应用等几何配点法求解电磁涡流场问题
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摘要
为简化等几何分析中多种场约束的添加问题,提出一种等几何配点法,并将其应用于电磁涡流场问题.针对磁矢位散度为零的条件、未来形状优化需求及电磁力后处理需求,在求解过程中取4个措施:(1)将高斯积分点选为配点;(2)应用BEM-FEM耦合方法的变形形式来处理电磁场外边界;(3)将多出的配点方程看做约束;(4)应用QR分解法求解所形成的过约束矩阵方程.数值算例结果表明,该方法适用于多种约束下的偏微分方程求解问题.
In order to simplify the process of imposing multiple constraints in Isogeometric analysis(IGA),an IGA collocation method for solving PDEs is established and applied on the eddy current problems. Considering the divergence-free property of the magnetic vector potential, the shape optimization in the future research and the electromagnetic force calculation in the post process, four technologies are used in the analysis process:(1) selecting the Gaussian points as the collocation points,(2) utilizing an equivalent form of the BEM-FEM coupling method to deal with the magnetic boundary conditions,(3) seeing the additional collocation equations as restrictions, and(4) applying the QR-decomposition method to solve the over-constrained matrix equations. Numerical results show that the presented IGA collocation method is available to solve the PDEs under over-constraints.
In order to simplify the process of imposing multiple constraints in Isogeometric analysis(IGA),an IGA collocation method for solving PDEs is established and applied on the eddy current problems. Considering the divergence-free property of the magnetic vector potential, the shape optimization in the future research and the electromagnetic force calculation in the post process, four technologies are used in the analysis process:(1) selecting the Gaussian points as the collocation points,(2) utilizing an equivalent form of the BEM-FEM coupling method to deal with the magnetic boundary conditions,(3) seeing the additional collocation equations as restrictions, and(4) applying the QR-decomposition method to solve the over-constrained matrix equations. Numerical results show that the presented IGA collocation method is available to solve the PDEs under over-constraints.
引文
[1]Austin Cottrell J, Hughes T J R, Bazilevs Y. Isogeometric analysis:towardintegrationofCADandFEA[M].NewYork:John Wiley and Sons, Ltd, 2009
[2]Bazilevs Y, Beir?o da Veiga L, Cottrell J A, et al. Isogeometric analysis:approximation,stabilityanderrorestimatesfor h-refined meshes[J]. Mathematical Models and Methods in Applied Sciences, 2006, 16(7):1031-1090
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[14]Schillinger D, Evans J A, Reali A, et al. Isogeometric collocation:cost comparison with Galerkin methods and extension to adaptivehierarchicalNURBSdiscretizations[J].Computer MethodsinAppliedMechanicsandEngineering,2013,267:170-232
[15]de Lorenzis L, Evans J A, Hughes T J R, et al. Isogeometric collocation:Neumannboundaryconditionsandcontact[J].ComputerMethodsinAppliedMechanicsandEngineering,2015, 284:21-54
[16]ZhangHH,MoR,WanN.AnIGAdiscontinuousGalerkin methodontheunionofoverlappedpatches[J].Computer MethodsinAppliedMechanicsandEngineering,2017,326:446-480
[17]Xu Gang, Li Xin, Huang Zhangjin, et al. Geometric computing forisogeometricanalysis[J].JournalofComputerAidedDesign&Computer Graphics, 2015, 27(4):570-581(in Chinese)(徐岗,李新,黄章进,等.面向等几何分析的几何计算[J].计算机辅助设计与图形学报, 2015, 27(4):570-581)
[18]Anitescu C, Jia Y, Zhang Y J, et al. An isogeometric collocation method using superconvergent points[J]. Computer Methods in Applied Mechanics and Engineering, 2015, 284:1073-1097
[19]de Boor C, Swartz B. Collocation at Gaussian points[J]. SIAM Journal on Numerical Analysis, 1973, 10(4):582-606
[20]Zhang X, Liu X H, Song K Z, et al. Least-squares collocation meshlessmethod[J].InternationalJournalforNumerical Methods in Engineering, 2001, 51(9):1089-1100
[21]Kurz S, Fetzer J, Lehner G, et al. A novel formulation for 3D eddy current problems with moving bodies using a Lagrangian description and BEM-FEM coupling[J]. IEEE Transactions on Magnetics, 1998, 34(5):3068-3073
[22]Liu Shoubao, Ruan Jiangjun, Zhang Yu, et al. Application of finite element and boundary element coupling method in movingconductoreddycurrentproblem[J].Proceedingsofthe CSEE, 2010, 30(9):123-128(in Chinese)(刘守豹,阮江军,张宇,等.有限元-边界元耦合法在运动导体涡流场中的应用[J].中国电机工程学报,2010,30(9):123-128)
[23]LinQinghua,LiBaoming.Finite-elementboundary-element coupling method for 3D transient eddy current field calculation in electromagnetic railgun[J]. Journal of Nanjing University of ScienceandTechnology:NaturalScience,2010,34(2):217-221(in Chinese)(林庆华,栗保明.有限元/边界元耦合法计算电磁轨道炮三维瞬态涡流场[J].南京理工大学学报:自然科学版,2010,34(2):217-221)
[24]Leonard P J, Lai H C, Hainswort G, et al.Analysis of the performanceoftubularpulsedcoilinductionlaunchers[J].IEEE Transactions on Magnetics, 1993, 29(1):686-690
