Discontinuous Petrov-Galerkin Methods for Topology Optimization

Discontinuous Petrov–Galerkin (DPG) methods constitute a modern class of finite element methods, which present several advantages when compared with traditional Bubnov–Galerkin methods, especially when the latter is applied to indefinite or non-symmetric problems. Ourobjective is to utilize the advantages of DPG methods in the context of topology optimization.

The direct application of DPG discretizations to BVPs arising in topology optimization is hindered by the very unusual scaling of the residual, caused by the gigantic jumps in the coefficients of the governing differential equations. In the prototypical case of linearized elasticity with SIMP model the coefficient ratio between the “stiff” and “soft”phases is held at a billion, which is further squared by Petrov–Galerkin methods based on minimizing the squared residual.

We introduce a DPG method with appropriately scaled residual norm, which allows us to deal with big contrast ratios in the coefficients. The method is tested on benchmark topology optimization problems.