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Generalised tangent stabilised nonlinear elasticity: An automated framework for controlling material and geometric instabilities

Roman Poya Orcid Logo, Rogelio Ortigosa Orcid Logo, Antonio Gil Orcid Logo, Theodore Kim Orcid Logo, Javier Bonet Orcid Logo

Computer Methods in Applied Mechanics and Engineering, Volume: 436, Start page: 117701

Swansea University Authors: Antonio Gil Orcid Logo, Javier Bonet Orcid Logo

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DOI (Published version): 10.1016/j.cma.2024.117701

Abstract

Tangent stabilised large strain isotropic elasticity was recently proposed by Poya et al. (2023) wherein by working directly with principal stretches the entire eigenstructure of constitutive and geometric/initial stiffness terms were found in closed-form, giving fresh insights into exact convexity...

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Published in: Computer Methods in Applied Mechanics and Engineering
ISSN: 0045-7825
Published: Elsevier BV 2025
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URI: https://cronfa.swan.ac.uk/Record/cronfa68628
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(2023) wherein by working directly with principal stretches the entire eigenstructure of constitutive and geometric/initial stiffness terms were found in closed-form, giving fresh insights into exact convexity conditions of highly non-convex functions in discrete settings. Consequently, owing to these newly found tangent eigenvalues an analytic tangent stabilisation was proposed (for common non-convex strain energies that exhibit material and/or geometric instabilities) bypassing incumbent numerical approaches routinely used in nonlinear finite element analysis. This formulation appears to be extremely robust for quasi-static simulation of complex deformations even with no load increments and time stepping while still capturing instabilities (similar to dynamic analysis) automatically in ways that are infeasible for path-following techniques in practice. In this work, we generalise the notion of tangent stabilised elasticity to virtually all known invariant formulations of nonlinear elasticity. We show that, closedform eigen-decomposition of tangents is easily available irrespective of invariant formulation or integrity basis. In particular, we work out closed-form tangent eigensystems for isotropic Total Lagrangian deformation gradient ()-based and right Cauchy&#x2013;Green ()-based as well as Updated Lagrangian left Cauchy&#x2013;Green ()-based formulations and present their exact convexity conditions postulated in terms of their corresponding tangent and geometric stiffness eigenvalues. In addition, we introduce the notion of geometrically stabilised polyconvex large strain elasticity for models that are materially stable but exhibit geometric instabilities for whom we construct their geometric stiffness in a spectrally-decomposed form analytically. We further extend this framework to the case of transverse isotropy where once again, closedform tangent eigensystems are found for common transversely isotropic invariants. In this context, we augment the recent work on mixed variational formulations in principal stretches for deformable and rigid bodies, by presenting a mixed variational formulation for models with arbitrarily directed inextensible fibres. Since, tangent stabilisation unleashes an unparallelled capability for extreme deformations new numerical techniques are required to guarantee element-inversion-safe analysis. To this end, we propose a discretisation-aware load-stepping together with a line search scheme for a robust industry-grade implementation of tangent stabilised elasticity over general polyhedral meshes. Extensive comparisons with path-following techniques provide conclusive evidence that utilising tangent stabilised elasticity can offer both faster and automated results.</abstract><type>Journal Article</type><journal>Computer Methods in Applied Mechanics and Engineering</journal><volume>436</volume><journalNumber/><paginationStart>117701</paginationStart><paginationEnd/><publisher>Elsevier BV</publisher><placeOfPublication/><isbnPrint/><isbnElectronic/><issnPrint>0045-7825</issnPrint><issnElectronic/><keywords/><publishedDay>1</publishedDay><publishedMonth>3</publishedMonth><publishedYear>2025</publishedYear><publishedDate>2025-03-01</publishedDate><doi>10.1016/j.cma.2024.117701</doi><url>https://doi.org/10.1016/j.cma.2024.117701</url><notes/><college>COLLEGE NANME</college><department>Aerospace, Civil, Electrical, and Mechanical Engineering</department><CollegeCode>COLLEGE CODE</CollegeCode><DepartmentCode>ACEM</DepartmentCode><institution>Swansea University</institution><apcterm/><funders>Grant PID2022-141957OA-C22 funded by MICIU/AEI/10.13039/501100011033 and by &#x201C;ERDF A way of making Europe&#x201D;. Autonomous Community of the Region of Murcia, Spain through the programme for the development of scientific and technical research by competitive groups (21996/PI/22), included in the Regional Program for the Promotion of Scientific and Technical Research of Fundacion Seneca - Agencia de Ciencia Tecnologia de la Region de Murcia. 