Stress-based topology optimization of continuum structures for the elastic contact problems with friction

Structural problems have various nonlinearities in the real world and these nonlinearities should be accommodated in structural topology optimization. This work proposes a topology optimization method for minimizing the maximum von Mises stress of elastic continuum structures with frictional contact under material usage constraint, using an extended Bi-directional Evolutionary Structural Optimization (BESO) method. Stresses are treated as global performance (objective) function, the global von Mises stress is measured by the p-norm stress aggregation approach, and the friction behavior is governed by the Coulomb friction law regularized in analogy with the perfect elasto-plastic theory. BESO method based on discrete variables which can avoid the well-known stress singularity and the numerical instability issue in frictional contact problems. The adjoint sensitivity analysis method is adopted to derive the sensitivity numbers. The effectiveness of the proposed method is validated through a series of comparison studies including elastic-rigid and elastic-elastic contact problems. The influence of varying friction coefficient on the optimized results and the stress distributions are investigated in comparison with the maximum stiffness design. The effect of different parameters including p-norm, volume fraction and mesh density on the optimized results are discussed. The optimized results, for elastic-rigid contact, indicate that the maximum stress can be reduced compared with elastic-elastic contact. The optimized stress decreases as the friction coefficient increases because the friction behavior resists the tangential deformation at the contact interface. The results also show that the proposed approach can achieve a reasonable design that effectively controls the stress level and reduces the stress concentration effect at the critical stress areas.