一种改进的四面体网格质量优化算法
|
摘要
网格质量直接影响计算的精度和计算的效率,为了更好地提高离心泵边界网格质量,针对现有网格质量优化算法中网格质量衡量准则、目标函数以及求解算法不合理等问题,提出了一种基于优化光顺的网格质量优化算法.该算法在提出一种新的四面体质量衡量准则并验证其合理性的基础上,建立了一个新的基于优化光顺的目标函数,并在求解目标函数最小值过程中,采用点态松弛技术将求解全局最小值问题转化为求解一系列局部最小值问题,应用变尺度算法求解非线性函数的局部最小值,以提高算法效率.数值试验结果表明,优化前网格中单元质量最小值为0.005 2,平均值为0.510 5,且在叶轮的进口边存在大量的劣质单元;采用文中算法优化后,网格中单元质量最小值为0.058 4,平均值为0.585 0,网格中的最差单元质量得到了显著提高,该算法能够较好地提高边界处和整体网格质量.
Mesh quality influences numerical accuracy and computational efficiency directly in CFD.In order to improve the quality of mesh on solid boundaries in a centrifugal pump greatly,an optimization-based smoothing algorithm is proposed. In this algorithm,a new mesh quality metric is employed,and an error function is derived by transforming the metric. The total error function of the mesh is the sum of the error in every element,and adopted as the objective function of the optimization. Furthermore,in the objective function,the qualities of the mesh on and off the boundaries are involved,a weight coefficient is also included to control the quality of the mesh on the boundaries. The Broyden Fletcher Goldfarb Shanno algorithm is applied to transform the global minimization to a series of local minimization to enhance the computational efficiency. A practical example shows that the proposed algorithm can improve the mesh quality considerably; moreover not only the quality of the mesh on theboundaries but also the overall quality of all the mesh can be upgraded with the increasing weight coefficient as well.
Mesh quality influences numerical accuracy and computational efficiency directly in CFD.In order to improve the quality of mesh on solid boundaries in a centrifugal pump greatly,an optimization-based smoothing algorithm is proposed. In this algorithm,a new mesh quality metric is employed,and an error function is derived by transforming the metric. The total error function of the mesh is the sum of the error in every element,and adopted as the objective function of the optimization. Furthermore,in the objective function,the qualities of the mesh on and off the boundaries are involved,a weight coefficient is also included to control the quality of the mesh on the boundaries. The Broyden Fletcher Goldfarb Shanno algorithm is applied to transform the global minimization to a series of local minimization to enhance the computational efficiency. A practical example shows that the proposed algorithm can improve the mesh quality considerably; moreover not only the quality of the mesh on theboundaries but also the overall quality of all the mesh can be upgraded with the increasing weight coefficient as well.
引文
[1]Montenegro R,Cascon J M.An automatic strategy for adaptive tetrahedral mesh generation[J].Applied Numerical Mathematics,2009,59:2203-2217.
[2]Dobrzynski C,Frey P.Anisotropic delaunay mesh adaptation for unsteady simulations[C]//Proceedings of the17th International Meshing Roundtable.Pittsburgh,USA:[s.n.],2008:177-194.
[3]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 Houlin,Tan Minggao,et al.Quality measurement criteria and optimization algorithm of tetrahedral mesh for centrifugal pumps[J].Journal of Xi'an Jiaotong University,2011,45(11):31-36.(in Chinese)
[4]Xu K,Cheng Z Q.Quality encoding for tetrahedral mesh optimization[J].Computers&Graphics,2009,33(3):250-261.
[5]Tournois J,Wormser C.Interleaving delaunay refinement and optimization for practical isotropic tetrahedron mesh generation[J].ACM Transactions on Graphics,2010,28(3):1557-1571.
[6]陈涛,李光耀.STL格式文件的四边形网格剖分与网格光顺[J].中国机械工程,2009,20(5):522-528.Chen Tao,Li Guangyao.Quadrilateral mesh generation based on STL file format model and mesh smoothing[J].China Mechanical Engineering,2009,20(5):522-528.(in Chinese)
[7]Mao Zhihong,Ma Lizhuan.A modified Laplacian smoothing approach with mesh saliency[J].Lecture Notes in Computer Science,2006,47:105-113.
[8]Vartziotis D,Wipper J.The geometric element transformation method for tetrahedral mesh smoothing[J].Computer Methods in Applied Mechanics and Engineering,2010,199(5):169-182.
[9]Hetmaniuk U,Knupp P.A mesh optimization algorithm to decrease the maximum interpolation error of linear triangular finite elements[J].Engineering with Computers,2010,27:3-15.
