Sensitivity Analysis with the Simplified Hybrid Boundary Element Method

1. Abstract The paper describes a formulation for computing design sensitivities required in inverse problems as well as in shape or material property optimization in the frame of the hybrid boundary element method. Implicit differentiation of the discretized boundary integrals is performed. The resulting integrals present the same types of singularities of the basic hybrid formulation, which can be generally handled by means of well-established quadrature techniques, as developed for the conventional, collocation boundary element method. It is demonstrated that the spectral properties of the original matrices (as related to rigid body displacements) apply to the sensitivity matrices, thus yielding a general and efficient technique for the shape design sensitivity analysis of all structural quantities one may be interested in. The formulation is valid for three-dimensional solids, in general. Multiply-connected and unbounded domains may be handled, as well. Most important, however, is the outline of the sensitivity equations in the frame of the frequency-domain, simplified hybrid boundary element method, which with no loss of accuracy circumvents the necessity of the time-consuming evaluation of a flexibility matrix. Moreover, it is shown that the sensitivity analysis of transient problems on the basis of a generalized mode superposition technique as well as of structures in free vibration can be performed efficiently. The formulations developed apply directly to Pian’s hybrid finite element method, as a particular case. Comparisons with finite difference results are given in an academic example to demonstrate the effectiveness of the formulation. 2.