Bounds for effective conductivity of heterogeneous materials by the finite element method

Bounds for the effective conductivity σ* are derived using two standard variational principles established by Beran and the finite element method. We divide a two-dimensional material into triangular finite elements, use a linear local function as a trial function for each element, and apply the function to the variational principles. As a result a very simple method of computing the bounds is derived. As a numerical example, we apply the method to a model for heterogeneous materials and obtain very narrow gaps between bounds for σ*.