We implemented a two-step Asymmetric Perfectly Matched Layer (APML) model in High-Order Finite Difference Time Domain (FDTD) algorithm for solving two-dimensional Maxwell’s equations. Initially, we applied the APML m...We implemented a two-step Asymmetric Perfectly Matched Layer (APML) model in High-Order Finite Difference Time Domain (FDTD) algorithm for solving two-dimensional Maxwell’s equations. Initially, we applied the APML method to the standard second-order FDTD algorithm to derive a two-step time-staggered APML (APML-2SS) and a two-step time-centered APML (APML-2SC) formulation for these equations, afterwards, we extended these formulations in high-order FDTD algorithm in order to derive a APML high-order FDTD (APML-HOFDTD) formulation for our Maxwell’s equations. To examine the performance and check out the accuracy of APML model, we conducted a numerical study using a 2D fluid where the three derived formulations were to analyze selected phenomena in terahertz radiation production by the filamentation of two femtosecond laser beams in air plasma. Numerical results illustrated that the two-step APML model is sufficiently accurate for solving our 2D Maxwell’s equations in high-order FDTD discretization and it demonstrated a great performance in studying the THz radiation production.展开更多
We present a parallel numerical method of simulating the interaction of atoms with a strong laser field by solving the time-depending Schr?dinger equation(TDSE) in spherical coordinates. This method is realized by com...We present a parallel numerical method of simulating the interaction of atoms with a strong laser field by solving the time-depending Schr?dinger equation(TDSE) in spherical coordinates. This method is realized by combining constructing block diagonal matrices through using the real space product formula(RSPF) with splitting out diagonal sub-matrices for short iterative Lanczos(SIL) propagator. The numerical implementation of the solver guarantees efficient parallel computing for the simulation of real physical problems such as high harmonic generation(HHG) in these interaction systems.展开更多
In this paper we propose a collocation method for solving Lane-Emden type equation which is nonlinear or-dinary differential equation on the semi-infinite domain. This equation is categorized as singular initial value...In this paper we propose a collocation method for solving Lane-Emden type equation which is nonlinear or-dinary differential equation on the semi-infinite domain. This equation is categorized as singular initial value problems. We solve this equation by the generalized Laguerre polynomial collocation method based on Her-mite-Gauss nodes. This method solves the problem on the semi-infinite domain without truncating it to a fi-nite domain and transforming domain of the problem to a finite domain. In addition, this method reduces so-lution of the problem to solution of a system of algebraic equations.展开更多
The paper is to introduce a computational methodology that is based on ordinary differential equations(ODE)solver for the structural systems adopted by a super tall building in its preliminary design stage so as to fa...The paper is to introduce a computational methodology that is based on ordinary differential equations(ODE)solver for the structural systems adopted by a super tall building in its preliminary design stage so as to facilitate the designers to adjust the dynamic properties of the adopted structural system.The construction of the study is composed by following aspects.The first aspect is the modelling of a structural system.As a typical example,a mega frame-core-tube structural system adopted by some famous super tall buildings such as Taipei 101 building,Shanghai World financial center,is employed to demonstrate the modelling of a computational model.The second aspect is the establishment of motion equations constituted by a group of ordinary differential equations for the analyses of free vibration and resonant response.The solutions of the motion equations(that constitutes the third aspect)resorted to ODE-solver technique.Finally,some valuable conclusions are summarized.展开更多
