We construct an unconventional divergence preserving discretization of updated Lagrangian ideal magnetohydrodynamics(MHD)over simplicial grids.The cell-centered finite-volume(FV)method employed to discretize the conse...We construct an unconventional divergence preserving discretization of updated Lagrangian ideal magnetohydrodynamics(MHD)over simplicial grids.The cell-centered finite-volume(FV)method employed to discretize the conservation laws of volume,momentum,and total energy is rigorously the same as the one developed to simulate hyperelasticity equations.By construction this moving mesh method ensures the compatibility between the mesh displacement and the approximation of the volume flux by means of the nodal velocity and the attached unit corner normal vector which is nothing but the partial derivative of the cell volume with respect to the node coordinate under consideration.This is precisely the definition of the compatibility with the Geometrical Conservation Law which is the cornerstone of any proper multi-dimensional moving mesh FV discretization.The momentum and the total energy fluxes are approximated utilizing the partition of cell faces into sub-faces and the concept of sub-face force which is the traction force attached to each sub-face impinging at a node.We observe that the time evolution of the magnetic field might be simply expressed in terms of the deformation gradient which characterizes the Lagrange-to-Euler mapping.In this framework,the divergence of the magnetic field is conserved with respect to time thanks to the Piola formula.Therefore,we solve the fully compatible updated Lagrangian discretization of the deformation gradient tensor for updating in a simple manner the cell-centered value of the magnetic field.Finally,the sub-face traction force is expressed in terms of the nodal velocity to ensure a semi-discrete entropy inequality within each cell.The conservation of momentum and total energy is recovered prescribing the balance of all the sub-face forces attached to the sub-faces impinging at a given node.This balance corresponds to a vectorial system satisfied by the nodal velocity.It always admits a unique solution which provides the nodal velocity.The robustness and the accuracy of this unconventional FV scheme have been demonstrated by employing various representative test cases.Finally,it is worth emphasizing that once you have an updated Lagrangian code for solving hyperelasticity you also get an almost free updated Lagrangian code for solving ideal MHD ensuring exactly the compatibility with the involution constraint for the magnetic field at the discrete level.展开更多
In this study,the cylindrical finite-volume method(FVM)is advanced for the efficient and high-precision simulation of the logging while drilling(LWD)orthogonal azimuth electromagnetic tool(OAEMT)response in a three-di...In this study,the cylindrical finite-volume method(FVM)is advanced for the efficient and high-precision simulation of the logging while drilling(LWD)orthogonal azimuth electromagnetic tool(OAEMT)response in a three-dimensional(3 D)anisotropic formation.To overcome the ill-condition and convergence problems arising from the low induction number,Maxwell’s equations are reformulated into a mixed Helmholtz equation for the coupled potentials in a cylindrical coordinate system.The electrical fi eld continuation method is applied to approximate the perfectly electrical conducting(PEC)boundary condition,to improve the discretization accuracy of the Helmholtz equation on the surface of metal mandrels.On the base,the 3 D FVM on Lebedev’s staggered grids in the cylindrical coordinates is employed to discretize the mixed equations to ensure good conformity with typical well-logging tool geometries.The equivalent conductivity in a non-uniform element is determined by a standardization technique.The direct solver,PARDISO,is applied to efficiently solve the sparse linear equation systems for the multi-transmitter problem.To reduce the number of calls to PARDISO,the whole computational domain is divided into small windows that contain multiple measuring points.The electromagnetic(EM)solutions produced by all the transmitters per window are simultaneously solved because the discrete matrix,relevant to all the transmitters in the same window,is changed.Finally,the 3 D FVM is validated against the numerical mode matching method(NMM),and the characteristics of both the coaxial and coplanar responses of the EM field tool are investigated using the numerical results.展开更多
A finite-volume charge method has been proposed to simulate PIN diodes and insulated-gate bipolar transistor(IGBT)devices using SPICE simulators by extending the lumped-charge method.The new method assumes local quasi...A finite-volume charge method has been proposed to simulate PIN diodes and insulated-gate bipolar transistor(IGBT)devices using SPICE simulators by extending the lumped-charge method.The new method assumes local quasi-neutrality in the undepleted N^(-)base region and uses the total collector current,the nodal hole density and voltage as the basic quantities.In SPICE implementation,it makes clear and accurate definitions of three kinds of nodes—the carrier density nodes,the voltage nodes and the current generator nodes—in the undepleted N^(-)base region.It uses central finite difference to approximate electron and hole current generators and sets up the current continuity equation in a control volume for every carrier density node in the undepleted N^(-)base region.It is easy to increase the number of nodes to describe the fast spatially varying carrier density in transient processes.We use this method to simulate IGBT devices in SPICE simulators and get a good agreement with technology computer-aided design simulations.展开更多
