基于无积分节点间断有限元的二维水沙模型:(1)水动力
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摘要
通过采用节点间断有限元方法对二维浅水方程进行离散,考虑了科氏力、风应力、底摩阻等作用,最终建立了一套高精度二维水动力模型。模型可采用任意四边形网格计算,并应用节点基函数和无积分数值离散方法,有效地减少了计算量。建立的模型通过理想算例对各源项求解、干湿与和谐性进行了验证。最后将模型应用于三亚市红塘湾实际潮流的模拟中,得到结果与全潮水文观测数据吻合良好。
A two-dimensional hydrodynamic model was developed based on the nodal discontinuous Galerkin method. The 2 D shallow water equations were numerically solved in the model with the source terms including Coriolis force, wind stress and bottom friction. The arbitrary quadrilateral meshes were used and a quadrature-free scheme without solving the numerical solution and function value at the integral node, thus effectively reducing the amount of calculation, was adopted. Numerical examples were adopted to verify the ability of the algorithm for description of source terms, wetting and drying and the well-balanced property. Finally, the model was applied to simulate the tide flow of Hongtang bay in Sanya, and the simulation results are in good agreement with the observed data in field.
A two-dimensional hydrodynamic model was developed based on the nodal discontinuous Galerkin method. The 2 D shallow water equations were numerically solved in the model with the source terms including Coriolis force, wind stress and bottom friction. The arbitrary quadrilateral meshes were used and a quadrature-free scheme without solving the numerical solution and function value at the integral node, thus effectively reducing the amount of calculation, was adopted. Numerical examples were adopted to verify the ability of the algorithm for description of source terms, wetting and drying and the well-balanced property. Finally, the model was applied to simulate the tide flow of Hongtang bay in Sanya, and the simulation results are in good agreement with the observed data in field.
引文
[1]王施恩,张征.洋山港四期工程水域水沙环境及港区增深方案研究[J].水道港口,2018,39(1):5-11.WANG S E,ZHANG Z.Study on flow-sediment condition and deepening scheme of harbor district in the sea area of Phase IV project of Yangshan Deep-water Port[J].Journal of Waterway and Harbor,2018,39(1):5-11.
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[3]Xing Y,Shu C W.High order well-balanced finite volume WENO schemes and discontinuous Galerkin methods for a class of hyperbolic systems with source terms[M].United States:Academic Press Professional,Inc.2006.
[4]Xing Y,Zhang X.Positivity-preserving well-balanced discontinuous Galerkin methods for the shallow water equations on unstructured triangular meshes[J].Journal of Scientific Computing,2013,57(1):19-41.
[5]Schwanenberg D,Harms M.Discontinuous Galerkin finite-element method for transcritical two-dimensional shallow water flows[J].Journal of Hydraulic Engineering,2004,130(5):412-421.
[6]Trahan C J,Dawson C.Local time-stepping in Runge-Kutta discontinuous Galerkin finite element methods applied to the shallow-water equations[J].Computer Methods in Applied Mechanics & Engineering,2012,217-220:139-152.
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[10]李龙翔,张庆河.一种无积分任意四边形非结构化网格节点间断Galerkin方法[J].天津大学学报:自然科学与工程技术版,2018,51(6):575-582.LI L X,ZHANG Q H.A quadrature-free scheme for nodal discontinuous Galerkin method on Arbitrary quadrilateral unstructured meshes[J].Journal of Tianjin University:Science and Technology,2018,51(6):575-582.
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[14]Khan A A,Lai W.Modeling shallow water flows using the discontinuous Galerkin method[M].Boca Raton:CRC Press,2014.
[15]Toro E F.Riemann solvers and numerical methods for fluid dynamics:a practical introduction[M].Berlin:Springer,1999.
[16]艾丛芳,金生.求解具有复杂地形二维浅水方程的修正HLL格式[J].大连理工大学学报,2009,49(6):926-931.AI C F,JIN S.Solution of 2D shallow water equations with complicated geometry using modified HLL scheme[J].Journal of Dalian University of Technology,2009,49(6):926-931.
[17]Atkins H L,Shu C W.Quadrature-free implementation of discontinuous Galerkin method for hyperbolic equations[J].AIAA Journal,1998,36(5):775-782.
[18]Karniadakis G,Sherwin S.Spectral/hp element methods for computational fluid dynamics[M].New York:Oxford University Press,2013.
[19]赵旭东,孙家文,孙昭晨,等.基于CC-CV混合有限体积法的干湿边界处理技术及其在浅水问题中的应用[J].水道港口,2017,38(6):548-554.ZHAO X D,SUN J W,SUN Z C,et al.A novel wet-dry fronts treatment based on CC-CV hybrid finite volume method and the application in shallow water problems[J].Journal of Waterway and Harbor,2017,38(6):548-554.
