Reduced basis isogeometric mortar approximations for eigenvalue problems in vibroacoustics

TL;DR

This paper develops reduced basis isogeometric mortar methods to efficiently solve eigenvalue problems in vibroacoustics, specifically for simulating violin bridge vibrations with complex geometries and multiple parameters.

Contribution

It introduces a novel combination of reduced basis techniques with isogeometric mortar methods for eigenvalue problems involving complex geometries and multiple parameters.

Findings

Efficient multi-query eigenvalue solutions for vibroacoustic models.

Effective snapshot selection using a multi-query greedy strategy.

Accurate eigenvalue approximations with reduced computational cost.

Abstract

We simulate the vibration of a violin bridge in a multi-query context using reduced basis techniques. The mathematical model is based on an eigenvalue problem for the orthotropic linear elasticity equation. In addition to the nine material parameters, a geometrical thickness parameter is considered. This parameter enters as a 10th material parameter into the system by a mapping onto a parameter independent reference domain. The detailed simulation is carried out by isogeometric mortar methods. Weakly coupled patch-wise tensorial structured isogeometric elements are of special interest for complex geometries with piecewise smooth but curvilinear boundaries. To obtain locality in the detailed system, we use the saddle point approach and do not apply static condensation techniques. However within the reduced basis context, it is natural to eliminate the Lagrange multiplier and formulate a reduced eigenvalue problem for a symmetric positive definite matrix. The selection of the snapshots is controlled by a multi-query greedy strategy taking into account an error indicator allowing for multiple eigenvalues.