Frontal solver: Difference between revisions
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A '''frontal solver''', due to [[Bruce Irons (engineer)|Bruce Irons]]<ref>Irons, B. M., 1970. A frontal solution scheme for finite element analysis. Int. J. Numer. Methods Eng. 2, 5--32</ref> is an approach to solving [[sparse matrix|sparse linear systems]] which is used extensively in [[finite element analysis]].<ref>Renaud Sizaire, keyFE2 User Manual, 2005, Sec. I.4.2 Solving_linear_system [http://users.skynet.be/keyFE2/manual/I_4_2_Solving_linear_system.html online] {{ |
A '''frontal solver''', due to [[Bruce Irons (engineer)|Bruce Irons]]<ref>Irons, B. M., 1970. A frontal solution scheme for finite element analysis. Int. J. Numer. Methods Eng. 2, 5--32</ref> is an approach to solving [[sparse matrix|sparse linear systems]] which is used extensively in [[finite element analysis]].<ref>Renaud Sizaire, keyFE2 User Manual, 2005, Sec. I.4.2 Solving_linear_system [http://users.skynet.be/keyFE2/manual/I_4_2_Solving_linear_system.html online] {{wayback|url=http://users.skynet.be/keyFE2/manual/I_4_2_Solving_linear_system.html |date=20061008212805 }}</ref> It is a variant of [[Gauss elimination]] that automatically avoids a large number of operations involving zero terms.<ref>Hayrettin Kardestuncer, Ed. ''Finite Element Handbook''.</ref> |
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A frontal solver builds a [[LU decomposition|LU]] or [[Cholesky decomposition]] of a sparse matrix given as the assembly of element matrices by assembling the matrix and eliminating equations only on a subset of elements at a time. However, elements can be stored in-core in a clique sequence as recently proposed by Areias.<ref>P. Areias, T. Rabczuk, J.I. Barbosa, The extended unsymmetric frontal solution for multiple-point constraints, Engineering Computations, v.31 n.7, pp. 1582 - 1607, Nov. 2014 [http://dx.doi.org/10.1108/EC-10-2013-0263]</ref> This subset is called the front and it is essentially the transition region between the part of the system already finished and the part not touched yet. The whole sparse matrix is never created explicitly. Only parts of the matrix are assembled as they enter the front. Processing the front involves [[dense matrix]] operations, which use the CPU efficiently. In a typical implementation, only the front is in [[computer memory|memory]], while the factors in the decomposition are written into [[computer file|files]]. The element matrices are read from files or created as needed and discarded. |
A frontal solver builds a [[LU decomposition|LU]] or [[Cholesky decomposition]] of a sparse matrix given as the assembly of element matrices by assembling the matrix and eliminating equations only on a subset of elements at a time. However, elements can be stored in-core in a clique sequence as recently proposed by Areias.<ref>P. Areias, T. Rabczuk, J.I. Barbosa, The extended unsymmetric frontal solution for multiple-point constraints, Engineering Computations, v.31 n.7, pp. 1582 - 1607, Nov. 2014 [http://dx.doi.org/10.1108/EC-10-2013-0263]</ref> This subset is called the front and it is essentially the transition region between the part of the system already finished and the part not touched yet. The whole sparse matrix is never created explicitly. Only parts of the matrix are assembled as they enter the front. Processing the front involves [[dense matrix]] operations, which use the CPU efficiently. In a typical implementation, only the front is in [[computer memory|memory]], while the factors in the decomposition are written into [[computer file|files]]. The element matrices are read from files or created as needed and discarded. |
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Revision as of 19:48, 18 January 2016
A frontal solver, due to Bruce Irons[1] is an approach to solving sparse linear systems which is used extensively in finite element analysis.[2] It is a variant of Gauss elimination that automatically avoids a large number of operations involving zero terms.[3]
A frontal solver builds a LU or Cholesky decomposition of a sparse matrix given as the assembly of element matrices by assembling the matrix and eliminating equations only on a subset of elements at a time. However, elements can be stored in-core in a clique sequence as recently proposed by Areias.[4] This subset is called the front and it is essentially the transition region between the part of the system already finished and the part not touched yet. The whole sparse matrix is never created explicitly. Only parts of the matrix are assembled as they enter the front. Processing the front involves dense matrix operations, which use the CPU efficiently. In a typical implementation, only the front is in memory, while the factors in the decomposition are written into files. The element matrices are read from files or created as needed and discarded.
A multifrontal solver of Duff and Reid[5] is an improvement of the frontal solver that uses several independent fronts at the same time. The fronts can be worked on by different processors, which enables parallel computing.
See [6] for a monograph exposition.
See also
References
- ^ Irons, B. M., 1970. A frontal solution scheme for finite element analysis. Int. J. Numer. Methods Eng. 2, 5--32
- ^ Renaud Sizaire, keyFE2 User Manual, 2005, Sec. I.4.2 Solving_linear_system online Archived 2006-10-08 at the Wayback Machine
- ^ Hayrettin Kardestuncer, Ed. Finite Element Handbook.
- ^ P. Areias, T. Rabczuk, J.I. Barbosa, The extended unsymmetric frontal solution for multiple-point constraints, Engineering Computations, v.31 n.7, pp. 1582 - 1607, Nov. 2014 [1]
- ^ I. S. Duff , J. K. Reid, The Multifrontal Solution of Indefinite Sparse Symmetric Linear, ACM Transactions on Mathematical Software (TOMS), v.9 n.3, p.302-325, Sept. 1983 DOI 10.1145/356044.356047
- ^ Iain S Duff , Albert M Erisman , John K Reid, Direct methods for sparse matrices, Oxford University Press, Inc., New York, NY, 1986
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