[20]Bonev B,Hesthaven J S,Giraldo F X,et al.Discontinuous Galerkin scheme for the spherical shallow water equations with applications to tsunami modeling and prediction[J].Journal of Computational Physics,2017,362:425-448
[21]Bunya S,Kubatko E J,Westerink J J,et al.A wetting and drying treatment for the Runge-Kutta discontinuous Galerkin solution to the shallow water equations[J].Computer Methods in Applied Mechanics & Engineering,2009,198(17-20):1 548-1 562.
[22]XING Y,ZHANG X,SHU C W.Positivity-preserving high order well-balanced discontinuous Galerkin methods for the shallow water equations[J].Advances in Water Resources,2010,33(12):1 476-1 493.
[23]LI L,ZHANG Q.A new vertex-based limiting approach for nodal discontinuous Galerkin methods on arbitrary unstructured meshes[J].Computers & Fluids,2017,159:316-326.
[24]潘存鸿.浅水间断流动数值模拟研究进展[J].水利水电科技进展,2010,30(5):77-84.PAN C H.Advances in numerical simulation of discontinuous shallow water flows[J].Advances in Science and Technology of Water Resources,2010,30(5):77-84.
[25]于守兵.计算二维浅水方程中静水压力项与底坡项的积分平衡法[J].水利水电科技进展,2009,29(4):32-35.YU S B.Integral balance method for computing stationary hydraulic pressure term and bed slope term in two-dimensional shallow water equations[J].Advances in Science and Technology of Water Resources,2009,29(4):32-35.
[26]Bollermann A,Chen G,Kurganov A,et al.A well-balanced reconstruction of wet/dry fronts for the shallow water equations[J].Journal of Scientific Computing,2013,56(2):267-290.
[27]Eskilsson C,Sherwin S J.A triangular spectral/hp discontinuous Galerkin method for modelling 2D shallow water equations[J].International Journal for Numerical Methods in Fluids,2004,45(6):605-623.
[28]Ma H.A spectral element basin model for the shallow water equations[J].Journal of Computational Physics,1993,109(1):133-149.
[29]Wirasaet D,Kubatko E J,Michoski C E,et al.Discontinuous Galerkin methods with nodal and hybrid modal/nodal triangular,quadrilateral,and polygonal elements for nonlinear shallow water flow[J].Computer Methods in Applied Mechanics & Engineering,2014,270(2):113-149.
[30]Carrier G F,Tei W T,Yeh H.Tsunami run-up and draw-down on a plane beach[J].Journal of Fluid Mechanics,2003,475:79-99.
[2]Bokhove O.Flooding and drying in discontinuous Galerkin finite-element discretizations of shallow-water equations.Part 1:one dimension[J].Journal of Scientific Computing,2005,22-23(1-3):47-82.
[3]Xing Y,Shu C W.High order well-balanced finite volume WENO schemes and discontinuous Galerkin methods for a class of hyperbolic systems with source terms[M].United States:Academic Press Professional,Inc.2006.
[4]Xing Y,Zhang X.Positivity-preserving well-balanced discontinuous Galerkin methods for the shallow water equations on unstructured triangular meshes[J].Journal of Scientific Computing,2013,57(1):19-41.
[5]Schwanenberg D,Harms M.Discontinuous Galerkin finite-element method for transcritical two-dimensional shallow water flows[J].Journal of Hydraulic Engineering,2004,130(5):412-421.
[6]Trahan C J,Dawson C.Local time-stepping in Runge-Kutta discontinuous Galerkin finite element methods applied to the shallow-water equations[J].Computer Methods in Applied Mechanics & Engineering,2012,217-220:139-152.
[7]Dawson C,Trahan C J,Kubatko E J,et al.A parallel local timestepping Runge-Kutta discontinuous Galerkin method with applications to coastal ocean modeling[J].Computer Methods in Applied Mechanics & Engineering,2013,259(6):154-165.
[8]Dawson C.A Local timestepping Runge-Kutta discontinuous Galerkin method for hurricane storm surge modeling[J].The Institute of Mathematics and its Applications,2014,157:133-148.
[9]赵张益.一维浅水方程的Runge-Kutta间断有限元数值模拟与应用[D].天津:天津大学,2010.
