Abstract
This paper is concerned with the numerical implementation of the boundary element method for shape optimal design of two-dimensional linear elastic structures. The design objective is to minimize the structural compliance, subject to an area constraint. Sensitivities of objective and constraint functions, derived by means of Lagrangean approach and the material derivative concept with an adjoint variable technique, are computed through analytical expressions that arise from optimality conditions. The boundary element method, used for the discretization of the state problem, applies the stress boundary integral equation for collocation on the design boundary and the displacement boundary integral equation for collocation on other boundaries. The use of the stress boundary integral equation, discretized with discontinuous quadratic elements, allows an efficient and accurate computation of stresses on the design boundary. This discretization strategy not only automatically satisfies the necessary conditions for the existence of the finite-part integrals, which occur naturally in the stress boundary integral equation, but also circumvents the problem of collocation at kinks and corners. The perturbation field is described with linear continuous elements, in which the position of each node is defined by a design variable. Continuity along the design boundary is assured by forcing the end points of each discontinuous boundary element to be coincident with a design node.
The optimization problem is solved by the modified method of feasible directions available in the program ADS. Examples of a plate with a hole are analyzed with the present method, for different loading conditions. The accuracy and efficiency of the implementation described herein make this formulation ideal for the study of shape optimal design of structures.
Displacement and temperature discontinuity boundary integral equation and boundary element method for analysis of cracks in three-dimensional isotropic thermoelastic media
