Guaranteed lower bounds to effective stiffness

Abstract

We present a numerical scheme for obtaining guaranteed (reliable) and arbitrarily close two sided bounds to effective (homogenized) parameters of the linear elasticity problem. For the upper bounds, we use standard finite element (FE) discretization of the so‐called primal problem with preconditioning based on the fast discrete Fourier transformation (FFT). For the lower bounds, we use the dual formulation and some smoother FE approximation spaces. Moreover, instead of solving the discretized dual problem, we can only compute an L²‐orthogonal projection of an auxiliary field built from the primal solution. The projection can be computed easily by FFT and provides a lower bound of almost the same quality as that obtained as the exact solution of the discretized dual problem. In addition, a simple low‐dimensional optimization improves the projected solution. Numerical examples are presented to support the theoretical developments.