Mixed virtual element formulations for incompressible and inextensible problems

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Original languageEnglish
Pages (from-to)1141-1174
Number of pages34
JournalComputational mechanics
Volume72
Issue number6
Early online date4 May 2023
Publication statusPublished - Dec 2023

Abstract

Locking effects can be a major concern during the numerical modelling of elastic materials, especially for large strains. Those effects arise from volumetric constraints such as incompressibility or anisotropic effects of the underlying material class. One particular solution strategy is to employ mixed formulations, which provide solutions tailored to the specific locking phenomena at hand. While being in general powerful, one drawback of such solution strategies is that the provided strategy to overcome locking is often tied or limited to some specific topological type of finite elements and thus forfeiting generality. Contrary the Virtual Element Method (VEM) benefits by definition from allowing arbitrary element shapes and number of nodes at element level. A variety of approaches for the treatment of locking phenomena for hyperelastic material is formulated in this contribution for the Virtual Element Method with a low order ansatz. Key ingredient to the implementation of multi-field mixed principles in VEM is the consideration of only one constant variable per field and one corresponding Lagrange multiplier over the entire virtual element. Hereby the stabilization contribution utilizes the mixed formulation but shares the element-wise constant variable with the projection part of the virtual element. A direct consequence of this rather simple implementation strategy is the combination of powerful mixed formulations with a computational approach that is able to treat general element shapes. The proposed formulations are tested with regard to structured mesh at standard examples in computational mechanics as well as at specific computational engineering applications where also unstructured meshes are utilized.

Keywords

    Computational homogenization, Elastic anisotropy, Large deformations, Mixed principles, Sensitivity analysis, Virtual element method (VEM), Volumetric/anisotropic locking

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Mixed virtual element formulations for incompressible and inextensible problems. / Böhm, Christoph; Korelc, Jože; Hudobivnik, Blaž et al.

In: Computational mechanics, Vol. 72, No. 6, 12.2023, p. 1141-1174.

Research output: Contribution to journalArticleResearchpeer review

Böhm C, Korelc J, Hudobivnik B, Kraus A, Wriggers P. Mixed virtual element formulations for incompressible and inextensible problems. Computational mechanics. 2023 Dec;72(6):1141-1174. Epub 2023 May 4. doi: 10.1007/s00466-023-02340-9
Böhm, Christoph ; Korelc, Jože ; Hudobivnik, Blaž et al. / Mixed virtual element formulations for incompressible and inextensible problems. In: Computational mechanics. 2023 ; Vol. 72, No. 6. pp. 1141-1174.
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title = "Mixed virtual element formulations for incompressible and inextensible problems",
abstract = "Locking effects can be a major concern during the numerical modelling of elastic materials, especially for large strains. Those effects arise from volumetric constraints such as incompressibility or anisotropic effects of the underlying material class. One particular solution strategy is to employ mixed formulations, which provide solutions tailored to the specific locking phenomena at hand. While being in general powerful, one drawback of such solution strategies is that the provided strategy to overcome locking is often tied or limited to some specific topological type of finite elements and thus forfeiting generality. Contrary the Virtual Element Method (VEM) benefits by definition from allowing arbitrary element shapes and number of nodes at element level. A variety of approaches for the treatment of locking phenomena for hyperelastic material is formulated in this contribution for the Virtual Element Method with a low order ansatz. Key ingredient to the implementation of multi-field mixed principles in VEM is the consideration of only one constant variable per field and one corresponding Lagrange multiplier over the entire virtual element. Hereby the stabilization contribution utilizes the mixed formulation but shares the element-wise constant variable with the projection part of the virtual element. A direct consequence of this rather simple implementation strategy is the combination of powerful mixed formulations with a computational approach that is able to treat general element shapes. The proposed formulations are tested with regard to structured mesh at standard examples in computational mechanics as well as at specific computational engineering applications where also unstructured meshes are utilized.",
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author = "Christoph B{\"o}hm and Jo{\v z}e Korelc and Bla{\v z} Hudobivnik and Alex Kraus and Peter Wriggers",
note = "Funding Information: CB and PW gratefully acknowledge the German Research Foundation (DFG, Deutsche Forschungsgemeinschaft) for financial support to this work with the Collaborative Research Centre 1153 (CRC 1153) “Process chain for the production of hybrid high-performance components through tailored forming” with the subproject C4 “Modelling and Simulation of the Joining Zone”, project number 252662854. BH and PW gratefully acknowledge financial support to this work by the German Research Foundation (DFG) with the cluster of excellence PhoenixD (EXC 2122, Project ID 390833453). This work was supported by the compute cluster, which is funded by the Leibniz Universit{\"a}t Hannover, the Lower Saxony Ministry of Science and Culture (MWK) and the German Research Association (DFG). ",
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T1 - Mixed virtual element formulations for incompressible and inextensible problems

AU - Böhm, Christoph

AU - Korelc, Jože

AU - Hudobivnik, Blaž

AU - Kraus, Alex

AU - Wriggers, Peter

N1 - Funding Information: CB and PW gratefully acknowledge the German Research Foundation (DFG, Deutsche Forschungsgemeinschaft) for financial support to this work with the Collaborative Research Centre 1153 (CRC 1153) “Process chain for the production of hybrid high-performance components through tailored forming” with the subproject C4 “Modelling and Simulation of the Joining Zone”, project number 252662854. BH and PW gratefully acknowledge financial support to this work by the German Research Foundation (DFG) with the cluster of excellence PhoenixD (EXC 2122, Project ID 390833453). This work was supported by the compute cluster, which is funded by the Leibniz Universität Hannover, the Lower Saxony Ministry of Science and Culture (MWK) and the German Research Association (DFG).

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N2 - Locking effects can be a major concern during the numerical modelling of elastic materials, especially for large strains. Those effects arise from volumetric constraints such as incompressibility or anisotropic effects of the underlying material class. One particular solution strategy is to employ mixed formulations, which provide solutions tailored to the specific locking phenomena at hand. While being in general powerful, one drawback of such solution strategies is that the provided strategy to overcome locking is often tied or limited to some specific topological type of finite elements and thus forfeiting generality. Contrary the Virtual Element Method (VEM) benefits by definition from allowing arbitrary element shapes and number of nodes at element level. A variety of approaches for the treatment of locking phenomena for hyperelastic material is formulated in this contribution for the Virtual Element Method with a low order ansatz. Key ingredient to the implementation of multi-field mixed principles in VEM is the consideration of only one constant variable per field and one corresponding Lagrange multiplier over the entire virtual element. Hereby the stabilization contribution utilizes the mixed formulation but shares the element-wise constant variable with the projection part of the virtual element. A direct consequence of this rather simple implementation strategy is the combination of powerful mixed formulations with a computational approach that is able to treat general element shapes. The proposed formulations are tested with regard to structured mesh at standard examples in computational mechanics as well as at specific computational engineering applications where also unstructured meshes are utilized.

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KW - Sensitivity analysis

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