Examines the moving mesh methods for solving one-dimensional time dependent partial differential equations. Introduction of the differential-algebraic formulations based on geometrical variables; Investigation of the ...Examines the moving mesh methods for solving one-dimensional time dependent partial differential equations. Introduction of the differential-algebraic formulations based on geometrical variables; Investigation of the well-posedness of the numerical approach; Discussion of some detailed numerical procedures.展开更多
Adaptive moving mesh research usually focuses either on analytical deriva-tions for prescribed solutions or on pragmatic solvers with challenging physical appli-cations. In the latter case, the monitor functions that ...Adaptive moving mesh research usually focuses either on analytical deriva-tions for prescribed solutions or on pragmatic solvers with challenging physical appli-cations. In the latter case, the monitor functions that steer mesh adaptation are oftendefined in an ad-hoc way. In this paper we generalize our previously used moni-tor function to a balanced sum of any number of monitor components. This avoidsthe trial-and-error parameter fine-tuning that is often used in monitor functions. Thekey reason for the new balancing method is that the ratio between the maximum andaverage value of a monitor component should ideally be equal for all components.Vorticity as a monitor component is a good motivating example for this. Entropy alsoturns out to be a very informative monitor component. We incorporate the monitorfunction in an adaptive moving mesh higher-order finite volume solver with HLLCfluxes, which is suitable for nonlinear hyperbolic systems of conservation laws. Whenapplied to compressible gas flow it produces very sharp results for shocks and otherdiscontinuities. Moreover, it captures small instabilities (Richtmyer-Meshkov, Kelvin-Helmholtz). Thus showing the rich nature of the example problems and the effective-ness of the new monitor balancing.展开更多
We introduce adaptive moving mesh central-upwind schemes for one-and two-dimensional hyperbolic systems of conservation and balance laws.The proposed methods consist of three steps.First,the solution is evolved by sol...We introduce adaptive moving mesh central-upwind schemes for one-and two-dimensional hyperbolic systems of conservation and balance laws.The proposed methods consist of three steps.First,the solution is evolved by solving the studied system by the second-order semi-discrete central-upwind scheme on either the one-dimensional nonuniform grid or the two-dimensional structured quadrilateral mesh.When the evolution step is complete,the grid points are redistributed according to the moving mesh differential equation.Finally,the evolved solution is projected onto the new mesh in a conservative manner.The resulting adaptive moving mesh methods are applied to the one-and two-dimensional Euler equations of gas dynamics and granular hydrodynamics systems.Our numerical results demonstrate that in both cases,the adaptive moving mesh central-upwind schemes outperform their uniform mesh counterparts.展开更多
We consider an iterative algorithm of mesh optimization for finite element solution, and give an improved moving mesh strategy that reduces rapidly the complexity and cost of solving variational problems. A numerical ...We consider an iterative algorithm of mesh optimization for finite element solution, and give an improved moving mesh strategy that reduces rapidly the complexity and cost of solving variational problems. A numerical result is presented for a 2-dimensional problem by the improved algorithm.展开更多
This paper studies the convergence rates of a moving mesh implicit finite difference method with interpolation for partial differential equations (PDEs) with moving boundary arising in Asian option pricing. The movi...This paper studies the convergence rates of a moving mesh implicit finite difference method with interpolation for partial differential equations (PDEs) with moving boundary arising in Asian option pricing. The moving mesh scheme is based on Rnnacher timestepping approach whose idea is running the implicit Euler schemes in the initial few steps and continuing with Crank-Nicolson schemes. With graded meshes for time direction and moving meshes for space direction, the fully discretized scheme is constructed using quadratic interpolation between two consecutive time level for the PDEs with moving boundary. The second-order convergence rates in both time and space are proved and numerical examples are carried out to confirm the theoretical results.展开更多