In this paper, a stochastic finite-volume solver based on polynomial chaos expansion is developed. The upwind scheme is used to avoid the numerical instabilities. The Burgers’ equation subjected to deterministic boun...In this paper, a stochastic finite-volume solver based on polynomial chaos expansion is developed. The upwind scheme is used to avoid the numerical instabilities. The Burgers’ equation subjected to deterministic boundary conditions and random viscosity is solved. The solution uncertainty is quantified for different values of viscosity. Monte-Carlo simulations are used to validate and compare the developed solver. The mean, standard deviation and the probability distribution function (p.d.f) of the stochastic Burgers’ solution is quantified and the effect of some parameters is investigated. The large sparse linear system resulting from the stochastic solver is solved in parallel to enhance the performance. Also, Monte-Carlo simulations are done in parallel and the execution times are compared in both cases.展开更多
An unstructured finite-volume numerical algorithm was presented for solution of the two-dimensional shallow water equations, based on triangular or arbitrary quadrilateral meshes. The Roe type approximate Riemann solv...An unstructured finite-volume numerical algorithm was presented for solution of the two-dimensional shallow water equations, based on triangular or arbitrary quadrilateral meshes. The Roe type approximate Riemann solver was used to the system. A second-order TVD scheme with the van Leer limiter was used in the space discretization and a two-step Runge-Kutta approach was used in the time discretization. An upwind, as opposed to a pointwise, treatment of the slope source terms was adopted and the semi-implicit treatment was used for the friction source terms. Verification for two-dimension dam-break problems are carried out by comparing the present results with others and very good agreement is shown.展开更多
A finite-volume Total Variation Diminishing (TVD) scheme is presented formodeling dam-break flows in open channels. This method is used for solving the 2D shallow waterequations on arbitrary quadrilateral meshes, base...A finite-volume Total Variation Diminishing (TVD) scheme is presented formodeling dam-break flows in open channels. This method is used for solving the 2D shallow waterequations on arbitrary quadrilateral meshes, based upon a second-order hybrid TVD scheme with anoptimum-selected limiter in the space discretization and a two-step Runge-Kutta approach in the timediscretization. Verification for a circular dam-break problem is carried out by comparing thepresent results with others and very good agreement is shown. The present algorithm is then used topredict dam-break flow characteristics in open channels such as in furcated channels. Morecomplicated unsteady flow characteristics in these furcated channels than in the regular channelsstudied previously can observed in this work.展开更多
We propose a positivity preserving entropy decreasing finite volume scheme for nonlinear nonlocal equations with a gradient flow structure.These properties allow for accurate computations of stationary states and long...We propose a positivity preserving entropy decreasing finite volume scheme for nonlinear nonlocal equations with a gradient flow structure.These properties allow for accurate computations of stationary states and long-time asymptotics demonstrated by suitably chosen test cases in which these features of the scheme are essential.The proposed scheme is able to cope with non-smooth stationary states,different time scales including metastability,as well as concentrations and self-similar behavior induced by singular nonlocal kernels.We use the scheme to explore properties of these equations beyond their present theoretical knowledge.展开更多
We propose finite-volume schemes for the Cahn-Hilliard equation which unconditionally and discretely preserve the boundedness of the phase field and the dissipation of the free energy.Our numerical framework is applic...We propose finite-volume schemes for the Cahn-Hilliard equation which unconditionally and discretely preserve the boundedness of the phase field and the dissipation of the free energy.Our numerical framework is applicable to a variety of free-energy potentials,including Ginzburg-Landau and Flory-Huggins,to general wetting boundary conditions,and to degenerate mobilities.Its central thrust is the upwind methodology,which we combine with a semi-implicit formulation for the freeenergy terms based on the classical convex-splitting approach.The extension of the schemes to an arbitrary number of dimensions is straightforward thanks to their dimensionally split nature,which allows to efficiently solve higher-dimensional problems with a simple parallelisation.The numerical schemes are validated and tested through a variety of examples,in different dimensions,and with various contact angles between droplets and substrates.展开更多