Diffusion probabilistic models(DPMs)have achieved impressive success in high-resolution image synthesis,especially in recent large-scale text-to-image generation applications.An essential technique for improving the s...Diffusion probabilistic models(DPMs)have achieved impressive success in high-resolution image synthesis,especially in recent large-scale text-to-image generation applications.An essential technique for improving the sample quality of DPMs is guided sampling,which usually needs a large guidance scale to obtain the best sample quality.The commonly-used fast sampler for guided sampling is denoising diffusion implicit models(DDIM),a first-order diffusion ordinary differential equation(ODE)solver that generally needs 100 to 250 steps for high-quality samples.Although recent works propose dedicated high-order solvers and achieve a further speedup for sampling without guidance,their effectiveness for guided sampling has not been well-tested before.In this work,we demonstrate that previous high-order fast samplers suffer from instability issues,and they even become slower than DDIM when the guidance scale grows larger.To further speed up guided sampling,we propose DPM-Solver++,a high-order solver for the guided sampling of DPMs.DPM-Solver++solves the diffusion ODE with the data prediction model and adopts thresholding methods to keep the solution matches training data distribution.We further propose a multistep variant of DPM-Solver++to address the instability issue by reducing the effective step size.Experiments show that DPM-Solver++can generate high-quality samples within only 15 to 20 steps for guided sampling by pixel-space and latent-space DPMs.展开更多
研究舰艇附近水下爆炸问题对船体结构设计、爆炸冲击损害预测及人员安全保障至关重要。为此,提出改进扩散界面法中的六方程可压多相流模型以解决冲击波条件下热力学状态预测偏差,并为相关抗冲击机理研究与数值方法优化提供支撑。通过引...研究舰艇附近水下爆炸问题对船体结构设计、爆炸冲击损害预测及人员安全保障至关重要。为此,提出改进扩散界面法中的六方程可压多相流模型以解决冲击波条件下热力学状态预测偏差,并为相关抗冲击机理研究与数值方法优化提供支撑。通过引入混合能量校正方程及更精确的气体状态方程改进模型,在非结构网格系统构建数值算法程序,采用基于最小二乘重建和Barth-Jespersen限制器的二阶守恒定律的单调上游中心方案(Monotonic Upstream-centered Scheme for Conservation Laws,MUSCL)-Hancock格式、两相流带接触的Harten-Lax-van Leer(Harten-Lax-van Leer Contact,HLLC)黎曼求解器求解齐次双曲型方程,以Newton-Raphson迭代法求解瞬时压力松弛方程。研究结果表明:混合能量方程校正后,模型模拟流体冲击波速度和界面的结果与欧拉方程精确解高度吻合,解决界面附近数值振荡问题;相较于实验数据,改进型模型相对误差1.13%,准确度提升0.33%,且通过拟合冲击Hugoniot曲线获得更精确的刚性气体状态方程(Stiffened Gas Equation of State,SG-EOS)参数,同时可清晰呈现水下爆炸的冲击波传播、气泡胀缩及坍塌水射流现象,但在气泡界面清晰度、射流精细度上存在缺陷,主要受数值格式极端梯度下耗散特性限制。综上,改进型六方程可压多相流模型有效提升了舰艇附近水下爆炸模拟准确性,为深入研究舰艇抗冲击机理提供重要支撑,也为后续相关数值方法的优化奠定了坚实基础。展开更多
A neural network(NN) is a powerful tool for approximating bounded continuous functions in machine learning. The NN provides a framework for numerically solving ordinary differential equations(ODEs) and partial differe...A neural network(NN) is a powerful tool for approximating bounded continuous functions in machine learning. The NN provides a framework for numerically solving ordinary differential equations(ODEs) and partial differential equations(PDEs)combined with the automatic differentiation(AD) technique. In this work, we explore the use of NN for the function approximation and propose a universal solver for ODEs and PDEs. The solver is tested for initial value problems and boundary value problems of ODEs, and the results exhibit high accuracy for not only the unknown functions but also their derivatives. The same strategy can be used to construct a PDE solver based on collocation points instead of a mesh, which is tested with the Burgers equation and the heat equation(i.e., the Laplace equation).展开更多
In this paper,a number of ordinary differential equation(ODE)conversion techniques for trans- formation of nonstandard ODE boundary value problems into standard forms are summarised,together with their applications to...In this paper,a number of ordinary differential equation(ODE)conversion techniques for trans- formation of nonstandard ODE boundary value problems into standard forms are summarised,together with their applications to a variety of boundary value problems in computational solid mechanics,such as eigenvalue problem,geometrical and material nonlinear problem,elastic contact problem and optimal design problems through some simple and representative examples,The advantage of such approach is that various ODE bounda- ry value problems in computational mechanics can be solved effectively in a unified manner by invoking a stand- ard ODE solver.展开更多
A lattice Boltzmann flux solver(LBFS)is presented for simulation of fluid flows.Like the conventional computational fluid dynamics(CFD)solvers,the new solver also applies the finite volume method to discretize the gov...A lattice Boltzmann flux solver(LBFS)is presented for simulation of fluid flows.Like the conventional computational fluid dynamics(CFD)solvers,the new solver also applies the finite volume method to discretize the governing differential equations,but the numerical flux at the cell interface is not evaluated by the smooth function approximation or Riemann solvers.Instead,it is evaluated from local solution of lattice Boltzmann equation(LBE)at cell interface.Two versions of LBFS are presented in this paper.One is to locally apply one-dimensional compressible lattice Boltzmann(LB)model along the normal direction to the cell interface for simulation of compressible inviscid flows with shock waves.The other is to locally apply multi-dimensional LB model at cell interface for simulation of incompressible viscous and inviscid flows.The present solver removes the drawbacks of conventional lattice Boltzmann method(LBM)such as limitation to uniform mesh,tie-up of mesh spacing and time interval,limitation to viscous flows.Numerical examples show that the present solver can be well applied to simulate fluid flows with non-uniform mesh and curved boundary.展开更多
