A preconditioner for the finite element computation of incompressible, nonlinear elastic deformations
Large, incompressible elastic deformations are governed by a system of nonlinear partial differential equations. The finite element discretisation of these partial differential equations yields a system of nonlinear algebraic equations that are usually solved using Newton’s method. On each iteration...
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| Format: | Journal article |
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Springer Verlag
2017
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| _version_ | 1797081877118451712 |
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| author | Whiteley, J |
| author_facet | Whiteley, J |
| author_sort | Whiteley, J |
| collection | OXFORD |
| description | Large, incompressible elastic deformations are governed by a system of nonlinear partial differential equations. The finite element discretisation of these partial differential equations yields a system of nonlinear algebraic equations that are usually solved using Newton’s method. On each iteration of Newton’s method, a linear system must be solved. We exploit the structure of the Jacobian matrix to propose a preconditioner, comprising two steps. The first step is the solution of a relatively small, symmetric, positive definite linear system using the preconditioned conjugate gradient method. This is followed by a small number of multigrid V-cycles for a larger linear system. Through the use of exemplar elastic deformations, the preconditioner is demonstrated to facilitate the iterative solution of the linear systems arising. The number of GMRES iterations required has only a very weak dependence on the number of degrees of freedom of the linear systems. |
| first_indexed | 2024-03-07T01:20:16Z |
| format | Journal article |
| id | oxford-uuid:90125c78-f6d1-4b82-aced-ce5f572719eb |
| institution | University of Oxford |
| last_indexed | 2024-03-07T01:20:16Z |
| publishDate | 2017 |
| publisher | Springer Verlag |
| record_format | dspace |
| spelling | oxford-uuid:90125c78-f6d1-4b82-aced-ce5f572719eb2022-03-26T23:08:57ZA preconditioner for the finite element computation of incompressible, nonlinear elastic deformationsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:90125c78-f6d1-4b82-aced-ce5f572719ebSymplectic Elements at OxfordSpringer Verlag2017Whiteley, JLarge, incompressible elastic deformations are governed by a system of nonlinear partial differential equations. The finite element discretisation of these partial differential equations yields a system of nonlinear algebraic equations that are usually solved using Newton’s method. On each iteration of Newton’s method, a linear system must be solved. We exploit the structure of the Jacobian matrix to propose a preconditioner, comprising two steps. The first step is the solution of a relatively small, symmetric, positive definite linear system using the preconditioned conjugate gradient method. This is followed by a small number of multigrid V-cycles for a larger linear system. Through the use of exemplar elastic deformations, the preconditioner is demonstrated to facilitate the iterative solution of the linear systems arising. The number of GMRES iterations required has only a very weak dependence on the number of degrees of freedom of the linear systems. |
| spellingShingle | Whiteley, J A preconditioner for the finite element computation of incompressible, nonlinear elastic deformations |
| title | A preconditioner for the finite element computation of incompressible, nonlinear elastic deformations |
| title_full | A preconditioner for the finite element computation of incompressible, nonlinear elastic deformations |
| title_fullStr | A preconditioner for the finite element computation of incompressible, nonlinear elastic deformations |
| title_full_unstemmed | A preconditioner for the finite element computation of incompressible, nonlinear elastic deformations |
| title_short | A preconditioner for the finite element computation of incompressible, nonlinear elastic deformations |
| title_sort | preconditioner for the finite element computation of incompressible nonlinear elastic deformations |
| work_keys_str_mv | AT whiteleyj apreconditionerforthefiniteelementcomputationofincompressiblenonlinearelasticdeformations AT whiteleyj preconditionerforthefiniteelementcomputationofincompressiblenonlinearelasticdeformations |