High-resolution global non-hydrostatic gridded dynamic models have drawn significant attention in recent years in conjunction with the rising demand for improving weather forecasting and climate predictions.By far it ...High-resolution global non-hydrostatic gridded dynamic models have drawn significant attention in recent years in conjunction with the rising demand for improving weather forecasting and climate predictions.By far it is still challenging to build a high-resolution gridded global model,which is required to meet numerical accuracy,dispersion relation,conservation,and computation requirements.Among these requirements,this review focuses on one significant topic—the numerical accuracy over the entire non-uniform spherical grids.The paper discusses all the topic-related challenges by comparing the schemes adopted in well-known finite-volume-based operational or research dynamical cores.It provides an overview of how these challenges are met in a summary table.The analysis and validation in this review are based on the shallow-water equation system.The conclusions can be applied to more complicated models.These challenges should be critical research topics in the future development of finite-volume global models.展开更多
In this study,the Lattice Boltzmann Method(LBM)is implemented through a finite-volume approach to perform 2-D,incompressible,and turbulent fluid flow analyses on structured grids.Even though the approach followed in t...In this study,the Lattice Boltzmann Method(LBM)is implemented through a finite-volume approach to perform 2-D,incompressible,and turbulent fluid flow analyses on structured grids.Even though the approach followed in this study necessitates more computational effort compared to the standard LBM(the so called stream and collide scheme),using the finite-volume method,the known limitations of the stream and collide scheme on lattice to be uniform and Courant-Friedrichs-Lewy(CFL)number to be one are removed.Moreover,the curved boundaries in the computational domain are handled more accurately with less effort.These improvements pave the way for the possibility of solving fluid flow problems with the LBM using coarser grids that are refined only where it is necessary and the boundary layers might be resolved better.展开更多
We present our results by using a machine learning(ML)approach for the solution of the Riemann problem for the Euler equations of fluid dynamics.The Riemann problem is an initial-value problem with piecewise-constant ...We present our results by using a machine learning(ML)approach for the solution of the Riemann problem for the Euler equations of fluid dynamics.The Riemann problem is an initial-value problem with piecewise-constant initial data and it represents a mathematical model of the shock tube.The solution of the Riemann problem is the building block for many numerical algorithms in computational fluid dynamics,such as finite-volume or discontinuous Galerkin methods.Therefore,a fast and accurate approximation of the solution of the Riemann problem and construction of the associated numerical fluxes is of crucial importance.The exact solution of the shock tube problem is fully described by the intermediate pressure and mathematically reduces to finding a solution of a nonlinear equation.Prior to delving into the complexities of ML for the Riemann problem,we consider a much simpler formulation,yet very informative,problem of learning roots of quadratic equations based on their coefficients.We compare two approaches:(i)Gaussian process(GP)regressions,and(ii)neural network(NN)approximations.Among these approaches,NNs prove to be more robust and efficient,although GP can be appreciably more accurate(about 30\%).We then use our experience with the quadratic equation to apply the GP and NN approaches to learn the exact solution of the Riemann problem from the initial data or coefficients of the gas equation of state(EOS).We compare GP and NN approximations in both regression and classification analysis and discuss the potential benefits and drawbacks of the ML approach.展开更多
We analyze mimetic properties of a conservative finite-volume (FV) scheme on polygonal meshes used for modeling solute transport on a surface with variable elevation. Polygonal meshes not only provide enormous mesh ge...We analyze mimetic properties of a conservative finite-volume (FV) scheme on polygonal meshes used for modeling solute transport on a surface with variable elevation. Polygonal meshes not only provide enormous mesh generation flexibility, but also tend to improve stability properties of numerical schemes and reduce bias towards any particular mesh direction. The mathematical model is given by a system of weakly coupled shallow water and linear transport equations. The equations are discretized using different explicit cell-centered FV schemes for flow and transport subsystems with different time steps. The discrete shallow water scheme is well balanced and preserves the positivity of the water depth. We provide a rigorous estimate of a stable time step for the shallow water and transport scheme and prove a bounds-preserving property of the solute concentration. The scheme is second-order accurate over fully wet regions and first-order accurate over partially wet or dry regions. Theoretical results are verified with numerical experiments on rectangular, triangular, and polygonal meshes.展开更多
