Performance of Refined Isogeometric Analysis in Solving Quadratic Eigenvalue Problems

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.
This work has received funding from the European Union’s Horizon 2020 research and innovation program under the Marie Sklodowska-Curie grant agreement No. 777778 (MATHROCKS); the European Regional Development Fund (ERDF) through the Interreg V-A Spain-France-Andorra program POCTEFA 2014–2020 Project PIXIL (EFA362/19); the Spanish Ministry of Science and Innovation projects with references PID2019-108111RB-I00 (FEDER/AEI) and PDC2021-121093-I00, the “BCAM Severo Ochoa” accreditation of excellence, Spain (SEV-2017-0718); and the Basque Government, Spain through the BERC 2022–2025 program, the three Elkartek projects 3KIA, Spain (KK-2020/00049), EXPERTIA, Spain (KK-2021/00048), and SIGZE, Spain (KK-2021/00095), and the Consolidated Research Group MATHMODE, Spain (IT1294-19) given by the Department of Education. This publication was also made possible in part by the Professorial Chair in Computational Geoscience at Curtin University and the Deep Earth Imaging Enterprise Future Science Platforms of the Commonwealth Scientific Industrial Research Organisation, CSIRO, of Australia. The Curtin Corrosion Centre and the Curtin Institute for Computation kindly provide ongoing support. The authors also acknowledge the computer resources and technical support provided by Barcelona Supercomputing Center through the MareNostrum4 cluster (activity IDs IM-2020-3-0009 and IM-2021-2-0015).