一种验证网格质量与CFD计算精度关系的方法
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摘要
针对网格质量衡量准则与计算精度之间关系验证过程中的难题,提出一种准确验证它们之间关系的方法。该方法首先生成一系列不同单元数的网格,接着采用三维线性插值算法,获得相邻两套网格单元速度相对误差,当该误差小于某一阈值时,则认为此网格计算得到的数值解为网格独立解。此外,在网格独立解的基础之上,获得不完善网格中各单元的计算误差,并与各单元的度量值建立相应的数据库,采用不同的统计方法对其进行对比分析,获得它们之间的关系。研究结果表明:同质量衡量准则QEVS和QEAS相比,本文提出的质量衡量准则QNEW1和QNEW2在误差较大的区间内(E re>0.15),与计算精度有较好的关系。因此,在网格生成过程中,采用QNEW1和QNEW2能够正确地评判出对计算精度影响较大的单元,而QEVS和QEAS则不能。
In order to investigate the relationship between the mesh quality measures and CFD numerical accuracy,a new confirming method was proposed.Firstly,a series of meshes with different numbers of elements were generated.Then,based on the three-dimensional linear interpolation algorithm,the relative velocity errors between adjacent meshes were obtained.When the error was less than the threshold value,the solution was considered as the grid-independent solution.Meanwhile,the error and the quality of each element in imperfect meshes formed a database.The database was analyzed and the relationship between the mesh quality measures and numerical accuracy was acquired.The results show that compared with the metrics QEVS and QEAS,the metrics QNEW1 and QNEW2 have better relationship in the high-error interval(error E re>0.15).Therefore,the metric QNEW1 and QNEW2 can correctly evaluate the elements which have greater effects on the numerical accuracy,but the metric QNEW1 and QNEW2 can not.
In order to investigate the relationship between the mesh quality measures and CFD numerical accuracy,a new confirming method was proposed.Firstly,a series of meshes with different numbers of elements were generated.Then,based on the three-dimensional linear interpolation algorithm,the relative velocity errors between adjacent meshes were obtained.When the error was less than the threshold value,the solution was considered as the grid-independent solution.Meanwhile,the error and the quality of each element in imperfect meshes formed a database.The database was analyzed and the relationship between the mesh quality measures and numerical accuracy was acquired.The results show that compared with the metrics QEVS and QEAS,the metrics QNEW1 and QNEW2 have better relationship in the high-error interval(error E re>0.15).Therefore,the metric QNEW1 and QNEW2 can correctly evaluate the elements which have greater effects on the numerical accuracy,but the metric QNEW1 and QNEW2 can not.
引文
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[14]聂春戈,刘剑飞,孙树立.四面体网格质量度量准则的研究[J].计算力学学报,2003,20(5):579-582.NIE Chun-ge,LIU Jian-fei,SUN Shu-li.Study on qualitymeasures for tetrahedral mesh[J].Chinese Journal ofComputational Mechanics,2003,20(5):579-582.
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[2]黄晓东,杜群贵.二维有限元网格的局部加密方法[J].华南理工大学学报:自然科学版,2004,32(12):56-60.HUANG Xiao-dong,DU Qun-gui.Local refinement of 2D finiteelement meshes[J].Journal of South China University ofTechnology:Natural Science Edition,2004,32(12):56-60.
[3]Park J,Shontz S M.Two derivative-free optimization algorithmsfor mesh quality improvement[J].International Conference onComputational Science,2010,1(1):387-396.
[4]Hetmaniuk U,Knupp P.A mesh optimization algorithm todecrease the maximum interpolation error of linear triangularfinite elements[J].Engineering with Computers,2010,27(1):3-15.
[5]Knupp P M.Algebraic mesh quality metric[J].SIAM Journal onScientific Computing,2001,23(1):193-218.
[6]Babuska I,Flaherty J E,Henshaw W D,et al.Modeling,meshgeneration,and adaptive numerical methods for partialdifferential equations[M].New York:Springer,1995:97-127.
[7]Gosselin S,Ollivier-Gooch C.Tetrahedral mesh generation usingDelaunay refinement with non-standard quality measures[J].International Journal for Numerical Methods in Engineering,2011,87(8):795-820.
[8]Dompierre J,Vallet M G,LabbéP,et al.An analysis of simplexshape measures for anisotropic meshes[J].Computer Methods inApplied Mechanics and Engineering,2005,194(48/49):4895-4914.
[9]Freitag L A,Knupp P M.Tetrahedral element shape optimizationvia the Jacobian determinant and condition number[C]//Proceedings of the 8th International Meshing.California,1999:247-258.
[10]董亮,刘厚林,谈明高,等.离心泵四面体网格质量衡量准则及优化算法[J].西安交通大学学报,2011,45(11):31-36.DONG Liang,LIU Hou-lin,TAN Ming-gao,et al.Qualitymeasurement criteria and optimization algorithm of tetrahedralmesh for centrifugal pumps[J].Journal of Xi’an Jiao TongUniversity,2011,45(11):31-36.
[11]Parthasarathy V N,Graichen C M,Hathaway A F.A comparisonof tetrahedron quality measures[J].Finite Elements in Analysisand Design,1993,15(3):255-261.
[12]Liu A,Joe B.Relationship between tetrahedron shapemeasures[J].BIT Numerical Mathematics,1994,34(2):268-287.
[13]Lo S H.Optimization of tetrahedral meshes based on elementshape measures[J].Computers&Structures,1997,63(5):951-961.
[14]聂春戈,刘剑飞,孙树立.四面体网格质量度量准则的研究[J].计算力学学报,2003,20(5):579-582.NIE Chun-ge,LIU Jian-fei,SUN Shu-li.Study on qualitymeasures for tetrahedral mesh[J].Chinese Journal ofComputational Mechanics,2003,20(5):579-582.
[15]Fluent Inc,Gambit User’s Guide[EB/OL].[2003-12-30].http://combust.hit.edu.cn:8080/fluent/Gambit13_heop/users_guide.
[16]Taylor A,Whitelaw J H.Curved ducts with strong secondarymotion:Velocity measurements of developing laminar andturbulent flow[J].Journal of Fluids Engineering,1982,104:350-359.