Yuanlong Wang Department of Mechanical Engineering, University of Colorado Denver, Denver, CO 80204 e-mail: yuanlong.wang@ucdenver.ed Abdalkaleg Hamad Department of Mechanical Engineering, University of Colorado Denver, Denver, CO 80204 e-mail: abdalkaleg.hamad@ucdenver.edu Mohsen Tadi 1 Department of Mechanical Engineering, University of Colorado Denver, Campus Box 112, P.O. Box 17336, Denver, CO 80217 e-mail: mohsen.tadi@ucdenver.edu Evaluation of a Diffusion Coefficient Based on Proper Solution Space This note is concerned with the evaluation of the unknown diffusion coefficient in a steady-state heat conduction problem. The proposed method is iterative and, starting with an initial guess, updates the assumed value at every iteration. The updating stage is achieved by generating a set of functions that satisfy some of the required boundary con- ditions. The correction to the assumed value is then computed by imposing the remaining boundary conditions. Numerical examples are used to study the applicability of this method. [DOI: 10.1115/1.4041343] Keywords: elliptic problem, inverse problem, diffusion coefficient 1 Introduction In this note, we consider the evaluation of the diffusion coeffi- cient for an elliptic equation. This problem, which is often referred to as the Calderon problem [1], appears very naturally in various applications including impedance tomography [2], identi- fication of injection and extraction wells [3], and early detection of breast cancer [4]. It is well known that this problem is ill-posed [5] and various methods have been developed to overcome such difficulties. Recent results include inversion methods based on statistical approaches [6], conjugate gradients method [7], regularized Gauss–Newton method [8], and linear back projection algorithm based on measuring data decomposition [9]. The purpose of this note is to present an iterative method for the evaluation of the diffusion coefficient for an elliptic equation. For this work, we assume that the value of the unknown function can be measured at the boundaries, and for simplicity, we assume that it is equal to one. We also assume that the data can be col- lected at the boundaries. In Sec. 2, we present the iterative algo- rithm. In Sec. 3, we present the updating stage which is the new feature of the method. In Sec. 4, we use a number of numerical examples to study the applicability of this method. 2 Problem Statement and the Identification Algorithm Let X be a bounded domain in R 2 (or R 3 ) with smooth boundary @X. In the absence of sinks or sources, the temperature field u(x) is given by rðkðxÞruÞ¼ 0; x 2 X; uðxÞ¼ gðxÞ; x 2 @X (1) where, k(x) > c 0 > 0 is the positive unknown thermal conductivity. The boundary is accessible and is used to measure the temperature gradient given by r  u ¼ @u @ ¼ f x ðÞ; x 2 @X (2) where  is the outward normal. The inverse problem of interest is to recover the diffusion coefficient k(x) based on the additional data given in Eq. (2). The algorithm consists of three steps (1) Assume a value for the unknown function ^ k ðxÞ and, using the given Dirichlet boundary condition, obtain a back- ground field satisfying the system ^ k ðxÞD^ u þr ^ k ðxÞr^ u ¼ 0; x 2 X (3) for ^ uðxÞ¼ gðxÞ; 8x 2 @X. (2) Subtract the background field from Eq. (1) and obtain the error field, eðxÞ¼ uðxÞ ^ uðxÞ, given by kðxÞDu þrkðxÞru  ^ k ðxÞD^ u r ^ k ðxÞr^ u ¼ 0 (4) for x  X, where the error field is required to satisfy eðxÞ¼ 0; 8x 2 @X. The error field is also required to sat- isfy additional condition given by r  eðxÞ¼ f ðxÞr  ^ e ðxÞ¼ ^ f ðxÞ; x 2 @X (5) (3) The assumed value, ^ k ðxÞ, is related to the actual value according to kðxÞ¼ ^ k ðxÞþ qðxÞ, where q(x) is still an unknown function. Use the additional boundary conditions in Eq. (5) and obtain the unknown correction term q(x), update according to ^ k ðxÞ¼ ^ k ðxÞþ qðxÞ, and go to step [1]. We have applied this algorithm to an elliptic problem in Ref. [10] and a parabolic problem in Ref. [11]. 3 Proper Solution Space The third step of the algorithm involves the identification of the correction, i.e., q(x), to the assumed value of the diffusion coeffi- cient. We first proceed to linearize Eq. (4) around the background field and arrive at ^ k ðxÞDe þr ^ k ðxÞre þ qðxÞD^ u þrqðxÞr^ u ¼ 0; x 2 X (6) Note that uðxÞ¼ ^ uðxÞþ eðxÞ and the terms that are quadratic in unknowns are dropped. In order to recover q(x), we can proceed as follows. Consider a linearly independent set of functions c ‘ (x), ‘ ¼ 1, 2,…, N over x  X, and assume that the unknown function q(x) can be expressed as a linear combination of c ‘ (x)’s, i.e., q(x)  {c 1 , c 2 ,…, c N }. We next generate a set of functions that sat- isfy the linearized error field in Eq. (6) according to ^ k ðxÞDe ‘ þr ^ k ðxÞre ‘ þ c ‘ D^ u þrc ‘ r^ u ¼ 0; x 2 X (7) 1 Corresponding author. Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF THERMAL SCIENCE AND ENGINEERING APPLICATIONS. Manuscript received February 10, 2018; final manuscript received July 30, 2018; published online October 23, 2018. Assoc. Editor: Steve Q. Cai. Journal of Thermal Science and Engineering Applications FEBRUARY 2019, Vol. 11 / 011017-1 Copyright V C 2019 by ASME Downloaded From: https://thermalscienceapplication.asmedigitalcollection.asme.org on 12/05/2018 Terms of Use: http://www.asme.org/about-asme/terms-of-use