Enhanced balancing Neumann-Neumann preconditioning in computational fluid and solid mechanics
SUMMARYIn this work, we propose an enhanced implementation of balancing Neumann–Neumann (BNN) preconditioning together with a detailed numerical comparison against the balancing domain decomposition by constraints (BDDC) preconditioner. As model problems, we consider the Poisson and linear elasticit...
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| Vydáno v: | International journal for numerical methods in engineering Ročník 96; číslo 4; s. 203 - 230 |
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| Jazyk: | angličtina |
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Chichester
Blackwell Publishing Ltd
26.10.2013
Wiley Wiley Subscription Services, Inc |
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| ISSN: | 0029-5981, 1097-0207 |
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| Abstract | SUMMARYIn this work, we propose an enhanced implementation of balancing Neumann–Neumann (BNN) preconditioning together with a detailed numerical comparison against the balancing domain decomposition by constraints (BDDC) preconditioner. As model problems, we consider the Poisson and linear elasticity problems. On one hand, we propose a novel way to deal with singular matrices and pseudo‐inverses appearing in local solvers. It is based on a kernel identification strategy that allows us to efficiently compute the action of the pseudo‐inverse via local indefinite solvers. We further show how, identifying a minimum set of degrees of freedom to be fixed, an equivalent definite system can be solved instead, even in the elastic case. On the other hand, we propose a simple implementation of the algorithm that reduces the number of Dirichlet solvers to only one per iteration, leading to similar computational cost as additive methods. After these improvements of the BNN preconditioned conjugate gradient algorithm, we compare its performance against that of the BDDC preconditioners on a pair of large‐scale distributed‐memory platforms. The enhanced BNN method is a competitive preconditioner for three‐dimensional Poisson and elasticity problems and outperforms the BDDC method in many cases. Copyright © 2013 John Wiley & Sons, Ltd. |
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| AbstractList | SUMMARYIn this work, we propose an enhanced implementation of balancing Neumann–Neumann (BNN) preconditioning together with a detailed numerical comparison against the balancing domain decomposition by constraints (BDDC) preconditioner. As model problems, we consider the Poisson and linear elasticity problems. On one hand, we propose a novel way to deal with singular matrices and pseudo‐inverses appearing in local solvers. It is based on a kernel identification strategy that allows us to efficiently compute the action of the pseudo‐inverse via local indefinite solvers. We further show how, identifying a minimum set of degrees of freedom to be fixed, an equivalent definite system can be solved instead, even in the elastic case. On the other hand, we propose a simple implementation of the algorithm that reduces the number of Dirichlet solvers to only one per iteration, leading to similar computational cost as additive methods. After these improvements of the BNN preconditioned conjugate gradient algorithm, we compare its performance against that of the BDDC preconditioners on a pair of large‐scale distributed‐memory platforms. The enhanced BNN method is a competitive preconditioner for three‐dimensional Poisson and elasticity problems and outperforms the BDDC method in many cases. Copyright © 2013 John Wiley & Sons, Ltd. In this work, we propose an enhanced implementation of balancing Neumann–Neumann (BNN) preconditioning together with a detailed numerical comparison against the balancing domain decomposition by constraints (BDDC) preconditioner. As model problems, we consider the Poisson and linear elasticity problems. On one hand, we propose a novel way to deal with singular matrices and pseudo‐inverses appearing in local solvers. It is based on a kernel identification strategy that allows us to efficiently compute the action of the pseudo‐inverse via local indefinite solvers. We further show how, identifying a minimum set of degrees of freedom to be fixed, an equivalent definite system can be solved instead, even in the elastic case. On the other hand, we propose a simple implementation of the algorithm that reduces the number of Dirichlet solvers to only one per iteration, leading to similar computational cost as additive methods. After these improvements of the BNN preconditioned conjugate gradient algorithm, we compare its performance against that of the BDDC preconditioners on a pair of large‐scale distributed‐memory platforms. The enhanced BNN method is a competitive preconditioner for three‐dimensional Poisson and elasticity problems and outperforms the BDDC method in many cases. Copyright © 2013 John Wiley & Sons, Ltd. SUMMARY In this work, we propose an enhanced implementation of balancing Neumann-Neumann (BNN) preconditioning together with a detailed numerical comparison against the balancing domain decomposition by constraints (BDDC) preconditioner. As model problems, we consider the Poisson and linear elasticity problems. On one hand, we propose a novel way to deal with singular matrices and pseudo-inverses appearing in local solvers. It is based on a kernel identification strategy that allows us to efficiently compute the action of the pseudo-inverse via local indefinite solvers. We further show how, identifying a minimum set of degrees of freedom to be fixed, an equivalent definite system can be solved instead, even in the elastic case. On the other hand, we propose a simple implementation of the algorithm that reduces the number of Dirichlet solvers