Fast formation and assembly for spline‐based 3D fictitious domain methods. Issue 1 (24th March 2023) - Marussig, Benjamin
Abstract: 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, C p −1, the cost of setting up the system matrix is 𝒪( p 3 d ) 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( p d +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.