In this study, porosity was introduced into two-dimensional shallow water equations to reflect the effects of obstructions, leading to the modification of the expressions for the flux and source terms. An extra porosi...In this study, porosity was introduced into two-dimensional shallow water equations to reflect the effects of obstructions, leading to the modification of the expressions for the flux and source terms. An extra porosity source term appears in the momentum equation. The numerical model of the shallow water equations with porosity is presented with the finite volume method on unstructured grids and the modified Roe-type approximate Riemann solver. The source terms of the bed slope and porosity are both decomposed in the characteristic direction so that the numerical scheme can exactly satisfy the conservative property. The present model was tested with a dam break with discontinuous porosity and a flash flood in the Toce River Valley. The results show that the model can simulate the influence of obstructions, and the numerical scheme can maintain the flux balance at the interface with high efficiency and resolution.展开更多
By introducing a more general auxiliary ordinary differential equation （ODE）, a modified variable separated ODE method is developed for solving the mKdV-sinh-Gordon equation. As a result, many explicit and exact sol...By introducing a more general auxiliary ordinary differential equation （ODE）, a modified variable separated ODE method is developed for solving the mKdV-sinh-Gordon equation. As a result, many explicit and exact solutions including some new formal solutions are successfully picked up for the mKdV-sinh-Gordon equation by this approach.展开更多
The emerging push of the differentiable programming paradigm in scientific computing is conducive to training deep learning turbulence models using indirect observations.This paper demonstrates the viability of this a...The emerging push of the differentiable programming paradigm in scientific computing is conducive to training deep learning turbulence models using indirect observations.This paper demonstrates the viability of this approach and presents an end-to-end differentiable framework for training deep neural networks to learn eddy viscosity models from indirect observations derived from the velocity and pressure fields.The framework consists of a Reynolds-averaged Navier–Stokes(RANS)solver and a neuralnetwork-represented turbulence model,each accompanied by its derivative computations.For computing the sensitivities of the indirect observations to the Reynolds stress field,we use the continuous adjoint equations for the RANS equations,while the gradient of the neural network is obtained via its built-in automatic differentiation capability.We demonstrate the ability of this approach to learn the true underlying turbulence closure when one exists by training models using synthetic velocity data from linear and nonlinear closures.We also train a linear eddy viscosity model using synthetic velocity measurements from direct numerical simulations of the Navier–Stokes equations for which no true underlying linear closure exists.The trained deep-neural-network turbulence model showed predictive capability on similar flows.展开更多
It is crucial to predict the outputs of a thickening system,including the underflow concentration(UC)and mud pressure,for optimal control of the process.The proliferation of industrial sensors and the availability of ...It is crucial to predict the outputs of a thickening system,including the underflow concentration(UC)and mud pressure,for optimal control of the process.The proliferation of industrial sensors and the availability of thickening-system data make this possible.However,the unique properties of thickening systems,such as the non-linearities,long-time delays,partially observed data,and continuous time evolution pose challenges on building data-driven predictive models.To address the above challenges,we establish an integrated,deep-learning,continuous time network structure that consists of a sequential encoder,a state decoder,and a derivative module to learn the deterministic state space model from thickening systems.Using a case study,we examine our methods with a tailing thickener manufactured by the FLSmidth installed with massive sensors and obtain extensive experimental results.The results demonstrate that the proposed continuous-time model with the sequential encoder achieves better prediction performances than the existing discrete-time models and reduces the negative effects from long time delays by extracting features from historical system trajectories.The proposed method also demonstrates outstanding performances for both short and long term prediction tasks with the two proposed derivative types.展开更多
