[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-124516-en":3,"doc-seo-124516-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":27,"seo_description":14,"update_tm":28,"read_time":29},124516,7971461741311,"Ophelia","https://ap-avatar.wpscdn.com/avatar/74000253aff267980c6?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779345379180704826",8,"Research & Report","Large deformation analysis of the inhomogeneous hyperelastic thick-walled sphere under internal/external pressure by Physics-Informed Neural Networks","Modeling the mechanical response of heterogeneous hyperelastic structural elements under complex loading remains difficult due to nonlinear material behavior, geometric constraints, and spatially varying properties. A physics-informed neural network framework is presented to predict the response of a heterogeneous neo-Hookean hyperelastic thick-walled sphere under simultaneous internal and external pressure. By embedding governing nonlinear PDEs, boundary conditions, and constitutive laws directly into the loss function, the approach avoids mesh generation and reduces computational burden. Coordinate-dependent material parameters capture spatial heterogeneity, and validation against analytical solutions shows relative errors below 1% for displacement and stress. Stress distributions reveal strong dependence on pressure gradients and heterogeneity, enabling localized effects and efficient parametric studies for design of soft, heterogeneous engineering systems.","Machine Learning for Computational Science and Engineering (2025) 1:35  \n[https://doi.org/10.1007/s44379-025-00037-9](https://doi.org/10.1007/s44379-025-00037-9)  \nLarge deformation analysis ofthe inhomogeneous hyperelastic thick‑walled sphere under internal/external pressure by Physics‑Informed Neural Networks  \nNasser Firouzi1 · Marco Amabili2 · Fadi Dohnal3 · Xiaoying Zhuang4 · Timon Rabczuk1  \nReceived: 27 July 2025 / Accepted: 15 September 2025 © The Author(s) 2025  \nAbstract  \nModeling the mechanical response of heterogeneous hyperelastic structural elements under complex loading conditions presents significant challenges, primarily due to nonlinear material behavior, geometric constraints, and spatially varying properties. This study introduces a novel framework based on Physics-Informed Neural Networks (PINNs) to predict the behavior of a heterogeneous neo-Hookean hyperelastic thick-walled sphere subjected simultaneously to internal and external pressures. In contrast to conventional finite element methods (FEM) which are often limited by challenges in mesh generation and high computational costs when addressing material heterogeneity the proposed PINNs approach incorporates the governing nonlinear partial differential equations, boundary conditions, and constitutive laws directly into the neural network’s loss function. Notably, the framework integrates spatial heterogeneity by embedding coordinate-dependent material parameters, such as the shear modulus, into the network architecture. Validation against analytical solutions confirms the method's high accuracy, with relative errors in displacement and stress predictions remaining below 1% . Further case studies demonstrate that radial and hoop stress distributions are significantly influenced by pressure gradients and material heterogeneity, thereby capturing localized mechanical effects. Moreover, the mesh-free nature of the PINNs method obviates the need for domain discretization, enhancing computational efficiency in parametric studies. This work bridges the gap between machine learning and continuum mechanics by providing a robust computational tool for the design of engineering systems that involve soft, heterogeneous materials; applications include biomedical implants, soft robotics, and pressurized energy storage devices. The versatility of the framework also paves the way for its extension to dynamic problems, multiphysics coupling, and the integration of experimental data.  \nKeywords Physics-Informed Neural Networks (PINNs) · Heterogeneous hyperelastic material · Thick-walled sphere · NeoHookean material · Nonlinear elasticity · Internal/external pressure  \n1 Introduction  \nThe computational modeling of hyperelastic materials has undergone significant evolution over the past decade, driven by the need to address anisotropy, compressibility, and spatial heterogeneity in soft, deformable systems [10, 11] . The neo-Hookean model, long regarded as a foundational framework for isotropic, incompressible materials, has been progressively refined to overcome limitations in capturing complex mechanical behaviors. Models by Gasser et al.  \n[13], Breslavsky et al. [5] and Amabili et al. [2] introduced fiber-reinforced hyperelastic models for arterial tissues, incorporating collagen fiber orientation to predict anisotropic stress–strain responses, while Ehret et al. [7] developed invariant-free formulations to resolve directional distinctions  \nin strain energy, enabling the integration of micropolar effects and material asymmetry. By 2020, compressible neo-Hookean models became widely adopted in commercial finite element software, though challenges persisted in ensuring numerical stability under large compressive strains (Moranon et al. 2016) . Recent innovations, such as polyconvex neural networks (pICNNs) proposed by [15], enforce thermodynamic consistency while accommodating process-dependent parameters like 3D printing conditions, achieving ± 3% error in stress pred","cbCaisrd1cLsjshP","https://ap.wps.com/l/cbCaisrd1cLsjshP","pdf",2191058,1,17,"English","en",105,"# Abstract\n# 1 Introduction","[{\"question\":\"What is the proposed modeling framework in this study?\",\"answer\":\"The study introduces a physics-informed neural network (PINN) framework to predict the large deformation response of a heterogeneous neo-Hookean hyperelastic thick-walled sphere under internal and external pressures.\"},{\"question\":\"How does the PINN approach handle heterogeneity and nonlinear mechanics?\",\"answer\":\"The method incorporates governing nonlinear partial differential equations, boundary conditions, and constitutive laws into the neural network loss, and it embeds coordinate-dependent material parameters (e.g., shear modulus) to represent spatial heterogeneity.\"},{\"question\":\"What accuracy and computational advantages are reported?\",\"answer\":\"Validation against analytical solutions shows relative errors in displacement and stress predictions below 1%, and the mesh-free PINN formulation avoids explicit domain discretization, improving efficiency for parametric studies.\"}]","Large deformation analysis of the inhomogeneous hyperelastic thick-walled sphere under internal/external pressure by Physics-Informed Neural Networks | 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is the proposed modeling framework in this study?","Question",{"text":75,"@type":76},"The study introduces a physics-informed neural network (PINN) framework to predict the large deformation response of a heterogeneous neo-Hookean hyperelastic thick-walled sphere under internal and external pressures.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the PINN approach handle heterogeneity and nonlinear mechanics?",{"text":80,"@type":76},"The method incorporates governing nonlinear partial differential equations, boundary conditions, and constitutive laws into the neural network loss, and it embeds coordinate-dependent material parameters (e.g., shear modulus) to represent spatial heterogeneity.",{"name":82,"@type":73,"acceptedAnswer":83},"What accuracy and computational advantages are reported?",{"text":84,"@type":76},"Validation against analytical solutions shows relative errors in displacement and stress predictions below 1%, and the mesh-free PINN formulation avoids 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