Optimum Shape Design of Composite Structures Using Boundary-Element Method.

Abstract

The boundary element method, combined with a numerical optimisation algorithm, has been employed for the shape optimisation of two-dimensional anisotropic structures. To find the optimum shape of a structure with the highest stiffness, the elastic compliance of the structure has been minimized subject to constraints upon stresses, weight and geometry.The optimum shapes of a series of anisotropic structures are obtained for maximum stiffness and minimum weight and stress, for specified loading conditions. The results are compared with the optimum shapes which were already created by the minimization of the structural weight while satisfying certain constraints upon stresses and geometry. A directly differentiated form of boundary integral equation with respect to geometric design variables is used to calculate shape design sensitivities of anisotropic materials. Because of the nonlinear nature of the mean compliance, weight and stresses, the numerical optimisation algorithm used is the feasible direction method, together with the golden section method for the one-dimensional search. Hermitian cubic spline functions are used to represent boundary shapes which offer considerable advantages in fitting a wide range of curves, and in the automatic remeshing process. Five example problems with anisotropic materialproperties are presented to demonstrate the applications of this general-purpose program.

Shape design sensitivity analysis with respect to the positioning of features in composite structures using the boundary element method

“…In another study by the author [7], an advanced version of the aforementioned algorithm was produced to find the optimum shape of an anisotropic structure for the highest stiffness and minimum weight. In this case the elastic compliance of the structure was minimized while there were some constraints upon the stresses and weight.…”

“…The derivatives of the elastic compliance, with respect to the boundary point coordinates x h , can also be calculated analytically as shown below [7]  Equation (31) shows that in order to obtain the compliance derivatives, the derivatives of tractions and displacements must already be calculated. Therefore, during the analysis the compliance derivatives will be evaluated after the stresses, displacements and their derivatives are obtained.…”

“…In a recent study by the author [7], the optimal shape design of an anisotropic elastic body of maximum stiffness and minimum weight under specified loadings and using the boundary element method, was obtained. The elastic compliance of the structure was minimized while there were constraints on the maximum stress and weight of the structure.…”

Fracture Mechanics Analysis of Composite Structures Using the Boundary Element Shape Sensitivities

“…Equation 38 shows that during the analysis, the compliance derivatives are calculated after, the tractions, displacements and their derivatives are obtained [18].…”

“…In Ref. [18], the optimal shape design of an anisotropic elastic body of maximum stiffness and minimum weight under specified loadings and using the BEM, was obtained. Using the sensitivities of the elastic compliance, the stiffness of the structure was minimized while constraints were placed on the maximum stress and weight.…”