تجزیه و تحلیل حساسیت طراحی با عناصر مرزی بیش از حد مفرد

The finite difference load method for shape design sensitivity analysis requires the calculation of stress and stress gradient on the boundary. In the standard boundary element method, the basic state variables-displacement and traction are continuous, and are considered as very accurate. However, the boundary stress and stress gradient, derived from the differentiation of the state variables and Hooke's law, are discontinuous and have relatively lower accuracy than the basic state variables. The hypersingular boundary integral equation is introduced in this paper to determine the stress and stress gradient in the design sensitivity analysis. The numerical examples demonstrate the accuracy of the design sensitivity using the hypersingular boundary elements.

As a crucial factor in solving shape optimization problems, the issue of design sensitivity analyses has been extensively studied. The many different approaches to design sensitivity calculation can be grouped into three categories: the finite difference method (FDM), the adjoint structure method (ASM) and the direct differentiation method (DDM). The FDM employs the finite difference formulation to approximate the differentiation of the design variable by solving two boundary value problems [1], [2] and [3]. The FDM is easy in concept and implementation, but with two serious drawbacks: (a) the accuracy often depends on the perturbation step; (b) the computation cost is relatively higher as the matrices of the perturbed geometry are required. The ASM uses the concept of the material derivative to derive the total change of the structural response with respect to the change of the design variables. By defining an adjoint problem, the final shape design sensitivity can be expressed as an integral of the solutions of the initial problem and the adjoint problem [4], [5] and [6]. One difficulty related to the ASM is the modeling of the adjoint problem because of the appearance of the concentrated adjoint load, which often results in a poor solution near the adjoint load, thus decreases the accuracy of design sensitivity. The DDM derives the sensitivity by direct differentiation of the boundary integral equations either before or after the boundary discretization [7] and [8]. The main difficulty of employing the DDM comes from the differentiation of boundary element matrices which involves the differentiation of the boundary variables (displacement and traction), the boundary geometry, as well as the fundamental solutions. Many attempts have been made to overcome some of the difficulties in evaluating design sensitivities, such as using new boundary element formulations or employing new design variables [9], [10] and [11]. A finite difference based approach, named finite difference load method (FDLM), has been developed by the author [12], which does not depend on the perturbation step. By analyzing the perturbation procedure of the FDM, the variation of the state variables between the initial geometry and the perturbed geometry can be replaced by a set of perturbation loads which can be obtained from the stress field of the initial problem and the design boundary perturbation. However, the determination of the perturbation loads requires the evaluation of boundary stress and stress gradient. The standard BEM usually provides very accurate displacements and boundary stresses, but not the stress gradient. In this paper, the hypersingular integral boundary element is used to improve the accuracy of the boundary stress and stress gradient, thus to improve the accuracy of design sensitivities. The basic formulation of the FDLM and the hypersingular boundary element are presented and discussed. Three examples are used to demonstrate the improved accuracy of the design sensitivity calculation.

The FDLM overcomes the two major drawbacks of the conventional FDM, namely it does not depend on the perturbation step, and the perturbation load is acting on the same domain as the original problem. However, the accuracy of the FDLM depends on the accuracy of perturbation loads derived. As the standard BEM has relative poor accuracy on stress gradient evaluation, the hypersingular boundary element is introduced in this paper to improve the accuracy of boundary stress evaluation. The test examples demonstrate that the FDLM is very accurate for design sensitivity analysis. The accuracy of the design sensitivity can be improved by using hypersingular boundary elements for stress evaluation. A further study is needed to look into the possibility to derive stress gradient by further differentiating the hypersingular integral equations.