanevce G. H., Kanevce L. P. and Dulikravich G. S. (2000). In: Proceedings of
NHTC'OO, ASME paper NHTC2000-12296, 34^^ ASME National Heat Transfer
Conference, August 20-22, Pittsburgh, PA, U.S.A.
4. Kanevce, G., Kanevce, L., Mitrevski, V. and Dulikravich, G.S. (2000). In: Proceedings
of the 12^^ International Drying Symposium, IDS'2000, August 28-31, Noordwijkerhout,
The Netherlands.
5. Kanevce G. H., Kanevce L. P. and Dulikravich G. S. (2002). In: Inverse Problems in
Engineering Mechanics III- ISIP 2001, eds: Tanaka, M. and Dulikravich, G. S., Elsevier,
Amsterdam.
6. Dantas L. B., Orlande H. R. B., Gotta R. M., Souza R. and Lobo P. D. C. (1999). In:
Proceedings of Jf^ Brazilian Congress of Mechanical Engineering, November 22-26,
Sao Paulo, Brazil.
7. Dantas L. B., Orlande H. R. B. and Gotta R. M. (2000). In: Inverse Problems in
Engineering Mechanics II- ISIP 2000, eds: Tanaka, M. and Dulikravich, G. S., Elsevier,
Amsterdam.
8. Kanevce, H. G. (1998). In: Proceedings of IDS '98, Vol. A , pp. 256-263, Halkidiki,
Greece.
9. Marquardt, D.W. (1963). J. Soc. Ind Appl. Math 11, 431 -441.
10. Ozisik, M.N. and Orlande, H.R.B. (2000). Inverse Heat Transfer: Fundamentals and
Applications, Taylor and Francis, New York
11. Beck, J.v., and Arnold, K.J. (1977). Parameter Estimation in Engineering and Science,
Wiley, New York.
12. Pfafl, R.C., and Mitchel, J.B. (1969). AIAA Paper, No.69602.
13. Dulikravich, G.S., Martin, T.J., Dennis, B.H. and Foster, N.F.(1999) Chapter 12 in
EUROGEN'99 - Evolutionary Algorithms in Engineering and Computer Science: Recent
Advances and Industrial Applications, (eds: K. Miettinen, M. M. Makela, P.
Neittaanmaki and J. Periaux), John Wiley & Sons, Ltd., Jyvaskyla, Finland, May 30 -
June3, 1999, pp. 231-260.

INVERSE PROBLEMS IN ENGINEERING MECHANICS IV
M. Tanaka (Editor)
© 2003 Elsevier Ltd. All rights reserved. 21
BOUNDARY AND GEOMETRY INVERSE THERMAL PROBLEMS IN
CONTINUOUS CASTING
I. NOWAKt, A.J. NOWAK* and L.C. WROBEL*
t Institute of Mathematics, ^ Institute of Thermal Technology,
Technical University of Silesia, 44-101 Gliwice, Konarskiego 22, Poland
' Brunei University, Dept. of Mechanical Engineering, Uxbridge, Middlesex, UBS 3PH, United
Kingdom
* corresponding author: [email protected]
ABSTRACT
This paper discusses an algorithm for the boundary condition estimation and the phase change
front identification in continuous casting process. This kind of problems are formulated as the
inverse problems. Since both boundary conditions and front location are searched simultane
ously, the task resolved is called in this paper an inverse boundary-geometrical problem. The
solution procedure utilizes sensitivity coefficients and temperature measurements inside the solid
phase. The algorithms proposed make use of the Boundary Element Method (BEM), with cubic
boundary elements. The Bezier splines are employed for modelHng the interface between the
solid and liquid phase.
In order to demonstrate the main advantages of the developed formulation, continuous casting
of aluminium alloy was considered as a numerical example. The results obtained were compared
to the experimental data and results acquired from other authors.
KEYWORDS
inverse geometry problem, sensitivity coefficients, BEM solution of continuous casting, Bezier
splines.
INTRODUCTION
Growing demand for high-quality alloys possessing specific properties stimulate frequent appli
cation of the continuous casting process in the metallurgical industry. The solidification of metal
or alloy takes place in a mould (cristallizer) cooled by a flowing water. The liquid material flows
into the mould where the solidifying ingot is then pulled by withdrawal rolls and very intensively
cooled outside the crystallizer (by water sprayed over the surface).
Definition of the process conditions and an accurate determination of the interface location
between the liquid and solid phases is very important for the design of cooling which controls
the process and also influence the quality of the casted material. These problems were the topics
of works dealing with the boundary and the geometry inverse problems. The boundary inverse
problem consists in determination of heat flux distribution along outer boundary of the ingot.

22 /. Nowak et al.
The numerical procedures were proposed basing on sensitivity analysis and boundary element
method [11]. The influence of the number and accuracy of measurements were tested. In this
work, numerically simulated temperature measurements were used.
The geometry inverse problem concerns the estimation of phase change front location. The pub
lications [8,7] include details of the solving algorithms and the method of modeling the boundary
shape. Special attention was paid to the application of the Bezier splines for the phase change
boundary approximation [10]. The way of using sensitivity analysis taking into account the sen
sitivity coefficients determination (in case of quadratic or cubic boundary elements application)
is widely presented in works [8, 10].
It has to be stressed that mathematical model of boundary inverse problem as well as geom
etry inverse problem include solid ingot only. It means that the problem is solved for the solid
phase and the liquid part interaction is captured by the temperature measurements.
Presented paper proposes an algorithm being a combination of both boundary and geometri
cal inverse problems in one task. The heat flux distribution along outside surface of the ingot and
the phase change front location are estimated at the same time. The calculations are based on the
measurements obtained in a real experiment.
These measurement data were acquired in continuous casting of aluminum alloy. Although
the algorithms and experiences acquired during the previous research concern pure metals (mainly
copper) [10, 8,7,11] there is made an attempt to solve the boundary-geometrical inverse problem
on the strength on experimental measurements.
PROBLEM FORMULATION
This section starts with a brief description of the mathematical model of the direct heat transfer
problem for continuous casting. This model serves as a basis for the inverse problem discussed
in detail in the remainder of the section.
The mathematical description of the above phenomenon, defined as the 2D steady-state
diffusion-convection heat transfer, consists of:
• a convection-diffusion equation for the solid part of the ingot:
V'T(v)--v,^=(i (1)
a ox
where T(r) is the temperature at point r, v^ stands for the casting velocity (assumed to be
constant and in the x-direction) and a is the thermal diffusivity of the solid phase.
• boundary conditions defining the heat transfer process along the boundaries ABCDO (Fig
ure 1), including the specification of the melting temperature along the phase change front:
(2)
(3)
(4)
(5)
(6)
where Tm stands for the melting temperature, Ta is the ambient temperature, Tg is the ingot
temperature while leaving the system, A is the thermal conductivity, h stands for convective
T(r)
dT
dn
dT
dn
T(r)
dT
dn
=
-
=
=
=
Tm,
q{r),
MT(r)-
Ts,
0,
-TJ,
reTAB
reTBC
reTcD
re Too
reToA

Boundary and geometry inverse thermal problems 23
heat transfer coefficient and q is the heat flux. All symbols T^, T^, T^ represent constant
temperatures.
A
O
X
liquid
J
h-
solid
y
B
C
n
Figure 1: Scheme of the continuous casting system and the 2D domain under consideration.
In the inverse analysis the location of the phase change front TAB (where the temperature is
equal to the melting temperature) and the boundary conditions along TBCD are unknown. This
means that the mathematical description needs to be supplemented by measurements because it
is incomplete. In this work, it is assumed that, the temperatures Ui were measured inside the
ingot using the L-rod technique [3, 4] and collected in a vector U.
The objective is to estimate the identified values uniquely describing the phase change front
location and the heat flux distribution along TBCD- These values are collected in the vector
Y = bl, . - . , y2n+mY = [Vi, . . . , W2n, q'l, • • • , qmV•
Components vi are connected with the front location and they are the coordinates of the
Bezier spline control points, components qj are connected with the heat flux distribution. The
way of modefling both quantities is discussed in subsequent section.
Because of ill-conditioned nature of all inverse problems, the number of measurements should
be appropriate to make the problem overdetermined. This is achieved by using a number of mea
surement points greater than the number of design variables. Thus, in general, inverse analysis
leads to optimization procedures with least squares calculations of the objective functions A.
However, in the cases studied here, an additional term intended to improve stability is also intro
duced [9, 6], i.e.
{Y - YV Wy^ {Y -
U)4-
(7)
where vector Tcai contains temperatures calculated at the sensor locations, U stands for the vec
tor of temperature measurements and superscript T denotes transpose matrices. The symbol W
denotes the covariance matrix of measurements. Thus, the contribution of more accurately mea
sured data is stronger than data obtained with lower accuracy. Known prior estimates of design

