[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82001-en":3,"doc-seo-82001-105":30,"detail-sidebar-cat-0-en-105":92},{"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},82001,687197207639,"Asher","https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd",8,"Research & Report","Accelerating Industrial Finite Element Simulations of Electric Machines based on Runtime Analysis","Simulation of electric machines is central to designing efficient, competitive products, yet computational overhead directly limits iteration speed and time-to-market. Targeted optimization can reduce total runtime by addressing dominant bottlenecks inside nonlinear finite-element workflows. This work analyzes typical solver-step costs in industrial 2D and 3D environments, identifies the most time-consuming components of Newton-based solution procedures, and demonstrates acceleration measures that lower the contribution of residual, Jacobian, line search, factorization/preconditioning, and linear solves.","arXiv :2607 .075 14v 1 [ cs .CE] 8 Jul 2026  \nAccelerating Industrial Finite Element Simulations of Electric Machines based on Runtime Analysis  \nArvinth Shankar,  \nIryna Kulchytska-Ruchka [0000−0002−9853−8006], and Sebastian Schps [0000−0001−9150−0219]  \nFast solvers for faster e-machine designs.  \nAbstract The simulation of electric machines plays a significant role in the design of efficient and competitive products. Faster simulations reduce computational costs, such as CPU hours, and shorten development cycles, thereby enabling faster design iterationsand ultimately accelerating time-to-market. In this work, we analyze the dominant computational bottlenecks and demonstrate how targeted acceleration measures can significantly reduce the overall runtime of 2D and 3D finite element simulations of electric machines in an industrial environment.  \nKeywords finite element solver · Newton line-search · preconditioner · electric machines  \n1 Introduction  \nComputationally efficient Finite Element (FE) simulations of electric machines depend on the performance of its various building blocks: mesher, finite element discretization, time stepping,(non-)linear solvers, post-processing etc. The required efficiency, however, depends on the use case. For high-fidelity simulations, both parallelization and algorithmic speed-up strategies can be effectively combined to reduce the wall-clock  \nArvinth Shankar  \nRobert Bosch GmbH, Robert-Bosch-Campus 1, 71272 Renningen, Germany, e-mail: [arvinth.shankar@de.bosch.com](arvinth.shankar@de.bosch.com)  \nIryna Kulchytska-Ruchka  \nRobert Bosch GmbH, Robert-Bosch-Campus 1, 71272 Renningen, Germany, e-mail: iryna.kulchytska[ruchka@de.bosch.com](ruchka@de.bosch.com)  \nSebastian Schps  \nComputational Electromagnetics Group, Technische Universitt Darmstadt, Schloßgartenstraße 8, 64289 Darmstadt, Germany, e-mail: [sebastian.schoeps@tu-darmstadt.de](sebastian.schoeps@tu-darmstadt.de)  \n2 Arvinth Shankar et al.  \ntime. In contrast, when a large number of simulations are executed concurrently, parallelization resources are already utilized across runs. Thus the focus shifts to energy efficiency, where algorithmic speed-up is the key to reducing the computational cost per simulation. In practice, the overall runtime is driven by the repeated execution of several solver steps rather than any single operation. Here, we investigate the typical nonlinear solver steps: S1: Residual evaluation, S2: Jacobian evaluation, S3: Line search (e.g. using Wolfe-Powell conditions), S4: Matrix factorization or preconditioning, S5: Linear system solution (direct or iterative) . Note that Wolfe-Powell conditions [1] include within the line search further residual evaluations in Armijo’s condition S3􀀰 (used for sufficient decrease) and further Jacobian evaluations in the curvature condition S3􀀱(used for sufficient change in gradient) . This implies that the computational cost of residual evaluation S1 or Jacobian evaluation S2 also affect the cost of the line search procedure S3, since the line search may require multiple such evaluations within its iterative procedure. In the following sections, we analyze the computational time spent in each solver step for representative 2D and 3D simulations of e-machines, identify the main bottlenecks, and describe the acceleration measures implemented to reduce their contribution to the overall runtime.  \n2 A 2D static case  \nWe consider a 2D model of Permanent-Magnet Synchronous Machine (PMSM), shown in Fig. 1a, discretized with 10,000 elements. Both the stator and rotor are modeled using materials characterized by the nonlinear 􀀗-􀀝 curves shown in Fig. 1b. The stator coils are excited with 3-phase sinusoidal currents and a magnetostatic simulation was performed for 90 rotor positions. Figure 2 shows the percentage of total runtime spent in  \nB (T)  \n2  \n1  \n0  \nH (A/m) × 105 (b) Nonlinear 􀀗-􀀝 curve using a spline approximation for the material M330-35A  \nFig. 1 A 2D model and 􀀗-􀀝 curve ","cbCaicC5L5esmKe0","https://ap.wps.com/l/cbCaicC5L5esmKe0","pdf",1245177,5,1,12,"English","en",105,"# Introduction\n## Solver-step bottlenecks in industrial FE workflows\n# A 2D static case\n## Newton with Armijo-only line search","[{\"question\":\"为什么电机有限元仿真会成为产品开发的计算瓶颈？\",\"answer\":\"电机仿真需要反复执行多步求解流程，整体耗时并不由单一操作主导，而是由各求解步骤的重复执行累计形成，从而限制迭代速度与时间到达市场。\"},{\"question\":\"文中将有限元求解的主要非线性步骤分成哪些环节？\",\"answer\":\"文中将常见非线性求解步骤概括为：残差评估、雅可比评估、线搜索（如 Wolfe-Powell 条件）、矩阵分解或预处理，以及线性系统求解。\"},{\"question\":\"2D 静态算例中，哪些步骤是主要耗时瓶颈？\",\"answer\":\"在给定的2D PMSM 场景中，最耗时的步骤是 S2（雅可比评估）和 S3（线搜索子迭代），文中据此提出针对性的加速策略。\"}]",1784177510,30,{"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":87,"head_meta":89,"extra_data":91,"updated_unix":28},"accelerating-industrial-finite-element-simulations-of-electric-machines-based-on-runtime-analysis","",{"@graph":36,"@context":86},[37,54,69],{"@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":53},"https://docshare.wps.com/document/accelerating-industrial-finite-element-simulations-of-electric-machines-based-on-runtime-analysis/82001/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":24,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-07-29","2026-07-16",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72,78,82],{"name":73,"@type":74,"acceptedAnswer":75},"为什么电机有限元仿真会成为产品开发的计算瓶颈？","Question",{"text":76,"@type":77},"电机仿真需要反复执行多步求解流程，整体耗时并不由单一操作主导，而是由各求解步骤的重复执行累计形成，从而限制迭代速度与时间到达市场。","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"文中将有限元求解的主要非线性步骤分成哪些环节？",{"text":81,"@type":77},"文中将常见非线性求解步骤概括为：残差评估、雅可比评估、线搜索（如 Wolfe-Powell 条件）、矩阵分解或预处理，以及线性系统求解。",{"name":83,"@type":74,"acceptedAnswer":84},"2D 静态算例中，哪些步骤是主要耗时瓶颈？",{"text":85,"@type":77},"在给定的2D PMSM 场景中，最耗时的步骤是 S2（雅可比评估）和 S3（线搜索子迭代），文中据此提出针对性的加速策略。","https://schema.org",{"og:url":52,"og:type":88,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":90,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":93},[94,98,102,106,110,115,120,122,127,130,134],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":95,"show_sort_weight":96,"slug":97},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":99,"show_sort_weight":100,"slug":101},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":103,"show_sort_weight":104,"slug":105},"Exam",70,"exam",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},"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":29,"slug":121},"research-report",{"id":123,"doc_module":4,"doc_module_name":46,"category_name":124,"show_sort_weight":125,"slug":126},9,"Religion & Spirituality",20,"religion-spirituality",{"id":125,"doc_module":4,"doc_module_name":46,"category_name":128,"show_sort_weight":125,"slug":129},"World Cup","world-cup",{"id":131,"doc_module":4,"doc_module_name":46,"category_name":132,"show_sort_weight":131,"slug":133},10,"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":46,"category_name":136,"show_sort_weight":20,"slug":137},19,"General","general"]