Polygonal composite elements for stress-constrained topology optimization of nearly incompressible materials

Abstract

We in this paper propose an efficient and vigorous approach based on flexible polygonal meshes for solving stress-constrained topology optimization problems involving both compressible and nearly incompressible materials. The core idea is to employ a polygonal composite finite element technique to deal with volumetric locking phenomena in nearly incompressible material limit. Then, to effectively solve topology optimization problems with local stress constraints, an augmented Lagrangian technique that allows treating problems with a large number of constraints is exploited. By employing the augmented Lagrangian method, the solution of the original optimization problem with a lot of constraints could be obtained by solving a series of unconstrained optimization ones and updating the design variables based on the method of moving asymptotes. The significant contribution of this work is to promote a unified formulation for stress-constrained topology optimization possessing the following salient features: (a) being valid for both compressible and nearly incompressible materials; (b) being valid for triangular, quadrilateral and polygonal elements; (c) sole displacement field involved; and (d) no constraint aggregation technique required. Through several numerical examples, for the first time, distinguishing and intriguing features of the optimized topology for nearly incompressible materials with stress constraints are exhibited in the present work.