24 / Nowak et al
vector components are collected in vector Y, and Wy stands for the covariance matrix of prior
estimates. The coefficients of matrix Wy have to be large enough to catch the minimum (these
coefficients tend to infinity if prior estimates are not known). It was found that the additional term
in the objective function, containing prior estimates, plays a very important role in the inverse
analysis, since it considerably improves the stability and accuracy of the inverse procedure.
The present inverse problem is solved by building up a series of direct solutions which grad
ually approach the correct values of design variables. This procedure is split into boundary and
geometry parts and can be expressed by the following main steps:
1. make the boundary problem well-posed. This means that the mathematical description of
the thermal process is completed by assuming arbitrary but known values Y* (as required
by the direct problem).
2. geometrical part - solve the direct problem obtained above and calculate temperatures T*
at the sensor locations; compare these temperatures and measured values U and modify
the assumed data ^*, j = 1,2,..., 2n keeping ql, k = 1,2,..., m unchanged - this part is
solved like a typical inverse boundary problem [11]
3. point 2. should be repeated until Vj is converged
4. boundary part - ones the previous step is completed continue iterations and compare T*
and measured values U and modify the assumed data ql, k = 1,..., m keeping v*, j =
1,2,..., 2n unchanged - typical inverse geometry problem [8, 10]
5. if it is necessary the external loop (points 2. to 4.) can be repeated
In both parts of the above algorithm the sensitivity analysis is applied and minimization of
the objective function (7) leads to the following set of equations [9, 8]:
(Z^ W-* Z + WY^) Y = Z"^ W-^ (U - T*) +
(Z^W-'Z) Y* + WY'Y (8)
where the sensitivity coefficients (found at measurement points) are collected in the matrix Z.
The sensitivity coefficients as the main concept of sensitivity analysis are the derivatives of
the temperature at point i with respect to the identified values at point j, ie.
dT,
They provide a measure of each identified value and indicate how much it should be modified.
The sensitivity coefficients are obtained by solving a set of auxiliary direct problems in suc
cession. Each of these direct problems arises through differentiation of equation (1) and corre
sponding boundary conditions (2)—(6) with respect to the particular component of vector Y {Vj
in geometry and qk in boundary part). For the sake of different nature of both parts, it is necessary
to build and solve different direct problems connected with Vj or q^ values, respectively.
In this work, the BEM [1, 12] is appUed for solving both the direct thermal and the sensitivity
coefficient problems. The main advantage of using this method is the simpUfication in mesh
ing since only the boundary has to be discretized. This is particularly important in the inverse
geometry problems in which the geometry of the body is changed in each iteration step. Further
more, the location of the internal measurement sensors does not affect the discretization. Finally,
in the heat transfer analysis, the BEM solutions directiy provide temperatures and heat fluxes.

Boundary and geometry inverse thermal problems 25
which are both required by the inverse solutions. In other words, the numerical differentiation of
temperature (i.e., numerical calculations of the heat fluxes) is not needed.
The BEM system of equations has boundary-only form both for the thermal and the sensitiv
ity coefficient problems
HT = GQ (10)
HZ - GQz (11)
where H and G stand for the BEM influence matrices. Depending on the dimensionality of
the problem, the fundamental solution to the convection-diffusion equation is expressed by the
following formulae [1, 12]
27rA
(12)
X V xj I ^j^
2a
where KQ stands for the Bessel function of the second kind and zero order, r is the distance
between source and field points, with its component along the x-axis denoted by r^.
DETERMINATION OF IDENTIFIED VALUES
As noticed before, the ill-conditioned nature of all inverse problems requires that they have to
be made overdetermined. On the other hand, it is very important to limit the number of sensors,
mainly because of the difficulties with measurements acquisition. This is achieved by application
of the Bezier splines for modelling the phase change front and the approximation the heat flux
distribution by broken line or some spline functions.
Application of the Bezier splines allows to define the phase change front location with simul
taneous limitation of the number of identified values.
The Bezier curves are made up of cubic segments based on four control points VQ, Vi, V2
and V3. The following formula presents the definition of cubic Bezier segments:
P(n) = (1-w)^ Vo + 3(l-w)^ wVi 4-
3(l-«)ti'V2 + w'V3 (13)
where P{u) stands for a point on the Bezier curve, and u varies in the range (0,1).
Numerical experiments have shown that a Bezier curve composed of two cubic segments sat
isfactorily approximates the phase change front [2]. Apart from limitations of identified values,
the application of the Bezier curves (cubic polynomials) has to ensure smoothness of the bound
ary. The colinear location of control points makes the whole boundary smooth even at points
which are shared by neighbouring segments.
The vector Y = [yi,...,y2n+mV can be writen as Y = [vf,v\...,v^,vy^,qi,...,g^]^
where v^^vf are the x and y coordinate of the given control point. Actually, some of these coor
dinates are defined by additional conditions resulting from the physical nature of the problem.
In consequence, the number of identified values can be limited to ten [7, 10], which also means
a reduction in the number of required measurements. This reduction is essential, mainly because
of practical difficulties connected with measurement acquisition during the experiment.

26 /. Nowak et al
The calculation of the sensitivity coefficients in geometrical part consists of solving a set
of direct problems. Each of these direct problems arises through differentiation of equation (1)
and corresponding boundary conditions (3)—(6) with respect to the particular design variable yj.
Thus, the resulting field Zj is governed by an equation of the form:
V%(r)-itv#^ = 0 (14)
a dx
Differentiation of the boundary conditions (3)—(6) produces conditions of the same type as in
the original thermal problem, but homogenous. The boundary condition along the phase change
front TAB is also obtained by differentiation of equation (2) and after some calculations [7, 10]
leads to the final form:
^,= - - I [q-n cos(cv) + Qr sm(a)] — + [qn sm{a) - qr cos(a)] — > (15)
where qr, qn are the tangential and normal components of the heat flux and cos(a), sin(a) the
direction cosines of the outward normal vector.
The derivatives of x and y with respect to the design variable y^ depend on the particular
geometrical representation of the phase change front. In presented calculations there are obtained
by differentiation the formula (13) with respect to design variable y^ (in fact coordinate of the
Bezier control point t'^ or v^-).
Solving the direct problems for all design variables, the whole sensitivity matrix Z can then
be constructed and used for building the set of equations (8) and calculate estimated values yj.
It has to be noted that the geometry inverse problems are always non-linear. It means that the
iteration procedure has to be applied. Usually the iteration procedure is conducted till the con
vergence criteria are satisfied. In this problem the interface determination makes only a part of
the whole algorithm, so the solution is realized in a different manner. Mainly, several iterations
of the geometric part are done, and despite not reaching the final solution, the algorithm skips
to the boundary problem. Afterwards the geometric problem is iterated again with estimating
boundary conditions. Such solution procedure is necessary to find the proper results.
In order to determine the boundary conditions along TBCD^ the heat flux distribution has to be
found. Preliminary calculations show that these distribution can be approximated by appropriate
spline function or broken line, based on m parameters. The number of these parameters do not
affect the way of calculations and leads only to increased number of the identified values. In the
continuous casting of copper[9,6], the heat flux varies linearly along the mould and exponentially
along the water spray. It means that the heat flux distribution is described only by three values. In
some different cases such approximation is unfeasible or too inaccurate. In the continuous casting
of alloy of aluminium, analyzed in the paper, the heat flux distribution along the boundary TBCD
is described by broken line based on nine points (xk, qk),k =^ 1,..., 9. In each interval [xk-i,Xk]
A: = 2,..., 9 the distribution is represented by a segment in form:
g(^) = ^^'^^-^(x - x,_0 + 9,_i (16)
Xk - Xk-l
The heat fluxes g^ are components of vector Y which estimation is an objective of the problem.
Similar approach concerning sensitivity analysis is applied for the boundary part and relevant
sensitivity analysis. The auxiliary direct problem arises through differentiation of the thermal
problem (1), the boundary conditions (2), (5), (6) and functions (16) (i.e. the boundary condition
along TBCD) with respect to q^. This leads to adequate governing equation:
V'Z,(v)--v,^ = 0 (17)
a OX

Boundary and geometry inverse thermal problems 27
with homogenous boundary condition along boundaries TABI ^DO, ^OA and conditions along the
surface TBCD in the form:
dn
{X - Xk-l)
{Xk - Xk-l) '
{x - Xk)
{Xk+l
0,
•Xk)'
X e [xk-i,Xk
X e [xk-i,Xk]
X^ [Xk-uXk+i]
-2,.. (18)
Calculated sensitivity coefficients are collected in matrix Z and introduced into the system of
equations (8). It has to be noted that the inverse boundary problems (to which boundary part is
reduced) is a linear one. It means that the iteration procedure is not needed in this part.
ALGORITHM VERIFICATION
In presented work the temperature measurements acquired from an experiment were used. Drezet
et al. [3] applied the L-Rod Technique schematically shown in Figure 2. This technique was
employed for the measurement of temperatures along the ingot surface as well as inside the
ingot. Five thermocouples were placed within the tube (one of them being at the very surface
of the ingot and remaining 5, 10, 15 and 20mm under exterior boundary). Additionally, the five
beads were located a few millimeters below the L-rod tube. Their final position was checked
after solidification by X-ray inspection.
Figure 2: Scheme of the experimental setup used for temperature measurement in continuous
casting (L-Rod Technique) (according to Drezet et al. [3])
During the measurement trial, the L-rod first drops into the liquid metal under its own weight
before being "swallowed" by the mushy zone. From that instant, the L-rod moves downward at
the casting speed of the ingot.
NUMERICAL RESULTS
In presented work the boundary conditions and the phase change front location were esti
mated by using measurements obtained in experiment presented above. In order to compare

28 /. Nowak et al.
results with those obtained by the authors of experiment, the size of the ingot and the thermal
conditions were analogous as in [3].
It has to be noted that the aims of presented work were different from the aims assumed by
Drezet et al. In consequence the location of measurement points were not optimal with respect
to problem formulated in paper [10]. Additionally, there were some differences between math
ematical model used in the work and by the authors of experiment. First of all, the material
casted in experiment was an aluminum alloy. It means that the solidification occurs along mushy
zone between solidus and liquidus temperatures. An algorithms, applied to solve the problem,
formulated in the paper were built for continuous casting of pure metals (e.g. copper - Cu).
In this situation, one assumes that the phase change front location corresponds to the melting
temperature isotherm and it is estimated like a curve.
In the first step the temperature calculated in the model (in some sensor points) was compared
to the measured values, which is presented in Figure 3. It has to be noted that differences between
Teal and U compose the main part of the objective function (7).
OOB
—I—
0.11 0.12
Figure 3: Measured (figures) and calculated (lines) temperatures at sensor points
Similar comparison, concerning measured and calculated temperature distribution along the
ingot boundary is shown in Figure 4. The vertical da^h Une specifies point where the water spray
starts.
The authors of experiment used the measurements to find, among others, the boundary con
ditions (the heat flux distribution) along the boundary TBCD- This distribution is also a solution
of the presented inverse boundary-geometrical problem. Appropriate comparison of heat flux
distribution found in this work and calculated by Drezet at al. is presented in Figure 5.
It is easy to note that, in spite of the pitch appeared, the character of distribution is preserved.
Observed discrepancies are an effect of some differences in both models.

