[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83748-en":3,"doc-seo-83748-105":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":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},83748,137441390410,"Hazel","https://ap-avatar.wpscdn.com/avatar/2000252f4ab5702993?_k=1776741390130283984",8,"Research & Report","Discretisation of Eulerian nonlinear elasticity and diffusion using gradient flows","This study presents a general energy-based modelling framework for viscous poroelastic materials with diffusive transport in both Lagrangian and Eulerian descriptions. Weak formulations are refined through the reference map concept in the Eulerian configuration, enabling a structure-preserving discretisation via mixed finite elements. Numerical experiments assess spatial and temporal convergence by comparing Lagrangian and Eulerian schemes, demonstrating robustness. For Eulerian quasi-static Euler–Euler multiphase settings, the work proves the existence of solitary fluid waves in poroviscoelastic media and motivates thermodynamical coupling and diffusion discretisation strategies.","arXiv :2607 .03829v1 [math .NA] 4 Jul 2026  \nDiscretisation of Eulerian nonlinear elasticity and diffusion  \nusing gradient flows  \nAndrea Zafferi *† Dirk Peschka ‡§  \nAbstract  \nIn this study, we introduce a general energy-based modelling approach for viscous poroelastic materials that feature diffusive transport in both Lagrangian and Eulerian frames. Our research produces refined weak formulations by using the reference map concept within the Eulerian configuration. We propose and implement a novel structure-preserving discretisation strategy, utilising mixed finite element methods. This paper highlights the spatial and temporal numerical convergence of our methods through a comparative analysis of Lagrangian and Eulerian schemes, thereby proving the robustness and usability of our approach. Furthermore, in the context of Eulerian multiphase flow, specifically of the quasi-static Euler-Euler type, our study demonstrates the existence of solitary fluid waves within poroviscoelastic media. This energy-based approach forms a basis for a deeper understanding of thermodynamical modelling and corresponding discretisation schemes for coupled poroelasticity, flow, and diffusion.  \nMSC (2020): 35Q74, 74L05, 35A15, 76S05  \nKeywords: viscoelasticity, gradient flow, nonlinear diffusion, Lagrangian, Eulerian  \n1 Introduction  \nMany physical, biological, and geoscientific phenomena involving viscoelastic materials can be described by thermodynamic models formulated in either Lagrangian or Eulerian coordinates [1] . Such models naturally consider diffusion-reaction processes, interfacial effects such as capillarity and phase separation, and external fields such as gravity and electrostatic fields, leading to complex interactions across multiple spatial and temporal scales [2, 3, 4, 5] . When large elastic deformations are present, the resulting partial differential equations are highly nonlinear, and even their rigorous formulation in the spatial (Eulerian) frame poses fundamental mathematical and modelling challenges, e.g., the consistent coupling of elastic and inelastic effects [6], the enforcement of Onsager symmetries [7], the incorporation of order parameters for phase-field models [8, 9, 10, 11, 12], the treatment of fluid–structure interaction [13, 14], and the choice of appropriate state variables, such as strain measures, mappings, or stresses [15, 16] . A key feature of most of these nonlinearly coupled models is their energy-variational formulation stated in either a Lagrangian or in an Eulerian frame.  \nTraditionally, Lagrangian approaches have been used for the mathematical treatment of large deformations, e.g., [17, 18, 19] . Alternatively, Eulerian descriptions are often based on formulations that use covariant Liederivatives of deformation gradients (F), left or right Cauchy green tensor (B = FFT or C = FT F) or related stress tensors, e.g., [20, 15, 21, 22] . These approaches have been effectively employed in modelling complex fluid-structure interactions [23] and extended to nonlinear elastic biological tissues, capturing their inherent compressibility and growth dynamics [24] . In a similar fashion, regarding geological applications an Eulerian large strain model for porous materials was developed in [25], where the energetics of the system is obtained as part of the mathematical analysis. Alternatively, the use of the reference map has emerged asa valuable tool for handling finite-strain elasticity within Eulerian frameworks with heterogeneous materials, combining the spatial and material viewpoints [26] . Therefore, such an approach might be particularly suitable for geophysical applications and porosity waves.  \nPorosity waves are relevant for various natural processes by governing the fluid transport in deformable porous media. Porosity waves typically emerge and evolve due to gradients in the pore pressure, mechanical  \n*Freie Universitt Berlin, Department of Applied and Computer Science, Arnimalle 9, 14195 Berlin","cbCailgx5Zbj5XZ7","https://ap.wps.com/l/cbCailgx5Zbj5XZ7","pdf",9268608,4,1,26,"English","en",105,"# Abstract\n# Keywords\n# Introduction","[{\"question\":\"What modelling framework is proposed for viscous poroelastic materials?\",\"answer\":\"An energy-based modelling approach is introduced to describe diffusive transport for viscous poroelastic materials in both Lagrangian and Eulerian frames.\"},{\"question\":\"How is the Eulerian weak formulation refined?\",\"answer\":\"The reference map concept within the Eulerian configuration is used to obtain refined weak formulations.\"},{\"question\":\"What theoretical and numerical results are highlighted for Eulerian multiphase flow and discretisation?\",\"answer\":\"The paper demonstrates spatial and temporal numerical convergence by comparing Lagrangian and Eulerian schemes, and it shows existence of solitary fluid waves in poroviscoelastic media for a quasi-static Euler–Euler Eulerian multiphase setting.\"}]",1784190194,66,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"discretisation-of-eulerian-nonlinear-elasticity-and-diffusion-using-gradient-flows","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":20},"https://docshare.wps.com/document/discretisation-of-eulerian-nonlinear-elasticity-and-diffusion-using-gradient-flows/83748/",{"url":52,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-27","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What modelling framework is proposed for viscous poroelastic materials?","Question",{"text":75,"@type":76},"An energy-based modelling approach is introduced to describe diffusive transport for viscous poroelastic materials in both Lagrangian and Eulerian frames.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How is the Eulerian weak formulation refined?",{"text":80,"@type":76},"The reference map concept within the Eulerian configuration is used to obtain refined weak formulations.",{"name":82,"@type":73,"acceptedAnswer":83},"What theoretical and numerical results are highlighted for Eulerian multiphase flow and discretisation?",{"text":84,"@type":76},"The paper demonstrates spatial and temporal numerical convergence by comparing Lagrangian and Eulerian schemes, and it shows existence of solitary fluid waves in poroviscoelastic media for a quasi-static Euler–Euler Eulerian multiphase setting.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]