Optimal Local Approximation Spaces for Component-Based Static Condensation Procedures
In this paper we introduce local approximation spaces for component-based static condensation (sc) procedures that are optimal in the sense of Kolmogorov. To facilitate simulations for large structures such as aircraft or ships, it is crucial to decrease the number of degrees of freedom on the inter...
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Society for Industrial and Applied Mathematics
2017
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| Online Access: | http://hdl.handle.net/1721.1/108624 https://orcid.org/0000-0003-4245-6586 https://orcid.org/0000-0002-2631-6463 |
| _version_ | 1826206100238630912 |
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| author | Smetana, Kathrin Patera, Anthony T |
| author2 | Massachusetts Institute of Technology. Department of Mechanical Engineering |
| author_facet | Massachusetts Institute of Technology. Department of Mechanical Engineering Smetana, Kathrin Patera, Anthony T |
| author_sort | Smetana, Kathrin |
| collection | MIT |
| description | In this paper we introduce local approximation spaces for component-based static condensation (sc) procedures that are optimal in the sense of Kolmogorov. To facilitate simulations for large structures such as aircraft or ships, it is crucial to decrease the number of degrees of freedom on the interfaces, or “ports,” in order to reduce the size of the statically condensed system. To derive optimal port spaces we consider a (compact) transfer operator that acts on the space of harmonic extensions on a two-component system and maps the traces on the ports that lie on the boundary of these components to the trace of the shared port. Solving the eigenproblem for the composition of the transfer operator and its adjoint yields the optimal space. For a related work in the context of the generalized finite element method, we refer the reader to [I. Babuška and R. Lipton, Multiscale Model. Simul., 9 (2011), pp. 373--406]. We further introduce a spectral greedy algorithm to generalize the procedure to the parameter-dependent setting and to construct a quasi-optimal parameter-independent port space. Moreover, it is shown that, given a certain tolerance and an upper bound for the ports in the system, the spectral greedy constructs a port space that yields an sc approximation error on a system of arbitrary configuration which is smaller than this tolerance for all parameters in a rich train set. We present our approach for isotropic linear elasticity, although the idea may be readily applied to any linear coercive problem. Numerical experiments demonstrate the very rapid and exponential convergence both of the eigenvalues and of the sc approximation based on spectral modes for nonseparable and irregular geometries such as an I-beam with an internal crack. |
| first_indexed | 2024-09-23T13:24:05Z |
| format | Article |
| id | mit-1721.1/108624 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T13:24:05Z |
| publishDate | 2017 |
| publisher | Society for Industrial and Applied Mathematics |
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| spelling | mit-1721.1/1086242022-09-28T13:55:50Z Optimal Local Approximation Spaces for Component-Based Static Condensation Procedures Smetana, Kathrin Patera, Anthony T Massachusetts Institute of Technology. Department of Mechanical Engineering Smetana, Kathrin Patera, Anthony T In this paper we introduce local approximation spaces for component-based static condensation (sc) procedures that are optimal in the sense of Kolmogorov. To facilitate simulations for large structures such as aircraft or ships, it is crucial to decrease the number of degrees of freedom on the interfaces, or “ports,” in order to reduce the size of the statically condensed system. To derive optimal port spaces we consider a (compact) transfer operator that acts on the space of harmonic extensions on a two-component system and maps the traces on the ports that lie on the boundary of these components to the trace of the shared port. Solving the eigenproblem for the composition of the transfer operator and its adjoint yields the optimal space. For a related work in the context of the generalized finite element method, we refer the reader to [I. Babuška and R. Lipton, Multiscale Model. Simul., 9 (2011), pp. 373--406]. We further introduce a spectral greedy algorithm to generalize the procedure to the parameter-dependent setting and to construct a quasi-optimal parameter-independent port space. Moreover, it is shown that, given a certain tolerance and an upper bound for the ports in the system, the spectral greedy constructs a port space that yields an sc approximation error on a system of arbitrary configuration which is smaller than this tolerance for all parameters in a rich train set. We present our approach for isotropic linear elasticity, although the idea may be readily applied to any linear coercive problem. Numerical experiments demonstrate the very rapid and exponential convergence both of the eigenvalues and of the sc approximation based on spectral modes for nonseparable and irregular geometries such as an I-beam with an internal crack. United States. Air Force Office of Scientific Research. Multidisciplinary University Research Initiative (Grant FA9550-09-1-0613) United States. Office of Naval Research (Grant N00014-11-1-0713) 2017-05-03T13:43:02Z 2017-05-03T13:43:02Z 2016-10 2015-02 Article http://purl.org/eprint/type/JournalArticle 1064-8275 1095-7197 http://hdl.handle.net/1721.1/108624 Smetana, Kathrin, and Anthony T. Patera. “Optimal Local Approximation Spaces for Component-Based Static Condensation Procedures.” SIAM Journal on Scientific Computing 38, no. 5 (January 2016): A3318–A3356. © SIAM https://orcid.org/0000-0003-4245-6586 https://orcid.org/0000-0002-2631-6463 en_US http://dx.doi.org/10.1137/15M1009603 SIAM Journal on Scientific Computing Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. application/pdf Society for Industrial and Applied Mathematics SIAM |
| spellingShingle | Smetana, Kathrin Patera, Anthony T Optimal Local Approximation Spaces for Component-Based Static Condensation Procedures |
| title | Optimal Local Approximation Spaces for Component-Based Static Condensation Procedures |
| title_full | Optimal Local Approximation Spaces for Component-Based Static Condensation Procedures |
| title_fullStr | Optimal Local Approximation Spaces for Component-Based Static Condensation Procedures |
| title_full_unstemmed | Optimal Local Approximation Spaces for Component-Based Static Condensation Procedures |
| title_short | Optimal Local Approximation Spaces for Component-Based Static Condensation Procedures |
| title_sort | optimal local approximation spaces for component based static condensation procedures |
| url | http://hdl.handle.net/1721.1/108624 https://orcid.org/0000-0003-4245-6586 https://orcid.org/0000-0002-2631-6463 |
| work_keys_str_mv | AT smetanakathrin optimallocalapproximationspacesforcomponentbasedstaticcondensationprocedures AT pateraanthonyt optimallocalapproximationspacesforcomponentbasedstaticcondensationprocedures |