In this paper,we develop new high-order numerical methods for hyperbolic systems of nonlinear partial differential equations(PDEs)with uncertainties.The new approach is realized in the semi-discrete finite-volume fram...In this paper,we develop new high-order numerical methods for hyperbolic systems of nonlinear partial differential equations(PDEs)with uncertainties.The new approach is realized in the semi-discrete finite-volume framework and is based on fifth-order weighted essentially non-oscillatory(WENO)interpolations in(multidimensional)random space combined with second-order piecewise linear reconstruction in physical space.Compared with spectral approximations in the random space,the presented methods are essentially non-oscillatory as they do not suffer from the Gibbs phenomenon while still achieving high-order accuracy.The new methods are tested on a number of numerical examples for both the Euler equations of gas dynamics and the Saint-Venant system of shallow-water equations.In the latter case,the methods are also proven to be well-balanced and positivity-preserving.展开更多
A novel numerical scheme to solve two coupled systems of conservation laws is introduced.The scheme is derived based on a relaxation approach and does not require information on the Lax curves of the coupled systems,w...A novel numerical scheme to solve two coupled systems of conservation laws is introduced.The scheme is derived based on a relaxation approach and does not require information on the Lax curves of the coupled systems,which simplifies the computation of suitable coupling data.The coupling condition for the underlying relaxation system plays a crucial role as it determines the behaviour of the scheme in the zero relaxation limit.The role of this condition is discussed,a consistency concept with respect to the original problem is introduced,the well-posedness is analyzed and explicit,nodal Riemann solvers are provided.Based on a case study considering the p-system of gas dynamics,a strategy for the design of the relaxation coupling condition within the new scheme is provided.展开更多
This paper presents a high-order discontinuous Galerkin(DG)finite-element method to solve the barotropic version of the conservative symmetric hyperbolic and thermodynamically compatible(SHTC)model of compressible two...This paper presents a high-order discontinuous Galerkin(DG)finite-element method to solve the barotropic version of the conservative symmetric hyperbolic and thermodynamically compatible(SHTC)model of compressible two-phase flow,introduced by Romenski et al.in[59,62],in multiple space dimensions.In the absence of algebraic source terms,the model is endowed with a curl constraint on the relative velocity field.In this paper,the hyperbolicity of the system is studied for the first time in the multidimensional case,showing that the original model is only weakly hyperbolic in multiple space dimensions.To restore the strong hyperbolicity,two different methodologies are used:(i)the explicit symmetrization of the system,which can be achieved by adding terms that contain linear combinations of the curl involution,similar to the Godunov-Powell terms in the MHD equations;(ii)the use of the hyperbolic generalized Lagrangian multiplier(GLM)curl-cleaning approach forwarded.The PDE system is solved using a high-order ADER-DG method with a posteriori subcell finite-volume limiter to deal with shock waves and the steep gradients in the volume fraction commonly appearing in the solutions of this type of model.To illustrate the performance of the method,several different test cases and benchmark problems have been run,showing the high order of the scheme and the good agreement when compared to reference solutions computed with other well-known methods.展开更多
For a complex flow about multi-element airfoils a mixed grid method is set up. C-type grids are produced on each element′s body and in their wakes at first, O-type grids are given in the outmost area, and H-type grid...For a complex flow about multi-element airfoils a mixed grid method is set up. C-type grids are produced on each element′s body and in their wakes at first, O-type grids are given in the outmost area, and H-type grids are used in middle additional areas. An algebra method is used to produce the initial grids in each area. And the girds are optimized by elliptical differential equation method. Then C-O-H zonal patched grids around multi-element airfoils are produced automatically and efficiently. A time accurate finite-volume integration method is used to solve the compressible laminar and turbulent Navier-Stokes (N-S) equations on the grids. Computational results prove the method to be effective.展开更多
A new unified macro- and micro-mechanics failure analysis method for composite structures was developed in order to take the effects of composite micro structure into consideration. In this method, the macro stress di...A new unified macro- and micro-mechanics failure analysis method for composite structures was developed in order to take the effects of composite micro structure into consideration. In this method, the macro stress distribution of composite structure was calculated by commercial finite element analysis software. According to the macro stress distribution, the damage point was searched and the micro-stress distribution was calculated by reformulated finite-volume direct averaging micromechanics （FVDAM）, which was a multi-scale finite element method for composite. The micro structure failure modes were estimated with the failure strength of constituents. A unidirectional composite plate with a circular hole in the center under two kinds of loads was analyzed with the traditional macro-mechanical failure analysis method and the unified macro- and micro-mechanics failure analysis method. The results obtained by the two methods are consistent, which show this new method＇s accuracy and efficiency.展开更多
基金support by Fondazione Cariplo and Fondazione CDP(Italy)under the project No.2022-1895.
