A numerical iterative scheme for computing finite order rank-one convex envelopes

Abstract. It is known that the i-th order laminated microstructures can be resolved by the k-th order rank-one convex envelopes with k ≥ i. So the requirement of establishing an efficient numerical scheme for the computation of the finite order rank-one convex envelopes arises. In this paper, we develop an iterative scheme for such a purpose. The 1-st order rank-one convex envelope R1f is approximated by evaluating its value on matrixes at each grid point in R and then extend to non-grid points by interpolation. The approximate k-th order rank-one convex envelope Rkf is obtained iteratively by computing the approximate 1-st order rank-one convex envelope of the numerical approximation of Rk−1f . Compared with O(h) obtained so far for other methods, the optimal convergence rate O(h) is established for our scheme, and numerical examples illustrate the computational efficiency of the scheme.