This paper presents the application of the boundary element method to the shape design sensitivity analysis of composite structures with holes and cutouts. A two-dimensional (2D) anisotropic domain, which contains a number of voids of arbitrary shapes will be considered. The objective is to perform the design sensitivity analysis of the structure with respect to the translation and rotation of the voids using the boundary element method. A directly differentiated form of the boundary integral equation, with respect to geometric design variables is used to calculate the shape design sensitivities for anisotropic materials. The response sensitivity analysis, with respect to the design variables such as feature positions and orientations, is achieved by the definition of appropriate design velocity fields for these variables. To find the optimum positions of the features within an anisotropic structure with the highest stiffness, the elastic compliance of the structure has been minimized subject to constraints upon stresses and geometry. Due to the non-linear nature of the mean compliance and stresses, the numerical optimisation algorithm used is the feasible direction method, together with the golden section method for the 1D search. A couple of test cases have been performed to verify the proposed method.
Laminated composites are gaining importance in aircraft structural applications because of their attractive performance characteristics, e.g. high strength-to-weight ratio, high stiffness-to-weight ratio, superior fatigue properties, and high corrosion resistance. The phenomenon of progressive failure in laminated composite structures is yet to be understood, and as a result, reliable strategies for designing optimal composite structures for desired life and strength are in demand. The analytical formulation of two- and three-dimensional (3D) anisotropic elasticity, using the finite element method (FEM) or boundary integral equation techniques (BIE), has been well developed in the last three decades. This paper discusses the shape design sensitivities of 2D anisotropic structures with respect to the positioning and orientation of their holes and cutouts.
In a project sponsored by the United Kingdom Atomic Energy Authority (UKAEA), two general purpose computer programs using the BEM [1], [2], [3] and [4] for shape optimisation of isotropic structures; weight minimization and maximum stress minimization, respectively, were developed. In 1990 [1], it represented a novel application of the boundary element method to practical design optimisation problems, and showed great potential for further development in the field of design optimisation.
The BEM, being a boundary-oriented technique, can overcome a number of the difficulties associated with its main rival, the FEM. In respect of the continuously changing geometry, the accuracy of the FE analysis using an initial mesh of elements may become inadequate during the optimisation process. If during the optimisation process, the finite element mesh has to be re-generated, the cost is relatively high. The sensitivity analysis in the calculation of the derivatives, with respect to the design variables, may be obtained directly [1] in the boundary element approach rather than by approximate methods, such as finite difference schemes.
In a study by the author [5], a directly differentiated form of the BIE, with respect to boundary point coordinates, was used to calculate stress and displacement derivatives for 2D anisotropic structures. The accuracy was compared with the results of the finite difference applied to the boundary element analysis. Not surprisingly, the results obtained by analytical differentiation are much more accurate. In another study by the author [6], the weight minimization of anisotropic structures with stress constraints using the BEM is presented. The design sensitivity analysis using the BEM was combined with an optimisation algorithm to form an optimum shape design program for anisotropic structures. Different materials were analysed to investigate the effect of engineering constants on the optimum shape design of the components.
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. To demonstrate the effectiveness of this procedure, a series of design problems were analysed and discussed in detail. These results were compared with those results, which were obtained with just minimizing the weight subject to stress and geometrical constraints [6].
It should be noted that to the author's knowledge, no other publications are available on the shape optimisation of composite materials using the boundary element method.
The objective of this work is directed towards the optimal positioning of features in anisotropic structures, using the boundary element method, for maximum stiffness while the weight remains unchanged. The elastic compliance will be minimized while there are constraints on the maximum stress and the geometry of the structure. Design sensitivity analysis, which is the calculation of quantitative information for how the response of a structure is affected by changes in the design variables that define its shape, has been validated through test cases with known solutions. To demonstrate the effectiveness of this procedure, a couple of design problems will be analysed and discussed in detail.
Following a brief review of the mathematical basis of the BIE method for 2D elastic anisotropic materials, analytical differentiation of the BIE was carried out with respect to the positions of the boundary nodes. By the application of design velocity fields, the shape variables, such as feature positions, were related to the relevant boundary nodes. Shape design sensitivity analysis was performed to compute the derivatives of displacements, stresses, and elastic compliance with respect to changes of shape variables associated with the positions and orientations of the voids of arbitrary shapes within an anisotropic structure. To maximize the stiffness, the elastic compliance was minimized while satisfying some constraints upon stresses and geometry. The weight of the structure remained constant throughout the optimisation procedure. The numerical optimisation algorithm used was the feasible direction method, together with the golden section method for the 1D search. The sensitivity analysis and optimization algorithm were validated using the test cases with known solutions. A couple of examples have been analysed and the results are presented. Five different anisotropic materials were employed for the analysis. It is shown that the optimum locations and orientations of voids in a composite structure, for maximum stiffness, not only depend on the loading conditions but also on the material properties.
