Control in the coefficients with variational crimes: Application to topology optimization of Kirchhoff plates

We study convergence of discontinuous Galerkin-type discretizations of the problems of control in the coefficients of uniformly elliptic partial differential equations (PDEs). As a model problem we use that of the optimal design of thin (Kirchhoff) plates, where the governing equations are of the fourth order.

Methods which do not require approximation subspaces to conform to the smoothness requirements dictated by the PDE are very attractive for such problems. However, variational formulations of such methods normally contain boundary integrals whose dependence on the small, with respect to “volumetric” Lebesgue norm, changes of the coefficients is generally speaking not continuous.

We utilize the lifting formulation of the discontinuous Galerkin method to deal with this issue.Our main result is that limit points of sequences of designs verifying discrete versions of stationarity can also be expected to satisfy stationarity for the limiting continuum mechanics problem. We illustrate the practical behaviour of our discretization strategy on some benchmark-type examples.