文摘We construct an unconventional divergence preserving discretization of updated Lagrangian ideal magnetohydrodynamics(MHD)over simplicial grids.The cell-centered finite-volume(FV)method employed to discretize the conservation laws of volume,momentum,and total energy is rigorously the same as the one developed to simulate hyperelasticity equations.By construction this moving mesh method ensures the compatibility between the mesh displacement and the approximation of the volume flux by means of the nodal velocity and the attached unit corner normal vector which is nothing but the partial derivative of the cell volume with respect to the node coordinate under consideration.This is precisely the definition of the compatibility with the Geometrical Conservation Law which is the cornerstone of any proper multi-dimensional moving mesh FV discretization.The momentum and the total energy fluxes are approximated utilizing the partition of cell faces into sub-faces and the concept of sub-face force which is the traction force attached to each sub-face impinging at a node.We observe that the time evolution of the magnetic field might be simply expressed in terms of the deformation gradient which characterizes the Lagrange-to-Euler mapping.In this framework,the divergence of the magnetic field is conserved with respect to time thanks to the Piola formula.Therefore,we solve the fully compatible updated Lagrangian discretization of the deformation gradient tensor for updating in a simple manner the cell-centered value of the magnetic field.Finally,the sub-face traction force is expressed in terms of the nodal velocity to ensure a semi-discrete entropy inequality within each cell.The conservation of momentum and total energy is recovered prescribing the balance of all the sub-face forces attached to the sub-faces impinging at a given node.This balance corresponds to a vectorial system satisfied by the nodal velocity.It always admits a unique solution which provides the nodal velocity.The robustness and the accuracy of this unconventional FV scheme have been demonstrated by employing various representative test cases.Finally,it is worth emphasizing that once you have an updated Lagrangian code for solving hyperelasticity you also get an almost free updated Lagrangian code for solving ideal MHD ensuring exactly the compatibility with the involution constraint for the magnetic field at the discrete level.
基金supported jointly by Strategic Pilot Science and Technology Project of Chinese Academy of Sciences (No. XDA14020102)National key research and development plan (No. 2017YFC0601805)+5 种基金National Natural Science Foundation of China (No. 41574110)Youth Foundation of Hebei Educational Committee (No. QN2018217)Hebei Higher Education Teaching Reform Research and Practice(No. 2018GJJG328)Zhangjiakou science and technology bureau(No. 1821011B)Doctoral Fund of Hebei Institute of Architecture and Civil Engineering (No. B-201606)Academic Team Innovation Ability Improvement Project of Hebei Institute of Architecture and Civil Engineering(TD202011)。
文摘In this study,the cylindrical finite-volume method(FVM)is advanced for the efficient and high-precision simulation of the logging while drilling(LWD)orthogonal azimuth electromagnetic tool(OAEMT)response in a three-dimensional(3 D)anisotropic formation.To overcome the ill-condition and convergence problems arising from the low induction number,Maxwell’s equations are reformulated into a mixed Helmholtz equation for the coupled potentials in a cylindrical coordinate system.The electrical fi eld continuation method is applied to approximate the perfectly electrical conducting(PEC)boundary condition,to improve the discretization accuracy of the Helmholtz equation on the surface of metal mandrels.On the base,the 3 D FVM on Lebedev’s staggered grids in the cylindrical coordinates is employed to discretize the mixed equations to ensure good conformity with typical well-logging tool geometries.The equivalent conductivity in a non-uniform element is determined by a standardization technique.The direct solver,PARDISO,is applied to efficiently solve the sparse linear equation systems for the multi-transmitter problem.To reduce the number of calls to PARDISO,the whole computational domain is divided into small windows that contain multiple measuring points.The electromagnetic(EM)solutions produced by all the transmitters per window are simultaneously solved because the discrete matrix,relevant to all the transmitters in the same window,is changed.Finally,the 3 D FVM is validated against the numerical mode matching method(NMM),and the characteristics of both the coaxial and coplanar responses of the EM field tool are investigated using the numerical results.
文摘A finite-volume charge method has been proposed to simulate PIN diodes and insulated-gate bipolar transistor(IGBT)devices using SPICE simulators by extending the lumped-charge method.The new method assumes local quasi-neutrality in the undepleted N^(-)base region and uses the total collector current,the nodal hole density and voltage as the basic quantities.In SPICE implementation,it makes clear and accurate definitions of three kinds of nodes—the carrier density nodes,the voltage nodes and the current generator nodes—in the undepleted N^(-)base region.It uses central finite difference to approximate electron and hole current generators and sets up the current continuity equation in a control volume for every carrier density node in the undepleted N^(-)base region.It is easy to increase the number of nodes to describe the fast spatially varying carrier density in transient processes.We use this method to simulate IGBT devices in SPICE simulators and get a good agreement with technology computer-aided design simulations.
文摘In this paper, a stochastic finite-volume solver based on polynomial chaos expansion is developed. The upwind scheme is used to avoid the numerical instabilities. The Burgers’ equation subjected to deterministic boundary conditions and random viscosity is solved. The solution uncertainty is quantified for different values of viscosity. Monte-Carlo simulations are used to validate and compare the developed solver. The mean, standard deviation and the probability distribution function (p.d.f) of the stochastic Burgers’ solution is quantified and the effect of some parameters is investigated. The large sparse linear system resulting from the stochastic solver is solved in parallel to enhance the performance. Also, Monte-Carlo simulations are done in parallel and the execution times are compared in both cases.
