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Stretch-based hyperelastic electromechanical constitutive metamodels via gradient enhanced Gaussian predictors using hierarchical structure discovery

Nathan Ellmer, Rogelio Ortigosa Orcid Logo, Jesús Martínez-Frutos Orcid Logo, Roman Poya Orcid Logo, Johann Sienz Orcid Logo, Antonio Gil Orcid Logo

Computer Methods in Applied Mechanics and Engineering, Volume: 448, Issue: Part A, Start page: 118349

Swansea University Authors: Nathan Ellmer, Johann Sienz Orcid Logo, Antonio Gil Orcid Logo

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DOI (Published version): 10.1016/j.cma.2025.118349

Abstract

This paper introduces a new approach to developing electromechanical constitutive metamodels via the use of Gradient Enhanced Gaussian Predictors (Kriging). The formulation uses principal stretches for the isotropic mechanics, invariants for the electrostatics and coupling terms, and accounts for an...

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Published in: Computer Methods in Applied Mechanics and Engineering
ISSN: 0045-7825 1879-2138
Published: Elsevier BV 2026
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The second novelty arises from the use of a hybrid formulation, namely the principal stretches for isotropic mechanics and the electromechanical anisotropic invariants. This is beneficial during the projection procedure in reducing the cases where the projection matrix becomes singular but requires careful development of the correlation function to maintain physical symmetry conditions. Thirdly, the electromechanical metamodels are calibrated upon the concentric styled data before being integrated within a Finite Element framework and tested upon a range of challenging simulations including bending actuators with induced torsion, frilling due to bending with selected electrode placement, as well as buckling plates tested with three rank-one laminate materials with increasing levels of anisotropy due to physical contrasts. 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spelling 2025-10-07T10:44:56.5207294 v2 70361 2025-09-15 Stretch-based hyperelastic electromechanical constitutive metamodels via gradient enhanced Gaussian predictors using hierarchical structure discovery da413556083b41e614a5d2264a0124dd Nathan Ellmer Nathan Ellmer true false 17bf1dd287bff2cb01b53d98ceb28a31 0000-0003-3136-5718 Johann Sienz Johann Sienz true false 1f5666865d1c6de9469f8b7d0d6d30e2 0000-0001-7753-1414 Antonio Gil Antonio Gil true false 2025-09-15 ACEM This paper introduces a new approach to developing electromechanical constitutive metamodels via the use of Gradient Enhanced Gaussian Predictors (Kriging). The formulation uses principal stretches for the isotropic mechanics, invariants for the electrostatics and coupling terms, and accounts for anisotropy through the relevant inclusion of anisotropic invariants associated with a respective symmetry integrity basis. Three novelties are presented in this paper. The first is the use of orthogonal projections to identify the most appropriate set of inputs - related to material anisotropy - for use in the metamodel. By projecting the stress and electric field data into several derivative bases - defined for each anisotropic class - and then reconstructing the quantities, the errors in reconstruction can be assessed thus inferring the most appropriate class of anisotropy. Furthermore, the procedure forms a pre-processing stage and is particularly useful when an underlying model is completely unknown as seen when modelling Relative Volume Elements. The second novelty arises from the use of a hybrid formulation, namely the principal stretches for isotropic mechanics and the electromechanical anisotropic invariants. This is beneficial during the projection procedure in reducing the cases where the projection matrix becomes singular but requires careful development of the correlation function to maintain physical symmetry conditions. Thirdly, the electromechanical metamodels are calibrated upon the concentric styled data before being integrated within a Finite Element framework and tested upon a range of challenging simulations including bending actuators with induced torsion, frilling due to bending with selected electrode placement, as well as buckling plates tested with three rank-one laminate materials with increasing levels of anisotropy due to physical contrasts. The successful calibration and implementation of the metamodels can be witnessed amongst the wide range of presented numerical examples. Journal Article Computer Methods in Applied Mechanics and Engineering 448 Part A 118349 Elsevier BV 0045-7825 1879-2138 Gradient Kriging; Constitutive modelling; Principal stretches; Electromechanics; Anisotropy 1 1 2026 2026-01-01 10.1016/j.cma.2025.118349 COLLEGE NANME Aerospace, Civil, Electrical, and Mechanical Engineering COLLEGE CODE ACEM Swansea University SU Library paid the OA fee (TA Institutional Deal) N. Ellmer and A. J. Gil wish to acknowledge the financial support provided by the Defence, science and technology laboratory (Dstl). Additionally, A. J. Gil acknowledges the support of The Luverhulme Trust through a Leverhulme Fellowship. R. Ortigosa and J. Martínez-Frutos acknowledge the support of grant PID2022-141957OA-C22 funded by MICIU/AEI/10.13039/501100011033 and by “RDF A way of making Europe”. 