Performance of Refined Isogeometric Analysis in Solving Quadratic Eigenvalue Problems

TL;DR

This paper demonstrates that refined isogeometric analysis (rIGA) enhances the efficiency of solving quadratic eigenvalue problems by reducing LU factorization time, especially for large-scale problems, through strategic basis function continuity modifications.

Contribution

The study introduces a novel rIGA approach with specific basis function continuity adjustments that significantly speeds up eigenproblem solutions compared to traditional IGA.

Findings

rIGA reduces LU factorization time by up to O((p-1)^2) for large problems.

Numerical tests show rIGA accelerates quadratic eigensystem solutions by O(p-1) for moderate-sized problems.

The computational cost benefits diminish as the number of eigenvalues increases due to additional matrix-vector operations.

Abstract

Certain applications that analyze damping effects require the solution of quadratic eigenvalue problems (QEPs). We use refined isogeometric analysis (rIGA) to solve quadratic eigenproblems. rIGA discretization, while conserving desirable properties of maximum-continuity isogeometric analysis (IGA), reduces the interconnection between degrees of freedom by adding low-continuity basis functions. This connectivity reduction in rIGA's algebraic system results in faster matrix LU factorizations when using multifrontal direct solvers. We compare computational costs of rIGA versus those of IGA when employing Krylov eigensolvers to solve quadratic eigenproblems arising in 2D vector-valued multifield problems. For large problem sizes, the eigencomputation cost is governed by the cost of LU factorization, followed by costs of several matrix-vector and vector-vector multiplications, which correspond to Krylov projections. We minimize the computational cost by introducing C^0 and C^1 separators at specific element interfaces for our rIGA generalizations of the curl-conforming Nedelec and divergence-conforming Raviart-Thomas finite elements. Let p be the polynomial degree of basis functions; the LU factorization is up to O((p-1)^2) times faster when using rIGA compared to IGA in the asymptotic regime. Thus, rIGA theoretically improves the total eigencomputation cost by O((p-1)^2) for sufficiently large problem sizes. Yet, in practical cases of moderate-size eigenproblems, the improvement rate deteriorates as the number of computed eigenvalues increases because of multiple matrix-vector and vector-vector operations. Our numerical tests show that rIGA accelerates the solution of quadratic eigensystems by O(p-1) for moderately sized problems when we seek to compute a reasonable number of eigenvalues.