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spelling 2025-01-02T09:40:03.9946395 v2 68628 2025-01-02 Generalised tangent stabilised nonlinear elasticity: An automated framework for controlling material and geometric instabilities 1f5666865d1c6de9469f8b7d0d6d30e2 0000-0001-7753-1414 Antonio Gil Antonio Gil true false b7398206d59a9dd2f8d07a552cfd351a 0000-0002-0430-5181 Javier Bonet Javier Bonet true false 2025-01-02 ACEM Tangent stabilised large strain isotropic elasticity was recently proposed by Poya et al. (2023) wherein by working directly with principal stretches the entire eigenstructure of constitutive and geometric/initial stiffness terms were found in closed-form, giving fresh insights into exact convexity conditions of highly non-convex functions in discrete settings. Consequently, owing to these newly found tangent eigenvalues an analytic tangent stabilisation was proposed (for common non-convex strain energies that exhibit material and/or geometric instabilities) bypassing incumbent numerical approaches routinely used in nonlinear finite element analysis. This formulation appears to be extremely robust for quasi-static simulation of complex deformations even with no load increments and time stepping while still capturing instabilities (similar to dynamic analysis) automatically in ways that are infeasible for path-following techniques in practice. In this work, we generalise the notion of tangent stabilised elasticity to virtually all known invariant formulations of nonlinear elasticity. We show that, closedform eigen-decomposition of tangents is easily available irrespective of invariant formulation or integrity basis. In particular, we work out closed-form tangent eigensystems for isotropic Total Lagrangian deformation gradient ()-based and right Cauchy–Green ()-based as well as Updated Lagrangian left Cauchy–Green ()-based formulations and present their exact convexity conditions postulated in terms of their corresponding tangent and geometric stiffness eigenvalues. In addition, we introduce the notion of geometrically stabilised polyconvex large strain elasticity for models that are materially stable but exhibit geometric instabilities for whom we construct their geometric stiffness in a spectrally-decomposed form analytically. We further extend this framework to the case of transverse isotropy where once again, closedform tangent eigensystems are found for common transversely isotropic invariants. In this context, we augment the recent work on mixed variational formulations in principal stretches for deformable and rigid bodies, by presenting a mixed variational formulation for models with arbitrarily directed inextensible fibres. Since, tangent stabilisation unleashes an unparallelled capability for extreme deformations new numerical techniques are required to guarantee element-inversion-safe analysis. To this end, we propose a discretisation-aware load-stepping together with a line search scheme for a robust industry-grade implementation of tangent stabilised elasticity over general polyhedral meshes. Extensive comparisons with path-following techniques provide conclusive evidence that utilising tangent stabilised elasticity can offer both faster and automated results. Journal Article Computer Methods in Applied Mechanics and Engineering 436 117701 Elsevier BV 0045-7825 1 3 2025 2025-03-01 10.1016/j.cma.2024.117701 https://doi.org/10.1016/j.cma.2024.117701 COLLEGE NANME Aerospace, Civil, Electrical, and Mechanical Engineering COLLEGE CODE ACEM Swansea University Grant PID2022-141957OA-C22 funded by MICIU/AEI/10.13039/501100011033 and by “ERDF A way of making Europe”. Autonomous Community of the Region of Murcia, Spain through the programme for the development of scientific and technical research by competitive groups (21996/PI/22), included in the Regional Program for the Promotion of Scientific and Technical Research of Fundacion Seneca - Agencia de Ciencia Tecnologia de la Region de Murcia. Financial support received through the UK Defence, Science and Technology Laboratory, grants: DSTLX1000160466R and RQ0000028640. 2025-01-02T09:40:03.9946395 2025-01-02T09:35:22.2162782 Faculty of Science and Engineering School of Aerospace, Civil, Electrical, General and Mechanical Engineering - Mechanical Engineering Roman Poya 0000-0003-2350-4933 1 Rogelio Ortigosa 0000-0002-4542-2237 2 Antonio Gil 0000-0001-7753-1414 3 Theodore Kim 0000-0002-1131-8685 4 Javier Bonet 0000-0002-0430-5181 5
title Generalised tangent stabilised nonlinear elasticity: An automated framework for controlling material and geometric instabilities
