直角坐标网格下LSFD方法的显式公式和性能研究

蔡庆东

直角坐标网格下LSFD方法的显式公式和性能研究

蔡庆东^1,2

1.
北京大学工学院力学与空天技术系,湍流与复杂系统国家重点实验室,北京 100871; 2.北京大学工学院应用物理与技术研究中心,北京 100871

基金项目: 国家自然科学基金资助项目(10872005;10532010)

详细信息

作者简介:
蔡庆东(1968- ),男,吉林松原人,副教授,博士(Tel:+86-10-62757942;E-mail:caiqd@pku.edu.cn).

中图分类号: O241.82；O242
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- 文章访问数: 2695
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- 被引次数: 0
出版历程
- 收稿日期: 2008-06-05
- 修回日期: 2008-10-20
- 刊出日期: 2009-02-15

Explicit Formulations and Performance Study of LSFD Method on Cartesian Mesh

CAI Qing-dong^1,2

1.
LTCS and CAPT; Department of Mechanics and Aerospace Engineering, Peking University, Beijing 100871, P. R. China;

摘要

摘要: 重点讨论了LSFD(least square-based finite difference)方法和传统的FD(finite difference)方法在性能上的对比问题．对于传统的中心差分格式，一阶导数和二阶导数在二维情况的数值格式基架点有9个点，三维情况有27个点．在同样的基架点下，给出了LSFD方法近似一阶导数和二阶导数的显式公式，并指出LSFD方法在这种情况下实质上就是在不同网格线上的传统中心差分格式的组合．在数值模拟中，LSFD方法达到收敛所需要的迭代步数比传统差分格式少，并且x和y方向的网格纵横尺度比在LSFD方法中是一个非常重要的参数，对计算的稳定性有重要影响．
- LSFD方法 /
- 无网格方法 /
- 直角坐标网格 /
- 网格纵横尺度比
Abstract: The performance of the ISFD (least square-based flute difference) method is compared with the conventional FD(fiute difference) schemes. For the approximation of the first and second order derivatives by the conventional central difference schemes, 9-point stencils for the 2D case and 27-point stencils for the 3D case are usually used. When the same stencils are used, the explicit ISFD formulations for apprrntimation of the fist and second order derivatives were present. the ISFD formutations are actually the combination of conventional central difference schemes along relevant mesh lines. It is found that ISFD formulations need much less iteration steps than the conventional FD schemes to get the converged solution, and the ratio of mesh spacing in the x and y directions is an important parameter in the ISFD application, which has a great effect on the stability of ISFD computation.
- LSFD method /
- meshfree method /
- Cartesian Mesh /
- aspect ratio

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参考文献(26)

