Fast formation and assembly for spline-based 3D fictitious domain methods

Standard finite element methods employ an element-wise assembly strategy. The element's contribution to the system matrix is formed by a loop over quadrature points. This concept is also used in fictitious domain methods, which perform simulations on a simple tensor-product background mesh cut by a boundary representation that defines the domain of interest.

Considering such d-dimensional background meshes based on splines of degree p with maximal smoothness, Cp−1, the cost of setting up the system matrix is 𝒪(p3d) per degree of freedom. Alternative assembly and formation techniques can significantly reduce this cost. In particular, the combination of (1) sum factorization, (2) weighted quadrature, and (3) row-based assembly yields a cost of 𝒪(pd+1) for non-cut background meshes. However, applying this fast approach to cut background meshes is an open challenge since they do not have a tensor-product structure.

This work presents techniques that allow the treatment of cut background meshes and thus the application of fast formation and assembly to fictitious domain methods. First, a discontinuous version of weighted quadrature is presented, which introduces a discontinuity into a cut test function's support. The cut region can be treated separately from the non-cut counterpart; the latter can be assembled by the fast concepts. A three-dimensional example investigates the accuracy and efficiency of the proposed concept and demonstrates its speed-up compared to conventional formation and assembly.