Boundary and geometry inverse thermal problems 29
Figure 4: Measured and calculated temperatures along the boundary of ingot
0-
-1EH)06-|
Figure 5: The heat flux distribution calculated in the work and by Drezet et al. [3]
The most difficult was to verify the determination of the phase change front location, which
was a part of the solution of the presented inverse problem. The phase change front was not
considered Drezet et al. work [3] therefore there is no information about location of interface
between the liquid and solid phases. Figure 6 shows the phase change boundary found in the
boundary-geometry inverse problem and area in which temperature between solidus and liquidus
temperature was measured (20mm under outside surface). In the model used for computations
the melting temperature was assumed equal to the liquidus one, which can explain the location
of the boundary found (calculated location is situated close to boundary of mushy zone).

30 /. Nowak et al.
- cslojistBd phas8 Changs Ihont
- boundaries of the ireishy zone
points with measued tenp. eq. Tm
(inside tte mushy zone)
points Mffth maasured tenp. eq. Tm
(outside Ihe nnushy zone)
Figure 6: The phase change front location
CONCLUSIONS
Results presented in the paper show that the combination of the geometry and the boundary
inverse algorithms are useful for solving the inverse boundary-geometrical problem. In this kind
of problems the location of the phase change boundary and the boundary conditions are deter
mined simultaneously. The work proves effectiveness of the developed method also in case of
using experimentally measured temperatures.
In order to limit the number of identified values, the phase change front was modelled by two
Bezier curves and the heat flux distribution was approximated by a broken line, based on 9 pa
rameters. It permits to reduce the number of estimated values to 19.
The results obtained show that presented in the paper algorithms can be used in an industrial
calculations. Despite of the mathematical model imperfection, obtained results are compara
ble to those results presented by other authors. Additional verification was possible, thanks to
comparison of measurements and temperature calculated within the model.
ACKNOWLEDGEMENTS
The financial assistance of the National Committee for Scientific Research, Poland, grant no. 8
TlOB 019 20 is gratefully acknowledged herewith.
Special thanks are expressed to J.-M. Drezet and co-workers who provided us with experimental
data.
REFERENCES
[1] Brebbia, C.A., Telles, J.C.F. and Wrobel, L.C.: Boundary Element Techniques - Theory and
Applications in Engineering, Springer-Verlag, Berlin, 1984.
[2] Cholewa R., Nowak A.J., Biaecki R., Wrobel L.C.: Application of Cubic Elements and
Bezier Splines for BEM Heat Transfer Analysis of the Continuous Casting Problem, in
Carino A.: lABEM 2000, Brescia, Italy, Kluver Academic Publishers, in press.
[3] Drezet J.-M., Rappaz M., Grn G.-U., Gremaud M.: Determination of Thermophysical

Boundary and geometry inverse thermal problems 31
Properties and Boundary Conditions of Direct Chill-Calst Aluminium Alloys Using Inverse
Methods, MetalL and Materials Trans. A, 2000, vol. 31A, pp.1627-1634.
[4] Drezet J.-M., Rappaz M., Carrupt B., Plata M., Determination of Thermophysical Proper
ties and Boundary Conditions of Direct Chill-Cast Aluminium Alloys Using Inverse Meth
ods, Metall. and Materials Trans. B, 1995,vol.26B, pp.821-829
[5] Draus, A. and Mazur T.: Corel DRAW Version 2.0 User Handbook, PLJ Publishing House,
Warsaw, 1991 (in Polish).
[6] Kurpisz, K. and Nowak, A.J.. Inverse Thermal Problems, Computational Mechanics Publi
cations, Southampton, 1995.
[7] Nowak, I., Nowak, A.J. and Wrobel, L.C.: Solution of inverse geometry problems using
Bezier splines and sensitivity coefficients, in Inverse Problems in Engineering Mechanics
III, Proc. of ISIP2001, Nagano, Japan, Tanaka, M. and Dulikravich, G.S. (eds), pp. 87-97,
Elsevier, 2001.
[8] Nowak, I., Nowak, A.J. and Wrobel, L.C.: Tracking of phase change front for continuous
casting - Inverse BEM solution, in Inverse Problems in Engineering Mechanics II, Proc. of
ISIP2000, Nagano, Japan, Tanaka, M. and Dulikravich, G.S. (eds), pp. 71 - 80, Elsevier,
2000.
[9] Nowak A.J.: BEM approach to inverse thermal problems, Chapter 10 in Ingham, D.B. and
Wrobel, L.C. (eds). Boundary Integral Formulations for Inverse Analysis, Computational
Techanics Publications, Southampton, 1997.
[10] Nowak, I., Nowak, A. J. and Wrobel, L.C: Identification of phase change fronts by Bezier
splines and BEM, Int. Joumal of Thermal Sciences, vol. 41(6), 2002, pp.492-499
[11] Nowak, I., Nowak, A.J. : Applications of sensitivity coefficients and boundary element
method for solving inverse boundary problems in continuous casting , Materiay XVII
Zjazdu Termodynamikw, Zakopane 1999, pp.999-1008.
[12] Wrobel, L.C. and Aliabadi, M.H.: The Boundary Element Method, Wiley, Chichester, 2002.

INVERSE PROBLEMS IN ENGINEERING MECHANICS IV
M. Tanaka (Editor)
© 2003 Elsevier Ltd. All rights reserved. 33
BOUNDARY VALUE IDENTIFICATION ANALYSIS
IN THE UNSTEADY HEAT CONDUCTION PROBLEM
I.GUSHIKEN ^^ and N.TOSAKA^^
1) Graduate School ofNihon University, Narashino, Chiha, 275-8575, Japan
2) Department of Mathematical Information Engineering, Nihon University, Narashino,
Chiba, 275-8575, Japan
ABSTRACT
In some special fields in thermal engineering, we encounter with a thermal conduction
problem in which the heat flux on the boundary is prescribed step-wisely. In this paper we
identify this kind of boundary condition from several temperatures at interior points in
one-dimensional thermal body. This inverse problem of one-dimensional heat conduction
equation is discussed with the finite difference method in solving the governing equation and
the filtering algorithm in detecting step-wisely time depending function. Numerically detected
results by using the Kalman filter and the projection filter are compared with the prescribed
thermal flux.
KEYWORDS
Unsteady heat conduction problem, step-wise condition, boundary heat flux, Kalman filter,
projection filter
INTRODUCTION
The inverse problem discussed in this paper concerns with unsteady heat conduction
phenomena. In order to solve this inverse problem we must determine exactly temperatures
which satisfy the time-dependent heat equation. We would like to identify the unknown
quantities with temperatures determined exactly. Consequently, our inverse problem can be
formulated mathematically as inverse problems of unsteady heat conduction equation. In
solving such kind of inverse problem, we are required two solution procedures which are an
identification method to determine unknown quantity and a solution method to solve a
governing differential equation.
The inverse problems in general must be solved under the consideration of stochastic
properties of the mathematical model because observations on the problem of engineering are
usually measured in presence of noise. The Kalman filter algorithm has been well known as the
analytical method that is able to consider with stochastic properties [1,2]. There exist several
solution algorithms excepting the Kalman filter [3]. Applicability and efficiency of the filtering
algorithm with any numerical solution procedure of the continuous field equation were shown
in paper [4,5].

34 / Gushiken andN. Tosaka
The inverse problem of one-dimensional heat conduction equation is discussed with the
finite difference method in solving the governing equation and the filtering algorithm in
identifying step-wisely time depending function. Numerically identified results by using the
Kalman filter and the projection filter compared with the prescribed thermal flux in our model.
Moreover, we mention to identification of the heat transfer coefficient.
FILTERING ALGORITHM
Generally, since the noise is added to the amount of observation and measurement in many
cases, solution procedure in consideration of the noise is required in solving inverse problem in
engineering. The filtering algorithm is one of powerful procedures. In this study, filtering
algorithms based on the Kalman filter and the projection filter are employed as procedures to
solve the inverse problem. This inverse problem of one-dimensional heat conduction equation
is discussed with the filtering algorithm in detecting step-wisely time depending function. The
dynamic system considered in our inverse problem can be written as discrete time model in the
following form:
State equation:
^M =^k^k+^k'^k (1)
Observation equation:
yit=M^z^+v^ (2)
where the stochastic variable vectors, z^ , y^ , w^^ and v^ are the state vector, the
observation vector, the system noise vector and measurement error vector, respectively. And
F^, D^ and M;^ are the state transition matrix, the driving matrix and the observation matrix,
respectively. The stochastic variable vectors are assumed to possess the following stochastic
properties:
^[wJ = 0,£[w^wf] = J^S^,£[w^zf] = O
£[v,] = 0,£[v,v[] = ^^Q,,£[v,zn = O (3)
£[w^vn = 0,£[(zo-Zo)(zo-Zo)^] = Ro
where £[] denotes expectation, 5j^ is Kronecher delta, S^ and Q^ are the covariance
matrices of the system noise and of the observation noise, respectively. The Kalman and
matrices projection filtering algorithms can be constructed with the recursive procedure based
on the Wiener filter and projection filter, respectively, as follows:
Filtering equation:
''^k+Mk -^k^k/k W
^k/k = Zyt/yt-1 +^k(yk -^k^kik-) (5)

