Topology optimization of softening structures under displacement constraints as an MPEC

This paper presents a mathematical programming based technique for the minimum weight (volume) topology optimization of truss-like structures such that strain softening material properties, that can lead to severe physical instability, can be accommodated. In addition, satisfaction of such serviceability criteria as limited displacements at some specific points is ensured. The problem is formulated in terms of truss member cross-sectional areas. This leads to a challenging nonconvex and nonsmooth optimization problem, known as a mathematical program with equilibrium constraints (MPEC). A two-step optimization algorithm is proposed to overcome the problems typically associated with nondefiniteness of some key matrices and at the same time nondifferentiability of the mathematical system. Each step involves updating the ground structure and solving the MPEC using a penalized nonlinear programming (NLP) approach. Some numerical examples are provided to illustrate application, robustness and efficiency of the proposed scheme. The safety and integrity of the designed topologically optimal structures are validated using appropriate stepwise holonomic elastoplastic analyses.