Fast isogeometric solvers for hyperbolic wave propagation problems

TL;DR

This paper introduces a fast isogeometric solver for hyperbolic wave problems using Kronecker product approximations, enabling linear-time implicit time stepping and demonstrating stability and efficiency in wave and elasticity simulations.

Contribution

It develops a novel algebraic approach using Kronecker products for efficient implicit isogeometric analysis of hyperbolic problems, including wave and elasticity simulations.

Findings

Achieves linear (O(N)) computational complexity per time step.

Demonstrates unconditional stability of the proposed methods.

Validates performance through wave and elasticity problem simulations.

Abstract

We use the alternating direction method to simulate implicit dynamics. ur spatial discretization uses isogeometric analysis. Namely, we simulate a (hyperbolic) wave propagation problem in which we use tensor-product B-splines in space and an implicit time marching method to fully discretize the problem. We approximate our discrete operator as a Kronecker product of one-dimensional mass and stiffness matrices. As a result of this algebraic transformation, we can factorize the resulting system of equations in linear (i.e., O(N)) time at each step of the implicit method. We demonstrate the performance of our method in the model P-wave propagation problem. We then extend it to simulate the linear elasticity problem once we decouple the vector problem using alternating triangular methods. We proof theoretically and experimentally the unconditional stability of both methods.