Material Optimization with a Penalty Barrier Multiplier Method

Abstract

Free material design deals with the question of finding the stiffest elastic structure with respect to a given load. Starting with the well known formulation as a control problem, we motivate the need for a reformulation of the problem. We do not use the standard saddle point approach for a further treatment, but use methods from Lagrange duality theory. We obtain a constrained convex programming problem, which is open to standard discretization schemes. After using a finite element ansatz we get a sparse convex quadratic system which needs due to its high dimension a suitable choice of the optimization routine. We propose the use of a penalty barrier multiplier method which takes care of the sparsity structure. For this reason the Newton system is solved by a very efficient sparse Cholesky factorization. Some examples computed with the design tool MOPED illustrate the effectiveness of our approach.