A mathematical and numerical study on regularization of an inverse boundary value problem in elasto-static field

Abstract

In an inverse boundary-value problem, boundary values on the incompletely prescribed boundaries can be estimated when the excessively prescribed boundaries are introduced. Application of the boundary element method to this inverse boundary value problem reduces to the solution of a matrix equation. This matrix equation is severely ill-conditioned because of the ill-posedness of the problem. Errors included in values on the excessively prescribed boundaries are magnified tremendously, when an inverse analysis scheme without regularization is applied. Regularization is then necessary to obtain a good solution of this matrix equation. The singular value decomposition with rank reduction can be applied for the regularization. If the rank of the coefficient matrix is reduced to the effective rank of the respective inverse problem, a good estimation of the boundary values can be made. The study presented in this chapter treats the inverse boundary value problem for two-dimensional elastic body. Mathematical structure of error magnification in this inverse analysis is investigated. An approximate equation describing the relation between the condition number with the rank is theoretically introduced to examine the error magnification behavior. The effective rank is estimated by using the optimum condition number method. It is found that the approximate equation describing the behavior of the condition number works well. It is shown that the optimum condition number method is useful for estimating the effective rank.