Fast immersed boundary method based on weighted quadrature

TL;DR

This paper introduces a fast immersed boundary method that combines sum factorization and weighted quadrature to efficiently handle higher-order tensor product spline computations on cut domains, with demonstrated benefits in elasticity simulations.

Contribution

It presents a novel approach to divide cut basis function supports and apply specialized quadrature rules, improving computational efficiency in immersed boundary methods.

Findings

Confirmed computational cost estimates for integration routines.

Speed-up decreases with mesh refinement.

Effective for linear elasticity benchmarks.

Abstract

Combining sum factorization, weighted quadrature, and row-based assembly enables efficient higher-order computations for tensor product splines. We aim to transfer these concepts to immersed boundary methods, which perform simulations on a regular background mesh cut by a boundary representation that defines the domain of interest. Therefore, we present a novel concept to divide the support of cut basis functions to obtain regular parts suited for sum factorization. These regions require special discontinuous weighted quadrature rules, while Gauss-like quadrature rules integrate the remaining support. Two linear elasticity benchmark problems confirm the derived estimate for the computational costs of the different integration routines and their combination. Although the presence of cut elements reduces the speed-up, its contribution to the overall computation time declines with h-refinement.