文摘An unstructured finite-volume numerical algorithm was presented for solution of the two-dimensional shallow water equations, based on triangular or arbitrary quadrilateral meshes. The Roe type approximate Riemann solver was used to the system. A second-order TVD scheme with the van Leer limiter was used in the space discretization and a two-step Runge-Kutta approach was used in the time discretization. An upwind, as opposed to a pointwise, treatment of the slope source terms was adopted and the semi-implicit treatment was used for the friction source terms. Verification for two-dimension dam-break problems are carried out by comparing the present results with others and very good agreement is shown.
文摘A finite-volume Total Variation Diminishing (TVD) scheme is presented formodeling dam-break flows in open channels. This method is used for solving the 2D shallow waterequations on arbitrary quadrilateral meshes, based upon a second-order hybrid TVD scheme with anoptimum-selected limiter in the space discretization and a two-step Runge-Kutta approach in the timediscretization. Verification for a circular dam-break problem is carried out by comparing thepresent results with others and very good agreement is shown. The present algorithm is then used topredict dam-break flow characteristics in open channels such as in furcated channels. Morecomplicated unsteady flow characteristics in these furcated channels than in the regular channelsstudied previously can observed in this work.
基金JAC acknowledges support from projects MTM2011-27739-C04-02,2009-SGR-345 from Agencia de Gestio d’Ajuts Universitaris i de Recerca-Generalitat de Catalunya,and the Royal Society through a Wolfson Research Merit AwardJAC and YH were supported by Engineering and Physical Sciences Research Council(UK)grant number EP/K008404/1+1 种基金The work of AC was supported in part by the NSF Grant DMS-1115682The authors also acknowledge the support by NSF RNMS grant DMS-1107444.
文摘We propose a positivity preserving entropy decreasing finite volume scheme for nonlinear nonlocal equations with a gradient flow structure.These properties allow for accurate computations of stationary states and long-time asymptotics demonstrated by suitably chosen test cases in which these features of the scheme are essential.The proposed scheme is able to cope with non-smooth stationary states,different time scales including metastability,as well as concentrations and self-similar behavior induced by singular nonlocal kernels.We use the scheme to explore properties of these equations beyond their present theoretical knowledge.
基金supported by Labex CEMPI(ANR-11-LABX-0007-01).RBJAC were supported by the ERC Advanced Grant No.883363(Nonlocal PDEs for Complex Particle Dynamics(Nonlocal-CPD):Phase Transitions,Patterns and Synchronization)under the European Union’s Horizon 2020 research and innovation programme+2 种基金JAC was partially supported by EPSRC Grants No.EP/V051121/1(Stability analysis for non-linear partial differential equations across multiscale applications)under the EPSRC lead agency agreement with the NSF,and EP/T022132/1(Spectral element methods for fractional differential equations,with applications in applied analysis and medical imaging)SK was partially supported by EPSRC Platform Grant No.EP/L020564/1(Multiscale Analysis of Complex Interfacial Phenomena(MACIPh):Coarse graining,Molecular modelling,stochasticity,and experimentation)EPSRC Grant No.EP/L027186/1(Fluid processes in smart microengineered devices:Hydrodynamics and thermodynamics in microspace).SPP acknowledges financial support from the Imperial College President’s PhD Scholarship scheme.
文摘We propose finite-volume schemes for the Cahn-Hilliard equation which unconditionally and discretely preserve the boundedness of the phase field and the dissipation of the free energy.Our numerical framework is applicable to a variety of free-energy potentials,including Ginzburg-Landau and Flory-Huggins,to general wetting boundary conditions,and to degenerate mobilities.Its central thrust is the upwind methodology,which we combine with a semi-implicit formulation for the freeenergy terms based on the classical convex-splitting approach.The extension of the schemes to an arbitrary number of dimensions is straightforward thanks to their dimensionally split nature,which allows to efficiently solve higher-dimensional problems with a simple parallelisation.The numerical schemes are validated and tested through a variety of examples,in different dimensions,and with various contact angles between droplets and substrates.
基金Supported by the National Key Research and Development Program of China(2017YFC1502201)Basic Scientific Research and Operation Fund of Chinese Academy of Meteorological Sciences(2017Z017)。
文摘High-resolution global non-hydrostatic gridded dynamic models have drawn significant attention in recent years in conjunction with the rising demand for improving weather forecasting and climate predictions.By far it is still challenging to build a high-resolution gridded global model,which is required to meet numerical accuracy,dispersion relation,conservation,and computation requirements.Among these requirements,this review focuses on one significant topic—the numerical accuracy over the entire non-uniform spherical grids.The paper discusses all the topic-related challenges by comparing the schemes adopted in well-known finite-volume-based operational or research dynamical cores.It provides an overview of how these challenges are met in a summary table.The analysis and validation in this review are based on the shallow-water equation system.The conclusions can be applied to more complicated models.These challenges should be critical research topics in the future development of finite-volume global models.