spellingShingle Generalised tangent stabilised nonlinear elasticity: An automated framework for controlling material and geometric instabilities
Antonio Gil
Javier Bonet
title_short Generalised tangent stabilised nonlinear elasticity: An automated framework for controlling material and geometric instabilities
title_full Generalised tangent stabilised nonlinear elasticity: An automated framework for controlling material and geometric instabilities
title_fullStr Generalised tangent stabilised nonlinear elasticity: An automated framework for controlling material and geometric instabilities
title_full_unstemmed Generalised tangent stabilised nonlinear elasticity: An automated framework for controlling material and geometric instabilities
title_sort Generalised tangent stabilised nonlinear elasticity: An automated framework for controlling material and geometric instabilities
author_id_str_mv 1f5666865d1c6de9469f8b7d0d6d30e2
b7398206d59a9dd2f8d07a552cfd351a
author_id_fullname_str_mv 1f5666865d1c6de9469f8b7d0d6d30e2_***_Antonio Gil
b7398206d59a9dd2f8d07a552cfd351a_***_Javier Bonet
author Antonio Gil
Javier Bonet
author2 Roman Poya
Rogelio Ortigosa
Antonio Gil
Theodore Kim
Javier Bonet
format Journal article
container_title Computer Methods in Applied Mechanics and Engineering
container_volume 436
container_start_page 117701
publishDate 2025
institution Swansea University
issn 0045-7825
doi_str_mv 10.1016/j.cma.2024.117701
publisher Elsevier BV
college_str Faculty of Science and Engineering
hierarchytype
hierarchy_top_id facultyofscienceandengineering
hierarchy_top_title Faculty of Science and Engineering
hierarchy_parent_id facultyofscienceandengineering
hierarchy_parent_title Faculty of Science and Engineering
department_str School of Aerospace, Civil, Electrical, General and Mechanical Engineering - Mechanical Engineering{{{_:::_}}}Faculty of Science and Engineering{{{_:::_}}}School of Aerospace, Civil, Electrical, General and Mechanical Engineering - Mechanical Engineering
url https://doi.org/10.1016/j.cma.2024.117701
document_store_str 0
active_str 0
description Tangent stabilised large strain isotropic elasticity was recently proposed by Poya et al. (2023) wherein by working directly with principal stretches the entire eigenstructure of constitutive and geometric/initial stiffness terms were found in closed-form, giving fresh insights into exact convexity conditions of highly non-convex functions in discrete settings. Consequently, owing to these newly found tangent eigenvalues an analytic tangent stabilisation was proposed (for common non-convex strain energies that exhibit material and/or geometric instabilities) bypassing incumbent numerical approaches routinely used in nonlinear finite element analysis. This formulation appears to be extremely robust for quasi-static simulation of complex deformations even with no load increments and time stepping while still capturing instabilities (similar to dynamic analysis) automatically in ways that are infeasible for path-following techniques in practice. In this work, we generalise the notion of tangent stabilised elasticity to virtually all known invariant formulations of nonlinear elasticity. We show that, closedform eigen-decomposition of tangents is easily available irrespective of invariant formulation or integrity basis. In particular, we work out closed-form tangent eigensystems for isotropic Total Lagrangian deformation gradient ()-based and right Cauchy–Green ()-based as well as Updated Lagrangian left Cauchy–Green ()-based formulations and present their exact convexity conditions postulated in terms of their corresponding tangent and geometric stiffness eigenvalues. In addition, we introduce the notion of geometrically stabilised polyconvex large strain elasticity for models that are materially stable but exhibit geometric instabilities for whom we construct their geometric stiffness in a spectrally-decomposed form analytically. We further extend this framework to the case of transverse isotropy where once again, closedform tangent eigensystems are found for common transversely isotropic invariants. In this context, we augment the recent work on mixed variational formulations in principal stretches for deformable and rigid bodies, by presenting a mixed variational formulation for models with arbitrarily directed inextensible fibres. Since, tangent stabilisation unleashes an unparallelled capability for extreme deformations new numerical techniques are required to guarantee element-inversion-safe analysis. To this end, we propose a discretisation-aware load-stepping together with a line search scheme for a robust industry-grade implementation of tangent stabilised elasticity over general polyhedral meshes. Extensive comparisons with path-following techniques provide conclusive evidence that utilising tangent stabilised elasticity can offer both faster and automated results.
published_date 2025-03-01T20:37:03Z
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