Code verification of immersed boundary techniques using the method of manufactured solutions

Code verification plays a crucial role for all finite element applications, especially for non-standard ones, such as immersed boundary approaches, which are typically based on novel algorithms and often error-prone in-house implementations. Instead of relying on rarely available analytical solutions or overkill FEM simulations, in this article, the capabilities of the method of manufactured solutions (MoMS) are explored, enabling an easy and straightforward derivation of closed-form reference solutions. The focus is kept on immersed problems, in particular, on the finite cell method (FCM), and manufactured solutions are derived for 2D and 3D problems involving voids and single/multiple inclusions. We propose several approaches for the construction of the manufactured solutions, where zero traction conditions for void regions and continuous normal stresses along material interfaces are directly fulfilled. Thus, no weak boundary conditions are required for reproducing the manufactured solution via FCM. This not only enables code verification for FCM implementations that lack the option of applying weak boundary conditions, but also keeps the simulation complexity low, when testing other relevant features, e.g., different integration schemes or the implementation of enrichment functions. The flexibility and wide application range of the MoMS in the context of immersed boundary simulations is demonstrated using static, quasi-static, and transient problems in the context of linear elasticity. Finally, the analytical derivations of the manufactured solutions used in this paper are provided as supplementary material.

Automated summary

Code verification is crucial for finite element applications, especially for non-standard approaches. In this article, the method of manufactured solutions (MoMS) is explored to derive closed-form reference solutions for immersed boundary problems. Various approaches are proposed for constructing manufactured solutions, enabling code verification without weak boundary conditions and keeping the simulation complexity low.