Otimização topológica de estruturas contínuas de polipropileno com restrições de tensão baseadas em critérios de escoamento

Abstract

This paper develops a stress-based topology optimization method for structures with mechanical properties of polypropylene. This method compares a yield criteria proposed for polypropylene with the yield criteria of the distortion energy theory (von Mises). The objective of the proposed optimization problem is to minimize the amount of material to be used under local stress constraints based on a yield criteria. We make a stress analysis by using a Finite Element Method. The domain is discretized using four node square elements with bilinear shape functions. Values of the stress tensor are computed in the element center. We apply an artificial density function with design variables varying continuously between 0 and 1 and the design penalization is performed through the SIMP (Solid Isotropic Microstructure with Penalization). We have basically three challenges to solve stress constrained topology optimization problems. The first challenger is the stress singularity. We adopt the -relaxation technique to modify the feasible domain. The second challenger is the large number of nonlinear constraints; therefore the Kreisselmeier-Steinhauser aggregation function is employed. And the ultimate problem is that the stress is highly nonlinear with respect to the design. The sensitivities of the objective function and stress constraints were analytically derived and a filter of sensibility was used to produce a mesh independent design and avoid the checkerboard problem. We used the method of moving asymptotes (MMA) and a primal-dual Newton method algorithm to solve the topology optimization problem. We used three benchmark problems to test the proposed formulation and solution strategy to compare a yield criteria proposed for polypropylene with the yield criteria of the distortion energy theory (von Mises) results. In the first case is a sheet subject to shear. In the second case, a cornersupported square sheet with a load in the center was optimized. In the third case, a sheet loaded in the extremity was optimized. The obtained solutions show the validity of the proposed approach to satisfy the yield criteria with polypropylene. The results reveals that the design results with a pressure-dependent yield criteria exhibit a different topology from one obtained with a pressure-independent yield criteria (von Mises). The obtained solutions also reveal that a non-symmetric optimal layout appears when using a material that has different compression and tensile strengths. Structures with less amount of material can be obtained when applying different yield criteria.