On advantages of the Kelvin mapping in finite element implementations of deformation processes
Classical continuum mechanical theories operate on three-dimensional Euclidian space using scalar, vector, and tensor-valued quantities usually up to the order of four. For their numerical treatment, it is common practice to transform the relations into a matrix–vector format. This transformation is...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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Springer Berlin Heidelberg
2016
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| Online Access: | http://hdl.handle.net/1721.1/105251 https://orcid.org/0000-0003-3768-4269 |
| _version_ | 1811088893182738432 |
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| author | Nagel, Thomas Görke, Uwe-Jens Kolditz, Olaf Moerman, Kevin M |
| author2 | Massachusetts Institute of Technology. Media Laboratory |
| author_facet | Massachusetts Institute of Technology. Media Laboratory Nagel, Thomas Görke, Uwe-Jens Kolditz, Olaf Moerman, Kevin M |
| author_sort | Nagel, Thomas |
| collection | MIT |
| description | Classical continuum mechanical theories operate on three-dimensional Euclidian space using scalar, vector, and tensor-valued quantities usually up to the order of four. For their numerical treatment, it is common practice to transform the relations into a matrix–vector format. This transformation is usually performed using the so-called Voigt mapping. This mapping does not preserve tensor character leaving significant room for error as stress and strain quantities follow from different mappings and thus have to be treated differently in certain mathematical operations. Despite its conceptual and notational difficulties having been pointed out, the Voigt mapping remains the foundation of most current finite element programmes. An alternative is the so-called Kelvin mapping which has recently gained recognition in studies of theoretical mechanics. This article is concerned with benefits of the Kelvin mapping in numerical modelling tools such as finite element software. The decisive difference to the Voigt mapping is that Kelvin’s method preserves tensor character, and thus the numerical matrix notation directly corresponds to the original tensor notation. Further benefits in numerical implementations are that tensor norms are calculated identically without distinguishing stress- or strain-type quantities, and tensor equations can be directly transformed into matrix equations without additional considerations. The only implementational changes are related to a scalar factor in certain finite element matrices, and hence, harvesting the mentioned benefits comes at very little cost. |
| first_indexed | 2024-09-23T14:09:27Z |
| format | Article |
| id | mit-1721.1/105251 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T14:09:27Z |
| publishDate | 2016 |
| publisher | Springer Berlin Heidelberg |
| record_format | dspace |
| spelling | mit-1721.1/1052512022-10-01T19:33:22Z On advantages of the Kelvin mapping in finite element implementations of deformation processes Nagel, Thomas Görke, Uwe-Jens Kolditz, Olaf Moerman, Kevin M Massachusetts Institute of Technology. Media Laboratory Program in Media Arts and Sciences (Massachusetts Institute of Technology) Moerman, Kevin M Classical continuum mechanical theories operate on three-dimensional Euclidian space using scalar, vector, and tensor-valued quantities usually up to the order of four. For their numerical treatment, it is common practice to transform the relations into a matrix–vector format. This transformation is usually performed using the so-called Voigt mapping. This mapping does not preserve tensor character leaving significant room for error as stress and strain quantities follow from different mappings and thus have to be treated differently in certain mathematical operations. Despite its conceptual and notational difficulties having been pointed out, the Voigt mapping remains the foundation of most current finite element programmes. An alternative is the so-called Kelvin mapping which has recently gained recognition in studies of theoretical mechanics. This article is concerned with benefits of the Kelvin mapping in numerical modelling tools such as finite element software. The decisive difference to the Voigt mapping is that Kelvin’s method preserves tensor character, and thus the numerical matrix notation directly corresponds to the original tensor notation. Further benefits in numerical implementations are that tensor norms are calculated identically without distinguishing stress- or strain-type quantities, and tensor equations can be directly transformed into matrix equations without additional considerations. The only implementational changes are related to a scalar factor in certain finite element matrices, and hence, harvesting the mentioned benefits comes at very little cost. Germany. Federal Ministry of Education and Research (Grant 03EK3022) Project Management Jülich Helmholtz Initiating and Networking Fund. NUMTHECHSTORE Project 2016-11-07T23:11:29Z 2017-03-01T16:14:48Z 2016-05 2015-10 2016-08-18T15:37:08Z Article http://purl.org/eprint/type/JournalArticle 1866-6280 1866-6299 http://hdl.handle.net/1721.1/105251 Nagel, Thomas et al. “On Advantages of the Kelvin Mapping in Finite Element Implementations of Deformation Processes.” Environmental Earth Sciences 75.11 (2016): n. pag. https://orcid.org/0000-0003-3768-4269 en http://dx.doi.org/10.1007/s12665-016-5429-4 Environmental Earth Sciences 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. Springer-Verlag Berlin Heidelberg application/pdf Springer Berlin Heidelberg Springer Berlin Heidelberg |
| spellingShingle | Nagel, Thomas Görke, Uwe-Jens Kolditz, Olaf Moerman, Kevin M On advantages of the Kelvin mapping in finite element implementations of deformation processes |
| title | On advantages of the Kelvin mapping in finite element implementations of deformation processes |
| title_full | On advantages of the Kelvin mapping in finite element implementations of deformation processes |
| title_fullStr | On advantages of the Kelvin mapping in finite element implementations of deformation processes |
| title_full_unstemmed | On advantages of the Kelvin mapping in finite element implementations of deformation processes |
| title_short | On advantages of the Kelvin mapping in finite element implementations of deformation processes |
| title_sort | on advantages of the kelvin mapping in finite element implementations of deformation processes |
| url | http://hdl.handle.net/1721.1/105251 https://orcid.org/0000-0003-3768-4269 |
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