74P05 Compliance or weight optimization
A strictly feasible sequential convex programming method
- In free material optimization (FMO), one tries to find the best mechanical structure by minimizing the weight or by maximizing the stiffness with respect to given load cases. Design variables are the material properties represented by elasticity tensors or elementary material matrices, respectively, based on a given finite element discretization. Material properties are as general as possible, i.e., anisotropic, leading to positive definite elasticity tensors, which may be arbitrarily small in case of vanishing material. To guarantee a positive definite global stiffness matrix for computing design constraints, it is required that all iterates of an optimization algorithm retain positive definite tensors. Otherwise, some constraints, e.g., the compliance, cannot be evaluated and the algorithm fails. FMO problems are generalizations of topology optimization problems. The goal of topology optimization is to find the stiffest structure subject to given loads and a limited amount of material. In contrast to FMO the material is explicitly given and cannot vary. Based on a finite element discretization, in each element it is decided whether to use material or not. The regions with vanishing material are interpreted as void. The resulting optimization problem can be solved by numerous efficient nonlinear optimization methods, for example sequential convex programming methods. Sequential convex programming (SCP) formulates separable and strictly convex nonlinear subproblems iteratively by approximating the objective and the constraints. Lower and upper asymptotes are introduced to truncate the feasible region. Due to the special structure, the resulting subproblems can be solved efficiently by appropriate methods, e.g., interior point methods. To ensure global convergence, a line search procedure is introduced. Moreover, an active set strategy is applied to reduce computation time. The iterates of SCP are not guaranteed to be inside the corresponding feasible region described by the constraints. As a consequence it is not able to solve free material optimization problems as the compliance function is only well-defined on the feasible region of some of the constraints. We propose a modification of a SCP method that ensures feasibility with respect to a given set of inequality constraints. The resulting procedure is called feasible sequential convex programming method (SCPF). SCPF expands the resulting subproblems by additional nonlinear constraints, that are passed to the subproblem directly to ensure their feasibility in each iteration step. They are referred as feasibility constraints. In addition, other constraints may be violated within the optimization process. As globalization technique a line search procedure is used to ensure convergence. The resulting subproblems can be solved efficiently taking the sparse structure into account. Moreover, semidefinite constraints have to be replaced by nonlinear ones, such that SCPF is applicable. SCPF successfully solved FMO problems with up to 120.000 variables and 60.000 constraints. Within a theoretical analysis global convergence of SCPF is shown for convex feasibility constraints.