تجزیه و تحلیل حساسیت طراحی شکل سازه ناهمسانگرد 2D با استفاده از روش المان مرزی

A directly differentiated form of boundary integral equation with respect to geometric design variables is used to calculate shape design sensitivities for anisotropic materials. An optimum shape design algorithm in two dimensions is developed by the coupling of an optimising technique and a boundary element stress analyser for stress minimisation of anisotropic structures. Applications of this general-purpose program to the optimum shape design of bars and holes in plates with anisotropic material properties are presented.

In the previous study [1], [2], [3] and [4], numerical optimisation techniques and the boundary element method were combined to optimise the shape of 2D isotropic linear elastic structures subjected to static loading. The steps that were required were as follows: shape representation, boundary element analysis to calculate stresses and displacements, design sensitivity analysis for calculating derivatives, numerical optimisation to find the optimum solution iteratively, and boundary element mesh re-generation as the optimisation proceeds. For shape representation Hermitian cubic spline functions were employed. The Hermitian cubic spline has two continuous derivatives everywhere, minimum mean curvature and is axis independent. It posses globally controlled properties, such that moving any design variable point on the curve will change the shape globally rather than just locally. Complex geometries can be modelled by a small set of design variables. Design sensitivity analysis to calculate derivatives of displacements and stresses with respect to the design variables were carried out by implicit differentiation of the corresponding boundary element elasticity kernels. For verification purposes, the derivatives of displacements and stresses were calculated both by this direct analytical differentiation method and by the finite difference method. Not surprisingly, results obtained by analytical differentiation were much more accurate. Two main types of problems were considered. Firstly, those problems involving the minimisation of the deviation of the von Mises equivalent stresses on the boundary of the region of interest from a desired uniform mean stress. In this case the objective function is highly non-linear, while the constraints are linear. The optimisation method used was the extended interior penalty function approach combined with the BFGS method [5] for unconstrained minimisation. The second main type of problem was the minimisation of structural weight, while satisfying certain constraints upon stresses and geometry. Since both the objective function and constraints are non-linear, the feasible direction method was employed. In the present study, using a similar procedure, a general-purpose computer program for shape optimal design of 2D anisotropic structures in order to smooth stress peaks is presented. The developed optimisation programme has three main components: an optimiser, a stress analyser and a design sensitivity analyser. Therefore, the main differences between the new programme and the one used for the isotropic materials [1], [2], [3] and [4] are in the stress analyser and design sensitivity analyser. The analytical formulation of the direct boundary integral equation (BIE) for plane anisotropic elasticity may be well developed by following the same steps as in the isotropic case. For the details of these, the reader is referred to the references [6], [7], [8], [9], [10] and [11]. Here the implicit differentiation of the BIE for 2D anisotropic linear elastic materials is carried out and then stress and displacement derivatives are calculated. The accuracy is compared against the results of the finite difference applied to the boundary element analysis. Applications of the program to the optimum shape design of an infinite plate with a central circular hole under a biaxial stress field and a cantilever beam under uniform shear load at one end are presented. In each case isotropic and anisotropic materials are employed and the results are compared.

Shape design sensitivity analysis using the boundary element method has been presented for 2D anisotropic elastic problems. A differentiated form of the BIE is used directly to determine the derivatives of the objective and constraint functions. The results are compared with the corresponding results obtained using the finite difference method. The analytical differentiation shows an excellent accuracy. The stress and design sensitivity analyser using the boundary element method have been combined with an optimisation program to form an optimum shape design algorithm in two-dimensions for anisotropic structures. The objective has been to smooth the stress peaks. Couple of test cases have been analysed and the results were presented.