Multi-Material Topology Optimization for Problems Dominated by Boundary and Interface Phenomena

Traditional topology optimization methods use some form of an Ersatz (or fictitious) material approach to model the mechanical response, smearing boundaries (void-material) and interfaces (material-material) across one or more elements in a finite element model. This approach is frequently used in both density and level set methods because of its simplicity and ease of implementation. However, Ersatz material approaches lead to fundamental challenges when the design performance requires accounting for boundary and interface phenomena, such as design-dependent surface loads, mechanical contact, and interface resistance for heat transfer. Often, rather ad-hoc and/or overly complex methods are used to approximate boundary and interface phenomena, without guarantee of convergence with mesh refinement.

Combining a level set method and an immersed boundary method with a sharp boundary and interface description overcomes the challenges outlined above. In this work, we present an explicit multi-material level set method where complex boundary and interface phenomena are predicted by the eXtended Finite Element Method (XFEM). Multiple level set fields are used in a hierarchical fashion to describe the layout and shape of multiple materials. Within each material domain, standard constitutive models are applied. At the material domain boundaries, boundary and interface conditions are imposed using weak formulations, namely Nitsche’s method. The weak form of the governing equations is augmented by the face-oriented ghost stabilization method to mitigate ill-conditioning issues arising in the XFEM. The level set fields and the state variable fields are discretized by B-splines on hierarchically refined tensor meshes. A generalized Heaviside enrichment is used to interpolate test and trial functions within the different material domains, avoiding spurious coupling between disconnected subdomains of the same material. In this work, the enrichment strategy is generalized for arbitrary enrichment functions.

Details of the computational strategies and techniques for creating XFEM models of multi-material configurations are presented. The accuracy and stability of the XFEM analyses for problems in linear and nonlinear elasticity, fluid mechanics, and heat transfer are presented, considering various types of boundary and interface conditions. The results show that explicitly tracking the interface geometry and the use of higher-order B-splines provide distinct computational advantages in terms of accuracy and convergence rates over analysis methods traditionally used in topology optimization.

The level set / XFEM framework outlined above is applied to topology optimization problems considering contact in linear elasticity, heat transfer problems with interface resistance, and turbulent flow problems where boundary layers need to be resolved. The problem-specific objective function is augmented by penalty terms that promote smooth shapes and level set functions with a uniform spatial gradient along the iso-contour that defines the boundaries and interfaces. The optimization problems are solved by gradient-based optimization algorithms. The gradients of objective and constraints with respect to the optimization variables are computed by the adjoint method; its accuracy is demonstrated by comparison against finite difference results. The numerical studies demonstrate the ability to predict the mechanical response and design performance on moderately refined meshes for rather complex material layouts in 2D and 3D. The optimization results highlight the importance of accurately capturing interface and boundary phenomena on the optimized designs.