The preconditioned generalized minimal residual(GMRES) method is a common method for solving non-symmetric,large and sparse linear systems which originated in discrete ordinary differential equations by Boundary value...The preconditioned generalized minimal residual(GMRES) method is a common method for solving non-symmetric,large and sparse linear systems which originated in discrete ordinary differential equations by Boundary value methods.In this paper,we propose a new circulant preconditioner to speed up the convergence rate of the GMRES method, which is a convex linear combination of P-circulant and Strang-type circulant preconditioners. Theoretical and practical arguments are given to show that this preconditioner is feasible and effective in some cases.展开更多
A heuristic technique is developed for a nonlinear magnetohydrodynamics （MHD） Jeffery-Hamel problem with the help of the feed-forward artificial neural net- work （ANN） optimized with the genetic algorithm （GA） a...A heuristic technique is developed for a nonlinear magnetohydrodynamics （MHD） Jeffery-Hamel problem with the help of the feed-forward artificial neural net- work （ANN） optimized with the genetic algorithm （GA） and the sequential quadratic programming （SQP） method. The twodimensional （2D） MHD Jeffery-Hamel problem is transformed into a higher order boundary value problem （BVP） of ordinary differential equations （ODEs）. The mathematical model of the transformed BVP is formulated with the ANN in an unsupervised manner. The training of the weights of the ANN is carried out with the evolutionary calculation based on the GA hybridized with the SQP method for the rapid local convergence. The proposed scheme is evaluated on the variants of the Jeffery-Hamel flow by varying the Reynold number, the Hartmann number, and the an- gles of the walls. A large number of simulations are performed with an extensive analysis to validate the accuracy, convergence, and effectiveness of the scheme. The comparison of the standard numerical solution and the analytic solution establishes the correctness of the proposed designed methodologies.展开更多
A formulation of a differential equation as projection and fixed point pi-Mem alloivs approximations using general piecnvise functions. We prone existence and uniqueness of the up proximate solution* convergence in th...A formulation of a differential equation as projection and fixed point pi-Mem alloivs approximations using general piecnvise functions. We prone existence and uniqueness of the up proximate solution* convergence in the L2 norm and nodal supercnnvergence. These results generalize those obtained earlier by Hulme for continuous piecevjise polynomials and by Delfour-Dubeau for discontinuous pieceuiise polynomials. A duality relationship for the two types of approximations is also given.展开更多
Solute transport simulations are important in water pollution events.This paper introduces a finite volume Godunovtype model for solving a 4×4 matrix form of the hyperbolic conservation laws consisting of 2D shal...Solute transport simulations are important in water pollution events.This paper introduces a finite volume Godunovtype model for solving a 4×4 matrix form of the hyperbolic conservation laws consisting of 2D shallow water equations and transport equations.The model adopts the Harten-Lax-van Leer-contact(HLLC)-approximate Riemann solution to calculate the cell interface fluxes.It can deal well with the changes in the dry and wet interfaces in an actual complex terrain,and it has a strong shock-wave capturing ability.Using monotonic upstream-centred scheme for conservation laws(MUSCL)linear reconstruction with finite slope and the Runge-Kutta time integration method can achieve second-order accuracy.At the same time,the introduction of graphics processing unit(GPU)-accelerated computing technology greatly increases the computing speed.The model is validated against multiple benchmarks,and the results are in good agreement with analytical solutions and other published numerical predictions.The third test case uses the GPU and central processing unit(CPU)calculation models which take 3.865 s and 13.865 s,respectively,indicating that the GPU calculation model can increase the calculation speed by 3.6 times.In the fourth test case,comparing the numerical model calculated by GPU with the traditional numerical model calculated by CPU,the calculation efficiencies of the numerical model calculated by GPU under different resolution grids are 9.8–44.6 times higher than those by CPU.Therefore,it has better potential than previous models for large-scale simulation of solute transport in water pollution incidents.It can provide a reliable theoretical basis and strong data support in the rapid assessment and early warning of water pollution accidents.展开更多
