Topology optimization of elastic contact problems with friction using efficient adjoint sensitivity analysis with load increment reduction

This work presents a topology optimization method for the stiffness maximization design of elastic structures with frictional contact. The friction behavior is assumed to be governed by the Coulomb friction law regularized in analogy with the perfect elasto-plastic theory. To facilitate efficient adjoint sensitivity analysis, two load increment reduction (LIR) rules are proposed for sticking friction and pure sliding friction, and are validated by theoretical proof and numerical verification. The first rule is that any load increment whose set of sticking contact node pairs contains that of the load increment after can be skipped in the adjoint sensitivity analysis without influencing the final calculated sensitivities. The second one is that all load increments before the load increment in pure sliding friction can also be skipped. The implemented topology optimization procedure is validated by numerical examples. It is found that only the final load increment is needed in the overwhelming majority of iterations for all the tested cases based on the LIR rules. The computational burden of sensitivity analysis is thereby greatly reduced. Results also show that the optimized compliance generally decreases as the friction coefficient increases because the friction behavior helps to resist tangential deformations at the contact interface.