Design sensitivity analysis of boundary-element substructures

A design sensitivity analysis (DSA) formulation using substructuring is presented for the boundary-element method (BEM). A reduced set of DSA equations is obtained from the implicit differentiation of reduced BEM equations. Relationships for the expansion of the reduced DSA expressions to determine the sensitivities of all condensed quantities that are present in the analysis are also given. These reduced and expansion sensitivity equations involve the sensitivity of the inverse of a system submatrix. A procedure is developed to obviate the need for the determination of this submatrix inverse sensitivity. The present formulation allows for arbitrary condensing and noncondensing of zones in multiple zone models, and such zones can simultaneously exist in the reduced set of analysis equations. A set of numerical examples for two-dimensional plane and axisymmetric continua are presented. Since the condensation formulation developed is exact, the accuracies of these examples are the same as those for the uncondensed model. It is shown, however, from the CPU timings presented for these examples, that the substructuring technique can dramatically economize the numerical shape optimization of models with partially sensitive geometries.