An alternative truly-mixed formulation to solve pressure load problems in topology optimization

Abstract

The paper deals with an alternative formulation for the topology optimization of structures acted upon by pressure loads, exploiting finite elements techniques that are able to handle incompressible materials. The method, firstly presented in [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 (2007) 1874–1889, [33]], consists in exploiting the modeling of fluid incompressibility within a topology optimization framework. The implementation of a fluid phase enables to transfer pressure loads from the domain boundaries to the evolving edges of optimal design, without relying on more complex techniques traditionally employed to recover the load application surfaces at each step of the minimization process. In this context, the main numerical trouble is therefore the application of finite elements techniques to solve incompressible materials analysis. This topic in fact cannot be tackled using most of the approaches of the current literature that are mainly based on displacement finite elements which are well known to be affected by the locking phenomenon. While Sigmund and Clausen (2007) uses a displacement–pressure finite element discretization to solve the problem, the approach herein presented consists in the adoption of a “truly-mixed” variational formulation coupled to a discretization based on the Johnson and Mercier finite element, that both pass the inf–sup conditions of the problem even in the presence of incompressible materials. The well-known method of moving asymptotes (MMA) is adopted in the numerical studies presented, along with a particular density interpolation to model the presence of a fluid and solid phase within the same design. The adopted scheme is especially conceived to avoid the arising of numerical instabilities that may arise within the optimization procedure when handling incompressible material. Moreover, the accuracy and stability in stress evaluation provided by the “truly-mixed” setting are herein exploited to introduce an alternative procedure that implements pressure constraints to avoid optimal designs that present cavities filled by fluid.