Topology optimization of incompressible media using mixed finite elements

Abstract

We present an alternative formulation for the topology optimization of structures made of incompressible materials, a topic that cannot be tackled using most of the approaches of the current literature that are mainly based on displacement finite elements that are well known to be affected by the locking phenomenon. A way out of the problem has recently been proposed in Sigmund and Clausen (2007) [O. Sigmund, P.M. Clausen, Topology optimization using a mixed formulation: an alternative way to solve pressure load problems, Comput. Methods Appl. Mech. Engrg. 196 (13–16) (2007) 1874–1889], based on a displacement–pressure finite element discretization. The approach presented here consists in a truly-mixed variational formulation coupled to a mixed-FEM discretization that uses the composite element of Johnson and Mercier for the discretization of the stress field. By so doing, the continuous and discrete inf–sup conditions of the problem are automatically met even in the presence of incompressible materials. A few numerical studies are presented to validate the theoretical framework for which the well-known method of moving asymptotes (MMA) is adopted for the numerical optimization of the problem. Different topologies in plane stress and plane strain conditions are presented, with particular attention to the convergence of the last ones to pure 0–1 designs. Some forthcoming investigations are eventually highlighted including the solution of stress-constrained topology-optimization problems that find in the truly-mixed setting their natural environment and the extension of the present formulation to deal with pressure–load structural problems.