This paper develops and analyzes a moving mesh finite difference method for solving partial integro-differential equations. First, the time-dependent mapping of the coordinate transformation is approximated by a a pie...This paper develops and analyzes a moving mesh finite difference method for solving partial integro-differential equations. First, the time-dependent mapping of the coordinate transformation is approximated by a a piecewise linear function in time. Then, piecewise quadratic polynomial in space and an efficient method to discretize the memory term of the equation is designed using the moving mesh approach. In each time slice, a simple piecewise constant approximation of the integrand is used, and thus a quadrature is constructed for the memory term. The central finite difference scheme for space and the backward Euler scheme for time are used. The paper proves that the accumulation of the quadrature error is uniformly bounded and that the convergence of the method is second order in space and first order in time. Numerical experiments are carried out to confirm the theoretical predictions.展开更多
In this paper, we present an adaptive moving mesh algorithm for meshesof unstructured polyhedra in three space dimensions. The algorithm automaticallyadjusts the size of the elements with time and position in the phys...In this paper, we present an adaptive moving mesh algorithm for meshesof unstructured polyhedra in three space dimensions. The algorithm automaticallyadjusts the size of the elements with time and position in the physical domain to resolvethe relevant scales in multiscale physical systems while minimizing computationalcosts. The algorithm is a generalization of the moving mesh methods basedon harmonic mappings developed by Li et al. [J. Comput. Phys., 170 (2001), pp. 562-588, and 177 (2002), pp. 365-393]. To make 3D moving mesh simulations possible,the key is to develop an efficient mesh redistribution procedure so that this part willcost as little as possible comparing with the solution evolution part. Since the meshredistribution procedure normally requires to solve large size matrix equations, wewill describe a procedure to decouple the matrix equation to a much simpler blocktridiagonaltype which can be efficiently solved by a particularly designed multi-gridmethod. To demonstrate the performance of the proposed 3D moving mesh strategy,the algorithm is implemented in finite element simulations of fluid-fluid interface interactionsin multiphase flows. To demonstrate the main ideas, we consider the formationof drops by using an energetic variational phase field model which describesthe motion of mixtures of two incompressible fluids. Numerical results on two- andthree-dimensional simulations will be presented.展开更多
This paper extends the adaptive moving mesh method developed by Tang and Tang[36]to two-dimensional(2D)relativistic hydrodynamic(RHD)equations.The algorithm consists of two“independent”parts:the time evolution of th...This paper extends the adaptive moving mesh method developed by Tang and Tang[36]to two-dimensional(2D)relativistic hydrodynamic(RHD)equations.The algorithm consists of two“independent”parts:the time evolution of the RHD equations and the(static)mesh iteration redistribution.In the first part,the RHD equations are discretized by using a high resolution finite volume scheme on the fixed but nonuniform meshes without the full characteristic decomposition of the governing equations.The second part is an iterative procedure.In each iteration,the mesh points are first redistributed,and then the cell averages of the conservative variables are remapped onto the new mesh in a conservative way.Several numerical examples are given to demonstrate the accuracy and effectiveness of the proposed method.展开更多
In this paper, we present an adaptive moving mesh technique for solvingthe incompressible viscous flows using the vorticity stream-function formulation. Themoving mesh strategy is based on the approach proposed by Li ...In this paper, we present an adaptive moving mesh technique for solvingthe incompressible viscous flows using the vorticity stream-function formulation. Themoving mesh strategy is based on the approach proposed by Li et al. [J. Comput. Phys.,170 (2001), pp. 562–588] to separate the mesh-moving and evolving PDE at each timestep. The Navier-Stokes equations are solved in the vorticity stream-function form bya finite-volume method in space, and the mesh-moving part is realized by solving theEuler-Lagrange equations to minimize a certain variation in conjunction with a moresophisticated monitor function. A conservative interpolation is used to redistributethe numerical solutions on the new meshes. This paper discusses the implementationof the periodic boundary conditions, where the physical domain is allowed to deformwith time while the computational domain remains fixed and regular throughout. Numericalresults demonstrate the accuracy and effectiveness of the proposed algorithm.展开更多
The numerical solution of the harmonic heat map flow problems with blowup in finite or infinite time is considered using an adaptive moving mesh method.A properly chosen monitor function is derived so that the moving ...The numerical solution of the harmonic heat map flow problems with blowup in finite or infinite time is considered using an adaptive moving mesh method.A properly chosen monitor function is derived so that the moving mesh method can be used to simulate blowup and produce accurate blowup profiles which agree with formal asymptotic analysis.Moreover,the moving mesh method has finite time blowup when the underlying continuous problem does.In situations where the continuous problem has infinite time blowup,the moving mesh method exhibits finite time blowup with a blowup time tending to infinity as the number of mesh points increases.The inadequacy of a uniform mesh solution is clearly demonstrated.展开更多
