Numerical study of a relaxed variational problem from optimal design

We revisit a well-known problem of optimal design, the placement of two elastic materials in the cross-section of a rod for maximum torsional rigidity. Another interpretation is the arrangement of two viscous fluids in a pipe for maximum flux under Poiseuille flow. The existence theory allows mixing on a microscopic scale, producing composite materials, and solving a relaxed version of the original design problem. This paper demonstrates that relaxation is as important for calculation as it is for existence. We minimize a discretized version of the relaxed problem using Newton's method; each quadratic approximation is solved by a multigrid method. This allows for greater resolution than previously published calculations, which were based on gradient flow.