Structural inverse problems solved by the T -element approach

Abstract

Any direct boundary-value problem is defined in a certain area Ω by a system of differential equations and respective set of boundary conditions. In structural inverse problems the above conditions can be partly unknown. Instead, we can measure certain quantities inside the investigated structure and then approximately define the whole boundary-value problem. Usually, the solutions of inverse problems are connected with the minimization of a certain functionals, which results in optimization procedures. The applications of the trial functions identically fulfilling governing partial differential equations of a discussed problem (the Trefftz approach) can considerably improve these procedures. The original idea of Erich Trefftz was based on modelling objects of simple geometry. In the case of more complex structures the division of the whole object into sub-regions (Trefftz elements) is necessary. This kind of formulation is presented in this paper and is illustrated by numerical examples. The properties of the Trefftz finite elements allow the formulation of effective algorithms, which considerably shorten the time of computer calculations in comparison to standard finite element solutions.