The five-equation model of multi-component flows has been attracting much attention among researchers during the past twenty years for its potential in the study of the multi-component flows.In this paper,we employ a ...The five-equation model of multi-component flows has been attracting much attention among researchers during the past twenty years for its potential in the study of the multi-component flows.In this paper,we employ a second order finite volume method with minmod limiter in spatial discretization,which preserves local extrema of certain physical quantities and is thus capable of simulating challenging test problems without introducing non-physical oscillations.Moreover,to improve the numerical resolution of the solutions,the adaptive moving mesh strategy proposed in[Huazhong Tang,Tao Tang,Adaptive mesh methods for one-and two-dimensional hyperbolic conservation laws,SINUM,41:487-515,2003]is applied.Furthermore,the proposed method can be proved to be capable of preserving the velocity and pressure when they are initially constant,which is essential in material interface capturing.Finally,several classical numerical examples demonstrate the effectiveness and robustness of the proposed method.展开更多
The Doi-Hess equation that describes the evolution of an orientational dis-tribution function is capable of predicting several rheological features of nematic poly-mers.Since the orientational distribution function be...The Doi-Hess equation that describes the evolution of an orientational dis-tribution function is capable of predicting several rheological features of nematic poly-mers.Since the orientational distribution function becomes sharply peaked as poten-tial intensity increases,powerful numerical methods become necessary in the relevant numerical simulations.In this paper,a numerical scheme based on the moving grid techniques will be designed to solve the orientational distribution functions with high potential intensities.Numerical experiments are carried out to demonstrate the effec-tiveness and robustness of the proposed scheme.展开更多
There have been several recent papers on developingmovingmeshmethodsfor solving phase-field equations. However, it is observed that some of these movingmesh solutions are essentially different from the solutions on ve...There have been several recent papers on developingmovingmeshmethodsfor solving phase-field equations. However, it is observed that some of these movingmesh solutions are essentially different from the solutions on very fine fixed meshes.One of the purposes of this paper is to understand the reason for the differences. Wecarried out numerical sensitivity studies systematically in this paper and it can be concludedthat for the phase-field equations, the numerical solutions are very sensitive tothe starting mesh and the monitor function. As a separate issue, an efficient alternatingCrank-Nicolson time discretization scheme is developed for solving the nonlinearsystem resulting from a finite element approximation to the phase-field equations.展开更多
A distributed Lagrangian moving-mesh finite element method is applied to problems involving changes of phase.The algorithm uses a distributed conservation principle to determine nodal mesh velocities,which are then us...A distributed Lagrangian moving-mesh finite element method is applied to problems involving changes of phase.The algorithm uses a distributed conservation principle to determine nodal mesh velocities,which are then used to move the nodes.The nodal values are obtained from an ALE(Arbitrary Lagrangian-Eulerian)equation,which represents a generalization of the original algorithm presented in Applied Numerical Mathematics,54:450–469(2005).Having described the details of the generalized algorithm it is validated on two test cases from the original paper and is then applied to one-phase and,for the first time,two-phase Stefan problems in one and two space dimensions,paying particular attention to the implementation of the interface boundary conditions.Results are presented to demonstrate the accuracy and the effectiveness of the method,including comparisons against analytical solutions where available.展开更多