Boundary value identification analysis 35
Filter gain for Kalman filter algorithm:
B^ =R,i/t-iM[(MiR,/,.iM[ +Q)-' (6)
Filter gain for projection filter algorithm:
B,={MIQI'M,)-'MIQ-,' (7)
Estimation error covariance matrix equation for Kalman filter algorithm:
^k/k = ^k/k-i -^k^k^k/k-\ (^)
Estimation error co variance matrix equation for projection filter algorithm:
R^.i/.t=F,R,/,F[+D,S,D[ (10)
R^/^ =R)tM-i-R)t/)t-iM[B[-B,M,R,/,_i+B,(M,R,/,_iM[+Q,)B[ (11)
Initial values:
zoM=Zo , Ro/_i=Ro (12)
where z^/^ denotes the estimation of z^j. by using {yo^y\^"'^yk} ^^^ ^k/k is the
estimation error covariance matrix of z^/^.
In many cases where we apply the dynamic system theory to identification problem of
distributed-parameter systems, we assume that the state vector to be identified can be satisfied
the following state equation:
^k+i iTk 03)
The above state equation expresses a stationary condition that the parameters to be identified
should be kept constant in time. Thus, the state transition matrix F^ reduces to a unit matrix
I and the system noise W;^ is not included.
The inverse problem such that heat fluxes on the boundary points will be identified from
interior measurement data is discussed in this paper. The mathematical model in solving our
inverse problem is given by the following equation and conditions:
Heat conduction equation:
— (X,0 = A:^(X,0 (0<X</,0<0 (14)
Initial condition:
u{x,0) = g (15)

36 /. Gushiken and N. Tosaka
Boundary conditions:
ox
-k~(l,t) = h{u(l,t)-fj
ox
(16)
(17)
Measurement conditions:
w(jc^,/) = w^(/), w(jc^,/) = w^(/) (0<x^,x^ </) (18)
where K = klpC is the thermal conductivity, h,T^,k,p,C and f(t) are the heat transfer
coefficient, bulk temperature, heat conduction coefficient, density, specific heat and heat flux
function, respectively. The quantities with " A " denotes the prescribed values.
Let us solve numerically the governing filed equation (14). Equation (14) can be discretized
by the implicit scheme in the finite difference method as follows:
- 2z/,«.V + (1 + ^^)^i - K-\ = <
(19)
where wf =w(x^,r„) and X^xAtllSx^ . If we divide the unknowns in the system of
equations resulting from Eq. (19) into the temperature quantity z""^ ={ul'^ ,u^ ) on
boundary, the interior temperature quantity u""^^ and the measurement quantity
y"^^ = (w;''\w;'^^)^, Eq. (19) can be rewritten as follows:
a
0
b"
d
^z"^'"
>"".
"f"
g
+
P
.q_
(20)
where the components a,b,d,p and q are the known quantity, and f" and g" are the
known q
follows:
known quantity at n time step. From Eq. (20) we can get each vectors u'''^ and z""^ as
u-'=d-'(g"+qy-')
z"^' =a-'(p-bd-'q)y""'+(f"-bd-V)
(21)
(22)
By comparing Eq. (2) with Eq. (22), the observation matrix M^ in the filtering algorithm can
be constructed as
M4={a-'(p-bd-'q)}- (23)
Since the known components a,b,d,p and q are fixed in each time step, the above
observation matrix M ^ is also fixed.

Boundary value identification analysis 37
NUMERICAL RESULTS
A heat conduction body with length / is set up as shown in Figure 1. The measurement
points are indicated with two black circles in Figure 1. The unknown quantities /" and h^
can be calculated in considering with the given boundary condition from temperatures as finite
difference solutions. The quantities to be identified from measured temperatures in the interior
points are the time dependent function f{t) of condition (16) and heat transfer coefficient h
of condition (17) as follows:
Wv -w, 0
Ax:
^{U^N-Ta)
(24)
(25)
where f^ = f(t^) and /z" =Kt^) are f(t) and h in n time step, respectively. In this
study, two kinds of sample problem are used as our heat conduction body. The condition of
calculation is shown in Table 1.
high temperature side low temperature side
I I I I I I I I I I I I I I I !•!• I
/ (m) ^
• measurement point
Fig.l Boundary value identification analysis
in the unsteady heat conduction field
Table 1 Condition of calculation
Model A Model B
Length of body
The number of division
Specific heat
Density
Heat conduction coefficient
Heat transfer coefficient
Bulk temperature
Maximum of heat flux
Minimum of heat flux
Time cycle
Initial condition
/ (m)
C (J/kgK)
p (kg/m^)
k (W/mK)
h (W/m^K)
Ta (K)
/max (W/m2)
/min (W^rn2)
fc (S)
g (K)
1.00
712
2300
21.2
300
30000
10000
36000
0.01
20
398
8960
378
20000
290
3000000
300000
36
300

38 /. Gushiken andN. Tosaka
The heat flux /(/) is given by step-wisely function as shown Figure 2. In Model A the
maximum value of f{t) is 30,000 and the minimum value is 10,000. In Model B the
maximum of /(/) is 3,000,000 and the minimum is 300,000.
In our numerical implementation of the inverse problem, we adopt numerical results given
by the implicit scheme in the finite difference method of forward problem of the model instead
of experimental data because we can not utilize any measured displacement data in this study.
In two models, we assume that the diagonal element of Q is determined as the distribution of
noise which results in standard deviation corresponding to the ten percent of absolute value of
the maximum value of measured displacements in order to take consideration of the influence
noise included in the measurement. And the estimation error covariance matrix is assumed
Ro = I with the Kalman filter algorithm. The initial temperature distribution is given as
constant value, g = 300.0 and then two initial temperatures on the boundary are prescribed as
zO = (300.0,300.0).
S ^max
S JvMn
Y^
Time (s)
Fig.2 Step-wise condition
The identification results of time dependent function /(/) and heat transfer coefficient h
in Model A are shown in Figure 3 and Figure 4, respectively. From Figure 3-(a) and Figure
4-(a), the results by using the Kalman filter algorithm can not produce correctness. Especially,
the identification of f{t) is very bad. A step-wiseness of f{t) as shovm in Figure 2 can not
be attained. However, as shown in Figure 3-(b) and Figure 4-(b), the result by using the
projection filter algorithm is complete identify time dependent function f{t) and heat transfer
coefficient h.
The identification results of /(/) and h in Model B are shown in Figure 5 and Figure 6,
respectively. From Figure 5-(a) and Figure 6-(a), it can be seen that the identified results by
using the Kalman filter algorithm can not produce correctness. As shown in Figure 5-(b) and
Figure 6-(b), the results by using the projection filter algorithm attain a good identification with
high accuracy in a similar way as Model A.

Boundary value identification analysis 39
- Prescribed value
j2 30000
g 20000
# 200
S -400
I -600
I -800
21.6 28.8
Time (s)
(a) Kalman filter algorithm
21.6 28.8
Time (s)
(b) Projection filter algorithm
Fig.3 The result of f{t) in Model A
•Kalman Prescribed value
Time (s)
(a) Kalman filter algorithm
21.6 28.8
Time (s)
(b) Projection filter algorithm
Fig.4 The result of h in Model A
xlO'^
xlO'*
7.2 14.4 21.6 28.8 36 43.2
xlO^

40 /. Gushiken and N. Tosaka
- Prescribed value 1
5- 50000
E
i 40000
i 30000
I 20000
I 10000
Time (s)
(a) Kalman filter algorithm
144.0 216.0 288.0 360.0
Time (s)
(b) Projection filter algorithm
Fig.5 The result of f{t) in Model B
Time (s)
(a) Kalman filter algorithm
•Projection - - Prescribed value
I r^—ir^-^r^-^r'\^^r''T^nr^~^
216.0 288.0
Time (s)
(b) Projection filter algorithm
Fig.6 The result of h in Model B

Boundary value identification analysis 41
CONCLUSION
In this paper we discuss some special inverse problem concerned with unsteady heat
conduction phenomena. Our efficient solution method in solving inverse problem is filtering
algorithm based on the Kalman filter and the projection filter in conjunction with the implicit
finite difference scheme.
We can show the identification of not only the step-wise time dependent boundary heat flux
but also heat transfer coefficient from two temperatures at interior points in one-dimensional
thermal body. The Kalman filter algorithm can not be identified time dependent function f{t)
on the high temperature side with high accuracy. On the other hand, the projection filter
algorithm can be identified not only f{t) but also heat transfer coefficient h on the low
temperature side with high accuracy. Finally, we can show the applicability and efficiency of
our solution method to boundary value identification problem in unsteady heat conduction
body.
REFERENCES
1. R.E.Kalman (1960) : A new approach to linear filtering and prediction problems,
Trans.ASME, J.Basic Eng., 82D(1), pp.34-45
2. R.E.Kalman and R.S.Bucy (1961) : New results in linear filtering and prediction theory,
Trans.ASME, J.Basic Eng., 83D(1), pp.95-108
3. H.Ogawa and E.Oja (1986): Projection filter, Wiener filter and Karhunen-Loeve subspaces
in digital image restoration, J.Math.Anal.Appl. 114, pp.37-51
4. N.Tosaka, A.Utani and H.Takahashi (1995) : Unknown defect identification in elastic field
by boundary element method .with filtering procedure. Engineering Analysis with Boundary
Elements, Vol.15, Elsvier, pp.207-215
5. N.Tosaka and A.Utani (1997) : Inverse Analysis of Continuous Fields using the BEM with
a Filtering Procedure, In Boundary Integral Formulations for Inverse Problem (eds.
D.B.Ingham and L.C.Wrobel), pp.299-325

