JOURNAL OF COMPUTATIONAL PHYSICS 126, 410–420 (1996) ARTICLE NO. 0146 Identification of the Thermal Conductivity and Heat Capacity in Unsteady Nonlinear Heat Conduction Problems Using the Boundary Element Method D. LESNIC, L. ELLIOTT, AND D. B. INGHAM Department of Applied Mathematical Studies, University of Leeds, Leeds LS2 9JT, England Received November 21, 1994; revised October 24, 1995 duction equations with linear and/or nonlinear boundary conditions can be formulated for the treatment of all these In this study the inverse problem of the identification of tempera- ture dependent thermal properties of a heat conducting body is problems. However, it should be said that simultaneous investigated. The solution of the corresponding direct problem is requests resulting from combining these inverse problems obtained using a time marching boundary element method (BEM), are yet to be investigated. which allows, without any need of interpolation and solution do- Inverse problems are more difficult than their corre- main discretisation, efficient and accurate evaluation of the temper- ature everywhere inside the space–time dependent domain. Since sponding direct formulated problems because they are ill- the inverse problem, which requires the determination of the ther- posed; i.e., either existence, uniqueness, or continuous de- mal conductivity and heat capacity from a finite set of temperature pendence upon the data (stability) are violated. In general, measurements taken inside the body, possesses poor uniqueness both the IHCP and BHCP violate stability when unique- features, additional information is achieved by assuming that the ness is satisfied, whilst the IDHCP is more difficult since thermal properties belong to a set of polynomials. Thus the inverse problem reduces to a parameter system estimation problem which the uniqueness problem has to be addressed. is solved using the nonlinear least-squares method. Convergent and It is the purpose of this study to investigate one of the stable numerical results are obtained for the finite set of parameters identification problems which requires the simultaneous which characterise the thermal properties for various test examples. estimation of the thermal conductivity and the heat capac- Once the thermal properties are accurately obtained then the BEM determines automatically the temperature inside the solution do- ity, which are temperature dependent, from boundary and main and the remaining unspecified boundary values and the nu- initial data and additional interior temperature measure- merically obtained results show good agreement with the corre- ments. Also, the unknown temperature solution and the sponding analytical solutions.  1996 Academic Press, Inc. remaining unspecified boundary values are required to be determined. Reports of analysis of inverse, nonlinear IDHCP are 1. INTRODUCTION limited in the literature. Theoretical studies, both in steady and unsteady, linear or nonlinear cases have been investi- Inverse problems in heat conduction have been the point gated in [9–11]. Numerically, the first step in the inverse of interest for many researchers in recent years. The deter- analysis is the development of the solution of the corre- mination of the unknown temperature and heat flux at an inaccessible portion of the boundary, i.e. the inverse heat sponding direct problem and previous works on the subject (see [7, 12]) have used finite differences. However, the conduction problem (IHCP) (see [1]) and the determina- tion of the unknown initial temperature, i.e., the backward identification of the thermal conductivity temperature de- pendence has been investigated using the BEM in [8] only heat conduction problem (BHCP) (see [2, 3]) are examples of typical boundary inverse problems which arise when for the steady case and it is the purpose of this study to investigate the unsteady nonlinear identification situation. analysing a heat conducting material. Another type of in- verse problem in heat conduction requires the estimation The advantages of the BEM, in comparison with finite- difference or finite element methods, are that the BEM of the thermal properties and/or heat source time, spatially and/or temperature dependent, i.e., the identification heat does not require any solution domain discretisation and, in addition, no need of interpolation is required when conduction problem (IDHCP) (see [4–8]). In these inverse formulations the determination of the boundary or coeffi- evaluating the interior estimated temperature values. Fur- thermore, the BEM gives in a straightforward manner both cient unknowns is obtained provided that additional boundary and/or interior temperature measurements are the remaining unspecified boundary values and the temper- ature inside the solution domain. All these advantages, available. Steady or transient, linear or nonlinear heat con- 410 0021-9991/96 $18.00 Copyright  1996 by Academic Press, Inc. All rights of reproduction in any form reserved.