Boundary element design sensitivity analysis formulations for coupled problems

Abstract

Concepts associated with the use of the implicit differentiation technique to derive Boundary Element theoretical formulations for Design Sensitivity Analysis (DSA) of coupled problems are presented. Within the context of a coupled thermal/structural problem, it is shown that the thermoelastic response DSA can be formulated even for transient temperature distributions without the need for explicit domain integrations and without the requirement to factor perturbed matrices. For the case of a coupled structural/fluid problem, a coupled shell structural/fluid acoustic DSA formulation is presented. In this second problem the coupling is twofold. First, there is the obvius coupling of the two physical phenomena governed by different differential equations. Secondly, the shell structural behavior is numerically simulated using the Finite Element Method (FEM) while the fluid acoustic behavior is treated by the employment of the BEM. It is shown that implicit differentiation of the coupled, discretized equations can lead to the formulation of a two step procedure for shape or property DSA for all response quantities involved in the coupled analysis. In both steps of this procedure, it is shown that the DSA approach allows for the reuse of matrix factorizations performed in the earlier coupled analysis step to once again produce a DSA methodology that obviates the need to factor perturbed matrices. These two applications can be considered typical of a much wider class of coupled problems for which effective DSA formulations can be derived via the implicit differentiation approach.