شکل تجزیه و تحلیل حساسیت طراحی در کشش با استفاده از روش المان مرزی

This paper deals with sensitivity analysis of the different functionals appearing in optimum shape design in elasticity using boundary element method (BEM). First, a general review concerning sensitivity analysis of the most usual functionals in elasticity is presented, based on the continuum approach. The accuracy in sensitivity analysis depends on the accuracy in evaluating strains and stresses on the boundary. A general procedure for strain calculation based upon some results of differential geometry of surfaces is shown. Another essential aspect in sensitivity analysis is the definition of the design velocity on the boundary, which defines the change in the geometry of the elastic solid. A computational treatment independent of the design variables used is presented, defining nodal values of the design velocity and taking advantage of the boundary element approximation. Finally, the feasibility and accuracy of the proposed procedures are assessed through several example problems.

Both the finite element method (FEM) [3], [4] and [10] and the boundary element method (BEM) [16], [1], [9] and [12] have been extensively used to solve the optimum shape design problem in elasticity. However, most researchers tend to use BEM because of its advantages in such a problem, both for the discretization and remeshing along the optimization process and its better accuracy in the evaluation of strains and stresses on the boundary, which is an essential aspect in sensitivity calculations [10] and [2]. Traditionally, two approaches have been used in sensitivity analysis of the different functionals in elasticity. In the discrete approach, an implicit derivation of the discretized set of equations is performed and then a new problem is solved with the same matrix but with a different load vector [9]. Obviously, obtaining the derivatives of the matrix and load vector with respect to each design variable could be difficult depending on the specific problem. An alternative option is the continuum approach, which uses the material derivative of continuum mechanics and the concept of adjoint problem in order to get explicit expressions in terms of the design velocity for the sensitivities of the different functionals [11]. The continuum approach, together with BEM, has been used in recent research to solve the shape design sensitivity analysis problem. Meric [14] analyzed the problem of shape optimization in 2D heat transfer problems, using specific design variables with a semi-analytical treatment of design velocity on the boundary. Tai and Fenner [16] used geometrical design variables with a specific definition of design velocity in shape optimization in 2D elasticity. Erman and Fenner [8] described the shape optimization problem in 3D elasticity based upon an implicit derivation of the BEM integral equations. Burczynski et al. [1] considered the shape optimization problem in 3D elasticity, using different functionals to define both objective and constraint functions, and defining specific adjoint problems. Finally, Kocandrle and Koska [13] studied the shape optimization problem in 3D elasticity, obtaining an explicit expression to evaluate the sensitivity of stress functionals, considering the design variables as fictitious loads applied on the boundary. However, the different applications make use of either specific functionals or ‘ad hoc’ design variables for each problem. This paper presents a revision of the sensitivity analysis for the different functionals appearing in shape optimization in elasticity based on the continuum approach, and after that the paper describes a general method to calculate strains and stresses on the boundary and a treatment of the design velocity on the boundary in terms of nodal values, taking advantage of the BEM discretization, both for 2D and 3D problems.

A complete revision of the continuum approach for the sensitivity analysis of the different functionals appearing in elasticity, based upon material derivative and adjoint problem concepts, has been presented. Both the expression for evaluating the sensitivity and the corresponding adjoint problem has been included for the different functionals. For the functionals depending on stresses in 3D elasticity a unified adjoint problem has been obtained. Under certain conditions, it can be used in most practical applications and it represents an improvement in comparison with previous techniques, which require several adjoint problems in order to get an unique sensitivity [1]. A systematic procedure to calculate the strains and stresses on the boundary, based upon the results of differential geometry of surfaces has been developed, considering the approximation defined in the boundary elements as a surface parameterization. The procedure is absolutely general and can be used with any type of element and approximation, both in 2D and 3D models. In fact, the same technique could be used with finite elements, considering the edges corresponding to the boundary. A smooth procedure has been developed to define the design velocity on the design boundary using the boundary element approximation. The shape optimization problem can be treated in the same way independent of the design variables used. If the design variables are nodal coordinates, then the definition of the design velocity is very simple. If the design variables are of global type (such as parameters that define curves or surfaces), then it is only necessary to relate the change in those parameters with the changes in the coordinates of nodes on the design boundary. With this relation it is easy to define the design velocity at each node. Finally, several examples have been presented in order to verify the feasibility and accuracy of the proposed procedures. The comparison with analytical results and numerical results obtained by different researchers (using alternative techniques) allows to ensure the reliability of the proposed procedures, which permit to obtain very accurate results even with meshes coarser than the ones used by other researchers.