This paper deals with the application of a moving mesh method for kinetic/hydrodynamic coupling model in two dimensions.With some criteria,the domain is dynamically decomposed into three parts:kinetic regions where fl...This paper deals with the application of a moving mesh method for kinetic/hydrodynamic coupling model in two dimensions.With some criteria,the domain is dynamically decomposed into three parts:kinetic regions where fluids are far from equilibrium,hydrodynamic regions where fluids are near thermodynamical equilibrium and buffer regions which are used as a smooth transition.The Boltzmann-BGK equation is solved in kinetic regions,while Euler equations in hydrodynamic regions and both equations in buffer regions.By a well defined monitor function,our moving mesh method smoothly concentrate the mesh grids to the regions containing rapid variation of the solutions.In each moving mesh step,the solutions are conservatively updated to the new mesh and the cut-off function is rebuilt first to consist with the region decomposition after the mesh motion.In such a framework,the evolution of the hybrid model and the moving mesh procedure can be implemented independently,therefore keep the advantages of both approaches.Numerical examples are presented to demonstrate the efficiency of the method.展开更多
We develop an efficient one-dimensional moving mesh algorithm for solving partial differential equations.The main contribution of this paper is to design an effective interpolation scheme based on L2-projection for th...We develop an efficient one-dimensional moving mesh algorithm for solving partial differential equations.The main contribution of this paper is to design an effective interpolation scheme based on L2-projection for the moving mesh method.The proposed method preserves not only the mass-conservation but also the first order momentum of the underlying numerical solution at each mesh redistribution step.Numerical examples are presented to demonstrate the effectiveness of the new interpolation technique.展开更多
Computational mesh is an important ingredient that affects the accuracy and efficiency of CFD numerical simulation.In light of the introduced large amount of computational costs for many adaptive mesh methods,moving m...Computational mesh is an important ingredient that affects the accuracy and efficiency of CFD numerical simulation.In light of the introduced large amount of computational costs for many adaptive mesh methods,moving mesh methods keep the number of nodes and topology of a mesh unchanged and do not increase CFD computational expense.As the state-of-the-art moving mesh method,the variational mesh adaptation approach has been introduced to CFD calculation.However,quickly estimating the flow field on the updated meshes during the iterative algorithm is challenging.A mesh optimization method,which embeds a machine learning regression model into the variational mesh adaptation,is proposed.The regression model captures the mapping between the initial mesh nodes and the flow field,so that the variational method could move mesh nodes iteratively by solving the mesh functional which is built from the estimated flow field on the updated mesh via the regression model.After the optimization,the density of the nodes in the high gradient area increases while the density in the low gradient area decreases.Benchmark examples are first used to verify the feasibility and effectiveness of the proposed method.And then we use the steady subsonic and transonic flows over cylinder and NACA0012 airfoil on unstructured triangular meshes to test our method.Results show that the proposed method significantly improves the accuracy of the local flow features on the adaptive meshes.Our work indicates that the proposed mesh optimization approach is promising for improving the accuracy and efficiency of CFD computation.展开更多
This study proposes a method for modelling the flow interaction of multiple moving objects where the flow field variables are communicated between multiple separate moving computational domains.Instead of using the co...This study proposes a method for modelling the flow interaction of multiple moving objects where the flow field variables are communicated between multiple separate moving computational domains.Instead of using the conventional approach with a single fixed computational domain covering the whole flow field,this method advances the moving computational domain(MCD)method in which the computational domain itself moves in line with the motions of an object inside.The computational domains created around each object move independently,and the flow fields of each domain interact where the flows cross.This eliminates the spatial restriction for simulating multiple moving objects.Firstly,a shock tube test verifies that the overset implementation and grid movement do not adversely affect the results and that there is communication between the grids.A second test case is conducted in which two spheres are crossed,and the forces exerted on one object due to the other’s crossing at a short distance are calculated.The results verify the reliability of this method and show that it is applicable to the flow interaction of multiple moving objects.展开更多
基金This research was supported by Hong Kong Baptist University, Hong Kong Research Grants Council,Special Funds for Major State B
文摘Examines the moving mesh methods for solving one-dimensional time dependent partial differential equations. Introduction of the differential-algebraic formulations based on geometrical variables; Investigation of the well-posedness of the numerical approach; Discussion of some detailed numerical procedures.
