A boundary-penalized isogeometric analysis for second-order hyperbolic equations

TL;DR

This paper introduces a boundary-penalized isogeometric analysis method that enhances the stability and allows larger time steps for solving second-order hyperbolic equations, especially effective with high-order elements.

Contribution

The paper proposes a novel boundary penalization technique to reduce system stiffness, thereby increasing the critical time step size in explicit isogeometric analysis for hyperbolic equations.

Findings

Critical step size increases by a factor of √((p^2 - 3p + 6)/4) for p-th order elements.

Method is validated through numerical examples in 1D, 2D, and 3D.

High-order elements benefit significantly from the proposed boundary penalization.

Abstract

Explicit time-marching schemes are popular for solving time-dependent partial differential equations; one of the biggest challenges these methods suffer is increasing the critical time-marching step size that guarantees numerical stability. In general, there are two ways to increase the critical step size. One is to reduce the stiffness of the spatially discretized system, while the other is to design time-marching schemes with larger stability regions. In this paper, we focus on the recently proposed explicit generalized-$\alpha$ method for second-order hyperbolic equations and increase the critical step size by reducing the stiffness of the isogeometric-discretized system. In particular, we apply boundary penalization to lessen the system's stiffness. For $p$-th order $C^{p-1}$ isogeometric elements, we show numerically that the critical step size increases by a factor of $\sqrt{\frac{p^2-3p+6}{4}}$, which indicates the advantages of using the proposed method, especially for high-order elements. Various examples in one, two, and three dimensions validate the performance of the proposed technique.