[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-126927-en":3,"doc-seo-126927-105":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":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":27,"seo_description":14,"update_tm":28,"read_time":29},126927,2336474459895,"Aria","https://ap-avatar.wpscdn.com/avatar/22000baeef7a5ed0655?x-image-process=image/resize,m_fixed,w_180,h_180&k=1786071322749376916",8,"Research & Report","Fast Computation of Friction Stir Welding Process - Model Order Reduction and Machine Learning - Student Research Symposium","Friction Stir Welding (FSW) involves many operational parameters, and selecting optimal settings to avoid joint defects requires time-consuming simulation. This presentation proposes a fast computation framework by combining reduced order modeling with machine learning. The FSW process is described using coupled 2D Navier–Stokes and heat equations, then accelerated via POD (Proper Orthogonal Decomposition) and DEIM to handle nonlinearity efficiently. A 2D heat-equation example validates the approach and shows how low-dimensional dynamics can enable reuse of precomputed terms while improving simulation speed.","Fast Computation of Friction Stir Welding Process with Model Order Reduction and Machine Learning  \nJoshua Kay  \nResearch Mentor: Dr. Zilong Song  \n2023 Student Research Symposium  \nFriction Stir Welding (FSW)  \n. Advantages:  \n Joining method is largely defect free  No filler materials  \n Environmentally friendly (no fumes)  Automatable and reproducible  \nFriction Stir Welding  \nThere are many operational parameters in the FSW process.  Rotational Speed  \n Applied Axial Force  Tool Shoulder Radius  \n Rotational and Advancing Speeds  etc...  \nProblem:  \nFinding optimal parameters to avoid product defects in the joints is time consuming. Fast Simulation is Required!  \nFast Simulation Approach  \n FSW process  \n Modeled by coupled system of 2D Navier Stokes and Heat equations [1, 2] .  \n Numerical modeling  \n Extremely time consuming due to the complexity and non-linearity of the equations.  \n Reduced order modeling (ROM)  \n Proper Orthogonal Decomposition (POD) [3] .  \n Discrete Empirical Interpolation Method (DEIM) [4, 5] .  \n Machine learning  \n Predict solutions to FSW process when operating parameters vary, etc. . .  \nThe ROM techniques will now be used on a simple example.  \n2D Heat Equation Example  \nut = D 􀀐 2~~ ~~ux2 + 2~~ ~~uy2 􀀑 + u2 (1)  \nu(0, y , t) = u (a, y , t) = 0 u (x, 0 , t) = u (x, b, t) = 0 u (x, y , 0) = (500ififyy≥\u003C11  \nWhere 0 ≤ a ≤ 2 and 0 ≤ b ≤ 2.  \nDiscretization of equation (1):  \n⃗Uk+1 = A ⃗Uk + ∆t( ⃗Uk )2 (2)  \n⃗Uk+1 , ⃗Uk ∈ RN and A ∈ RNxN  \n.  \nNumerical Results:  \nThe dynamics are modeled up to time t = .25 on a 39x39 grid with 500 time steps.  \nComputationally expensive as each time step requires  \nN-dimensional computations.        \nPOD Implementation:  \nGoal: Construct an optimal, low dimensional, basis matrix Φ r to  \napproximate ⃗Uk as:  \n⃗Uk ≈ Φr⃗ak  \nΦr ∈ RNxr , ⃗ak ∈ Rr , r \u003C\u003C N  \nStep 1) Start by constructing snapshot matrix X ∈ RNxL:  \nX = 􀀲 ⃗U1 ⃗U2 ... ⃗UL􀀳􀀴    􀀵  \nWith L \u003C m. m is the number of time-steps.  \nStep 2) Compute the SVD of X:  \nX = ΦΣVT  \n. Φ ∈ RNxN is an orthogonal matrix that holds the spatial  \ndynamics of the problem.        \nSingular Values:  \n. Step 3) Truncate Φ to its first r columns. This gives Φr ∈ RNxr .  \nPOD Implementation:  \nApplying the POD method to equation (1) . This gives us:  \n⃗Uk+1 = A ⃗Uk + ∆t( ⃗Uk )2  \nΦr⃗ak+1 = AΦr⃗ak + ∆t(Φr⃗ak )2  \n⃗ak+1 = ΦTrAΦr⃗ak + ∆tΦTr(Φr⃗ak )2  \n(rxr)  \n.  \nEach ⃗ak ∈ Rr can be converted back to n-dimensional space by computing:  \nΦr⃗ak ∈ Rn  \nΦTrAΦr can be computed once and re-used. Non-linear term still  \nhigh dimensional. This problem motivates DE IM.       \nDEIM and POD Implementation:  \nApproximate the non-linearity with the DEIM approximation:  \nN ( ⃗Uk ) ≈ Ξp(PT Ξp)−1PT N( ⃗Uk )  \n= Ξp(PT Ξp)−1N (PT ⃗Uk )  \nN ( ⃗Uk ) non-linear function.  \nΞp ∈ RNxp is the optimal low rank basis for the non-linearity. P ∈ RNxp is an interpolation matrix.  \np \u003C\u003C N , p ∼ r  \nNon-linear approximation applied to the POD method:  \nΦTrN(Φr⃗ak ) ≈ ΦTrΞp(PT Ξp)−1N((PT Φr)⃗ak )  \n(rxr) (rxr)","cbCaicOJZ2KR158L","https://ap.wps.com/l/cbCaicOJZ2KR158L","pdf",1067261,1,18,"English","en",105,"# Friction Stir Welding (FSW) Overview\n## Key Operational Parameters\n## Simulation Bottleneck\n# Fast Simulation Approach\n## Reduced Order Modeling (ROM)\n### POD (Proper Orthogonal Decomposition)\n### DEIM (Discrete Empirical Interpolation Method)\n## Machine Learning for Parameter Variations\n# 2D Heat Equation Example\n## Governing Equation\n## Discretization\n## Numerical Results\n# POD Implementation Details\n## Snapshot Matrix and SVD\n## Low-Dimensional Basis Construction\n# DEIM and POD with Nonlinearity Approximation\n## DEIM Nonlinear Approximation\n## Coupled ROM Formulation","[{\"question\":\"Why is fast simulation important for Friction Stir Welding?\",\"answer\":\"Finding optimal operational parameters to prevent defects in welded joints is time-consuming. Fast simulation is required to explore parameter choices efficiently.\"},{\"question\":\"How does the approach reduce computational cost?\",\"answer\":\"It uses reduced order modeling: POD constructs a low-dimensional basis from solution snapshots, and DEIM approximates the nonlinear term to avoid high-dimensional computation at every step.\"},{\"question\":\"What role does machine learning play in the framework?\",\"answer\":\"Machine learning is used to predict FSW solutions when operating parameters vary, complementing the reduced order modeling strategy for faster evaluation.\"}]","Fast Computation of Friction Stir Welding Process - Model Order Reduction and Machine Learning - Student Research Symposium | PDF",1785935698,45,{"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},"fast-computation-of-friction-stir-welding-process-model-order-reduction-and-machine-learning-student-research-symposium","",{"@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/fast-computation-of-friction-stir-welding-process-model-order-reduction-and-machine-learning-student-research-symposium/126927/",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","2026-08-05",true,{"@type":65,"interactionType":66,"userInteractionCount":4},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"Why is fast simulation important for Friction Stir Welding?","Question",{"text":75,"@type":76},"Finding optimal operational parameters to prevent defects in welded joints is time-consuming. Fast simulation is required to explore parameter choices efficiently.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the approach reduce computational cost?",{"text":80,"@type":76},"It uses reduced order modeling: POD constructs a low-dimensional basis from solution snapshots, and DEIM approximates the nonlinear term to avoid high-dimensional computation at every step.",{"name":82,"@type":73,"acceptedAnswer":83},"What role does machine learning play in the framework?",{"text":84,"@type":76},"Machine learning is used to predict FSW solutions when operating parameters vary, complementing the reduced order modeling strategy for faster evaluation.","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"]