تجزیه و تحلیل حساسیت محدودیت های استرس در بهینه سازی توپولوژی ساختاری

Sensitivity Analysis is an essential issue in the structural optimization field. The calculation of the derivatives of the most relevant quantities (displacements, stresses, strains) in optimum design of structures allows to estimate the structural response when changes in the design variables are introduced. This essential information is used by the most frequent conventional optimization algorithms (SLP, MMA, Feasible directions) in order to reach the optimal solution. According to this idea, the Sensitivity Analysis of the stress constraints in Topology Optimization problems is a crucial aspect to obtain the optimal solution when stress constraints are considered. Maximum stiffness approaches usually involve one linear constraint and one non-linear objective function. Thus, the computation of the required sensitivity analysis does not mean a crucial limitation. However, in the topology optimization problem with stress constraints, efficient and accurate computation of the derivatives is needed in order to reach appropriate optimal solutions. In this paper, a complete analytic and efficient procedure to obtain the Sensitivity Analysis of the stress constraints in topology optimization of continuum structures is analyzed. First order derivatives and second order directional derivatives of the stress constraints are analyzed and included in the optimization procedure. In addition, topology optimization problems usually involve thousands of design variables and constraints. Thus, an efficient implementation of the algorithms used in the computation of the Sensitivity Analysis is developed in order to reduce the computational cost required. Finally, the sensitivity analysis techniques presented in this paper are tested by solving some application examples.

Since topology optimization of continuum structures problems was studied by Bendsøe and Kikuchi in 1988 [1] and [2], important tasks have been extensively analyzed (e.g. the Sensitivity Analysis and the Optimization Algorithms used to obtain the optimal solution). Although there are optimization algorithms that do not require the Sensitivity Analysis (e.g. GA, Optimality criteria [28] and [34]), most of the conventional optimization algorithms require specific information about the derivatives of the objective function and the constraints [8], [10], [18], [21], [22] and [31]. In fact, when the problem involves a large number of design variables and constraints, high order derivatives are more appropriate to reach the optimal solution [16], [17] and [18]. Structural Topology Optimization problems with stress constraints usually introduce a large number of design variables (usually one design variable per element of the mesh of finite elements: the relative density [1], [2], [3], [8], [19], [22], [26] and [33]) and a large number of non-linear constraints (usually one stress constraint in the central node of each element [4], [8], [9], [11], [21], [22], [26] and [33]). In addition, if several load cases are considered, the number of stress constraints increases considerably. In this paper, three different formulations of the stress constraints have been analyzed: the local approach [4], [8], [22], [26] and [33], the global approach [14], [22], [24], [25] and [27] and the block aggregation approach of the stress constraints [24] and [25]. According to that, the resulting optimization problems require efficient and specific optimization algorithms [16], [22], [30] and [31]. These algorithms usually require first order derivatives of the objective function and the constraints. In some cases, higher order derivatives need to be considered in the optimization process in order to avoid unexpected effects (e.g. zig-zag) [16]. In this paper we have used a Sequential Linear Programming with Quadratic Line Search optimization algorithm. This procedure requires first order derivatives of the stress constraints and the objective function, and first and second order directional derivatives of the stress constraints and the objective function [16] and [21]. The computation of these derivatives can be carried out by using different numerical algorithms (finite differences, analytical derivation). When it is possible and suitable, the most usual technique to obtain the sensitivity analysis is the theoretical analysis. Thus, the derivation algorithm consists in the implementation of the theoretical functions previously obtained by applying analytical differentiation techniques [17]. The analytical computation of the derivatives requires an important and rigorous mathematical study in order to obtain the expressions to be implemented in the optimization source code. In this paper we develop the whole derivation process of the stress constraints in topology optimization of structures. This theoretical analysis allows to obtain exact mathematical expressions of the derivatives of the stress constraints and good numerical approximations. This paper is divided into 10 sections where this introduction is the first one. The second section is dedicated to present and briefly explain the optimization procedure used to solve the topology optimization problem. Section 3 is devoted to state the structural analysis methodology by means of a FEM model. Section 4 corresponds to the study of the sensitivity analysis of constraints that depend on the structural displacements and the design variables. Section 5 presents the sensitivity analysis of constraints that depend on a reference stress and on the design variables by using the algorithms proposed in Section 4. 6, 7 and 8 are devoted to obtain the sensitivity analysis of three different formulations of stress constraints studied by the authors: the local approach, the global approach and the block aggregation of stress constraints. Section 9 presents some numerical examples of the topology optimization problem by using the techniques proposed in the previous sections. Finally, some conclusions are stated in Section 10.

In this paper we propose a complete analytical procedure to obtain high order sensitivity analysis of stress constraints in topology optimization of structures. The Sensitivity Analysis of the objective function and the stress constraints is essential to obtain adequate solutions since the resulting optimization problem is very complicated due to the large number of design variables and constraints involved. Consequently, it is necessary to use suitable optimization algorithms that involve, at least, first order derivatives of the objective function and, specially, the stress constraints. In this paper we have developed first order derivatives and second order directional derivatives of the stress constraints. This information is required by the Sequential Linear Programming with Quadratic Line Search algorithm that we have used. All the derivatives are developed analytically in order to reduce the computing time required by numerical approximations (finite differences) and in order to obtain reliable approximations. The minimum weight approach with stress constraints involves a very large number of constraints (contrary to the most usual maximum stiffness formulations) and consequently requires very large computing time. Full set of first order derivatives of the stress constraints can be obtained via an “Adjoint Variable” procedure in order to reduce the computing requirements. When the number of load cases is small, which is the most frequent situation, this technique considerably reduces the computing effort required in comparison with “Direct Differentiation” algorithms. Furthermore, this technique allows to compute the first order derivatives of a set of active constraints. Thus, it is possible to avoid the computation of the stress constraints derivatives in case they are not necessary. In addition, these calculations can be easily performed in parallel since the operations required to obtain each one of these derivatives are independent from a computational point of view. Thus, first order derivatives of the stress constraints can be easily computed in parallel [7]. On the other hand, full set of first and second order directional derivatives of the stress constraints can be easily obtained with small computing resources by applying a “Direct Differentiation” procedure. This technique allows to compute the full set of first and second order directional derivatives of the stress constraints although full set of first order derivatives were not previously obtained. This fact allows to include the full set of constraints of the problem in the Quadratic Line Search algorithm although the full set of first order derivatives were not obtained neither used in the SLP algorithm. The computation of these sensitivities requires to solve a large number of systems of linear equations, especially in the computation of the full set of first order derivatives of the stress constraints. However, the matrix of these systems of linear equations is the stiffness matrix of the structural problem. Thus, a Cholesky factorization procedure is the right choice in order to reduce the computing effort devoted to solve additional systems of linear equations. The whole sensitivity analysis procedure proposed in this paper has demonstrated to work properly even in large structural optimization problems with thousands of design variables and thousands of non-linear constraints. The algorithms proposed and implemented in a topology optimization code are exact from an analytical point of view. In addition, computational aspects have been addressed in order to improve and speed up the sensitivity analysis methodology. All the techniques proposed are devoted to reduce as much as possible the required computation effort without losing precision. Otherwise the computation of the sensitivity analysis will become unaffordable in practical applications due to the large number of stress constraints involved.