基金The first author performs his research in the project‘Adaptive moving mesh methods for higher-dimensional nonlinear hyperbolic conservation laws’,funded by the Netherlands Organisation for Scientific Research(NWO)under project number 613.002.055.
文摘Adaptive moving mesh research usually focuses either on analytical deriva-tions for prescribed solutions or on pragmatic solvers with challenging physical appli-cations. In the latter case, the monitor functions that steer mesh adaptation are oftendefined in an ad-hoc way. In this paper we generalize our previously used moni-tor function to a balanced sum of any number of monitor components. This avoidsthe trial-and-error parameter fine-tuning that is often used in monitor functions. Thekey reason for the new balancing method is that the ratio between the maximum andaverage value of a monitor component should ideally be equal for all components.Vorticity as a monitor component is a good motivating example for this. Entropy alsoturns out to be a very informative monitor component. We incorporate the monitorfunction in an adaptive moving mesh higher-order finite volume solver with HLLCfluxes, which is suitable for nonlinear hyperbolic systems of conservation laws. Whenapplied to compressible gas flow it produces very sharp results for shocks and otherdiscontinuities. Moreover, it captures small instabilities (Richtmyer-Meshkov, Kelvin-Helmholtz). Thus showing the rich nature of the example problems and the effective-ness of the new monitor balancing.
基金The work of A.Kurganov was supported in part by the National Natural Science Foundation of China grant 11771201by the fund of the Guangdong Provincial Key Laboratory of Computational Science and Material Design(No.2019B030301001).
文摘We introduce adaptive moving mesh central-upwind schemes for one-and two-dimensional hyperbolic systems of conservation and balance laws.The proposed methods consist of three steps.First,the solution is evolved by solving the studied system by the second-order semi-discrete central-upwind scheme on either the one-dimensional nonuniform grid or the two-dimensional structured quadrilateral mesh.When the evolution step is complete,the grid points are redistributed according to the moving mesh differential equation.Finally,the evolved solution is projected onto the new mesh in a conservative manner.The resulting adaptive moving mesh methods are applied to the one-and two-dimensional Euler equations of gas dynamics and granular hydrodynamics systems.Our numerical results demonstrate that in both cases,the adaptive moving mesh central-upwind schemes outperform their uniform mesh counterparts.
基金Supported by the National Natural Science Foundation of China( No.197710 6 2 )
文摘We consider an iterative algorithm of mesh optimization for finite element solution, and give an improved moving mesh strategy that reduces rapidly the complexity and cost of solving variational problems. A numerical result is presented for a 2-dimensional problem by the improved algorithm.
文摘This paper studies the convergence rates of a moving mesh implicit finite difference method with interpolation for partial differential equations (PDEs) with moving boundary arising in Asian option pricing. The moving mesh scheme is based on Rnnacher timestepping approach whose idea is running the implicit Euler schemes in the initial few steps and continuing with Crank-Nicolson schemes. With graded meshes for time direction and moving meshes for space direction, the fully discretized scheme is constructed using quadratic interpolation between two consecutive time level for the PDEs with moving boundary. The second-order convergence rates in both time and space are proved and numerical examples are carried out to confirm the theoretical results.
基金partly supported by SRF for ROCS, SEMsupported by a grant from the "project 211 (phase Ⅲ)" of the Southwestern University of Finance and Economics
文摘This paper develops and analyzes a moving mesh finite difference method for solving partial integro-differential equations. First, the time-dependent mapping of the coordinate transformation is approximated by a a piecewise linear function in time. Then, piecewise quadratic polynomial in space and an efficient method to discretize the memory term of the equation is designed using the moving mesh approach. In each time slice, a simple piecewise constant approximation of the integrand is used, and thus a quadrature is constructed for the memory term. The central finite difference scheme for space and the backward Euler scheme for time are used. The paper proves that the accumulation of the quadrature error is uniformly bounded and that the convergence of the method is second order in space and first order in time. Numerical experiments are carried out to confirm the theoretical predictions.