INVERSE PROBLEMS IN ENGINEERING MECHANICS IV
M. Tanaka (Editor)
© 2003 Elsevier Ltd. All rights reserved. 43
A HYPER SPEED BOUNDARY ELEMENT- BASED INVERSE CONVOLUTION
SCHEME FOR SOLUTION OF IHCP
A. BEHBAHANINIA and F. KOWSARY
Department of Mechanical Engineering, University of Tehran,
Tehran, Iran
ABSTRACT
In the present paper a new scheme is presented which combines the Sequential Function
Specification Method (SFSM) of Beck and the Dual Reciprocity Boundary Element Method
(DRBEM). In this scheme the unknown boundary condition is estimated sequentially by using
two transformation matrices. One of the matrices takes into account the initial condition, while
the other incorporates the measured temperature data. The matrices defined in the direct heat
conduction calculations by the Dual Reciprocity Boundary Element Method are used as a basis
for the definition of the transformation matrices, and the mathematical derivations for the
inverse estimation are in accordance with the Sequential Function Specification of Beck. A
two-dimensional problem with unknown heat flux components, both temporally and spatially,
is considered as a test case. The inactive surfaces are all adiabatic. In order to compare the
speed and the accuracy of the method with the existing SFS methods, the exact analytical
solution of a simulated test case is utilized. Results indicate an impressive improvement in the
efficiency as well as the accuracy of this scheme as compared with the conventional ones.
KEYWORDS
Inverse conduction, Sequential, Dual reciprocity. Transformation matrices, Boundary element
INTRODUCTION
Heat Conduction problems may be classified as direct and inverse problems. In direct
problems, which occur more frequently in practice, the geometry, thermophysical properties, as
well as boundary and initial conditions are known a priori and are used for the calculation of

44 A. Behbahaninia and F. Kowsary
the temperature distribution throughout the body. In inverse heat conduction, however, some
of the above data are unknown and are to be determined using the measured temperature data
taken from within or the boundary of the problem domain. The applications of inverse heat
conduction problem IHCP can, in turn, be classified into three broad categories of control,
design and identification [1]. In identification and control problems the ultimate objective is
determination of a parameter or a function, whose measurement by direct means is either
impractical or essentially impossible. In design problems, minimizing a target function which
models performance of a thermal system may define an inverse problem.
Inverse problems can in short be defined sis ones in which the "cause" is "estimated" from the
"effecf. These types of problems are generally characterized as being mathematically "ill-
posed"; i.e., being highly sensitive to measurement errors. In Inverse Heat Conduction
Problems where the unknown boundary conditions are to be estimated using measured
temperature data the, so called, lagging and damping effects cause the solution not to be readily
and continuously derived from the measured data [2]. As a result, exact solutions for these
types of problems do not in general exist, and the solutions are usually found on the basis of
minimizing a sum of squares of error function by trials and errors. The well-known example of
this practice is iterative methods such as the Conjugate Gradients Method in which the
unknown boundary condition is estimated by repetitive computation of the temperature at
sensor location, each time with an improved estimate of the unknown boundary condition
obtained by using the conjugate directions, until the desired minimum point of the sum of the
squares of the error function is reached [I].
One of the most efficient IHCP algorithms, however, is the Sequential Function Specification
Method. The underlying idea for this method stems from this fact that the temporal component
of the heat flux (i.e., the unknown boundary condition considered in this work) to be estimated
at time t^ affects only the measured temperatures starting from the time /M and the subsequent
r-l data, where r^At is the amount of time required for the sensor to detect the employed
thermal effect on the boundary. In this Sequential procedure, it is assumed that the heat flux
components up to the time step tM-i is estimated, along with the temperature distribution within
the body, and the heat flux components at time t^ are to be estimated using the following three-
steps algorithm[3]:
1-Calculate the temperature values at sensor locations for times (/A/+/-7, i=l,...,r). This is done
using the temperature distribution at time tM-i as the initial condition and a guessed value
(usually zero) for the unknown boundary condition.
2-Calculate the unknown boundary condition using
q'' =q*+Z''Z(Y-T), (1)
in which Y and T are, respectively, measured and calculated temperatures; also defined in
equations (27) and (28), q*is the initial gusted value for heat flux and Z is sensitivity
coefficient. Equation one is obtained from Taylor expansion of T(qM) about zero heat flux and
minimizing the sum of squares of the error function.
3-Calculate the temperature distribution at time /M, which is used subsequently as an initial
condition for the following time step.
As the outlined procedure demonstrates, the efficiency of the method owes itself to the fact that
it involves only two loops of trial and error, culminating precisely at the desired solution after

A hyper speed boundary element-based inverse convolution scheme 45
the second iteration. There is, however, this seemingly controversial assumption that the heat
flux component qM remains "temporarily" constant (or in a predetermined manner) during the
time period tu-i to tM+r-i- Nevertheless, experiences have shown that this assumption,
suggested for the first time by Beck [4], does not induce significant amount of errors in the
estimation.
The first stage of the algorithm expends the greatest amount of computation time as, for every
step of inverse computations; it requires r time steps of direct heat conduction calculations.
This is especially true for cases in which one has to choose a high value for r. Nonetheless this
method is clearly more computationally efficient as compared to its whole domain
counterparts[5].
In this paper a sequential method is presented which utilizes features of the Dual Reciprocity
Boundary Elements in order to eliminate the first step of the aforementioned procedure. Using
this method, estimation of the unknown boundary condition is accomplished simply by
muhiplication of two matrices: one of which takes into account the initial condition for the
domain, and the other incorporates measured temperature data. The abovementioned
advantage, when added to the well-known advantages of the Boundary Element Method over
the other discritization schemes, such as finite element and finite difference[6], as well as the
reported advantages of the Dual Reciprocity BEM over the other transient BEM methods[7],
are noted improvements of the present scheme over the existing IHCP algorithms.
PROBLEM STATMENT
In general, the domain boundary in an IHCP problem is divided into active boundaries, where
the boundary condition is unknown and inactive boundaries, where the boundary condition is
known. The inactive boundary condition may have any of the three commonly known forms,
i.e., Dirichlet, Neumann, or Robin; however, in practice it is commonly taken to be adiabatic.
This is due to the fact that usually the heat flux from (or to) the active surface is negligible as
compared to the heat flux imposed on the active surface. For simplicity the method presented
in this paper is formulated for the case in which the inactive boundary condition is adiabatic,
although the algorithm can easily be extended to the more general cases. The imposed
boundary condition on the active surface is considered to vary both temporally and spatially.
The governing differential equation along with boundary and initial conditions for the case of
unvarying thermophysical properties is given by
dx' dy^ dt'
T(x,y,0) = \.0, Q. (3)
1^ = 0. r, (4)
dn
^ = g(^,y,t), r, (5)
dn
^(0 = g,(0, (/ = lv..,Ar,) (6)
which r, and T^ represent inactive and active surfaces respectively and Ni is the number of
thermocouples. Heat flux imposed to the active surface ''q(x,y,t)" is unknown in the inverse
problems, and is to be estimated in a discrete form using the measured temperature data,
equation(6).

46 A. Behbahaninia and F. Kowsary
DISCRETIZATION OF THE DIFFUSION EQUATION
USING DRBEM
The Dual Reciprocity Boundary Elements Method is used for solving equation(2). In this
methodology, the time derivative of the temperature is treated as a source term and the
fundamental solution of Laplace equation is used as followed
G = ±^Lnir,). (7)
Using equation(7), the weighted integral of equation(2) can be given as
\Gi^^T}Kl = \G^dQ., (8)
Q ft "^
which Q defines the domain of integration. Applying Green's second identity, equation(8) can
be written as
which r represents the boundary of the domain Q, and X is the shape coefficient.
The right hand side of equation(9) can now be manipulated by using a secondary interpolation
to reduce it to a boundary only form. One may write
f=i::,/(o)^.(')- (10)
A few internal points known as the dual reciprocity points and boundary points are used for the
above interpolation.
By substitution of equation(lO) into equation(9) and performing some mathematical
manipulations, which can be found in Ref [8], one may obtain
[R][|;-]-[S][T] = [P][^]. (11)
Different methods are used to discretize the time derivative, including one-step, and multi-step
0 schemes, and a series of schemes known as the least square methods. The accuracy and
stability of these schemes as applied to the Dual Reciprocity BEM are compared in several
references such as [8] and [9]. Implicit 6 scheme was used in this work due to its simplicity
and reported adequate accuracy. Thus
Combining equations (11) and (12), one can obtain
[s][Tr^ -[R][—r^ +—[P][Tr^ -—mrrr =o. (i3)
d« A/ A/
Equation(13) can now be used for solving the direct problem.
INVERSE SOLUTION
Equation(13) may be rearranged as
[H][T^] + [G][q^] + [F][T^-'] = 0, (14)
in which q represents heat flux components over active surface and T contains temperatures on
boundary and internal points. One can gather all of the unknowns in a single vector "X" and
write equation(14) as

