Abstract
We propose the use of highly continuous finite element spaces interconnected with low continuity hyperplanes to maximize the performance of direct solvers. Starting from a highly continuous Isogeometric Analysis (IGA) discretization, we introduce C°-separators to reduce the interconnection between degrees of freedom in the mesh. By doing so, both the solution time and best approximation errors are simultaneously improved. We call the resulting method “refined Isogeometric Analysis (rIGA)”. To illustrate the impact of the continuity reduction, we analyze the number of Floating Point Operations (FLOPs), computational times, and memory required to solve the linear system obtained by discretizing the Laplace problem with structured meshes and uniform polynomial orders. Theoretical estimates demonstrate that an optimal continuity reduction may decrease the total computational time by a factor between p2 and p3, with p being the polynomial order of the discretization. Numerical results indicate that our proposed refined isogeometric analysis delivers a speed-up factor proportional to p2. In a 2D mesh with four million elements and p=5, the linear system resulting from rIGA is solved 22 times faster than the one from highly continuous IGA. In a 3D mesh with one million elements and p=3, the linear system is solved 15 times faster for the refined than the maximum continuity isogeometric analysis.
| Original language | English |
|---|---|
| Pages (from-to) | 586-605 |
| Number of pages | 20 |
| Journal | Computer Methods in Applied Mechanics and Engineering |
| Volume | 316 |
| DOIs | |
| State | Published - Apr 1 2017 |
Funding
David Pardo and Daniel Garcia have received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie Grant Agreement No. 644602 , the Projects of the Spanish Ministry of Economy and Competitiveness with reference MTM2013-40824-P and MTM2016-76329-R , the BCAM Severo Ochoa accreditation of excellence SEV-2013-0323 , and the Basque Government through the BERC 2014–2017 program, the Consolidated Research Group Grant IT649-13 on “Mathematical Modeling, Simulation, and Industrial Applications (M2SI)”, and the ICERMAR Project KK-2015/0000097 . The work of Maciej Paszyński has been supported by National Science Centre , Poland, Grant No. DEC-2015/17/B/ST6/01867 . This publication was made possible in part by a National Priorities Research Program grant 7-1482-1-278 from the Qatar National Research Fund (a member of The Qatar Foundation) and the Center for Numerical Porous Media at King Abdullah University of Science and Technology (KAUST). The J. Tinsley Oden Faculty Fellowship Research Program at the Institute for Computational Engineering and Sciences (ICES) of the University of Texas at Austin has partially supported the visits of VMC to ICES. The authors acknowledge the Texas Advance Computing Center (TACC) at The University of Texas at Austin for providing HPC resources that have contributed to the research results reported in the paper.
| Funders | Funder number |
|---|---|
| BCAM | SEV-2013-0323 |
| BERC | IT649-13 |
| ICERMAR | KK-2015/0000097 |
| Institute for Computational Engineering and Sciences | |
| Spanish Ministry of Economy and Competitiveness | MTM2013-40824-P, MTM2016-76329-R |
| Qatar National Research Fund | |
| Horizon 2020 Framework Programme | 644202, 644602 |
| Eusko Jaurlaritza | |
| King Abdullah University of Science and Technology | |
| Narodowe Centrum Nauki | 7-1482-1-278 |
Keywords
- Direct solvers
- Finite element analysis (FEA)
- Isogeometric analysis (IGA)
- Multi-frontal solvers
- Refined isogeometric analysis (rIGA)
- k-refinement