基金the Joint Applied Mathematics Research Institute of Peking University and Hong Kong Baptist University.Li was also partially supported by the National Basic Research Program of China under the grant 2005CB321701The research of Tang was supported by CERG Grants of Hong Kong Research Grant Council,FRG grants of Hong Kong Baptist University,and NSAF Grant#10476032 of National Science Foundation of China.He was supported in part by the Chinese Academy of Sciences while visiting its Institute of Computational Mathematics.
文摘In this paper, we present an adaptive moving mesh algorithm for meshesof unstructured polyhedra in three space dimensions. The algorithm automaticallyadjusts the size of the elements with time and position in the physical domain to resolvethe relevant scales in multiscale physical systems while minimizing computationalcosts. The algorithm is a generalization of the moving mesh methods basedon harmonic mappings developed by Li et al. [J. Comput. Phys., 170 (2001), pp. 562-588, and 177 (2002), pp. 365-393]. To make 3D moving mesh simulations possible,the key is to develop an efficient mesh redistribution procedure so that this part willcost as little as possible comparing with the solution evolution part. Since the meshredistribution procedure normally requires to solve large size matrix equations, wewill describe a procedure to decouple the matrix equation to a much simpler blocktridiagonaltype which can be efficiently solved by a particularly designed multi-gridmethod. To demonstrate the performance of the proposed 3D moving mesh strategy,the algorithm is implemented in finite element simulations of fluid-fluid interface interactionsin multiphase flows. To demonstrate the main ideas, we consider the formationof drops by using an energetic variational phase field model which describesthe motion of mixtures of two incompressible fluids. Numerical results on two- andthree-dimensional simulations will be presented.
基金supported by the National Natural Science Foundation of China(No.10925101,10828101)the Program for New Century Excellent Talents in University(NCET-07-0022)and the Doctoral Program of Education Ministry of China(No.20070001036).
文摘This paper extends the adaptive moving mesh method developed by Tang and Tang[36]to two-dimensional(2D)relativistic hydrodynamic(RHD)equations.The algorithm consists of two“independent”parts:the time evolution of the RHD equations and the(static)mesh iteration redistribution.In the first part,the RHD equations are discretized by using a high resolution finite volume scheme on the fixed but nonuniform meshes without the full characteristic decomposition of the governing equations.The second part is an iterative procedure.In each iteration,the mesh points are first redistributed,and then the cell averages of the conservative variables are remapped onto the new mesh in a conservative way.Several numerical examples are given to demonstrate the accuracy and effectiveness of the proposed method.
文摘In this paper, we present an adaptive moving mesh technique for solvingthe incompressible viscous flows using the vorticity stream-function formulation. Themoving mesh strategy is based on the approach proposed by Li et al. [J. Comput. Phys.,170 (2001), pp. 562–588] to separate the mesh-moving and evolving PDE at each timestep. The Navier-Stokes equations are solved in the vorticity stream-function form bya finite-volume method in space, and the mesh-moving part is realized by solving theEuler-Lagrange equations to minimize a certain variation in conjunction with a moresophisticated monitor function. A conservative interpolation is used to redistributethe numerical solutions on the new meshes. This paper discusses the implementationof the periodic boundary conditions, where the physical domain is allowed to deformwith time while the computational domain remains fixed and regular throughout. Numericalresults demonstrate the accuracy and effectiveness of the proposed algorithm.
基金supported in part by NSF(U.S.A.)under grants DMS-0712935 and DMS-1115118by NSERC(Canada)under discovery grant 311796.
