Refined isogeometric analysis for a preconditioned conjugate gradient solver

Handle URI:
http://hdl.handle.net/10754/627135
Title:
Refined isogeometric analysis for a preconditioned conjugate gradient solver
Authors:
Garcia, Daniel; Pardo, David ( 0000-0002-1101-2248 ) ; Dalcin, Lisandro ( 0000-0001-8086-0155 ) ; Calo, Victor M. ( 0000-0002-1805-4045 )
Abstract:
Starting from a highly continuous Isogeometric Analysis (IGA) discretization, refined Isogeometric Analysis (rIGA) introduces C0 hyperplanes that act as separators for the direct LU factorization solver. As a result, the total computational cost required to solve the corresponding system of equations using a direct LU factorization solver dramatically reduces (up to a factor of 55) Garcia et al. (2017). At the same time, rIGA enriches the IGA spaces, thus improving the best approximation error. In this work, we extend the complexity analysis of rIGA to the case of iterative solvers. We build an iterative solver as follows: we first construct the Schur complements using a direct solver over small subdomains (macro-elements). We then assemble those Schur complements into a global skeleton system. Subsequently, we solve this system iteratively using Conjugate Gradients (CG) with an incomplete LU (ILU) preconditioner. For a 2D Poisson model problem with a structured mesh and a uniform polynomial degree of approximation, rIGA achieves moderate savings with respect to IGA in terms of the number of Floating Point Operations (FLOPs) and computational time (in seconds) required to solve the resulting system of linear equations. For instance, for a mesh with four million elements and polynomial degree p=3, the iterative solver is approximately 2.6 times faster (in time) when applied to the rIGA system than to the IGA one. These savings occur because the skeleton rIGA system contains fewer non-zero entries than the IGA one. The opposite situation occurs for 3D problems, and as a result, 3D rIGA discretizations provide no gains with respect to their IGA counterparts when considering iterative solvers.
KAUST Department:
Extreme Computing Research Center; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division; Numerical Porous Media SRI Center (NumPor)
Citation:
Garcia D, Pardo D, Dalcin L, Calo VM (2018) Refined isogeometric analysis for a preconditioned conjugate gradient solver. Computer Methods in Applied Mechanics and Engineering. Available: http://dx.doi.org/10.1016/j.cma.2018.02.006.
Publisher:
Elsevier BV
Journal:
Computer Methods in Applied Mechanics and Engineering
Issue Date:
12-Feb-2018
DOI:
10.1016/j.cma.2018.02.006
Type:
Article
ISSN:
0045-7825
Sponsors:
David Pardo has received funding from the Project of the Spanish Ministry of Economy and Competitiveness with reference MTM2016-76329-R (AEI/FEDER, EU), and MTM2016-81697-ERC/AEI, the BCAM “Severo Ocho” accreditation of excellenceSEV-2013-0323, and the Basque Government through the BERC 2014-2017 program and the Consolidated Research Group Grant IT649-13 on “Mathematical Modeling, Simulation, and Industrial Applications (M2SI)”. This publication was also made possible in part by the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 644602, the CSIRO Professorial Chair in Computational Geoscience at Curtin University, the Deep Earth Imaging Enterprise Future Science Platforms of the Commonwealth Scientific Industrial Research Organisation, CSIRO, of Australia, the Mega-grant of the Russian Federation Government ( N14.Y26.31.0013) and the Curtin Institute for Computation. 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 Advanced Computing Center (TACC) at The University of Texas at Austin for providing HPC resources that have contributed to the research results reported within this paper.
Additional Links:
http://www.sciencedirect.com/science/article/pii/S004578251830077X
Appears in Collections:
Articles; Extreme Computing Research Center; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