2025-10-07T10:44:56.5207294 2025-09-15T16:33:17.1035323 Faculty of Science and Engineering School of Aerospace, Civil, Electrical, General and Mechanical Engineering - Civil Engineering Nathan Ellmer 1 Rogelio Ortigosa 0000-0002-4542-2237 2 Jesús Martínez-Frutos 0000-0002-7112-3345 3 Roman Poya 0000-0003-2350-4933 4 Johann Sienz 0000-0003-3136-5718 5 Antonio Gil 0000-0001-7753-1414 6 70361__35267__a0d0aed3af0f4aafb87e692636d34c30.pdf 70361.VOR.pdf 2025-10-07T10:40:13.6508028 Output 14656765 application/pdf Version of Record true © 2025 The Author(s). This is an open access article distributed under the terms of the Creative Commons CC-BY license. true eng http://creativecommons.org/licenses/by/4.0/
title Stretch-based hyperelastic electromechanical constitutive metamodels via gradient enhanced Gaussian predictors using hierarchical structure discovery
spellingShingle Stretch-based hyperelastic electromechanical constitutive metamodels via gradient enhanced Gaussian predictors using hierarchical structure discovery
Nathan Ellmer
Johann Sienz
Antonio Gil
title_short Stretch-based hyperelastic electromechanical constitutive metamodels via gradient enhanced Gaussian predictors using hierarchical structure discovery
title_full Stretch-based hyperelastic electromechanical constitutive metamodels via gradient enhanced Gaussian predictors using hierarchical structure discovery
title_fullStr Stretch-based hyperelastic electromechanical constitutive metamodels via gradient enhanced Gaussian predictors using hierarchical structure discovery
title_full_unstemmed Stretch-based hyperelastic electromechanical constitutive metamodels via gradient enhanced Gaussian predictors using hierarchical structure discovery
title_sort Stretch-based hyperelastic electromechanical constitutive metamodels via gradient enhanced Gaussian predictors using hierarchical structure discovery
author_id_str_mv da413556083b41e614a5d2264a0124dd
17bf1dd287bff2cb01b53d98ceb28a31
1f5666865d1c6de9469f8b7d0d6d30e2
author_id_fullname_str_mv da413556083b41e614a5d2264a0124dd_***_Nathan Ellmer
17bf1dd287bff2cb01b53d98ceb28a31_***_Johann Sienz
1f5666865d1c6de9469f8b7d0d6d30e2_***_Antonio Gil
author Nathan Ellmer
Johann Sienz
Antonio Gil
author2 Nathan Ellmer
Rogelio Ortigosa
Jesús Martínez-Frutos
Roman Poya
Johann Sienz
Antonio Gil
format Journal article
container_title Computer Methods in Applied Mechanics and Engineering
container_volume 448
container_issue Part A
container_start_page 118349
publishDate 2026
institution Swansea University
issn 0045-7825
1879-2138
doi_str_mv 10.1016/j.cma.2025.118349
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 - Civil Engineering{{{_:::_}}}Faculty of Science and Engineering{{{_:::_}}}School of Aerospace, Civil, Electrical, General and Mechanical Engineering - Civil Engineering
document_store_str 1
active_str 0
description This paper introduces a new approach to developing electromechanical constitutive metamodels via the use of Gradient Enhanced Gaussian Predictors (Kriging). The formulation uses principal stretches for the isotropic mechanics, invariants for the electrostatics and coupling terms, and accounts for anisotropy through the relevant inclusion of anisotropic invariants associated with a respective symmetry integrity basis. Three novelties are presented in this paper. The first is the use of orthogonal projections to identify the most appropriate set of inputs - related to material anisotropy - for use in the metamodel. By projecting the stress and electric field data into several derivative bases - defined for each anisotropic class - and then reconstructing the quantities, the errors in reconstruction can be assessed thus inferring the most appropriate class of anisotropy. Furthermore, the procedure forms a pre-processing stage and is particularly useful when an underlying model is completely unknown as seen when modelling Relative Volume Elements. The second novelty arises from the use of a hybrid formulation, namely the principal stretches for isotropic mechanics and the electromechanical anisotropic invariants. This is beneficial during the projection procedure in reducing the cases where the projection matrix becomes singular but requires careful development of the correlation function to maintain physical symmetry conditions. Thirdly, the electromechanical metamodels are calibrated upon the concentric styled data before being integrated within a Finite Element framework and tested upon a range of challenging simulations including bending actuators with induced torsion, frilling due to bending with selected electrode placement, as well as buckling plates tested with three rank-one laminate materials with increasing levels of anisotropy due to physical contrasts. The successful calibration and implementation of the metamodels can be witnessed amongst the wide range of presented numerical examples.
published_date 2026-01-01T05:32:28Z
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