文摘The numerical solution of the harmonic heat map flow problems with blowup in finite or infinite time is considered using an adaptive moving mesh method.A properly chosen monitor function is derived so that the moving mesh method can be used to simulate blowup and produce accurate blowup profiles which agree with formal asymptotic analysis.Moreover,the moving mesh method has finite time blowup when the underlying continuous problem does.In situations where the continuous problem has infinite time blowup,the moving mesh method exhibits finite time blowup with a blowup time tending to infinity as the number of mesh points increases.The inadequacy of a uniform mesh solution is clearly demonstrated.
基金The research of Yaguang Gu is funded by China Postdoctoral Science Foundation(2021M703040)The research of Dongmi Luo is supported by the National Natural Science Foundation of China(12101063)+3 种基金The research of Zhen Gao is supported by the National Natural Science Foundation of China(11871443)Shandong Provincial Qingchuang Science and Technology Project(2019KJI002)Fundamental Research Funds for the Central Universities(202042004)The research of Yibing Chen is supported by National Key Project(GJXM92579).
文摘The five-equation model of multi-component flows has been attracting much attention among researchers during the past twenty years for its potential in the study of the multi-component flows.In this paper,we employ a second order finite volume method with minmod limiter in spatial discretization,which preserves local extrema of certain physical quantities and is thus capable of simulating challenging test problems without introducing non-physical oscillations.Moreover,to improve the numerical resolution of the solutions,the adaptive moving mesh strategy proposed in[Huazhong Tang,Tao Tang,Adaptive mesh methods for one-and two-dimensional hyperbolic conservation laws,SINUM,41:487-515,2003]is applied.Furthermore,the proposed method can be proved to be capable of preserving the velocity and pressure when they are initially constant,which is essential in material interface capturing.Finally,several classical numerical examples demonstrate the effectiveness and robustness of the proposed method.
基金special funds for Major State Research Projects 2005CB1704National Science Foundation of China for Distinguished Young Scholars 10225103.
文摘The Doi-Hess equation that describes the evolution of an orientational dis-tribution function is capable of predicting several rheological features of nematic poly-mers.Since the orientational distribution function becomes sharply peaked as poten-tial intensity increases,powerful numerical methods become necessary in the relevant numerical simulations.In this paper,a numerical scheme based on the moving grid techniques will be designed to solve the orientational distribution functions with high potential intensities.Numerical experiments are carried out to demonstrate the effec-tiveness and robustness of the proposed scheme.
基金The authors are grateful to Professor Tao Tang for many helpful discussions.The research of the first author was supported by Hong Kong Baptist University through an RGC Grant.The second author was partially supported by the National Basic Research Program of China under the grant 2005CB321701 and the Joint Applied Mathematics Research Institute between Peking University and Hong Kong Baptist University.
文摘There have been several recent papers on developingmovingmeshmethodsfor solving phase-field equations. However, it is observed that some of these movingmesh solutions are essentially different from the solutions on very fine fixed meshes.One of the purposes of this paper is to understand the reason for the differences. Wecarried out numerical sensitivity studies systematically in this paper and it can be concludedthat for the phase-field equations, the numerical solutions are very sensitive tothe starting mesh and the monitor function. As a separate issue, an efficient alternatingCrank-Nicolson time discretization scheme is developed for solving the nonlinearsystem resulting from a finite element approximation to the phase-field equations.
基金This work was undertaken with the support of EPSRC Grant EP/D058791/1.R.Mahmood wishes to thank his employer PINSTECH for granting him study leave to carry out research work at Leeds.
文摘A distributed Lagrangian moving-mesh finite element method is applied to problems involving changes of phase.The algorithm uses a distributed conservation principle to determine nodal mesh velocities,which are then used to move the nodes.The nodal values are obtained from an ALE(Arbitrary Lagrangian-Eulerian)equation,which represents a generalization of the original algorithm presented in Applied Numerical Mathematics,54:450–469(2005).Having described the details of the generalized algorithm it is validated on two test cases from the original paper and is then applied to one-phase and,for the first time,two-phase Stefan problems in one and two space dimensions,paying particular attention to the implementation of the interface boundary conditions.Results are presented to demonstrate the accuracy and the effectiveness of the method,including comparisons against analytical solutions where available.
