تجزیه و تحلیل حساسیت شکلی از کامپوزیت های در تماس با استفاده از روش المان مرزی

This paper presents the application of the boundary element method to the shape sensitivity analysis of two-dimensional composite structures in contact. A directly differentiated form of boundary integral equation with respect to geometric design variables is used to calculate shape design sensitivities for anisotropic materials with frictionless contact. The selected design variables are the coordinates of the boundary points either in the contact or non-contact area. Three example problems with anisotropic material properties are presented to validate the applications of this formulation.

Shape sensitivity analysis, that is the calculation of quantitative information on how the response of a structure is affected by changes in the variables that define its shape, is a fundamental requirement for shape optimization. Shape optimization is an important area of current development in mechanical and structural design. Computerized procedures using optimization algorithms can iteratively determine the optimum shape of a component while satisfying some objectives, without at the same time violating the design constraints. The boundary element method being a surface-oriented technique is well suited for shape optimization problems [1], [2], [3], [4] and [5]. In the last two decades important advancements have been made in the analysis of contact problems using the finite element or boundary element methods. The latter seems to have proved advantageous in treating the contact between linear elastic solids [6], [7] and [8]. The contact surface design is usually the first requirement to reduce the stress peaks. Therefore, various efforts have been made to produce optimal designs which increase the performance and reliability of the structure in contact environments [9], [10], [11], [12] and [13]. The effect of material properties should next be considered in conjunction with the shape optimization to obtain the required performance of the component. However, in this field of research the analysis has been mostly concentrated on isotropic materials. The application of composites in aerospace, automobile, civil and marine industries is well established today due to the known benefits such as high specific stiffness or strength and the material's tailoring facilities for creating high-performance structures. An understanding of the interactions between the composite material components and their optimum contact surface design can further enhance their potential applications. The objective of this work is directed towards the shape sensitivity analysis of two-dimensional anisotropic structures in frictionless contact. This study continues the previous works of the author on the shape optimization of anisotropic structures using the boundary element method [1], [2] and [3], where the effect of material properties on the optimum shape design of structures was investigated. In Ref. [3], a directly differentiated form of the BIE, with respect to boundary point coordinates, was used to calculate stress and displacement derivatives for two-dimensional anisotropic structures. In Ref. [2], 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. The objective of the work in Ref. [1] was directed towards the optimal positioning of features in anisotropic structures for maximum stiffness while the weight remains unchanged. The elastic compliance was minimized while there were constraints on the maximum stress and the geometry of the structure. To the author's knowledge, no other publications are available on the shape optimization of composite materials using the boundary elements. Here, the design sensitivity analysis of composite structures in contact has been carried out by direct differentiation of the structural response rather than using the finite difference method. The design variables are taken as the coordinates of some nodes on the boundaries of either body which is in contact. The selection of the boundary points as the design variables is more general than selecting simple geometrical variables such as radii, etc. The advantage of the proposed method is that it can be applied to any geometry, not necessarily regular shapes. However, when entire segments of the boundary or domain are governed by a single variable such as radius, the relevant velocity terms are applied together in the sensitivity analysis with respect to that variable [1]. The formulation obtained in the present study may be employed in conjunction with any numerical optimization algorithm for the shape optimization of anisotropic components in contact.

Following a brief review of the mathematical basis of the BIE method for two elastic anisotropic materials in contact, analytical differentiation of the BIE was carried out with respect to the positions of the boundary nodes. Shape design sensitivity analysis was performed to compute the derivatives of displacements, stresses, elastic compliance, etc. with respect to changes of boundary point coordinates. In the case when entire segments of the boundary or domain are governed by a single variable such as radius, then each shape variable was associated with the coordinates of a series of boundary nodes, Thus the relevant velocity terms are applied together in the sensitivity analysis with respect to that variable to determine the gradients with respect to the design variables. Due to the general approach and flexibility in the selection of the design variables the proposed formulation can be employed in conjunction with the optimization algorithms for the shape optimization of anisotropic components with arbitrary shapes in contact. The sensitivity analysis algorithm was validated using the test cases with known analytical solutions. Three example problems have been analysed and the results are presented. Four different anisotropic materials were employed for the analysis. The isotropic materials were treated as if they were anisotropic. The results showed the influence of the material properties in the sensitivities of the anisotropic components in contact.