[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84682-en":3,"doc-seo-84682-105":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},84682,4398048949847,"Eliana","https://ap-avatar.wpscdn.com/avatar/400002536579ef2da7f?_k=1778318612642679267",8,"Research & Report","Accelerating droplet-laden Stokes flow simulations with hierarchical surrogate modeling","A multi-fidelity surrogate modeling strategy addresses Stokes flows with suspended liquid droplets in a carrier fluid. At the lowest fidelity, droplets act as passive tracers, and a PDE is derived for the resulting modeling error. The error equation is solved approximately to correct the flow field, iterating this correction. Two alternating fidelities use (i) droplet-free Stokes flow and (ii) free-space flow around a single droplet, capturing droplet-flow, boundary, and droplet-droplet effects. For geometrically similar droplets, an offline-online reuse of precomputed single-droplet solutions reduces computational cost while maintaining accuracy for cases up to 10^4 droplets.","arXiv :2607 .03301v1 [math .NA] 3 Jul 2026  \nAccelerating droplet-laden Stokes flow simulations with hierarchical  \nsurrogate modeling  \nDavide Pradovera∗ Thomas Frachon† Sara Zahedi†  \nJuly 7, 2026  \nAbstract  \nWe present a surrogate modeling strategy for Stokes flows with liquid droplets suspended in a carrier fluid. Our approach is based on a multi-fidelity framework. At the lowest fidelity, droplets are treated as passive tracers, neglecting their influence on the ambient flow field. Building on this approximation, we derive a PDE that represents the current modeling error. This error equation is then solved approximately to correct the flow field and the procedure is iterated. Two fidelities are employed in an alternating fashion: Stokes flow in the absence of droplets and flow around a single droplet in free space. By systematically combining these models, the method captures droplet-flow, droplet-boundary, and droplet-droplet interactions. For geometrically similar droplets, we further develop an efficient offline-online strategy that exploits this structure by reusing precomputed single-droplet solutions. Numerical experiments demonstrate the accuracy and efficiency of the proposed surrogate in a variety of tests, including scenarios with up to 104 droplets. Notably, we show that the proposed surrogate achieves substantially reduced computational cost compared to fully resolved multi-fluid simulations with state-of-the-art software.  \n1 Introduction  \nMultiphase flow systems arise in a wide range of applications, including microfluidics, porous media, ink-jet printing and biomedical applications. Accurate predictions for such systems increasingly rely on high-fidelity computer simulations [3,9,27,30] . In this work, we study incompressible multiphase Stokes flow involving a large number of immiscible droplets suspended in a carrier fluid.  \nFrom a computational point of view, a key difficulty arises when droplets are many and become highly concentrated or approach boundaries. Standard discretization techniques can only capture the dynamics in such regimes by resolving the narrow interfacial gaps and near-boundary interactions, making the resulting numerical schemes computationally demanding.  \nFor example, standard finite-element methods rely on fine meshes that conform to the interfaces and therefore require repeated remeshing as the interfaces evolve. Unfitted methods such as CutFEM avoid remeshing but still require fine local resolution to accurately capture narrow gaps [2,12] . Boundary integral methods face similar challenges, requiring specialized quadrature schemes and fine resolution around interfaces to accurately evaluate near-singular integrals arising from closely interacting droplets and boundaries [20] .  \nAlternative particle-based methods, such as smoothed particle hydrodynamics [31] and the moving particle semi-implicit method [5,28], eliminate the need for mesh generation and remeshing. However, accurately resolving near-contact interactions still requires sufficiently fine particle resolutions. Moreover, these methods typically achieve lower accuracy than high-order finite-element or boundary-integral methods and often require careful tuning of numerical parameters (such as kernel functions and stabilization terms) to ensure numerical stability.  \nAn additional challenge arises as the interfaces evolve in time, since the flow problem must be approximated and solved on a sequence of changing geometries. In scenarios with many droplets, the fine spatial resolution required to capture near-contact interactions usually leads to expensive simulations. In this work,  \n∗ Stockholm University, Stockholm, Sweden ([davide.pradovera@math.su.se](davide.pradovera@math.su.se)).  \n†KTH Royal Institute of Technology, Stockholm, Sweden ([frachon@kth.se](frachon@kth.se), [sara.zahedi@math.kth.se](sara.zahedi@math.kth.se)) .  \nFigure 1: Example: domain of the Stokes interface problem.  \nwe present a surrogate model tha","cbCaincInkBAvOYR","https://ap.wps.com/l/cbCaincInkBAvOYR","pdf",2470403,3,1,27,"English","en",105,"# Introduction\n## Target problem","[{\"question\":\"What is the core idea of the surrogate modeling strategy for droplet-laden Stokes flow?\",\"answer\":\"The method uses a multi-fidelity framework where a low-fidelity tracer approximation is used to derive a PDE for the modeling error. The error is then solved approximately to correct the flow field and the process is iterated.\"},{\"question\":\"Which two fidelity models are alternated in the approach?\",\"answer\":\"The approach alternates between Stokes flow without droplets (droplet-free fidelity) and Stokes flow around a single droplet in free space (single-droplet fidelity). Combined, these capture droplet-flow, droplet-boundary, and droplet-droplet interactions.\"},{\"question\":\"How does the offline-online strategy help for geometrically similar droplets?\",\"answer\":\"It reuses precomputed single-droplet solutions by exploiting structural similarity, enabling efficient prediction without repeating expensive computations for each configuration.\"}]",1784197645,68,{"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},"accelerating-droplet-laden-stokes-flow-simulations-with-hierarchical-surrogate-modeling","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,50],{"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":20},"https://docshare.wps.com/document/research-report/",{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/accelerating-droplet-laden-stokes-flow-simulations-with-hierarchical-surrogate-modeling/84682/",4,{"url":51,"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","2026-07-23","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 is the core idea of the surrogate modeling strategy for droplet-laden Stokes flow?","Question",{"text":75,"@type":76},"The method uses a multi-fidelity framework where a low-fidelity tracer approximation is used to derive a PDE for the modeling error. The error is then solved approximately to correct the flow field and the process is iterated.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Which two fidelity models are alternated in the approach?",{"text":80,"@type":76},"The approach alternates between Stokes flow without droplets (droplet-free fidelity) and Stokes flow around a single droplet in free space (single-droplet fidelity). Combined, these capture droplet-flow, droplet-boundary, and droplet-droplet interactions.",{"name":82,"@type":73,"acceptedAnswer":83},"How does the offline-online strategy help for geometrically similar droplets?",{"text":84,"@type":76},"It reuses precomputed single-droplet solutions by exploiting structural similarity, enabling efficient prediction without repeating expensive computations for each configuration.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"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":52,"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"]