基于弯曲网格的自适应间断有限元法研究
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- 英文论文题名：Adaptive Discontinuous Galerkin Method to Solve Euler Equations on High-order Approximative Boundary
- 论文作者：孙强 ; 吕宏强 ; 陈正武
- 年：2016
- 作者机构：南京航空航天大学航空宇航学院;中国空气动力学研究与发展中心;
- 论文关键词：高阶间断有限元 ; 弯曲网格 ; 网格自适应 ; Euler方程 ; N-S方程
- 英文论文关键词：High-order DGM ; Adaptive method ; Euler equations ; Artificial viscosity ; High-order boundary
- 会议召开时间：2016-11-04
- 会议录名称：2016年度全国气动声学学术会议论文摘要集
- 语种：中文
- 分类号：O241.82
- 学会代码：LTLX
- 会议名称：2016年度全国气动声学学术会议
- 会议地点：中国北京
- 主办单位：中国航空学会航空声学专业委员会、中国航空学会气动声学专业委员会、中国商飞上海飞机设计研究院、北京航空航天大学
- 学会名称：北京航空航天大学流体力学教育部重点实验室
- 页数：8
- 文件大小：1025k
- 原文格式：D
- 会议级别：全国
摘要
为保证应用间断有限元方法(Discontinuous Galerkin Method,DGM)收敛稳定并且得到高精度的数值解,物面网格必须能够精确表达物面信息。同时,为了捕捉更关注的守恒量变化以及降低计算量,网格自适应也势在必行。本文设计了一套只有物面附近单元弯曲的方法,二维情况以6阶多项式表达物面,三维情况以3阶多项式表达。在此基础上,网格数值解多项式的高阶项贡献作为自适应的指示器进行网格的加密或放粗。二维情况以欧拉方程的圆柱绕流算例、激波捕捉算例、层流算例进行验证,三维情况以圆球物面构造和计算进行了验证。数值结果表明,采用文章提出的方法可以保证以尽量小的计算量得到高精度结果。
A high-order discontinuous method(DGM) is integrated with adaptive method to solve Euler equations on unstructured mesh.Contribution of the polynomial's highest-order terms is quantified in the form of artificial viscous coefficient.The coefficient is regarded as the indicator of h-adaptivity.Elements where the coefficients are greater than the upper limit are refined.Those where the coefficients are less than the lower limit are coarsened if they have been refined.A high-order geometric approximation of curved boundaries is adopted to ensure the convergence.Numerical results of test cases are consistent with corresponding experimental ones.High accurate numerical results can be obtained with the h-adaptive method at low expense.
A high-order discontinuous method(DGM) is integrated with adaptive method to solve Euler equations on unstructured mesh.Contribution of the polynomial's highest-order terms is quantified in the form of artificial viscous coefficient.The coefficient is regarded as the indicator of h-adaptivity.Elements where the coefficients are greater than the upper limit are refined.Those where the coefficients are less than the lower limit are coarsened if they have been refined.A high-order geometric approximation of curved boundaries is adopted to ensure the convergence.Numerical results of test cases are consistent with corresponding experimental ones.High accurate numerical results can be obtained with the h-adaptive method at low expense.
引文
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[16]吴迪,蔚喜军.自适应间断有限元方法求解三维欧拉方程[J].计算物理,2010,27(004):492-500.WU Di,YU Xijun.Adaptive Discontinuous Galerkin Method for Euler Equations[J].Chinese Journal of Computational Physics,2010,27(004):492-500(in Chinese).
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[18]夏轶栋,伍贻兆,吕宏强,等.高阶间断有限元法的并行计算研究[J].空气动力学学报,2011,29(5):537-541.Xia Y D,Wu Y Z,Lv H Q,et al.Parallel computation of a high-order discontinuous Galerkin method on unstructured grids[J].Acta Aerodynamica Sinica,2011,29(5):537-541(in Chinese).
[19]Lu H,Berzins M,Goodyer C E,et al.Adaptive highorder Discontinuous Galerkin solution of elastohydrodynamic lubrication point contact problems[J].Advances in Engineering Software,2012,45(1):313-324.
[2]Bassi F,Rebay S.A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations[J].Journal of computational physics,1997,131(2):267-279.
[3]Bassi F,Rebay S.High-order accurate discontinuous finite element solution of the 2D Euler equations[J].Journal of Computational Physics,1997,138(2):251-285.
