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DOI 10.1007/s12665-016-5429-4
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Title (Primary) On advantages of the Kelvin mapping in finite element implementations of deformation processes
Author Nagel, T.; Görke, U.-J.; Moerman, K.M.; Kolditz, O. ORCID logo
Source Titel Environmental Earth Sciences
Year 2016
Department ENVINF
Volume 75
Issue 11
Page From art. 937
Language englisch
Keywords Kelvin mapping; Voigt mapping; Finite elements; Numerical algorithms; OpenGeoSys
UFZ wide themes RU5;
Abstract 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.
Persistent UFZ Identifier
Nagel, T., Görke, U.-J., Moerman, K.M., Kolditz, O. (2016):
On advantages of the Kelvin mapping in finite element implementations of deformation processes
Environ. Earth Sci. 75 (11), art. 937 10.1007/s12665-016-5429-4