We present here asymptotic solutions of equations of the type , where is a large parameter. The Bessel differential equation is considered as a typical example of the above and the solutions are provided as . Furtherm...We present here asymptotic solutions of equations of the type , where is a large parameter. The Bessel differential equation is considered as a typical example of the above and the solutions are provided as . Furthermore, the behaviour of the solutions as well as the stability of the Bessel ode is investigated numerically as the parameter grows indefinitely.展开更多
文摘We implemented a two-step Asymmetric Perfectly Matched Layer (APML) model in High-Order Finite Difference Time Domain (FDTD) algorithm for solving two-dimensional Maxwell’s equations. Initially, we applied the APML method to the standard second-order FDTD algorithm to derive a two-step time-staggered APML (APML-2SS) and a two-step time-centered APML (APML-2SC) formulation for these equations, afterwards, we extended these formulations in high-order FDTD algorithm in order to derive a APML high-order FDTD (APML-HOFDTD) formulation for our Maxwell’s equations. To examine the performance and check out the accuracy of APML model, we conducted a numerical study using a 2D fluid where the three derived formulations were to analyze selected phenomena in terahertz radiation production by the filamentation of two femtosecond laser beams in air plasma. Numerical results illustrated that the two-step APML model is sufficiently accurate for solving our 2D Maxwell’s equations in high-order FDTD discretization and it demonstrated a great performance in studying the THz radiation production.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11534004,11627807,11774131,and 11774130)the Scientific and Technological Project of Jilin Provincial Education Department in the Thirteenth Five-Year Plan,China(Grant No.JJKH20170538KJ)
文摘We present a parallel numerical method of simulating the interaction of atoms with a strong laser field by solving the time-depending Schr?dinger equation(TDSE) in spherical coordinates. This method is realized by combining constructing block diagonal matrices through using the real space product formula(RSPF) with splitting out diagonal sub-matrices for short iterative Lanczos(SIL) propagator. The numerical implementation of the solver guarantees efficient parallel computing for the simulation of real physical problems such as high harmonic generation(HHG) in these interaction systems.
文摘In this paper we propose a collocation method for solving Lane-Emden type equation which is nonlinear or-dinary differential equation on the semi-infinite domain. This equation is categorized as singular initial value problems. We solve this equation by the generalized Laguerre polynomial collocation method based on Her-mite-Gauss nodes. This method solves the problem on the semi-infinite domain without truncating it to a fi-nite domain and transforming domain of the problem to a finite domain. In addition, this method reduces so-lution of the problem to solution of a system of algebraic equations.
基金Acknowledgment The research work was financially supported both by the Natural Science Foundation of China(51178164)and the Priority Discipline Foundation of Henan Province(507909).
文摘The paper is to introduce a computational methodology that is based on ordinary differential equations(ODE)solver for the structural systems adopted by a super tall building in its preliminary design stage so as to facilitate the designers to adjust the dynamic properties of the adopted structural system.The construction of the study is composed by following aspects.The first aspect is the modelling of a structural system.As a typical example,a mega frame-core-tube structural system adopted by some famous super tall buildings such as Taipei 101 building,Shanghai World financial center,is employed to demonstrate the modelling of a computational model.The second aspect is the establishment of motion equations constituted by a group of ordinary differential equations for the analyses of free vibration and resonant response.The solutions of the motion equations(that constitutes the third aspect)resorted to ODE-solver technique.Finally,some valuable conclusions are summarized.
