[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-120824-en":3,"doc-seo-120824-105":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":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},120824,16904993612988,"Olivia Brown","https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd",8,"Research & Report","Machine learning aided multiscale magnetostatics","Computational material modeling for heterogeneous microstructures often demands heavy computation to achieve high accuracy in multiscale simulations. This research introduces a convolutional neural network (CNN) surrogate for linear magnetostatics at the microscale, mapping periodic and biphasic microstructure images to the apparent permeability of statistical volume elements (SVE). Training and testing data are generated via finite element two-scale asymptotic homogenization. Results in 2D and 3D show high prediction accuracy and substantially reduced computation time compared with the homogenization baseline.","Mechanics of Materials 184 (2023) 104726  \n| Research paper\u003Cbr>Machine learning aided multiscale magnetostatics\u003Cbr>Fadi Aldakheel a,∗, Celal Soyarslanb,c, Hari Subramani Palanisamy c, Elsayed Saber Elsayed aa Institute of Mechanics and Computational Mechanics, Leibniz Universität Hannover, 30167 Hannover, Germany\u003Cbr>b Chair of Nonlinear Solid Mechanics, University of Twente, 7522 NB Enschede, The Netherlands\u003Cbr>c Fraunhofer Innovation Platform, University of Twente, 7522 NB Enschede, The Netherlands |  |  |  |\n| --- | --- | --- | --- |\n| A R T I C L E I N F O |  | A B S T R A C T |  |\n| Keywords:\u003Cbr>Convolutional Neural Network (CNN) Magnetostatics\u003Cbr>Homogenization |  | Computational material modeling using advanced numerical techniques speeds up the design process and reduces the costs of developing new engineering products. In the field of multiscale modeling, huge computation efforts are expected for modeling heterogeneous materials while trying to reach high accuracy levels. In this work, a machine learning approach, namely the convolutional neural network (CNN), is developed as a solution providing a high level of accuracy while being computationally efficient. The input for the CNN model consists of two/three-dimensional images of artificial periodic and biphasic microstructures in the form of nonoverlapping and overlapping, mono- and polydisperse circular/spherical inclusion systems, which are generated by a random sequential inhibition process. These correspond to Statistical Volume Elements (SVE). Considering linear magnetostatics at the microscale, the output is the apparent permeability of the SVE. Training and testing data for the apparent properties is produced with finite element method-based two-scale asymptotic homogenization. The model efficiency is revealed by employing some representative examples in two and three-dimensional settings. In this regard, the performance of the CNN model is assessed with the applied computational homogenization method relating to the accuracy and computational efficiency. The results with the CNN model show high accuracy in predicting the homogenized permeability and a significant decrease in computation time. |  |\n\n1. Introduction  \nEngineering materials often have heterogeneous microstructures composed of constituents at various scales; see, e.g., Bargmann et al.(2018). The properties of these constituents, whether geometrical or physical, play a crucial role in determining the material’s macroscopic properties. Understanding the relationship between the microstructure and macroscopic physical properties is crucial for material design optimization and manufacturing control. In this regard, scientists revert to mathematical modeling approaches such as analytical bounds (Voigt, 1887; Reuss, 1929), effective medium theories, e.g., the Maxwell, selfconsistent, differential effective-medium approximations, e.g., Hashinand Shtrikman (1962, 1963), or hierarchical (Michel et al., 1999; Fish, 2014) and concurrent (Lloberas-Valls et al., 2012; Fish, 2014; Fish and Wagiman, 1993; Aldakheel, 2021; Aldakheel et al., 2020) multiscale computational homogenization techniques. The former analytical methods provide fast but usually inaccurate predictions as their formulation incorporates limited microstructural descriptors. While the latter numerical homogenization methods provide high-fidelity solutions, they require large computational times, making their use impossible in  \n∗ Corresponding author.  \nE-mail address: [fadi.aldakheel@ibnm.uni-hannover.de](fadi.aldakheel@ibnm.uni-hannover.de) (F. Aldakheel). URL: [https://www.ibnm.uni-hannover.de/en/](https://www.ibnm.uni-hannover.de/en/) (F. Aldakheel).  \nreal-time control. This motivates the development of computationally feasible approaches in this realm. A recently emerging third alternative in literature to this end is employing data-driven surrogate models devising machine learning, see, e.g., Zhang and Garikipati (2020), Lu et al. (2021a","cbCaisQAMb1rXhza","https://ap.wps.com/l/cbCaisQAMb1rXhza","pdf",6879312,1,14,"English","en",105,"# Introduction\n## Background and motivation\n## CNN for structure–property relations\n## Contribution and goal","[{\"question\":\"What problem does the paper address in multiscale magnetostatics modeling?\",\"answer\":\"It addresses the high computational cost of numerical multiscale homogenization for heterogeneous materials while still needing high accuracy for macroscopic properties.\"},{\"question\":\"How is the CNN model set up and what does it predict?\",\"answer\":\"The CNN takes 2D/3D images of artificial periodic and biphasic microstructures as input and predicts the apparent permeability of the statistical volume element (SVE) under linear magnetostatics.\"},{\"question\":\"How are the training and testing data for apparent properties generated?\",\"answer\":\"They are produced using a finite element method-based two-scale asymptotic homogenization procedure.\"}]","Machine learning aided multiscale magnetostatics | PDF",1785732201,35,{"code":4,"msg":31,"data":32},"ok",{"site_id":24,"language":23,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"machine-learning-aided-multiscale-magnetostatics","",{"@graph":36,"@context":85},[37,54,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":20},"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/machine-learning-aided-multiscale-magnetostatics/120824/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":23,"description":14,"dateModified":62,"datePublished":62,"encodingFormat":61,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-08-03",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 problem does the paper address in multiscale magnetostatics modeling?","Question",{"text":75,"@type":76},"It addresses the high computational cost of numerical multiscale homogenization for heterogeneous materials while still needing high accuracy for macroscopic properties.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How is the CNN model set up and what does it predict?",{"text":80,"@type":76},"The CNN takes 2D/3D images of artificial periodic and biphasic microstructures as input and predicts the apparent permeability of the statistical volume element (SVE) under linear magnetostatics.",{"name":82,"@type":73,"acceptedAnswer":83},"How are the training and testing data for apparent properties generated?",{"text":84,"@type":76},"They are produced using a finite element method-based two-scale asymptotic homogenization procedure.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"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":53,"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"]