An accurate strategy for computing reaction forces and fluxes on trimmed locally refined meshes

The finite element method is classically based on nodal Lagrange basis functions defined on conforming meshes. In this context, total reaction forces are commonly computed from the so-called "nodal forces", yielding higher accuracy and convergence rates than reactions obtained from the differentiated primal solution ("direct" method). The finite cell method and isogeometric analysis promise to improve the interoperability of computer-aided design and computer-aided engineering, enabling a direct approach to the numerical simulation of trimmed geometries. However, body-unfitted meshes preclude the use of classic nodal reaction algorithms. This work shows that the direct method can perform particularly poorly for immersed methods. Instead, conservative reactions can be obtained from equilibrium expressions given by the weak problem formulation, yielding superior accuracy and convergence rates typical of nodal reactions. This approach is also extended to non-interpolatory basis functions, such as the (truncated) hierarchical B-splines.