to only one per iteration, leading to similar computational cost as additive methods. After these improvements of the BNN preconditioned conjugate gradient algorithm, we compare its performance against that of the BDDC preconditioners on a pair of large-scale distributed-memory platforms. The enhanced BNN method is a competitive preconditioner for three-dimensional Poisson and elasticity problems and outperforms the BDDC method in many cases. Copyright © 2013 John Wiley & Sons, Ltd. [PUBLICATION ABSTRACT] SUMMARY In this work, we propose an enhanced implementation of balancing Neumann-Neumann (BNN) preconditioning together with a detailed numerical comparison against the balancing domain decomposition by constraints (BDDC) preconditioner. As model problems, we consider the Poisson and linear elasticity problems. On one hand, we propose a novel way to deal with singular matrices and pseudo-inverses appearing in local solvers. It is based on a kernel identification strategy that allows us to efficiently compute the action of the pseudo-inverse via local indefinite solvers. We further show how, identifying a minimum set of degrees of freedom to be fixed, an equivalent definite system can be solved instead, even in the elastic case. On the other hand, we propose a simple implementation of the algorithm that reduces the number of Dirichlet solvers to only one per iteration, leading to similar computational cost as additive methods. After these improvements of the BNN preconditioned conjugate gradient algorithm, we compare its performance against that of the BDDC preconditioners on a pair of large-scale distributed-memory platforms. The enhanced BNN method is a competitive preconditioner for three-dimensional Poisson and elasticity problems and outperforms the BDDC method in many cases. Copyright [copy 2013 John Wiley & Sons, Ltd. |
| Author | Príncipe, Javier Martín, Alberto F. Badia, Santiago |
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| Cites_doi | 10.1002/nme.2932 10.1002/cnm.1640090307 10.1007/s00607-013-0293-5 10.1002/nla.341 10.1137/S1064827595287997 10.1002/nme.3187 10.1017/CBO9780511618635 10.1137/050629902 10.1137/S1064829598339761 10.1016/j.apnum.2004.09.022 10.1002/nme.1620320604 10.1002/nme.3209 10.1016/j.future.2003.07.011 10.1016/j.cma.2006.03.010 10.1007/s11831-008-9020-3 10.1137/0710032 10.1002/1099-1506(200010/12)7:7/8<687::AID-NLA219>3.0.CO;2-S 10.1002/(SICI)1097-0207(19980228)41:4<675::AID-NME305>3.0.CO;2-8 10.1007/s10915-009-9272-6 10.1007/s00211-006-0022-5 10.1002/cpa.20156 10.1002/nme.76 10.1016/S0045-7825(01)00234-1 10.1007/s11831-013-9086-4 10.1137/S1064827502412887 10.1016/j.jcp.2008.08.025 10.1137/S0036142995291019 10.1016/S0045-7825(99)00234-0 10.1007/s002110050438 10.1007/s00607-008-0014-7 10.1093/oso/9780198528678.001.0001 10.1016/j.matcom.2011.06.007 10.1006/jcph.1999.6255 |
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| Keywords | Performance evaluation BNN Constraint elasticity Grid coarse-grid correction Modeling scalability Distributed memory Inverse matrix Linear elasticity Tridimensional elasticity Dirichlet problem Vibration control BDDC Conjugate gradient method Singular matrix Neumann problem Balancing Pseudoinverse Poisson equation parallelization Boundary value problem balancing domain decomposition Large scale Domain decomposition Shared memory Preconditioning Structural analysis |
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| References_xml | – reference: Farhat C, Géradin M. On the general solution by a direct method of a large-scale singular system of linear equations: application to the analysis of floating structures. International Journal for Numerical Methods in Engineering 1998; 41(4):675-696. – reference: Le Tallec P, Mandel J, Vidrascu M. A Neumann-Neumann domain decomposition algorithm for solving plate and shell problems. SIAM Journal on Numerical Analysis 1998; 35(2):836-867. – reference: Schenk O, Gärtner K. On fast factorization pivoting methods for sparse symmetric indefinite systems. ETNA. Electronic Transactions on Numerical Analysis [electronic only] 2006; 23:158-179. – reference: George A. Nested dissection of a regular finite element mesh. SIAM Journal on Numerical Analysis 1973; 10(2):345-363. – reference: Elman HC, Silvester DJ, Wathen AJ. Finite Elements and Fast Iterative Solvers: With Applications in Incompressible Fluid Dynamics. 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| Snippet | SUMMARYIn this work, we propose an enhanced implementation of balancing Neumann–Neumann (BNN) preconditioning together with a detailed numerical comparison... In this work, we propose an enhanced implementation of balancing Neumann–Neumann (BNN) preconditioning together with a detailed numerical comparison against... SUMMARY In this work, we propose an enhanced implementation of balancing Neumann-Neumann (BNN) preconditioning together with a detailed numerical comparison... |
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| SubjectTerms | Algebra Algorithms balancing domain decomposition BDDC BNN coarse-grid correction Computation Computational methods in fluid dynamics Dirichlet problem Domain decomposition Elasticity Exact sciences and technology Fluid dynamics Fundamental areas of phenomenology (including applications) Linear and multilinear algebra, matrix theory Mathematics Methods of scientific computing (including symbolic computation, algebraic computation) Numerical analysis. Scientific computation parallelization Physics Preconditioning scalability Sciences and techniques of general use Solid mechanics Solvers Static elasticity (thermoelasticity...) Structural and continuum mechanics Three dimensional |
| Title | Enhanced balancing Neumann-Neumann preconditioning in computational fluid and solid mechanics |
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