[2]Bazilevs Y, Beir?o da Veiga L, Cottrell J A, et al. Isogeometric analysis:approximation,stabilityanderrorestimatesfor h-refined meshes[J]. Mathematical Models and Methods in Applied Sciences, 2006, 16(7):1031-1090
[3]Bazilevs Y, Calo V M, Cottrell J A, et al. Isogeometric analysis usingT-splines[J].ComputerMethodsinAppliedMechanics and Engineering, 2010, 199(5-8):229-263
[4]AtroshchenkoE,TomarS,XuG,etal.Weakeningthetight couplingbetweengeometryandsimulationinisogeometric analysis:fromsub-andsuper-geometricanalysistogeometry-independentfieldapproximation(GIFT)[J].International JournalforNumericalMethodsinEngineering,2018,114(10):1131-1159
[5]Beir?o da Veiga L, Buffa A, Rivas J, et al. Some estimates for h-p-k-refinementinisogeometricanalysis[J].Numerische Mathematik, 2011, 118(2):271-305
[6]Cottrell J A, Hughes T J R, Reali A. Studies of refinement and continuityinisogeometricstructuralanalysis[J].Computer MethodsinAppliedMechanicsandEngineering,2007,196(41-44):4160-4183
[7]Hughes T J R, Cottrell J A, Bazilevs Y. Isogeometric analysis:CAD, finite elements, NURBS, exact geometry andmesh refinement[J]. Computer Methods in Applied Mechanics and Engineering, 2005, 194(39-41):4135-4195
[8]Vazquez R, Buffa A. Isogeometric analysis for electromagnetic problems[J].IEEETransactionsonMagnetics,2010,46(8):3305-3308
[9]Nguyen D M, Evgrafov A, Gravesen J. Isogeometric shape optimization for electromagnetic scattering problems[J]. Progress in Electromagnetics Research B, 2012, 45:117-146
[10]ZhangYong,LinGao,LiuJun,etal.Isogeometricanalysis methodforelectrostaticfieldproblems[J].JournalofDalian University of Technology, 2012, 52(6):870-877(in Chinese)(张勇,林皋,刘俊,等.静电场问题的等几何分析方法[J].大连理工大学学报, 2012, 52(6):870-877)
[11]Zhang Yong, Lin Gao, Liu Jun, et al. Isogeometric analysis and itsapplicationtoeccentricallyparallelcylinderselectrostatic problem[J].ChineseJournalofRadioScience,2012,27(1):177-183(in Chinese)(张勇,林皋,刘俊,等.等几何分析法应用于偏心柱面静电场问题[J].电波科学学报, 2012, 27(1):177-183)
[12]AuricchioF,Beir?odaVeigaL.Isogeometriccollocationfor elastostaticsandexplicitdynamics[J].ComputerMethodsin Applied Mechanics and Engineering, 2012, 249-252:2-14
[13]AuricchioF,Beir?odaVeigaL,HughesTJR,etal.Isogeometriccollocationmethods[J].MathematicalModelsand Methods in Applied Sciences, 2010, 20(11):2075-2107
[14]Schillinger D, Evans J A, Reali A, et al. Isogeometric collocation:cost comparison with Galerkin methods and extension to adaptivehierarchicalNURBSdiscretizations[J].Computer MethodsinAppliedMechanicsandEngineering,2013,267:170-232
[15]de Lorenzis L, Evans J A, Hughes T J R, et al. Isogeometric collocation:Neumannboundaryconditionsandcontact[J].ComputerMethodsinAppliedMechanicsandEngineering,2015, 284:21-54
[16]ZhangHH,MoR,WanN.AnIGAdiscontinuousGalerkin methodontheunionofoverlappedpatches[J].Computer MethodsinAppliedMechanicsandEngineering,2017,326:446-480
[17]Xu Gang, Li Xin, Huang Zhangjin, et al. Geometric computing forisogeometricanalysis[J].JournalofComputerAidedDesign&Computer Graphics, 2015, 27(4):570-581(in Chinese)(徐岗,李新,黄章进,等.面向等几何分析的几何计算[J].计算机辅助设计与图形学报, 2015, 27(4):570-581)
[18]Anitescu C, Jia Y, Zhang Y J, et al. An isogeometric collocation method using superconvergent points[J]. Computer Methods in Applied Mechanics and Engineering, 2015, 284:1073-1097
[19]de Boor C, Swartz B. Collocation at Gaussian points[J]. SIAM Journal on Numerical Analysis, 1973, 10(4):582-606
[20]Zhang X, Liu X H, Song K Z, et al. Least-squares collocation meshlessmethod[J].InternationalJournalforNumerical Methods in Engineering, 2001, 51(9):1089-1100
[21]Kurz S, Fetzer J, Lehner G, et al. A novel formulation for 3D eddy current problems with moving bodies using a Lagrangian description and BEM-FEM coupling[J]. IEEE Transactions on Magnetics, 1998, 34(5):3068-3073
[22]Liu Shoubao, Ruan Jiangjun, Zhang Yu, et al. Application of finite element and boundary element coupling method in movingconductoreddycurrentproblem[J].Proceedingsofthe CSEE, 2010, 30(9):123-128(in Chinese)(刘守豹,阮江军,张宇,等.有限元-边界元耦合法在运动导体涡流场中的应用[J].中国电机工程学报,2010,30(9):123-128)
[23]LinQinghua,LiBaoming.Finite-elementboundary-element coupling method for 3D transient eddy current field calculation in electromagnetic railgun[J]. Journal of Nanjing University of ScienceandTechnology:NaturalScience,2010,34(2):217-221(in Chinese)(林庆华,栗保明.有限元/边界元耦合法计算电磁轨道炮三维瞬态涡流场[J].南京理工大学学报:自然科学版,2010,34(2):217-221)
[24]Leonard P J, Lai H C, Hainswort G, et al.Analysis of the performanceoftubularpulsedcoilinductionlaunchers[J].IEEE Transactions on Magnetics, 1993, 29(1):686-690