[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83379-en":3,"doc-seo-83379-105":29,"detail-sidebar-cat-0-en-105":89},{"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":13,"seo_description":14,"update_tm":27,"read_time":28},83379,1099514068365,"Aurelia","https://ap-avatar.wpscdn.com/avatar/10000253d8d9f28188e?_k=1776742907772140068",8,"Research & Report","An Iterative Method for Transient Finite Element Simulations of Non-Linear Eddy Current Problems","An iterative numerical method enables transient finite element simulation of eddy current problems involving nonlinear magnetic materials. Voltage-driven coils are coupled to finite element magnetics using Biot–Savart fields, while the magnetic vector potential forms the FEM formulation. Time integration uses implicit Euler, leading to a nonlinear system split into FEM and circuit equations. Each subsystem is solved separately with Newton’s method augmented by line search, and the approach is benchmarked using inrush current behavior and flux densities for a laminated iron core in a cylindrical coil.","An Iterative Method for Transient Finite Element Simulations of  \nNon-Linear Eddy Current Problems  \nK. Hollaus 1 , and H. Silm 1  \n1Institute of Analysis and Scientific Computing, TU Wien, Vienna, A-1040 Austria  \nA method is presented to carry out a transient simulation of eddy current problems with nonlinear materials. Coils are voltagedriven. The magnetic field due to currents in coils are considered by their Biot-Savart-fields. The magnetic vector potential is used in the finite element formulation. The time stepping method is based on implicit Euler. The arising nonlinear equation system is split into two parts, the common finite element system and a circuit equation. Each part is solved separately by Newton’s method. Additionally, a line search is used to solve the nonlinear field equations. Inrush currents and average magnetic flux densities through cross sections of laminates of a nonlinear benchmark problem consisting of a laminated iron core inserted in a cylindrical coil are studied. All details of the numerical benchmark are given to evaluate the presented results. Numerical data describing the performance of the presented method are provided.  \nIndex Terms—Biot-Savart-field, circuit coupling, finite element method, nonlinear eddy current problem, voltage-driven coil.  \narXiv :2607 .08432v1 [math .NA] 9 Jul 2026  \nI. INTRODUCTION  \nTHE METHOD presented here facilitates a transient finite  \nelement simulation of nonlinear eddy current problems with the magnetic vector potential (MVP) A taking into account of voltage-driven coils, of currents in coils by BiotSavart-field (BSF) and of material properties by a magnetization curve. Such a method is required for instance to study inrush currents of electrical devices. Inrush currents are of great importance in the design of electrical devices due to the electrical and mechanical stability. Reference solutions may also support the development of approximate methods considering laminated iron cores efficiently.  \nAn early work of a coupling of a voltage-driven coil with the finite element method (FEM) based on A can be found in [1] . A method based on a current vector potential T anda reduced magnetic scalar potential Φ is presented in [2] . It can handle solid conductors. A comprehensive overview of various possible coupling techniques for different problems can be found in the work [3] . The solution of the arising nonlinear system is not discussed by the above works. A coupling of the finite element harmonic balance method (FEHBM) using T and Φ with a circuit to get the steady state solution has been introduced in [4] . The nonlinear system is iteratively solved with the aid of a fixed point method technique allowing a parallel processing of the resulting problem. The problem is solved combined and separated.  \nIterative solvers are required for large linear equation systems. To get a reasonable convergence rate pre-conditioners are required. The problem arises, what is the appropriate preconditioner for the present case? The system is composed of two parts, the common non-linear system due to the FEM with A extended by the non-linear equation due to the circuit coupling. To circumvent this problem the system has been split corresponding to the two parts.  \nThe arising nonlinear equation system is split into two parts, Newton’s method (NM) is applied to each part separately exploiting the fast convergence rate of NM. The two parts  \nare alternately solved until a stopping criterion is fulfilled. NM for the system due to the FEM is supplemented by a line search. The BSF caused by a current in a coil is calculated only once and then accordingly scaled to the course of time. Induction effects in the windings are neglected (stranded coils) . A solution for that can be found for linear materials and BSF in [5] .  \nThe paper presents the case of one coil in detail, the extension to several coils is straight forward and outlined at the end. A study of a numerical benchmark is","cbCaih6dZR6ScLrT","https://ap.wps.com/l/cbCaih6dZR6ScLrT","pdf",418266,1,4,"English","en",105,"# Introduction\n# Eddy Current Problem\n## Boundary Value Problem\n## Weak Form\n## Voltage-Driven Coil and Biot-Savart-Field","[{\"question\":\"What problem does the proposed method address?\",\"answer\":\"It performs transient finite element simulations of nonlinear eddy current problems with voltage-driven coils and nonlinear magnetic materials.\"},{\"question\":\"How is the coil coupling represented in the formulation?\",\"answer\":\"The magnetic field contribution from currents in the coils is incorporated through Biot–Savart fields, while the circuit equation is solved together with the FEM weak form.\"},{\"question\":\"How is the nonlinear system solved during time stepping?\",\"answer\":\"The nonlinear equation system is split into a common finite element system and a circuit equation; each part is solved with Newton’s method, with line search used for the nonlinear field equations.\"}]",1784187105,10,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":84,"head_meta":86,"extra_data":88,"updated_unix":27},"an-iterative-method-for-transient-finite-element-simulations-of-non-linear-eddy-current-problems","",{"@graph":35,"@context":83},[36,52,66],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":21},"https://docshare.wps.com/document/an-iterative-method-for-transient-finite-element-simulations-of-non-linear-eddy-current-problems/83379/",{"url":51,"name":13,"@type":53,"author":54,"headline":13,"publisher":56,"fileFormat":59,"inLanguage":23,"description":14,"dateModified":60,"datePublished":60,"encodingFormat":59,"isAccessibleForFree":61,"interactionStatistic":62},"DigitalDocument",{"name":9,"@type":55},"Person",{"url":40,"name":57,"@type":58},"DocShare","Organization","application/pdf","2026-07-16",true,{"@type":63,"interactionType":64,"userInteractionCount":4},"InteractionCounter",{"@type":65},"ViewAction",{"@type":67,"mainEntity":68},"FAQPage",[69,75,79],{"name":70,"@type":71,"acceptedAnswer":72},"What problem does the proposed method address?","Question",{"text":73,"@type":74},"It performs transient finite element simulations of nonlinear eddy current problems with voltage-driven coils and nonlinear magnetic materials.","Answer",{"name":76,"@type":71,"acceptedAnswer":77},"How is the coil coupling represented in the formulation?",{"text":78,"@type":74},"The magnetic field contribution from currents in the coils is incorporated through Biot–Savart fields, while the circuit equation is solved together with the FEM weak form.",{"name":80,"@type":71,"acceptedAnswer":81},"How is the nonlinear system solved during time stepping?",{"text":82,"@type":74},"The nonlinear equation system is split into a common finite element system and a circuit equation; each part is solved with Newton’s method, with line search used for the nonlinear field equations.","https://schema.org",{"og:url":51,"og:type":85,"og:title":13,"og:site_name":57,"og:description":14},"article",{"robots":87,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":90},[91,95,99,103,108,113,118,121,126,129,132],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":92,"show_sort_weight":93,"slug":94},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":96,"show_sort_weight":97,"slug":98},"Literature",80,"literature",{"id":21,"doc_module":4,"doc_module_name":45,"category_name":100,"show_sort_weight":101,"slug":102},"Exam",70,"exam",{"id":104,"doc_module":4,"doc_module_name":45,"category_name":105,"show_sort_weight":106,"slug":107},5,"Comic",60,"comic",{"id":109,"doc_module":4,"doc_module_name":45,"category_name":110,"show_sort_weight":111,"slug":112},6,"Technology",50,"technology",{"id":114,"doc_module":4,"doc_module_name":45,"category_name":115,"show_sort_weight":116,"slug":117},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":119,"slug":120},30,"research-report",{"id":122,"doc_module":4,"doc_module_name":45,"category_name":123,"show_sort_weight":124,"slug":125},9,"Religion & Spirituality",20,"religion-spirituality",{"id":124,"doc_module":4,"doc_module_name":45,"category_name":127,"show_sort_weight":124,"slug":128},"World Cup","world-cup",{"id":28,"doc_module":4,"doc_module_name":45,"category_name":130,"show_sort_weight":28,"slug":131},"Lifestyle","lifestyle",{"id":133,"doc_module":4,"doc_module_name":45,"category_name":134,"show_sort_weight":104,"slug":135},19,"General","general"]