A hyper speed boundary element-based inverse convolution scheme 47
AX = B, (15)
which unknowns are heat flux components on the active surface and temperatures associated to
all of internal and boundary nodes except at location of thermocouples. ^4 is a iVj xiVj matrix
for which N3 is the total number of unknowns and N2 is the number of equations. For the case
of N2=N3, equation(15) is regular but ill conditioned. In the general a solution may be found
for equation(15), if N2 >N., but such a solution is not satisfactory because of the damping
and lagging effects, and some kind of regularization would be required.
In order to solve equation(14), it may be written as
[T'']=IGV']+[FIT'^"'], (16)
in which
^J=-[H]-'[G], (17)
[Fj=-[Hr[F]. (18)
Equation(16) may be used for direct solution when ^ has been estimated. Rewriting
equation(16) for the time step "M+l" gives
[T^^^']=[Gy^^^]+[F|r^^]. (19)
Combining equations (16) and (19), and using assumption of temporary constant heat flux i.e.,
q'^^ =q^-i=... = q'^^'-', (20)
which was originally used by "Beck" [4,10], the following equation is obtained
|^M.ij^|^ijqA/J^|^pi|p./-iJ (21)
In a similar manner, it may be written
in which
[c']=tGMFlc'-'], (23)
[D']=[FID^-'], (24)
and
D'=F , C =G.
Temperatures are known at the location of thermocouples and are unknown in the other points.
The vector Tmay be split up into two vectors Ti and T2, and equation(22) may be rewritten as
rriM+l
k]H [T''-'], / = 0,...,r-l (25)
in which Ti and T2 respectively contain unknown and known temperature components.
The unknown heat flux vector ^ is estimated by minimizing the target function which is
defined as
S = (Y-T)'(Y-f), (26)
in which
Y = iY^^ Y'^+' ... Y'^'^''"'T (27)
T = (rf,Tf^',.--,Tf^''-7. (28)
The temperature vector T may be written directly as a function of unknown heat flux and
initial condition by using equation(25) as followed
T = [clq'^]+[D][T^-'], (29)
in which
C^=[C»,C',,....,Cr'], (30)

48 A. Behbahaninia andF. Kowsary
D^=[D«,D',....,Dr^]. (31)
Substituting equation(29) in equation(26), and differentiating sum of squares of errors with
respect to unknown heat flux vector and setting the resulting equation to zero gives
q''=G,Y + G2T'^'-', (32)
in which
G, =(C^C)-'C^ (33)
Gj =-(€''€)-'C^'D. (34)
In this method, which we identify it as the Transformation Matrix Method, or in short TMM,
the first step is computation of the transformation matrices as used in Equations (16) and (32).
As in the SFSM, this method is implemented sequentially. Thus, in a maimer similar to the
other sequential methods, it is first assumed that the solution is obtained up to the time level /M-
y; i.e., the unknown heat flux has been resolved up to this time level. Components of the
unknown heat flux at time /A/are then estimated using the following two-step algorithm:
1- Estimate the heat flux using equation(32).
2- Calculate the temperature distribution at time t^ using equation(16), which is used in turn as
the initial condition for the following time step.
It can be seen that the solution algorithm consists of merely four matrix multiplications. In
practice two separate computer routines: one, which builds the matrices and the other, which
performs inverse computation schemes, may be used. This feature of the method is quite
advantageous for the users as in a typical problem the transformation matrix may be calculated
and stored once and only once and used many times. Other noted advantages of this method
are its efficiency and accuracy as is discussed in the following sections.
ANALYTICAL SIMULATION
A simulated experiment based on an analytical solution is used to study reliability of the
method. The geometry and boundary conditions of the direct problem which is used to
simulate an experiment and its associated inverse problem is shown in figure(l). This problem
has already been used by some other researchers[7,ll]. The analytic solution of the direct
problem is given by
T(x,y,t) = w(x,t)w(yj), (35)
for which
>Kz,0 = 2A.«ifjM^exp(-^„^0, (36)
t;Nu(Nu + l) + p'„
and fin are the roots of fitan(fi)-Nu=0 .
In order to make the situation more realistic the direct temperatures given by(35) are pertubated
by a random function as followed
Y = T + d(l-2p), (37)
in which/? is a random number between zero and one and d is the amplitude of the errors.
RESULTS AND DISCUSSION
In order to assess the speed and accuracy of the present method, a comparison is made with
results obtained by the Sequential Function Specification Method (SFSM). Figure(l) shows
the geometry, mesh, and the thermocouple arrangement. In reference [11] this problem is

A hyper speed boundary element-based inverse convolution scheme 49
solved using the combined conventional SFSM and the Boundary Element Method based on
the time-dependent fundamental solution. The reference reports existence of instabilities in the
solution for location A (shown in figure(2)), and that a combined sequential function
estimation and time Tikhonov regularization was used as a remedy for treating these
instabilities. In the present
/
-^
/
/
/
/
/
/
/
/
/
/
/
/
/
Nu
^2
^3 ^1
^4
////////////////////J/
Direct problem
/
/
/
/ J
/
/
/
/
/
/
/
/
/
/
/
qp?
5r,
^3 £-,
s.
/J/////J//J//////J/J//
Inverse problem
q=?
Fig.l. Geometry and boundary conditions of the direct and inverse problems
work, however, this problem is treated by placing a high number of thermocouples nearby that
location. Listabilities in IHCP solutions at the comer formed by two active boundaries are
discussed in [12].
j 1 1 1
« « « a
O
• themio couple
O DRM point
1 1 1
•
•
•
*
*
• thermocouple
o DRM point
Fig,2. Mesh and location of thermocouples
Figure(3) compares the solution obtained by the present inverse algorithm with the exact
values of the heat flux. The comparison is made for two error amplitudes of ((^0.005, 0.01).
As it is seen the accuracy of the estimated results is within an acceptable level, although the
accuracy is reduced slightly as the noise level is increased. For higher noise levels, as is
suggested by [2] and [10], adding a regularization term to the sum of squares of errors function

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PUNCH,
OR THE LONDON CHARIVARI.

Volume 93

November 26, 1887.

PAPERS FROM PUMP-HANDLE
COURT.
A Recollection of the Long Vacation.
During the Long Vacation (now happily over) I
have been present at my chambers a great deal
more frequently than some of the men with
whom I share my rooms. In fact, I may say
that I have been constantly the sole occupant
of the entire set. Chuckbob, the well-known
authority on International Law, has spent
September and October in the Highlands, and
my other friends have been on the Continent.
Even Portington, my excellent and admirable
clerk, has taken a fortnight's rest at Eastend-
on-Mud (a pleasant watering-place not many
miles from Town), where I fancy he spent his well-earned holiday in
trying to get up a libel action against the Sanitary Board. It is just to
say that my presence at Pump-Handle Court has not been entirely
necessitated by my forensic labours. The fact is, that JowlÉr, a very
dear friend of mine, who has some mysterious supervisorship
(sanctioned by an eccentric will) over an Institution connected with
the Vegetarian Movement, was recently called away, by his duties as
a trustee, to Australia, to look after a number of sheep somehow
affected and inconvenienced by the increase of rabbits in that
favourite colony. Being thus for a season expatriated, he asked me
to look after the Institution connected with the Vegetarian
Movement, in his place during his absence.

"You will really find the work simple enough," he said on bidding me
farewell. "You hold my power of attorney, and all you have to do is
not to quarrel with the Committee of Inspection, who, as you know,
can play the very dickens with us."
"But what have the Committee of Inspection to do with the place?" I
asked rather anxiously, as I never like to accept responsibility, so to
speak, with my eyes blindfolded.
"Oh, you will soon find out," replied JowlÉr. "You will pick it up as
you go along. I shall soon be back—perhaps in six months."
The Institution connected with the Vegetarian Movement was within
easy distance of my chambers, so I came to the conclusion that I
could combine the vague superintendence it apparently required
with my ordinary legal engagements. I found, on a visit to the
Institution about a fortnight after JowlÉr had left, that all seemed to
be right, and the head employé assured me that if my services were
needed, he would send round to me.
"Fortunately since Mr. JowlÉr's departure, Sir," said the head
employé, "we have seen nothing of the Committee of Inspection."
He lowered his voice to a tone of the deepest awe as he spoke of
the mysterious body.
"I am very glad to have seen you, Sir," he continued; "the fact is,
there may be a number of things I should like to consult you about,
and I was loth to worry you."
"Oh, not in the least," I replied, airily; "consult me at any time; only
too glad to give you every assistance in my power."
Upon this, I took my leave, saying as I did, to show that I really
knew what I was about, that whoever had broken the hall-lamp,
which I noticed was damaged, should have been made to pay for it.

On my return to my chambers, I found Portington in a great state of
excitement. He had actually got a brief for me! A real brief marked
with a real fee and endorsed by a real firm of Solicitors! I was
actually retained! Mordaunt JonÉs, Brown and Snobkins ! Perhaps the
best firm in the profession! I was delighted!
"Portington," I observed when I had regained sufficient control over
my feelings to speak calmly, "I do not think you will find the names
in my fee-book?"
"I fancy not, Sir," replied Portington; "they wanted Mr. Chuckbob, only
I said he was in Scotland, and persuaded—I mean told them you
were in, and would be glad to look through the papers instead."
"Thank you, Portington," I answered, as I took the bundle into my
own special room; "thank you, if they come for them, let me know."
"Certainly, Sir; Mordaunt JonÉs, Brown and Snobkins seemed most
anxious to have them back."
Once alone I undid the tape and found the matter resolved itself into
a most delicate point of international usage. I went to my bookshelf
and hunted for authorities, and was soon deep in Mexican Maritime
Law. I was searching in its statutes for one dealing with a ship
detained by stress of weather in quarantine, when I was disturbed
by Portington ushering in the head employé from the Institution
connected with the Vegetarian Movement.
"Very sorry, Sir," said my visitor, "but we are in sad distress. We have
just received twelve dozen cases of ginger-beer, when the
Committee of Inspection particularly ordered that only soda-water
should be supplied, and I really don't know what we shall do."
"Can they not be exchanged for the required liquid?" I asked,
looking up from my work, a trifle annoyed at the interruption.