基金This work was partially supported by a grant of key program from the National Natural Science Foundation of China(No.10731060,10801120)National Basic Research Program of China(2011CB309704)Chinese Universities Scientific Fund No.2010QNA3019.
文摘This paper deals with the application of a moving mesh method for kinetic/hydrodynamic coupling model in two dimensions.With some criteria,the domain is dynamically decomposed into three parts:kinetic regions where fluids are far from equilibrium,hydrodynamic regions where fluids are near thermodynamical equilibrium and buffer regions which are used as a smooth transition.The Boltzmann-BGK equation is solved in kinetic regions,while Euler equations in hydrodynamic regions and both equations in buffer regions.By a well defined monitor function,our moving mesh method smoothly concentrate the mesh grids to the regions containing rapid variation of the solutions.In each moving mesh step,the solutions are conservatively updated to the new mesh and the cut-off function is rebuilt first to consist with the region decomposition after the mesh motion.In such a framework,the evolution of the hybrid model and the moving mesh procedure can be implemented independently,therefore keep the advantages of both approaches.Numerical examples are presented to demonstrate the efficiency of the method.
文摘We develop an efficient one-dimensional moving mesh algorithm for solving partial differential equations.The main contribution of this paper is to design an effective interpolation scheme based on L2-projection for the moving mesh method.The proposed method preserves not only the mass-conservation but also the first order momentum of the underlying numerical solution at each mesh redistribution step.Numerical examples are presented to demonstrate the effectiveness of the new interpolation technique.
基金co-supported by the Key Laboratory of Aerodynamic Noise Control,China(No.ANCL20190103)the State Key Laboratory of Aerodynamics,China(No.SKLA20180102)the Aeronautical Science Foundation of China(Nos.2018ZA52002 and 2019ZA052011)。
文摘Computational mesh is an important ingredient that affects the accuracy and efficiency of CFD numerical simulation.In light of the introduced large amount of computational costs for many adaptive mesh methods,moving mesh methods keep the number of nodes and topology of a mesh unchanged and do not increase CFD computational expense.As the state-of-the-art moving mesh method,the variational mesh adaptation approach has been introduced to CFD calculation.However,quickly estimating the flow field on the updated meshes during the iterative algorithm is challenging.A mesh optimization method,which embeds a machine learning regression model into the variational mesh adaptation,is proposed.The regression model captures the mapping between the initial mesh nodes and the flow field,so that the variational method could move mesh nodes iteratively by solving the mesh functional which is built from the estimated flow field on the updated mesh via the regression model.After the optimization,the density of the nodes in the high gradient area increases while the density in the low gradient area decreases.Benchmark examples are first used to verify the feasibility and effectiveness of the proposed method.And then we use the steady subsonic and transonic flows over cylinder and NACA0012 airfoil on unstructured triangular meshes to test our method.Results show that the proposed method significantly improves the accuracy of the local flow features on the adaptive meshes.Our work indicates that the proposed mesh optimization approach is promising for improving the accuracy and efficiency of CFD computation.
基金JKA through its promotion funds from KEIRIN RACE and by JSPS KAKENHI Grant Number 21K03856.
文摘This study proposes a method for modelling the flow interaction of multiple moving objects where the flow field variables are communicated between multiple separate moving computational domains.Instead of using the conventional approach with a single fixed computational domain covering the whole flow field,this method advances the moving computational domain(MCD)method in which the computational domain itself moves in line with the motions of an object inside.The computational domains created around each object move independently,and the flow fields of each domain interact where the flows cross.This eliminates the spatial restriction for simulating multiple moving objects.Firstly,a shock tube test verifies that the overset implementation and grid movement do not adversely affect the results and that there is communication between the grids.A second test case is conducted in which two spheres are crossed,and the forces exerted on one object due to the other’s crossing at a short distance are calculated.The results verify the reliability of this method and show that it is applicable to the flow interaction of multiple moving objects.