[4]Cockburn B and C Shu.The Runge-Kutta discontinuous Galerkin method for conservation laws V:multidimensional systems[J].Journal of Computational Physics,1998,141(2):199-224.
[5]Cockburn B and C Shu.The local discontinuous Galerkin method for time-dependent convection-diffusion systems[J].SIAM Journal on Numerical Analysis,1998,35(6):2440-2463.
[6]Nguyen N C,Persson P O,Peraire J.RANS solutions using high order discontinuous Galerkin methods[J].AIAA Paper,2007,914:2007.
[7]吕宏强,伍贻兆,周春华等.稀疏非结构网格上的亚音速流高精度数值模拟[J].航空学报,2009,30(2):200-204Lu Hongqiang,Wu Yizhao,Zhou Chunhua,et al.High resolution of subsonic flows on coarse grids[J].Acta Aeronautica et Astronautica Sinca,2009,30(2):200-204(in Chinese).
[8]吕宏强,朱国祥,宋江勇等.线化欧拉方程的高阶间断有限元数值解法研究[J].力学学报,2011,43(3):621-624Lu Hongqiang,Zhu Guoxiang,Song Jiangyong,et al.High-order discontinuous galerkin solution oflinearizedeuler equations[J].Chinese Journal of Theoretical and Applied Mechanics,2011,43(3):621-624(in Chinese).
[9]Lu H.High-Order Discontinuous Galerkin Solution of Low-Re Viscous Flows[J].Modern Physics Letters B,2009,23(03):309-312.
[10]张来平,刘伟,贺立新,邓小刚.基于静动态混合重构的DG/FV混合格式[J].力学学报,2010,42(6):1013-1022Zhang Laiping,Liu Wei,He Lixin,Deng Xiaogang.Aclass of discontinuous galerkin/finite volume hybrid schemes based on the“static re-construction”and“dynamic re-construction”[J].Chinese Journal of Theoretical and Applied Mechanics,2010,42(6):1013-1022(in Chinese).
[11]YU J and C YAN.An Artificial Diffusivity Discontinuous Galerkin Scheme for Discontinuous Flows.Computers&Fluids,2013,75(20):56-71.
[12]于剑,阎超.Navier-Stokes方程间断Galerkin有限元方法研究[J].力学学报,2010,42(5):962-969.Yu Jian,Yan Chao.Study on discontinuous galerkin method for navier-stokes equations[J].Chinese Journal of Theoretical and Applied Mechanics,2010,42(5):962-969(in Chinese).
[13]Yang X,James A J,Lowengrub J,et al.An adaptive coupled level-set/volume-of-fluid interface capturing method for unstructured triangular grids[J].Journal of Computational Physics,2006,217(2):364-394.
[14]Remacle J F,Li X,Shephard M S,et al.Anisotropic adaptive simulation of transient flows using discontinuous Galerkin methods[J].International Journal for Numerical Methods in Engineering,2005,62(7):899-923.
[15]徐云,蔚喜军.自适应间断有限元方法求解双曲守恒律方程[J].计算物理,2009,26(2):159-168.XU Yun,YU Xijun.Adaptive Discontinuous Galerkin Methods for Hyperbolic Conservation Laws[J].Chinese Journal of Computational Physics,2009,26(2):159-168(in Chinese).
[16]吴迪,蔚喜军.自适应间断有限元方法求解三维欧拉方程[J].计算物理,2010,27(004):492-500.WU Di,YU Xijun.Adaptive Discontinuous Galerkin Method for Euler Equations[J].Chinese Journal of Computational Physics,2010,27(004):492-500(in Chinese).
[17]Von Neumann J,Richtmyer R D.A method for the numerical calculation of hydrodynamic shocks[J].Journal of applied physics,1950,21(3):232-237.
[18]夏轶栋,伍贻兆,吕宏强,等.高阶间断有限元法的并行计算研究[J].空气动力学学报,2011,29(5):537-541.Xia Y D,Wu Y Z,Lv H Q,et al.Parallel computation of a high-order discontinuous Galerkin method on unstructured grids[J].Acta Aerodynamica Sinica,2011,29(5):537-541(in Chinese).
[19]Lu H,Berzins M,Goodyer C E,et al.Adaptive highorder Discontinuous Galerkin solution of elastohydrodynamic lubrication point contact problems[J].Advances in Engineering Software,2012,45(1):313-324.