[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84061-en":3,"doc-seo-84061-105":29,"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":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},84061,1374391975076,"Riley","https://ap-avatar.wpscdn.com/avatar/14000253ca4ec9f6853?x-image-process=image/resize,m_fixed,w_180,h_180&k=1783305029341752051",8,"Research & Report","Efficient Pareto-Front Generation for Electric Machines using IGA and Second Order Derivatives","Multiobjective optimization of electric machines requires managing trade-offs among competing goals such as performance and cost. Design decisions typically rely on comparing variants along a Pareto front produced from many simulations. The work presents an efficient Pareto-front generation approach using a continuation method based on homotopy, leveraging second-order derivative information for superlinear convergence and rapid computation of new Pareto-optimal points in a few iterations. Hessian formulas enable direct motor geometry updates within isogeometric analysis.","Efficient Pareto-Front Generation for Electric Machines using IGA and Second Order Derivatives  \nTheodor Komanna , Michael Wiesheub , Stefan Ulbricha , Sebastian Schöpsb , Peter Ganglc  \na Department of Mathematics, Technical University of Darmstadt, Darmstadt, Dolivostraße 15, 64297, Germany b Computational Electromagnetics Group, Technical University of Darmstadt, Schloßgartenstraße 8, Darmstadt, 64289, Germany c Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenberger Straße 69, Linz, 4040, Austria  \nAbstract  \nThe multiobjective optimization of electric machines always involves a trade-off caused by various competing objectives such as performance and cost. A suitable design is usually determined by comparing variants from the Pareto front, which has been generated by a large number of simulation runs. This paper addresses the efficient generation of the Pareto front using a continuation method based on a homotopy method that exploits second-order derivative information to achieve superlinear convergence, enabling the fast generation of new Pareto-optimal points within only a few iterations. A key contribution is the derivation of formulas to compute the Hessian with respect to geometry parameters and shape, thus enabling direct modifications of the motor geometry in the context of Isogeometric Analysis. We apply our method to nonlinear 2D magnetostatic simulations of a permanent magnet synchronous motor and demonstrate its effectiveness by optimizing the cost, mean torque and torque ripple of the motor. Compared to a first-order optimization method, this approach reduces the number of iterations and function evaluations needed, making the pareto optimization fast and efficient.  \nKeywords: Electric Motor, Isogeometric Analysis, Parameter Optimization, Second Order, Shape Optimization  \n7 Jul 2026  \n[Email address:](Email address: michael.wiesheu@tu-darmstadt.de)[ michael.wiesheu@tu-darmstadt.de](Email address: michael.wiesheu@tu-darmstadt.de) (Michael Wiesheu)  \net al., 2019 ; Schmidt et al., 2016) . Although shape optimization alone typically dominates the computational cost, especially when many degrees of freedom are involved, the simultaneous optimization of shape and parameters introduces a design space of higher dimension. This makes the overall optimization more time-consuming than optimizing either the shape or the parameters alone. Optimizing sequentially, on the other hand, leads to a bilevel problem of the form min min, which requires deciding the order of parameter versus shape optimization. This sequential approach often leads to suboptimal solutions, as the full design space is not effectively explored in a stepwise way (Dempe, 2002 ; Sinha et al., 2018) . By optimizing both simultaneously, we overcome these limitations and achieve better designs. A table of existing design optimization methods applied for electric machines can be found in (Brun, 2024, Table 1 .3)  \nHowever, the challenges go beyond single-objective optimization. Conflicting criteria such as torque maximization versus cost minimization require multi-objective approaches. From an industrial point of view, electric machines are evaluated by several key performance indicators (KPIs) at once, such as average torque, torque ripple, losses and efficiency over multiple operating points, as well as mechanical and thermal performance. Improving one KPI typically worsens another, so finding the best compromise efficiently is challenging (Liu et al., 2025 ; Sun et al., 2021) . In this context, Pareto-optimal solution sets offer a systematic approach to industrial decisionmaking. They identify the possible compromises between competing KPIs, enabling engineers to select designs that best fulfil application-specific and economic requirements. This applies  \nnot only to electrical machines, but to many other engineering applications. Despite this clear relevance, evolutionary algorithms (EAs), such as NSG","cbCaibWAVUsFW9Vt","https://ap.wps.com/l/cbCaibWAVUsFW9Vt","pdf",978418,1,14,"English","en",105,"# Abstract\n# Multiobjective Optimization and Pareto-Optimal Sets\n## Industrial KPIs and Trade-Offs\n## Evolutionary vs. Gradient-Based Methods","[{\"question\":\"What problem does the paper address in electric machine design?\",\"answer\":\"It addresses efficient multiobjective optimization where conflicting goals like cost and performance require selecting designs along a Pareto front.\"},{\"question\":\"How does the proposed method generate the Pareto front more efficiently?\",\"answer\":\"It uses a continuation method based on homotopy that exploits second-order derivative information to achieve superlinear convergence and compute new Pareto-optimal points in only a few iterations.\"},{\"question\":\"What capability does the paper add to isogeometric analysis for motor geometry changes?\",\"answer\":\"It derives formulas to compute the Hessian with respect to geometry parameters and shape, enabling direct modifications of motor geometry in the isogeometric analysis 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problem does the paper address in electric machine design?","Question",{"text":75,"@type":76},"It addresses efficient multiobjective optimization where conflicting goals like cost and performance require selecting designs along a Pareto front.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the proposed method generate the Pareto front more efficiently?",{"text":80,"@type":76},"It uses a continuation method based on homotopy that exploits second-order derivative information to achieve superlinear convergence and compute new Pareto-optimal points in only a few iterations.",{"name":82,"@type":73,"acceptedAnswer":83},"What capability does the paper add to isogeometric analysis for motor geometry changes?",{"text":84,"@type":76},"It derives formulas to compute the Hessian with respect to geometry parameters and shape, enabling direct modifications of motor geometry in the isogeometric analysis 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