文摘In this study,the Lattice Boltzmann Method(LBM)is implemented through a finite-volume approach to perform 2-D,incompressible,and turbulent fluid flow analyses on structured grids.Even though the approach followed in this study necessitates more computational effort compared to the standard LBM(the so called stream and collide scheme),using the finite-volume method,the known limitations of the stream and collide scheme on lattice to be uniform and Courant-Friedrichs-Lewy(CFL)number to be one are removed.Moreover,the curved boundaries in the computational domain are handled more accurately with less effort.These improvements pave the way for the possibility of solving fluid flow problems with the LBM using coarser grids that are refined only where it is necessary and the boundary layers might be resolved better.
基金This work was performed under the auspices of the National Nuclear Security Administration of the US Department of Energy at Los Alamos National Laboratory under Contract No.DE-AC52-06NA25396The authors gratefully acknowledge the support of the US Department of Energy National Nuclear Security Administration Advanced Simulation and Computing Program.The Los Alamos unlimited release number is LA-UR-19-32257.
文摘We present our results by using a machine learning(ML)approach for the solution of the Riemann problem for the Euler equations of fluid dynamics.The Riemann problem is an initial-value problem with piecewise-constant initial data and it represents a mathematical model of the shock tube.The solution of the Riemann problem is the building block for many numerical algorithms in computational fluid dynamics,such as finite-volume or discontinuous Galerkin methods.Therefore,a fast and accurate approximation of the solution of the Riemann problem and construction of the associated numerical fluxes is of crucial importance.The exact solution of the shock tube problem is fully described by the intermediate pressure and mathematically reduces to finding a solution of a nonlinear equation.Prior to delving into the complexities of ML for the Riemann problem,we consider a much simpler formulation,yet very informative,problem of learning roots of quadratic equations based on their coefficients.We compare two approaches:(i)Gaussian process(GP)regressions,and(ii)neural network(NN)approximations.Among these approaches,NNs prove to be more robust and efficient,although GP can be appreciably more accurate(about 30\%).We then use our experience with the quadratic equation to apply the GP and NN approaches to learn the exact solution of the Riemann problem from the initial data or coefficients of the gas equation of state(EOS).We compare GP and NN approximations in both regression and classification analysis and discuss the potential benefits and drawbacks of the ML approach.
基金This work was carried out under the auspices of the National Nuclear Security Administration of the U.S.Department of Energy at Los Alamos National Laboratory under Contract No.DE-AC52-06NA25396The Los Alamos unlimited release number is LA-UR-22-30864.
文摘We analyze mimetic properties of a conservative finite-volume (FV) scheme on polygonal meshes used for modeling solute transport on a surface with variable elevation. Polygonal meshes not only provide enormous mesh generation flexibility, but also tend to improve stability properties of numerical schemes and reduce bias towards any particular mesh direction. The mathematical model is given by a system of weakly coupled shallow water and linear transport equations. The equations are discretized using different explicit cell-centered FV schemes for flow and transport subsystems with different time steps. The discrete shallow water scheme is well balanced and preserves the positivity of the water depth. We provide a rigorous estimate of a stable time step for the shallow water and transport scheme and prove a bounds-preserving property of the solute concentration. The scheme is second-order accurate over fully wet regions and first-order accurate over partially wet or dry regions. Theoretical results are verified with numerical experiments on rectangular, triangular, and polygonal meshes.
基金supported in part by the NSF grant DMS-2208438.The work of M.Herty was supported in part by the DFG(German Research Foundation)through 20021702/GRK2326,333849990/IRTG-2379,HE5386/18-1,19-2,22-1,23-1under Germany’s Excellence Strategy EXC-2023 Internet of Production 390621612+1 种基金The work of A.Kurganov was supported in part by the NSFC grant 12171226the fund of the Guangdong Provincial Key Laboratory of Computational Science and Material Design,China(No.2019B030301001).
