Numerical solution of inverse problems in mechanics using the boundary element method

Abstract

<p>Due to the ill-posed nature of inverse problems, it is difficult to obtain solutions using well known analytical and numerical techniques. The use of boundary element method as a numerical technique to solve inverse problems is quite new. In this work, the algorithms for the solution of two kinds of inverse problems are examined in detail. For the first kind, the shape and location of a part of the boundary is unknown; and for the second kind, the boundary condition is not specified on a part of the boundary;Boundary value problems with partially unknown boundary are ill-posed. To solve these problems additional information is necessary. Over-specified boundary data in the form of experimentally measured quantities can be used as additional information for solving the problem. An algorithm, based on the boundary element method and non-linear optimization techniques, is proposed to solve this inverse problem. Using the overspecified boundary data, a functional is formed which involves parameters describing the unknown boundary. Minimization of this functional with respect to these parameters determines the unknown boundary. The performance of this scheme is examined through two problems. It is shown that the algorithm performs well even for complex shapes of the unknown boundary;For the problems in which the specified boundary conditions are insufficient, experimentally obtained data at some internal points are used as additional conditions. The boundary is divided into straight boundary elements and the unknown boundary conditions are represented as unknowns at the nodes of the boundary elements. It is shown that, for practical reasons, the number of nodes where the boundary condition is not specified is usually larger than the number of probes used for obtaining interior data. This results in an under-determined system of linear equations. A regularization method is used to solve these equations. The scheme, when applied to several example problems, showed satisfactory performance. Few guidelines for the placement of the temperature probes in the interior of the domain are developed through numerical experiments.</p>