Refined isogeometric analysis of quadratic eigenvalue problems

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 C0 and C1 separators at specific element interfaces for our rIGA generalizations of the curl-conforming Nédélec 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.