文摘In this paper,we develop new high-order numerical methods for hyperbolic systems of nonlinear partial differential equations(PDEs)with uncertainties.The new approach is realized in the semi-discrete finite-volume framework and is based on fifth-order weighted essentially non-oscillatory(WENO)interpolations in(multidimensional)random space combined with second-order piecewise linear reconstruction in physical space.Compared with spectral approximations in the random space,the presented methods are essentially non-oscillatory as they do not suffer from the Gibbs phenomenon while still achieving high-order accuracy.The new methods are tested on a number of numerical examples for both the Euler equations of gas dynamics and the Saint-Venant system of shallow-water equations.In the latter case,the methods are also proven to be well-balanced and positivity-preserving.
基金Funding Open Access funding enabled and organized by Projekt DEALthe Deutsche Forschungsgemeinschaft(DFG,German Research Foundation)for the financial support through 320021702/GRK2326,333849990/IRTG-2379,B04,B05,and B06 of 442047500/SFB1481,HE5386/18-1,19-2,22-1,23-1,25-1,ERS SFDdM035 and under Germany’s Excellence Strategy EXC-2023 Internet of Production 390621612 and under the Excellence Strategy of the Federal Government and the Länder.Support through the EU DATAHYKING is also acknowledged.
文摘A novel numerical scheme to solve two coupled systems of conservation laws is introduced.The scheme is derived based on a relaxation approach and does not require information on the Lax curves of the coupled systems,which simplifies the computation of suitable coupling data.The coupling condition for the underlying relaxation system plays a crucial role as it determines the behaviour of the scheme in the zero relaxation limit.The role of this condition is discussed,a consistency concept with respect to the original problem is introduced,the well-posedness is analyzed and explicit,nodal Riemann solvers are provided.Based on a case study considering the p-system of gas dynamics,a strategy for the design of the relaxation coupling condition within the new scheme is provided.
基金Initiative 2018–2027 attributed to DICAM of the University of Trento(grant L.232/2016)the PRIN 2022 project High-order structure-preserving semi-implicit schemes for hyperbolic equations and by the European Union-Next GenerationEU(PNRR,Spoke 7 CN HPC).
文摘This paper presents a high-order discontinuous Galerkin(DG)finite-element method to solve the barotropic version of the conservative symmetric hyperbolic and thermodynamically compatible(SHTC)model of compressible two-phase flow,introduced by Romenski et al.in[59,62],in multiple space dimensions.In the absence of algebraic source terms,the model is endowed with a curl constraint on the relative velocity field.In this paper,the hyperbolicity of the system is studied for the first time in the multidimensional case,showing that the original model is only weakly hyperbolic in multiple space dimensions.To restore the strong hyperbolicity,two different methodologies are used:(i)the explicit symmetrization of the system,which can be achieved by adding terms that contain linear combinations of the curl involution,similar to the Godunov-Powell terms in the MHD equations;(ii)the use of the hyperbolic generalized Lagrangian multiplier(GLM)curl-cleaning approach forwarded.The PDE system is solved using a high-order ADER-DG method with a posteriori subcell finite-volume limiter to deal with shock waves and the steep gradients in the volume fraction commonly appearing in the solutions of this type of model.To illustrate the performance of the method,several different test cases and benchmark problems have been run,showing the high order of the scheme and the good agreement when compared to reference solutions computed with other well-known methods.
文摘For a complex flow about multi-element airfoils a mixed grid method is set up. C-type grids are produced on each element′s body and in their wakes at first, O-type grids are given in the outmost area, and H-type grids are used in middle additional areas. An algebra method is used to produce the initial grids in each area. And the girds are optimized by elliptical differential equation method. Then C-O-H zonal patched grids around multi-element airfoils are produced automatically and efficiently. A time accurate finite-volume integration method is used to solve the compressible laminar and turbulent Navier-Stokes (N-S) equations on the grids. Computational results prove the method to be effective.
基金co-supported by National Basic Research Program of China, National Natural Science Foundation of China(No. 51075204)Aeronautical Science Foundation of China (No.2009ZB52028, No. 2012ZB52026)+1 种基金Research Fund for the Doctoral Program of Higher Education of China (No. 20070287039)NUAA Research Funding (No. NZ2012106)
文摘A new unified macro- and micro-mechanics failure analysis method for composite structures was developed in order to take the effects of composite micro structure into consideration. In this method, the macro stress distribution of composite structure was calculated by commercial finite element analysis software. According to the macro stress distribution, the damage point was searched and the micro-stress distribution was calculated by reformulated finite-volume direct averaging micromechanics （FVDAM）, which was a multi-scale finite element method for composite. The micro structure failure modes were estimated with the failure strength of constituents. A unidirectional composite plate with a circular hole in the center under two kinds of loads was analyzed with the traditional macro-mechanical failure analysis method and the unified macro- and micro-mechanics failure analysis method. The results obtained by the two methods are consistent, which show this new method＇s accuracy and efficiency.
