Topology Optimization of Orthotropic Elastic Design Domains with Mortar Contact Conditions

Abstract

In this paper we perform topology optimization of orthotropic elastic design domains in unilateral contact with non-matching meshes by adopting the mortar approach. The motivation is 3D printing of assemblies of parts, where the build direction as well as the contact interfaces between the parts strongly will influence the optimal solutions. Thus, topology optimization of a standard isotropic formulation with tied interfaces might not be a proper approach for this kind of design problems. This is studied in this work by maximizing the potential energy of orthotropic linear elastic bodies in unilateral frictionless contact. In such manner, no extra adjoint equation is needed to be solved and non-zero prescribed displacements are also included in the formulation properly. This will not be true if the compliance is minimized. The contact conditions of the non-matching meshes are treated by deriving the mortar integral from the principle of virtual power by representing the normal contact pressure with the trace of the global displacement shape functions. The mortar integral is then approximated with the Lobatto rule using a high number of integration points. In such manner, we obtain a set of Signorini conditions for each non-matching interface which we solve using the augmented Lagrangian approach and Newtons method. The design domains are formulated with orthotropic linear elasticity where intermediate density values are penalized using SIMP or RAMP, and the regularization is obtained by applying sensitivity or density filters following the approaches of Sigmund and Bourdin. The implementation of the methodology is efficient and produces reliable solutions. This is demonstrated for assemblies of design domains in unilateral contact which is rotated with respect to the build direction of a 3D printer. In particular, compliance as function of build direction is generated for different problems. This kind of curves might be most useful when orienting the parts in order to minimize the volume of support structures.