[10]Freitag L A.On combining Laplacian an optimizationbased mesh smoothing techniques[J].AMD Trends in Unstructured Mesh Generation,1997,220(11):37-43.
[11]Parthasarathy V,Kodiyalam S.A constrained optimization approach to finite element mesh smoothing[J].Finite Elements in Analysis and Design,1991,9(3):309-320.
[12]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 Chunge,Liu Jianfei,Sun Shuli.Study on quality measures for tetrahedral mesh[J].Chinese Journal of Computational Mechanics,2003,20(5):579-582.(in Chinese)
[13]董亮.离心泵非结构化四面体网格生成及应用研究[D].镇江:江苏大学流体机械工程技术研究中心,2012.
[14]Parthasarathy V N,Graichen C,Hathaway A.A comparison of tetrahedron quality measures[J].Finite Elements in Analysis and Design,1994,15(3):255-261.
[15]Shewchuk J R.What is a good linear element interpolation,conditioning,and quality measures[C]//Proceedings of the 11th International Meshing Roundtable,2002.
[16]Liu Dongxi,Jorge Nocedal.On the limited memory BFGS method for large scale optimization[J].Mathematical Programming,1989,45(1/2/3):503-528.
[17]Freitag L A,Carl O G.Tetrahedral mesh improvement using swapping and smoothing[J].International Journal for Numerical Methods in Engineering,1997,40(21):3979-4002.
[2]Dobrzynski C,Frey P.Anisotropic delaunay mesh adaptation for unsteady simulations[C]//Proceedings of the17th International Meshing Roundtable.Pittsburgh,USA:[s.n.],2008:177-194.
[3]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 Houlin,Tan Minggao,et al.Quality measurement criteria and optimization algorithm of tetrahedral mesh for centrifugal pumps[J].Journal of Xi'an Jiaotong University,2011,45(11):31-36.(in Chinese)
[4]Xu K,Cheng Z Q.Quality encoding for tetrahedral mesh optimization[J].Computers&Graphics,2009,33(3):250-261.
[5]Tournois J,Wormser C.Interleaving delaunay refinement and optimization for practical isotropic tetrahedron mesh generation[J].ACM Transactions on Graphics,2010,28(3):1557-1571.
[6]陈涛,李光耀.STL格式文件的四边形网格剖分与网格光顺[J].中国机械工程,2009,20(5):522-528.Chen Tao,Li Guangyao.Quadrilateral mesh generation based on STL file format model and mesh smoothing[J].China Mechanical Engineering,2009,20(5):522-528.(in Chinese)
[7]Mao Zhihong,Ma Lizhuan.A modified Laplacian smoothing approach with mesh saliency[J].Lecture Notes in Computer Science,2006,47:105-113.
[8]Vartziotis D,Wipper J.The geometric element transformation method for tetrahedral mesh smoothing[J].Computer Methods in Applied Mechanics and Engineering,2010,199(5):169-182.
[9]Hetmaniuk U,Knupp P.A mesh optimization algorithm to decrease the maximum interpolation error of linear triangular finite elements[J].Engineering with Computers,2010,27:3-15.
[10]Freitag L A.On combining Laplacian an optimizationbased mesh smoothing techniques[J].AMD Trends in Unstructured Mesh Generation,1997,220(11):37-43.
[11]Parthasarathy V,Kodiyalam S.A constrained optimization approach to finite element mesh smoothing[J].Finite Elements in Analysis and Design,1991,9(3):309-320.
[12]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 Chunge,Liu Jianfei,Sun Shuli.Study on quality measures for tetrahedral mesh[J].Chinese Journal of Computational Mechanics,2003,20(5):579-582.(in Chinese)
[13]董亮.离心泵非结构化四面体网格生成及应用研究[D].镇江:江苏大学流体机械工程技术研究中心,2012.
[14]Parthasarathy V N,Graichen C,Hathaway A.A comparison of tetrahedron quality measures[J].Finite Elements in Analysis and Design,1994,15(3):255-261.
[15]Shewchuk J R.What is a good linear element interpolation,conditioning,and quality measures[C]//Proceedings of the 11th International Meshing Roundtable,2002.
[16]Liu Dongxi,Jorge Nocedal.On the limited memory BFGS method for large scale optimization[J].Mathematical Programming,1989,45(1/2/3):503-528.
[17]Freitag L A,Carl O G.Tetrahedral mesh improvement using swapping and smoothing[J].International Journal for Numerical Methods in Engineering,1997,40(21):3979-4002.