"I am afraid that is impossible, Sir. You see that the Committee of
Inspection are so opposed to any alteration of procedure."
"Well, well, you must do the best you can," I replied. "You see I am
very much engaged at this moment."
The chief employé, seeming greatly surprised at my lack of
excitement, bowed, and withdrew. I was once more deep in my
Mexican Maritime Law, when Portington put in his head.
"Suppose that opinion isn't ready yet, Sir? Mordaunt, Brown, JonÉs and
Snobkins are waiting for it."
"Ready directly. My compliments, and they can call for it in half an
hour."
I had just got to the point where I thought I began to comprehend
the Mexican method of dealing with a fraudulent bill of lading, when
I was again interrupted. A small boy forced himself in.
"Please you are to come round at once. The chess-boards are out of
order, and want mending, and there is something wrong with the lift,
between the kitchen and the dining-room, and——"
"You had no right to intrude, sirrah!" I exclaimed, with haughty
impatience. "Begone!"
Murmuring something about the Committee of Inspection, "kicking
up a shindy" the urchin withdrew. Again I dived into Mexican
Maritime Law, and nearly got hold of the rules governing a sale of
cargo for the benefit of ship-repairs. I had jotted down a line or two
upon the brief-paper before me, when the door was again thrown
open, and a gentleman of immense presence entered.
"I believe you are Mr. JowlÉr's substitute?" he began, without
removing his hat. I inclined my head and made a gesture with my
pen which was intended to convey to him the joint ideas that he was

to take a chair and not to disturb me until I was less preoccupied.
He ignored my dumb-show. "And that being the case, it is my duty
to call your attention to the unsatisfactory condition of the chimney-
pots of your Institution, and to mention the fact that a pane of glass
in the pantry has been broken, and is still unrepaired."
"Really," I replied, "I am exceedingly busy with a matter of the
greatest importance, and I must ask you to be so very kind as to call
again on an occasion when my time is more my own."
The gentleman rose with an air of astonishment so profound that it
nearly approached an aspect of absolute terror. He gasped for a
moment, and then asked, in a bone-freezing whisper—
"Do you understand that I am a Member of the Committee of
Inspection?"
"I shall be delighted to make your acquaintance on some future
occasion," I replied, with that easy courtesy that I hope is one of my
characteristics, and I opened the door for him to pass out.
He got up and with the same expression of profound astonishment
left my chambers. Once more I dived into Mexican Maritime Law,
and was only disturbed by a letter sent by hand from the Institution,
which I did not open, but threw carelessly on the desk before me. I
had just got to the last point in my opinion when the door was again
dashed open and JowlÉr himself rushed in.
"Why, my dear fellow,——" I began.
"No time to explain," he cried, "Australian visit deferred.
Presentiment of evil. Came back. What about the Institution?"
I gave an account of my stewardship.
"And this is a letter I got a few minutes ago," I said, when I had
finished my story, handing the document to my friend who hurriedly

opened it.
"Good gracious!" he exclaimed, "why it is from a Member of the
Committee of Inspection complaining of the hall-lamp! Oh! what
have you been doing?"
"They are all there, Sir!" cried the urchin, returning at the moment
out of breath from running, "and there's a nice row at the
Institution!"
"What the Committee of Inspection!" exclaimed JowlÉr, seizing his
hat, "Oh, what have you been doing? Why the place will be ruined!"
And he hurried off followed by the urchin.
The next morning I got a letter from JowlÉr, saying that he would
never forgive me, as, by my "want of tact with the Committee of
Inspection, I had ruined a widow and five small children," and, to
make matters worse, I have been subsequently informed, in a
satirical communication signed "Mordaunt, Brown, JonÉs and Snobkins ,"
that my opinion is not one they can conscientiously adopt without
further advice, "as my knowledge of Mexican Law seems to be of a
superficial description."
It is a painful experience, and none the less painful because I have
to add it to a number of experiences of a not entirely dissimilar
character.
A. BriÉflÉss,
Junior.

"ThÉ Grand Old Man" in DÉcÉmbÉr .—Father
Christmas.
THE LETTER-BAG OF
TOBY M.P.
From QuiÉt QuartÉrs.
By-the-Sea, Saturday.
DÉar Toby,

I have been intending to write to you for some weeks past, but,
really, life passes so quickly here, with such gentle rotation of days
and nights, that a week is over before I realise that I have well
entered upon it. Besides, I find, in practical experience, that the
writing of a letter usually involves the receipt of one; and, though I
am not bound by any rule involving the necessity of reading, or even
opening the letters that reach me, it is as well to avoid, as far as
possible, little annoyances of that kind. I write to you because, in
your case, I make an exception to the rule of my epistolary conduct,
and really want to hear from you.
The occasion of this solicitude is, that I find chance references in the
local weekly paper (I never see a daily) to the Irish Question, which
seem to show that it is in a somewhat unusually perturbed state. I
daresay if I could make up my mind to open the pile of letters that
have been accumulating on my desk for the last month or so, I
should be able to inform myself on the subject? But, if I once began
that practice, whither would it lead me? I have found, in the course
of my public life, that the last thing to do with a letter received
through the post, is to open it. My correspondence, conducted in the
main upon that principle, answers itself, and thus much labour, and
possible friction, are saved.
From the source of intelligence already alluded to, I gather hints that
the Government are "being firm" in Ireland, that evictions have been
going on, that there have been conflicts between the police and the
people, and that even some of my colleagues in the Parliamentary
Party have been arrested. One paragraph goes so far as to mention
the really interesting circumstance, that W-ll-m O'Br-n, has been cast
into gaol, where he sleeps on a plank bed, and that Arth-r B-lf-r,
emulating a historic political feat, has stolen his clothes whilst he
was sleeping.
This thing is probably an allegory, but it serves to support an opinion
I have always had with respect to the future of the Conservative
Government, and which enables me from time to time to stand aside

from the hurly-burly of active politics. I suppose that what the
paragraphist really means by the story of stealing O'Br-n's clothes, is
that Arth-r B-lf-r, as representative of Lord S-l-sb-ry's Government,
is coming out as an advocate of Home Rule for Ireland. If I misread
the allegory, the error has but temporary effect. If it is not true to-
day it will be true to-morrow, or the day after, if only the Liberals
have the ill-luck to be deprived of precedence in the opportunity. If I
never stirred finger or raised voice again, Home Rule would be
granted to Ireland by whatever English Party chances to be in power
when the moment is ripe. The ball is set spinning, and it would be a
mere accident, of no great import to me or the Irish people, whether
it is the M-rk-ss or Gl-dst-nÉ that kicks it into goal.
Hence you will see that though it may strike a superficial observer as
odd that I, of all men, should, at such a juncture, absent myself
from the field of battle and hide no one knows where, the course is
not so unreasonable as it appears. Why should I run the risk of
burning my fingers by pulling chestnuts out of the fire, when the
foremost men in English politics vie with each other in the effort to
do it for me? Amongst the few people with whom I come in contact
here I pass for a curate of Evangelical views, who, for private
reasons, has quitted his family and congregation, and tries,
ineffectually they slily think, to disguise himself by dispensing with
clerical garb. I encourage this self-deception, and am left free to sit
in the sun when there is any—and there is really an astonishing
amount on this Southern coast in November—and when it rains I put
up my umbrella. Sometimes I hear on it the patter of distant
conflicts in Ireland, and open revolt in London. These echoes of wild
disturbance only make the sweeter my retirement. I know that I am
foolish to imperil my pastoral peace by inviting a communication
from you which may confirm the vague reports I have alluded to.
Still, I am a little curious to know is it really true that W-ll-m O'Br-n
sleeps on a plank bed; that W-lfr-d Bl-nt, wearied of the long repose
of Egyptian affairs, has had his head broken by the Royal Irish
Constabulary; and that, with a refined cruelty which testifies to the
innate fiendishness of the Saxon nature, the presiding Magistrate at

Bow Street Police Court has ruthlessly refused to commit for trial
that truculent, dangerous personage, Mr. S-nd-rs, whom I remember
in the House as formerly Member for Hull?
Yours serenely,
C. S. P-rn-ll.

THE WAIL OF THE WIRE.
(With apologies to the Poet.)
"It is stated that Mr. SwinburnÉ 's new poem was
cabled to New York."
Had I wist, wailed the wire in sea's hollow,
That thousands of lines I should list,
Pumped forth by a son of Apollo,
I would not have lain here, not I,
'Twixt Briton and Yankee a tie:
No messages through me should fly,
Had I wist.
Had I wist, they would make me swallow,
Huge poems all moonshine and mist,
In addition to "speeches" all hollow;
They shouldn't have cabled a thing,
They shouldn't have used me to wing,
Leagues of rhymes that the word-spinners sing,
Had I wist.
ValuablÉ Oéinion.—We understand that the Authorities have consulted
Mr. BriÉflÉss, Junior, Q.C., (Queer Counsel) on the right claimed by
indifferent passers-by to stand between the police and the mob, in
view of the Chief Commissioner's statement that such passers-by
cause the chief difficulty in quelling disturbance; The learned
Counsel has given a lucid opinion to the effect that any mere
sightseer may be arrested and imprisoned, unless he or she can

prove the having come to the spot for a riotous or other unlawful
object.