[1]	Steger J L, Warming R F.Flux vector splitting of the inviscid gasdynamic equations with application to finite-difference methods[J].J Comp Phys,1981,40(2):263-293. doi: 10.1016/0021-9991(81)90210-2
[2]	Chung T J.Finite Element Analysis in Fluid Dynamics[M].New York: McGraw-Hill Publ, 1978.
[3]	Baker A J, Kim J W.A Taylor weak-statement algorithm for hyperbolic conservation laws[J].Internat J Numer Methods Fluids,1987,7(5):489-520. doi: 10.1002/fld.1650070505
[4]	Patankar S V. Numerical Heat Transfer and Fluid Flow[M].Washington D C: Hemisphere, 1980.
[5]	Wang Z J, Srinivasan K.An adaptive Cartesian grid generation method for ‘dirty' geometry[J].Internat J Numer Methods Fluids, 2002,39(8):703-717. doi: 10.1002/fld.344
[6]	Monaghan J J. An introduction to SPH[J].Comput Physics Communications,1988,48(1):89-96. doi: 10.1016/0010-4655(88)90026-4
[7]	Aluru N R, Li G.Finite cloud method: a true meshless technique based on a fixed reproducing kernel approximation[J]. Internat J Numer Methods Engrg,2001,50(10):2373-2410. doi: 10.1002/nme.124
[8]	Liu W K, Jun S, Zhang Y F.Reproducing kernel particle methods[J].Internat J Numer Methods Fluids, 1995, 30(8/9):1081-1106.
[9]	Nayroles B, Touzot G, Villon P.Generalizing the FEM: diffuse approximation and diffuse elements[J].Computational Mechanics,1992,10(5):307-318. doi: 10.1007/BF00364252
[10]	Belytschko T, Lu Y Y, Gu L.Element-free Galerkin methods[J].Internat J Numer Methods Engrg,1994,37(2):229-256. doi: 10.1002/nme.1620370205
[11]	Duarte C A, Oden J T.A hp adaptive method using clouds[J]. Comput Methods Appl Mech Engrg,1996, 139(1/4):237-262. doi: 10.1016/S0045-7825(96)01085-7
[12]	Kansa E J.Multiquadrics —a scattered data approximation scheme with applications to computational fluid-dynamics —Ⅰ surface approximations and partial derivative estimates[J].Computers Math Appl,1990, 19(8/9):127-145.
[13]	Kansa E J.Multiquadrics —a scattered data approximation scheme with applications to computational fluid-dynamics —Ⅱ solutions to parabolic, hyperbolic, and elliptic partial differential equations[J]. Computers Math Appl,1990,19(8/9):147-161.
[14]	Shu C, Ding H, Yeo K S.Local radial basis function-based differential quadrature method and its application to solve two-dimensional incompressible Navier-Stokes equations[J].Comput Methods Appl Mech Engrg,2003,192(7/8):941-954. doi: 10.1016/S0045-7825(02)00618-7
[15]	Li J.A radial basis meshless method for solving inverse boundary value problems[J].Comm Num Meth Eng,2004, 20(1):51-61.
[16]	Ding H, Shu C, Yeo K S,et al.Simulation of incompressible viscous flows past a circular cylinder by hybrid FD scheme and meshless least square-based finite difference method[J].Comput Methods Appl Mech Engrg,2004,193(9/11):727-744. doi: 10.1016/j.cma.2003.11.002
[17]	Ding H, Shu C,Yeo K S,et al.Development of least-square-based two-dimensional finite-difference schemes and their application to simulate natural convection in a cavity[J]. Computers & Fluids, 2004,33(1):137-154.
[18]	Balakrishann N, Deshpande S M.Reconstruction on unstructured meshes with upwind solvers[A].In:Hui W H, Kwok Y K, Chasnov J R,Eds.Proc of the First Asian CFD Conference[C].Hong Kong:Hong Kong University of Science and Technology,1995,359-364.
[19]	Luo D, Joseph D B, Lhner R.A comparison of reconstruction schemes for compressible flows on unstructured grids[A]. In: Hui W H, Kwok Y K, Chasnov J R,Eds.Proc of the First Asian CFD Conference[C].Hong Kong:Hong Kong University of Science and Technology,1995,365-370.
[20]	蔡庆东.各种网络上统一的数值离散方法[J].力学学报，2004,36(4):393-400.
[21]	Schnauer W, Adolph T.How we solve PDEs[J].J Comp Appl Math,2001,131(1/2):473-492. doi: 10.1016/S0377-0427(00)00255-7
[22]	Sridar D, Balakrishnan N.An upwind finite difference scheme for meshless solvers[J].J Comp Phys,2003,189(1): 1-29. doi: 10.1016/S0021-9991(03)00197-9
[23]	Ding H, Shu C, Yeo K S,et al.Numerical simulation of flows around two circular cylinders by mesh-free least square-based finite difference methods[J].Internat J Numer Methods Fluids,2007,53(2):305-332. doi: 10.1002/fld.1281
[24]	Wu W X, Shu C, Wang C M.Computation of modal stress resultants for completely free vibrating plates by LSFD method[J].Journal of Sound and Vibration,2006, 297(3/5):704-726. doi: 10.1016/j.jsv.2006.04.019
[25]	Wang C M, Wu W X, Shu C,et al.LSFD method for accurate vibration modes and modal stress-resultants of freely vibrating plates that model VLFS[J].Computers and Structures,2006,84(31/32): 2329-2339. doi: 10.1016/j.compstruc.2006.08.055
[26]	Wu W X, Shu C,Wang C M.Mesh-free least-squares-based finite difference method for large amplitude free vibration analysis of arbitrarily shaped thin plates[J].Journal of Sound and Vibration, 2008,317(3/5): 955-974. doi: 10.1016/j.jsv.2008.03.050