Full metadata record

DC FieldValue Language
dc.contributor.authorGarcia, Danielen
dc.contributor.authorPardo, Daviden
dc.contributor.authorDalcin, Lisandroen
dc.contributor.authorCalo, Victor M.en
dc.date.accessioned2018-02-14T11:52:53Z-
dc.date.available2018-02-14T11:52:53Z-
dc.date.issued2018-02-12en
dc.identifier.citationGarcia D, Pardo D, Dalcin L, Calo VM (2018) Refined isogeometric analysis for a preconditioned conjugate gradient solver. Computer Methods in Applied Mechanics and Engineering. Available: http://dx.doi.org/10.1016/j.cma.2018.02.006.en
dc.identifier.issn0045-7825en
dc.identifier.doi10.1016/j.cma.2018.02.006en
dc.identifier.urihttp://hdl.handle.net/10754/627135-
dc.description.abstractStarting from a highly continuous Isogeometric Analysis (IGA) discretization, refined Isogeometric Analysis (rIGA) introduces C0 hyperplanes that act as separators for the direct LU factorization solver. As a result, the total computational cost required to solve the corresponding system of equations using a direct LU factorization solver dramatically reduces (up to a factor of 55) Garcia et al. (2017). At the same time, rIGA enriches the IGA spaces, thus improving the best approximation error. In this work, we extend the complexity analysis of rIGA to the case of iterative solvers. We build an iterative solver as follows: we first construct the Schur complements using a direct solver over small subdomains (macro-elements). We then assemble those Schur complements into a global skeleton system. Subsequently, we solve this system iteratively using Conjugate Gradients (CG) with an incomplete LU (ILU) preconditioner. For a 2D Poisson model problem with a structured mesh and a uniform polynomial degree of approximation, rIGA achieves moderate savings with respect to IGA in terms of the number of Floating Point Operations (FLOPs) and computational time (in seconds) required to solve the resulting system of linear equations. For instance, for a mesh with four million elements and polynomial degree p=3, the iterative solver is approximately 2.6 times faster (in time) when applied to the rIGA system than to the IGA one. These savings occur because the skeleton rIGA system contains fewer non-zero entries than the IGA one. The opposite situation occurs for 3D problems, and as a result, 3D rIGA discretizations provide no gains with respect to their IGA counterparts when considering iterative solvers.en
dc.description.sponsorshipDavid Pardo has received funding from the Project of the Spanish Ministry of Economy and Competitiveness with reference MTM2016-76329-R (AEI/FEDER, EU), and MTM2016-81697-ERC/AEI, the BCAM “Severo Ocho” accreditation of excellenceSEV-2013-0323, and the Basque Government through the BERC 2014-2017 program and the Consolidated Research Group Grant IT649-13 on “Mathematical Modeling, Simulation, and Industrial Applications (M2SI)”. This publication was also made possible in part by the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 644602, the CSIRO Professorial Chair in Computational Geoscience at Curtin University, the Deep Earth Imaging Enterprise Future Science Platforms of the Commonwealth Scientific Industrial Research Organisation, CSIRO, of Australia, the Mega-grant of the Russian Federation Government ( N14.Y26.31.0013) and the Curtin Institute for Computation. 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 Advanced Computing Center (TACC) at The University of Texas at Austin for providing HPC resources that have contributed to the research results reported within this paper.en
dc.publisherElsevier BVen
dc.relation.urlhttp://www.sciencedirect.com/science/article/pii/S004578251830077Xen
dc.rightsNOTICE: this is the author’s version of a work that was accepted for publication in Computer Methods in Applied Mechanics and Engineering. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Computer Methods in Applied Mechanics and Engineering, 12 February 2018. DOI: 10.1016/j.cma.2018.02.006. © 2018. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/en
dc.subjectIsogeometric analysis (IGA)en
dc.subjectFinite element analysis (FEA)en
dc.subjectRefined isogeometric analysis (rIGA)en
dc.subjectSolver-based discretizationen
dc.subjectIterative solversen
dc.subjectConjugate gradienten
dc.subjectIncomplete LU factorizationen
dc.subjectk-refinementen
dc.titleRefined isogeometric analysis for a preconditioned conjugate gradient solveren
dc.typeArticleen
dc.contributor.departmentExtreme Computing Research Centeren
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Divisionen
dc.contributor.departmentNumerical Porous Media SRI Center (NumPor)en
dc.identifier.journalComputer Methods in Applied Mechanics and Engineeringen
dc.eprint.versionPost-printen
dc.contributor.institutionBasque Center for Applied Mathematics, (BCAM), Bilbao, Spainen
dc.contributor.institutionIkerbasque (Basque Foundation for Sciences), Bilbao, Spainen
dc.contributor.institutionDepartment of Applied Mathematics, Statistics, and Operational Research, University of the Basque Country UPV/EHU, Leioa, Spainen
dc.contributor.institutionUniversidad Nacional del Litoral, Santa Fe, Argentinaen
dc.contributor.institutionConsejo Nacional de Investigaciones Científicas y Técnicas, Santa Fe, Argentinaen
dc.contributor.institutionCurtin Institute for Computation, Curtin University, Perth, Australiaen
dc.contributor.institutionMineral Resources, Commonwealth Scientific and Industrial Research Organisation (CSIRO), Perth, Australiaen
dc.contributor.institutionDepartment of Applied Geology, Western Australian School of Mines, Curtin University, Perth, Australiaen
kaust.authorDalcin, Lisandroen
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