May in November.
(At the Royalty Theatre.)
Pieces French they're playing,—
JanÉ's a pretty player,—
Come with me a-Maying,
Gaily sings the MayÉr.
ThÉ LÉsson for thÉ Day.—At Lowestoft Mr. MundÉlla spoke well and
wisely on certain fishery questions. "With regard to outrages," said
he, "in the North Sea, I counsel English fishermen to suffer wrong
rather than do wrong, as then they could demand the protection of
their industry by Government." Why not get the start of the
Hartington and GoschÉn Travelling Co. (Limited), and deliver these
excellent sentiments in Ireland?
"ThÉ GrosvÉnor 'Split,'" ought at once to be adopted by the
Restaurant of that establishment as a title for a special mixed drink.
Let Sir Coutts patent it.
"SéÉcial ConstablÉs."—Those belonging to the Collection in the
National Gallery.
"In thÉ PrÉss."—Mr. O'BriÉn's clothes.

'TWILL ILLUME.
(Poe applied.)
"Mr. Walt Whitman has just sent to Mr. ErnÉst Rhys, a preface and
some new material for a second 'popular' volume of prose, to consist
of 'Democratic Vistas' and other pieces."
Athenæum.
Then I pacified PsychÉ, and kissed her,
And tempted her out of her gloom,
With the latest Walt-Whitmanish "Vista,"
Which Democracy showed as our doom;
Our unwelcome but obvious doom.
And I said, "How's it written, sweet Sister?"
"Is it bosh? Will it be a big boom?"
She replied, "'Twill illume, 'twill illume.
It is bosh, but quidnuncs 'twill illume!"
*
*
*
Mr. PoÉ, and not Mr. Punch's Poet, is
responsible for this Cockney rhyme.
"Christmas Is Coming!"—"Tell me not in Christmas Numbers," that
Christmas is coming. We wish the good old gentleman would not
announce his intended arrival so long beforehand. Everybody knows,
that, like one of his own Christmas books, he is "bound to appear" at
a certain fixed date. Among the first of the heralds on the bookstalls
is the Christmas Number of the Penny Illustrated, price threepence,
and well worth the money. Mr. LatÉy, Junior, arranges a Christmas
Literary and Artistic Banquet, and every plate has a plateful of
Christmas fare. The picture entitled "Spoons" and representing two

persons in evening-dress slipping downstairs—"such a getting
downstairs"—in a sitting position, probably two amateur
Tobogganists, is distinctly humorous. The coloured illustration, called
The Christmas Ball, will be a great favourite with boys. If the Early
Bird still catches the worm, the Latey one who is first in the field
with this Christmas number ought to pick up the three-pennies.
LitÉrary.—It is announced that Mr. Snodgrass has "thoroughly revised
his translations from HÉinÉ." We expect next to hear that Mr. Tracy
Tupman has "Englished" Catullus, and that Mr. Winkle is preparing a
new edition of the Book of Sports.
Floral AééÉal To NovÉmbÉr .—"Fog-get-me-not!"
THE NE PLUS ULTRA.
Jeames I. "VÉry dangÉrous
PartiÉs thÉsÉ HunÉméloyÉd! Why,

"'Twas in
Trafalgar's
Square."
Nov. 20, 1887.
Nelson (as Special
Constable) sings:—
"England expects
that every man
This day will go
on Duty!"
thÉy'rÉ a bÉginnin ' to dÉnouncÉ Hus!"
Jeames II. "No!"
ThÉ Last of thÉ SolomonsÉs.—The final
knockdown blow was given to poor TuééÉr's
Proverbial Philosophy by Mr. John MorlÉy,
who, in his admirable discourse on
Aphorisms, described it as a "too famous
volume," which "had immense vogue, but it
is so vapid, so wordy, so futile, as to have a
place among the books that dispense with
parody." Alas! poor TuééÉr! Mr. Punch bids
thee adieu for ever!
Will Mr. LockyÉr turn his attention
Eastwards, and inform us if the Corporation
of the City of London is a "Self-luminous
Body"? If so, couldn't it be utilised in a fog?
Describing the state of mind her Nephew
was in on not being able to find a stud at
the last moment to put in his shirt-front,
Mrs. Ram said, "Oh, he was awfully
iterated."
A MÉss.—What's on the tapis in France? Grévy. M. Wilson, who
speaks Latin with English pronunciation, throws all the blame on his
father-in-law, and says it's a "Grévy delictum."

"SPECIAL" REASONS:
Or, Why They were "Sworn In."
Paterfamilias. "Because I think it's my duty, as a law-abiding citizen,
to set a good example."
Mister Tom (his son). "Because I must look after the old Governor,
and see he doesn't come to grief."
Mr. Brown, Q.C. "Because I'm not going to let those fellows, JonÉs
and Robinson , think that I shirk the responsibility."
Messrs. Jones, M.D., and Robinson, R.A. "Because we don't mean to
be outdone by that fellow Brown."
The West-end Young Man. "Because, you know, I think, on the
whole, it's the correct thing to do."
The Primrose-League Young Man. "Because I should very much like
to have a real chance of giving a Social Democrat a good
whack on the head."
'Arry. "Because it's such a prime lark."
The General Person. "Because everybody seems to be doing it."

Mem. by a Hater of Premature
"Christmassing."
"Christmas comes but once a year"—
But it lasts three months at a stretch, that's clear.
I should like to pass the whole quarter in slumbers,
To dodge the infliction of—Christmas Numbers!
The Great Ochipaway Chief says that he intends to continue selling
his chips. But he has a log by him with which, as he has kept it for
many years, he will not part on any account.
ON A RECENT CASTING VOTE.
What! How did Lytton get into the chair!
The usual way—he mounted by the Stair.
ThÉ RÉéort on thÉ FirÉ at thÉ ExÉtÉr ThÉatrÉ.—"Slow, but Shaw."

OUR BOOKING-OFFICE.
For the library shelves of those whom
"Providence has not blessed with affluence,"
and who cannot afford first editions or
expensive bindings, and for the working
Journalist's library, the most useful books, the
most handy, though not belonging to the
regular "Handy Volume Series," and the best
adapted to the pockets of most men, specially
of the class above mentioned, are those
forming Morley's Universal Library; published by
RoutlÉdgÉ and Sons, which now number about fifty-five volumes.
ButlÉr, Bacon, CavÉndish , CobbÉtt, DantÉ, GoÉthÉ, Goldsmith , Thomas-à-
KÉméis, SoéhoclÉs , and DÉ QuincÉy , are all well represented; and,
following the fashion of the day, were I asked to provide "the young
man just beginning active life" with a list of the best set of books for
his study and perusal, I should have no hesitation in referring him to
Morley's Universal Library; and I know of no more useful present at
this Christmas time, or at any other time, than the neat and
convenient oak cases, a guinea each, made on purpose to contain
fifteen of the MorlÉy volumes. I trust they will go on from year to
year, and so continue to deserve the title first given them by Mr.
Punch, of the "More-and-Morely Series," which fully expresses a
constant supply to meet a growing demand.
Long expected come at last! The HÉnry Irving and Frank Marshall
Shakspeare, Vol. I., produced by Messrs. BlackiÉ (one of which Firm
ought evidently to come out as Othello) as the Manager of the
Lyceum always gets up his plays "regardless of expense." The
prefaces and introductions will delight everyone who acknowledges
the force of the common-sense opinion, emphatically expressed

"Hist, Romeo,
hist!"
R. & J., Act II., Sc.
2.
more than once in Mr. Punch's pages, that ShakséÉarÉ if acted just "as
he is wrote" would not suit the taste of an audience of the present
day. The taste of the modern audience is corrupted by
Sensationalism and Materialism in every shape and form—and at
some theatres Materialism in shape and form is one of the main
attractions—and so impatient is it of anything like development of
character by means of dialogue, that it would have most plays, no
matter whether comedies or melodramas (there are no tragedies
now, except ShakséÉarÉ 's), reduced as nearly as may be to mere
ballets of action. For the maxim of our audiences in this last quarter
of the "so-called" Nineteenth Century, as regards the drama, is Facta
non verba; before which imperious command those "who live to
please," and who "must please to live," are compelled, be they
authors or actors, to bow, and do their best, speaking as little as
possible, so as not to give offence.
"Break, break, my heart, for I must hold my tongue,"
is the cry of any author nowadays who aims at
writing a true Comedy. Mr. Irving marks clearly
enough all the passages usually omitted in
representation, which of themselves would make a
small volume, but we are not shown the
arrangement of scenes necessitated by the
exigences of the stage, or rather by the taste of the
audience, and so in this respect the plays remain
pretty much as their author left them. Some stage-
directions have been introduced, but as Mr. Frank Marshall denies
that this is in any sense an "acting edition"—while Mr. Irving in his
preface rather seems to imply that in some sense it is so,—I should
be inclined to describe the work as "a contribution in aid of an acting
edition," and I am delighted to add, a most valuable contribution it
is, at least so far. Ex uno disce omnes, and if the other volumes are
only on a par with this first instalment, Irving and Marshall 's—it
wouldn't do to put Marshall first in the Firm, because it would at
once suggest, "and SnÉlgrovÉ" to follow—or this HÉnry and Frank's

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Inverse Problems In Engineering Mechanics Iv International Symposium On Inverse Problems In Engineering Mechanics 2003 Isip 2003 Nagano Japan Masa Tanaka Eds

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Inverse Problems In Engineering Mechanics Iv International Symposium On Inverse Problems In Engineering Mechanics 2003 Isip 2003 Nagano Japan Masa Tanaka Eds

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