[10]李龙翔,张庆河.一种无积分任意四边形非结构化网格节点间断Galerkin方法[J].天津大学学报:自然科学与工程技术版,2018,51(6):575-582.LI L X,ZHANG Q H.A quadrature-free scheme for nodal discontinuous Galerkin method on Arbitrary quadrilateral unstructured meshes[J].Journal of Tianjin University:Science and Technology,2018,51(6):575-582.
[11]Luettich R A,Westerink J J.Formulation and numerical implementation of the 2D/3D ADCIRC finite element model version 44.XX[R].Chapel Hill:University of North Carolina,2004.
[12]刘亚坤.水力学[M].北京:中国水利水电出版社,2008.
[13]Hesthaven J S,Warburton T.Nodal discontinuous Galerkin methods:algorithms,analysis,and applications[M].Berlin:Springer Science & Business Media,2007.
[14]Khan A A,Lai W.Modeling shallow water flows using the discontinuous Galerkin method[M].Boca Raton:CRC Press,2014.
[15]Toro E F.Riemann solvers and numerical methods for fluid dynamics:a practical introduction[M].Berlin:Springer,1999.
[16]艾丛芳,金生.求解具有复杂地形二维浅水方程的修正HLL格式[J].大连理工大学学报,2009,49(6):926-931.AI C F,JIN S.Solution of 2D shallow water equations with complicated geometry using modified HLL scheme[J].Journal of Dalian University of Technology,2009,49(6):926-931.
[17]Atkins H L,Shu C W.Quadrature-free implementation of discontinuous Galerkin method for hyperbolic equations[J].AIAA Journal,1998,36(5):775-782.
[18]Karniadakis G,Sherwin S.Spectral/hp element methods for computational fluid dynamics[M].New York:Oxford University Press,2013.
[19]赵旭东,孙家文,孙昭晨,等.基于CC-CV混合有限体积法的干湿边界处理技术及其在浅水问题中的应用[J].水道港口,2017,38(6):548-554.ZHAO X D,SUN J W,SUN Z C,et al.A novel wet-dry fronts treatment based on CC-CV hybrid finite volume method and the application in shallow water problems[J].Journal of Waterway and Harbor,2017,38(6):548-554.
[20]Bonev B,Hesthaven J S,Giraldo F X,et al.Discontinuous Galerkin scheme for the spherical shallow water equations with applications to tsunami modeling and prediction[J].Journal of Computational Physics,2017,362:425-448
[21]Bunya S,Kubatko E J,Westerink J J,et al.A wetting and drying treatment for the Runge-Kutta discontinuous Galerkin solution to the shallow water equations[J].Computer Methods in Applied Mechanics & Engineering,2009,198(17-20):1 548-1 562.
[22]XING Y,ZHANG X,SHU C W.Positivity-preserving high order well-balanced discontinuous Galerkin methods for the shallow water equations[J].Advances in Water Resources,2010,33(12):1 476-1 493.
[23]LI L,ZHANG Q.A new vertex-based limiting approach for nodal discontinuous Galerkin methods on arbitrary unstructured meshes[J].Computers & Fluids,2017,159:316-326.
[24]潘存鸿.浅水间断流动数值模拟研究进展[J].水利水电科技进展,2010,30(5):77-84.PAN C H.Advances in numerical simulation of discontinuous shallow water flows[J].Advances in Science and Technology of Water Resources,2010,30(5):77-84.
[25]于守兵.计算二维浅水方程中静水压力项与底坡项的积分平衡法[J].水利水电科技进展,2009,29(4):32-35.YU S B.Integral balance method for computing stationary hydraulic pressure term and bed slope term in two-dimensional shallow water equations[J].Advances in Science and Technology of Water Resources,2009,29(4):32-35.
[26]Bollermann A,Chen G,Kurganov A,et al.A well-balanced reconstruction of wet/dry fronts for the shallow water equations[J].Journal of Scientific Computing,2013,56(2):267-290.
[27]Eskilsson C,Sherwin S J.A triangular spectral/hp discontinuous Galerkin method for modelling 2D shallow water equations[J].International Journal for Numerical Methods in Fluids,2004,45(6):605-623.
[28]Ma H.A spectral element basin model for the shallow water equations[J].Journal of Computational Physics,1993,109(1):133-149.
[29]Wirasaet D,Kubatko E J,Michoski C E,et al.Discontinuous Galerkin methods with nodal and hybrid modal/nodal triangular,quadrilateral,and polygonal elements for nonlinear shallow water flow[J].Computer Methods in Applied Mechanics & Engineering,2014,270(2):113-149.
[30]Carrier G F,Tei W T,Yeh H.Tsunami run-up and draw-down on a plane beach[J].Journal of Fluid Mechanics,2003,475:79-99.