{"title":"Learning nonlinear constitutive models in finite strain electromechanics with Gaussian process predictors","authors":"","doi":"10.1007/s00466-024-02446-8","DOIUrl":null,"url":null,"abstract":"<h3>Abstract</h3> <p>This paper introduces a metamodelling technique that employs gradient-enhanced Gaussian process regression (GPR) to emulate diverse internal energy densities based on the deformation gradient tensor <span> <span>\\(\\varvec{F}\\)</span> </span> and electric displacement field <span> <span>\\(\\varvec{D}_0\\)</span> </span>. The approach integrates principal invariants as inputs for the surrogate internal energy density, enforcing physical constraints like material frame indifference and symmetry. This technique enables accurate interpolation of energy and its derivatives, including the first Piola-Kirchhoff stress tensor and material electric field. The method ensures stress and electric field-free conditions at the origin, which is challenging with regression-based methods like neural networks. The paper highlights that using invariants of the dual potential of internal energy density, i.e., the free energy density dependent on the material electric field <span> <span>\\(\\varvec{E}_0\\)</span> </span>, is inappropriate. The saddle point nature of the latter contrasts with the convexity of the internal energy density, creating challenges for GPR or Gradient Enhanced GPR models using invariants of <span> <span>\\(\\varvec{F}\\)</span> </span> and <span> <span>\\(\\varvec{E}_0\\)</span> </span> (free energy-based GPR), compared to those involving <span> <span>\\(\\varvec{F}\\)</span> </span> and <span> <span>\\(\\varvec{D}_0\\)</span> </span> (internal energy-based GPR). Numerical examples within a 3D Finite Element framework assess surrogate model accuracy across challenging scenarios, comparing displacement and stress fields with ground-truth analytical models. Cases include extreme twisting and electrically induced wrinkles, demonstrating practical applicability and robustness of the proposed approach.</p>","PeriodicalId":55248,"journal":{"name":"Computational Mechanics","volume":"126 1","pages":""},"PeriodicalIF":3.7000,"publicationDate":"2024-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Computational Mechanics","FirstCategoryId":"5","ListUrlMain":"https://doi.org/10.1007/s00466-024-02446-8","RegionNum":2,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, INTERDISCIPLINARY APPLICATIONS","Score":null,"Total":0}
引用次数: 0
Abstract
This paper introduces a metamodelling technique that employs gradient-enhanced Gaussian process regression (GPR) to emulate diverse internal energy densities based on the deformation gradient tensor \(\varvec{F}\) and electric displacement field \(\varvec{D}_0\). The approach integrates principal invariants as inputs for the surrogate internal energy density, enforcing physical constraints like material frame indifference and symmetry. This technique enables accurate interpolation of energy and its derivatives, including the first Piola-Kirchhoff stress tensor and material electric field. The method ensures stress and electric field-free conditions at the origin, which is challenging with regression-based methods like neural networks. The paper highlights that using invariants of the dual potential of internal energy density, i.e., the free energy density dependent on the material electric field \(\varvec{E}_0\), is inappropriate. The saddle point nature of the latter contrasts with the convexity of the internal energy density, creating challenges for GPR or Gradient Enhanced GPR models using invariants of \(\varvec{F}\) and \(\varvec{E}_0\) (free energy-based GPR), compared to those involving \(\varvec{F}\) and \(\varvec{D}_0\) (internal energy-based GPR). Numerical examples within a 3D Finite Element framework assess surrogate model accuracy across challenging scenarios, comparing displacement and stress fields with ground-truth analytical models. Cases include extreme twisting and electrically induced wrinkles, demonstrating practical applicability and robustness of the proposed approach.
期刊介绍：
The journal reports original research of scholarly value in computational engineering and sciences. It focuses on areas that involve and enrich the application of mechanics, mathematics and numerical methods. It covers new methods and computationally-challenging technologies.
Areas covered include method development in solid, fluid mechanics and materials simulations with application to biomechanics and mechanics in medicine, multiphysics, fracture mechanics, multiscale mechanics, particle and meshfree methods. Additionally, manuscripts including simulation and method development of synthesis of material systems are encouraged.
Manuscripts reporting results obtained with established methods, unless they involve challenging computations, and manuscripts that report computations using commercial software packages are not encouraged.
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