Applications of the Boundary Element Method to Some Inverse Problems in Engineering Mechanics

The paper presents a brief review of current research on computational approach to inverse problems in engineering mechanics. Emphasis is placed on applications of the boundary element method, and the paper reports on some recent applications of such approach to a few classes of the inverse problems which are of practical importance. INTRODUCTION Computational methods of analysis based on the finite difference, finite element, and boundary element methods have been so well developed that we can easily solve the initialand boundary-value problems, which are to be called the direct problems. It has been increasingly attracting the attention of scientists and engineers to apply the computer analysis software well established for the direct problems to the corresponding inverse problems [1-5]. There are many inverse problems around the world, in which we should estimate the reasons from the observed results. However, from the engineering point of view, the inverse problem can be stated such that some information on the initial and/or boundary conditions, domain shapes, material constants, etc. are not known, and this lacking information should be identified by using additional information which is usually provided as measured data. If we consider a system which is modeled as an initialand boundary-value problem, we may classify the corresponding inverse problems into the following: 1. Estimation of the initial and/or boundary conditions 2. Determination of domain shapes 3. Estimation of sources 4. Estimation of material constants 5. Estimation of governing differential equations In this article, we shall first explain fundamentals of the computational approach to the solution of inverse problems using the methods of analysis for the direct problems, and then show several investigations on the inverse problems by author's group. Finally, the paper is concluded by some remarks toward Copyright © 1999 by ASME -1more fruitful and more successful analysis of the inverse problems. METHODS OF INVERSE ANALYSIS The inverse problem under consideration is modeled as a parameter identification problem. In a computational approach to this inverse problem, we first assume the values of parameters in an appropriate manner and then carry out analysis of the direct problem. The results obtained are compared with the measured data given as additional information, and the parameter values are then modified so that an appropriate cost function is minimized in an iterative manner. The cost function is usually defined as a square sum of differences between the measured and computed data. We may express the cost function as follows: