Explicit high-order generalized-$\alpha$ methods for isogeometric analysis of structural dynamics

TL;DR

This paper introduces a family of high-order explicit generalized-$\alpha$ methods for isogeometric analysis of structural dynamics, enabling dissipation control, maintaining stability, and reducing computational costs through specialized preconditioners.

Contribution

The paper develops a new high-order explicit generalized-$\alpha$ method with dissipation control and efficient preconditioning for isogeometric analysis, improving accuracy and computational efficiency.

Findings

Achieves $2k$ order accuracy in time with explicit matrix solves.

Provides dissipation control in high-frequency spectral regions.

Demonstrates robustness and efficiency through numerical examples.

Abstract

We propose a new family of high-order explicit generalized-$\alpha$ methods for hyperbolic problems with the feature of dissipation control. Our approach delivers $2k,\, \left(k \in \mathbb{N}\right)$ accuracy order in time by solving $k$ matrix systems explicitly and updating the other $2k$ variables at each time-step. The user can control the numerical dissipation in the discrete spectrum's high-frequency regions by adjusting the method's coefficients. We study the method's spectrum behaviour and show that the CFL condition is independent of the accuracy order. The stability region remains invariant while we increase the accuracy order. Next, we exploit efficient preconditioners for the isogeometric matrix to minimize the computational cost. These preconditioners use a diagonal-scaled Kronecker product of univariate parametric mass matrices; they have a robust performance with respect to the spline degree and the mesh size, and their decomposition structure implies that their application is faster than a matrix-vector product involving the fully-assembled mass matrix. Our high-order schemes require simple modifications of the available implementations of the generalized-$\alpha$ method. Finally, we present numerical examples demonstrating the methodology's performance regarding single- and multi-patch IGA discretizations.