积分精度对三角形求积元收敛性的影响
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摘要
该文通过提高积分精度改进了非均匀格点三角形求积元的收敛性。以球面映射点为初值,以能够精确积分的多项式最高阶数为目标,采用数值优化方法,得到了一系列新的积分点位置和积分权系数。这样的积分法则所能够精确积分的多项式最高阶数为插值多项式阶数的1.6倍左右。初步的数值实验表明,对于平面位势问题和平面弹性问题,改进的单元具有更好的收敛性。此外还讨论了进一步改进单元性质的方法以及这些方法可能存在的上限。
An improvement of the convergence of triangular quadrature elements is achieved via the increase of quadrature precision. Taking a planar triangular grid points mapped from equi-areal-point-set on a spherical triangle as initial values and the highest order of polynomials that can be accurately integrated as the objective, a new set of quadrature points and weights in a triangle is obtained using numerical optimization methods. The new quadrature rule raises the quadrature precision from the order of the interpolation by approximately 1.6 times. Preliminary numerical examples show that the new triangular quadrature element exhibits a higher convergence rate. Further improvements and possible limitations of this strategy are also discussed.
An improvement of the convergence of triangular quadrature elements is achieved via the increase of quadrature precision. Taking a planar triangular grid points mapped from equi-areal-point-set on a spherical triangle as initial values and the highest order of polynomials that can be accurately integrated as the objective, a new set of quadrature points and weights in a triangle is obtained using numerical optimization methods. The new quadrature rule raises the quadrature precision from the order of the interpolation by approximately 1.6 times. Preliminary numerical examples show that the new triangular quadrature element exhibits a higher convergence rate. Further improvements and possible limitations of this strategy are also discussed.
引文
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[5]Zhang Run,Zhong Hongzhi.Weak form quadrature element analysis of geometrically exact shells[J].International Journal of Non-Linear Mechanics,2015,71:63―71.
[6]Zhang Run,Zhong Hongzhi.A quadrature element formulation of an energy-momentum conserving algorithm for dynamic analysis of geometrically exact beams[J].Computers&Structures,2016,165:96―106.
[7]Zhong H.Triangular differential quadrature[J].Communications in Numerical Methods in Engineering,2000,16(6):401―408.
[8]Zhong H,Hua Y,He Y.Localized triangular differential quadrature[J].Numerical Methods for Partial Differential Equations,2003,19(5):682―692.
[9]徐嘉,钟宏志.一种三角形求积元[EB].中国科技论文在线,http://www.paper.edu.cn/releasepaper/content/201202-150,2012-02-08.Xu Jia,Zhong Hongzhi.A triangular quadrature element[EB].Sciencepaper Online,http://www.paper.edu.cn/releasepaper/content/201202-150,2012-02-08.(in Chinese)
[10]Zhong H,Xu J.A non-uniform grid for triangular differential quadrature[J].Science China Physics,Mechanics&Astronomy,2016,59(12):124611.
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[14]Van Oosterom A,Strackee J.The solid angle of a plane triangle[J].IEEE Transactions on Biomedical Engineering,1983,BME-30(2):125―126.
[15]Proriol J.Sur une famille de polynomesàdeux variables orthogonaux dans un triangle[J].Comptes Rendus Mathématique.Académie des Sciences.Paris,1957,245:2459―2461.
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[17]Lyness J,Cools R.A survey of numerical cubature over triangles[M]//Mathematics of Computation 1943-1993:a half-century of computational mathematics.Proceedings of Symposia in Applied Mathematics,American Mathematical Society,Providence,RI,1995,48:12―150.
[18]Zhang L,Cui T,Liu H.A set of symmetric quadrature rules on triangles and tetrahedra[J].Journal of Computational Mathematics,2009,27(1):89―96.
[19]喻畑.二维问题的弱形式求积元分析[D].北京:清华大学,2007:31―35.Yu Tian.Weak-form quadrature element analysis of two-dimensional problems[D].Beijing:Tsinghua University,2007:31―35.(in Chinese)
[2]Bellman R,Casti J.Differential quadrature and long term integration[J].Journal of Mathematical Analysis and Applications,1971,34(2):235―238.
[3]Shuai Yuan,Hongzhi Zhong,Consolidation analysis of non-homogeneous soil by the weak form quadrature element method[J].Computers and Geotechnics,2014,62:1―10.
[4]申志强,钟宏志.界面滑移组合梁的几何非线性求积元分析[J].工程力学,2013,30(3):270―275.Shen Zhiqiang,Zhong Hongzhi.Geometically nonlinear quadrature element analysis of composite beams with partial interaction[J].Engineering Mechanics,2013,30(3):270―275.(in Chinese)
[5]Zhang Run,Zhong Hongzhi.Weak form quadrature element analysis of geometrically exact shells[J].International Journal of Non-Linear Mechanics,2015,71:63―71.
[6]Zhang Run,Zhong Hongzhi.A quadrature element formulation of an energy-momentum conserving algorithm for dynamic analysis of geometrically exact beams[J].Computers&Structures,2016,165:96―106.
[7]Zhong H.Triangular differential quadrature[J].Communications in Numerical Methods in Engineering,2000,16(6):401―408.
[8]Zhong H,Hua Y,He Y.Localized triangular differential quadrature[J].Numerical Methods for Partial Differential Equations,2003,19(5):682―692.
[9]徐嘉,钟宏志.一种三角形求积元[EB].中国科技论文在线,http://www.paper.edu.cn/releasepaper/content/201202-150,2012-02-08.Xu Jia,Zhong Hongzhi.A triangular quadrature element[EB].Sciencepaper Online,http://www.paper.edu.cn/releasepaper/content/201202-150,2012-02-08.(in Chinese)
[10]Zhong H,Xu J.A non-uniform grid for triangular differential quadrature[J].Science China Physics,Mechanics&Astronomy,2016,59(12):124611.
[11]Trefethen L.Spectral methods in Matlab[M].Philadelphia:SIAM,2000.
[12]Ciarlet P.The finite element method for elliptic problems[M].Amsterdam,North-Holland,1978.
[13]Helenbrook B.On the existence of explicit hp-finite element method using Gauss-Lobatto integration on the triangle[J].SIAM Journal of Numerical Analysis,2009,47(2):1304―1318.
[14]Van Oosterom A,Strackee J.The solid angle of a plane triangle[J].IEEE Transactions on Biomedical Engineering,1983,BME-30(2):125―126.
[15]Proriol J.Sur une famille de polynomesàdeux variables orthogonaux dans un triangle[J].Comptes Rendus Mathématique.Académie des Sciences.Paris,1957,245:2459―2461.
[16]Koornwinder T.Two-variable analogues of the classical orthogonal polynomials[M]//Askey R A.Theory and application of special functions.New York:Academic Press,1975:435―495.
[17]Lyness J,Cools R.A survey of numerical cubature over triangles[M]//Mathematics of Computation 1943-1993:a half-century of computational mathematics.Proceedings of Symposia in Applied Mathematics,American Mathematical Society,Providence,RI,1995,48:12―150.
[18]Zhang L,Cui T,Liu H.A set of symmetric quadrature rules on triangles and tetrahedra[J].Journal of Computational Mathematics,2009,27(1):89―96.
[19]喻畑.二维问题的弱形式求积元分析[D].北京:清华大学,2007:31―35.Yu Tian.Weak-form quadrature element analysis of two-dimensional problems[D].Beijing:Tsinghua University,2007:31―35.(in Chinese)
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