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A general class of enriched methods for the simplicial linear finite elements

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  • Dell’Accio, Francesco
  • Di Tommaso, Filomena
  • Guessab, Allal
  • Nudo, Federico

Abstract

Low-order elements are widely used and preferred for finite element analysis, specifically the three-node triangular and four-node tetrahedral elements, both based on linear polynomials in barycentric coordinates. They are known, however, to under-perform when nearly incompressible materials are involved. The problem may be circumvented by the use of higher degree polynomial elements, but their application become both more complex an computationally expensive. For this reason, non-polynomial enriched finite element methods have been proposed for solving engineering problems. In line with previous researches, the main contribution of this paper is to present a general strategy for enriching the standard simplicial linear finite element by non-polynomial functions. A key role is played by a characterization result, given in terms of the non-vanishing of a certain determinant, which provides necessary and sufficient conditions, on the enrichment functions and functionals, that guarantee the existence of families of such enriched elements. We show that the enriched basis functions admit a closed form representation in terms of enrichment functions and functionals. Finally, we provide concrete examples of admissible enrichment functions and perform some numerical tests.

Suggested Citation

  • Dell’Accio, Francesco & Di Tommaso, Filomena & Guessab, Allal & Nudo, Federico, 2023. "A general class of enriched methods for the simplicial linear finite elements," Applied Mathematics and Computation, Elsevier, vol. 456(C).
    • Handle: RePEc:eee:apmaco:v:456:y:2023:i:c:s0096300323003181
    DOI: 10.1016/j.amc.2023.128149
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    References listed on IDEAS

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    1. Allal Guessab & Yassine Zaim, 2017. "A Unified and General Framework for Enriching Finite Element Approximations," Springer Optimization and Its Applications, in: Narendra Kumar Govil & Ram Mohapatra & Mohammed A. Qazi & Gerhard Schmeisser (ed.), Progress in Approximation Theory and Applicable Complex Analysis, pages 491-519, Springer.
    2. Achchab, Boujemâa & Bouihat, Khalid & Guessab, Allal & Schmeisser, Gerhard, 2015. "A general approach to the construction of nonconforming finite elements on convex polytopes," Applied Mathematics and Computation, Elsevier, vol. 268(C), pages 916-923.
    3. Dell’Accio, Francesco & Di Tommaso, Filomena & Guessab, Allal & Nudo, Federico, 2023. "Enrichment strategies for the simplicial linear finite elements," Applied Mathematics and Computation, Elsevier, vol. 451(C).
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    Cited by:

    1. Shaofeng Yao & Liang Yue & Wei Xie & Sen Zheng & Shuo Tang & Jinglong Liu & Wenkai Wang, 2024. "Investigating the Influence of Non-Uniform Characteristics of Layered Foundation on Ground Vibration Using an Efficient 2.5D Random Finite Element Method," Mathematics, MDPI, vol. 12(10), pages 1-17, May.

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