Boundary-value recovery by the Trefftz approach in structural inverse problems

Abstract

The main idea of the Trefftz approach to numerical modelling consists in the application of trial functions identically fulfilling governing partial differential equations of a considered problem. When boundary conditions of the problem are defined a priori (direct formulation), they can be used to calculate the unknown solution coefficients. In structural inverse problems, the above conditions can be partly unknown (its shape is assumed to be unchanged). Instead, we can measure certain quantities inside the investigated structure and then approximately define the whole boundary-value problem. Usually, solutions of inverse problems are connected with minimization of certain functionals, which results in optimization procedures. This kind of formulation is presented in detail and illustrated by numerical examples. The properties of the Trefftz approach allow to formulate alternative, much more effective, simple direct algorithms, which considerably shorten the time of computer calculations. This is clearly shown in several computational benchmarks for 2D elastic inverse problems. The proposed algorithms can be applied to any inverse boundary-value problem, for which the complete T-function sets are known. Copyright © 2006 John Wiley & Sons, Ltd.