文摘Diffusion probabilistic models(DPMs)have achieved impressive success in high-resolution image synthesis,especially in recent large-scale text-to-image generation applications.An essential technique for improving the sample quality of DPMs is guided sampling,which usually needs a large guidance scale to obtain the best sample quality.The commonly-used fast sampler for guided sampling is denoising diffusion implicit models(DDIM),a first-order diffusion ordinary differential equation(ODE)solver that generally needs 100 to 250 steps for high-quality samples.Although recent works propose dedicated high-order solvers and achieve a further speedup for sampling without guidance,their effectiveness for guided sampling has not been well-tested before.In this work,we demonstrate that previous high-order fast samplers suffer from instability issues,and they even become slower than DDIM when the guidance scale grows larger.To further speed up guided sampling,we propose DPM-Solver++,a high-order solver for the guided sampling of DPMs.DPM-Solver++solves the diffusion ODE with the data prediction model and adopts thresholding methods to keep the solution matches training data distribution.We further propose a multistep variant of DPM-Solver++to address the instability issue by reducing the effective step size.Experiments show that DPM-Solver++can generate high-quality samples within only 15 to 20 steps for guided sampling by pixel-space and latent-space DPMs.
文摘研究舰艇附近水下爆炸问题对船体结构设计、爆炸冲击损害预测及人员安全保障至关重要。为此,提出改进扩散界面法中的六方程可压多相流模型以解决冲击波条件下热力学状态预测偏差,并为相关抗冲击机理研究与数值方法优化提供支撑。通过引入混合能量校正方程及更精确的气体状态方程改进模型,在非结构网格系统构建数值算法程序,采用基于最小二乘重建和Barth-Jespersen限制器的二阶守恒定律的单调上游中心方案(Monotonic Upstream-centered Scheme for Conservation Laws,MUSCL)-Hancock格式、两相流带接触的Harten-Lax-van Leer(Harten-Lax-van Leer Contact,HLLC)黎曼求解器求解齐次双曲型方程,以Newton-Raphson迭代法求解瞬时压力松弛方程。研究结果表明:混合能量方程校正后,模型模拟流体冲击波速度和界面的结果与欧拉方程精确解高度吻合,解决界面附近数值振荡问题;相较于实验数据,改进型模型相对误差1.13%,准确度提升0.33%,且通过拟合冲击Hugoniot曲线获得更精确的刚性气体状态方程(Stiffened Gas Equation of State,SG-EOS)参数,同时可清晰呈现水下爆炸的冲击波传播、气泡胀缩及坍塌水射流现象,但在气泡界面清晰度、射流精细度上存在缺陷,主要受数值格式极端梯度下耗散特性限制。综上,改进型六方程可压多相流模型有效提升了舰艇附近水下爆炸模拟准确性,为深入研究舰艇抗冲击机理提供重要支撑,也为后续相关数值方法的优化奠定了坚实基础。
基金Project supported by the National Natural Science Foundation of China(No.11521091)
文摘A neural network(NN) is a powerful tool for approximating bounded continuous functions in machine learning. The NN provides a framework for numerically solving ordinary differential equations(ODEs) and partial differential equations(PDEs)combined with the automatic differentiation(AD) technique. In this work, we explore the use of NN for the function approximation and propose a universal solver for ODEs and PDEs. The solver is tested for initial value problems and boundary value problems of ODEs, and the results exhibit high accuracy for not only the unknown functions but also their derivatives. The same strategy can be used to construct a PDE solver based on collocation points instead of a mesh, which is tested with the Burgers equation and the heat equation(i.e., the Laplace equation).
基金The project is supported by National Natural Science Foundation of China
文摘In this paper,a number of ordinary differential equation(ODE)conversion techniques for trans- formation of nonstandard ODE boundary value problems into standard forms are summarised,together with their applications to a variety of boundary value problems in computational solid mechanics,such as eigenvalue problem,geometrical and material nonlinear problem,elastic contact problem and optimal design problems through some simple and representative examples,The advantage of such approach is that various ODE bounda- ry value problems in computational mechanics can be solved effectively in a unified manner by invoking a stand- ard ODE solver.
基金Supported by the National Natural Science Foundation of China(11272153)
文摘A lattice Boltzmann flux solver(LBFS)is presented for simulation of fluid flows.Like the conventional computational fluid dynamics(CFD)solvers,the new solver also applies the finite volume method to discretize the governing differential equations,but the numerical flux at the cell interface is not evaluated by the smooth function approximation or Riemann solvers.Instead,it is evaluated from local solution of lattice Boltzmann equation(LBE)at cell interface.Two versions of LBFS are presented in this paper.One is to locally apply one-dimensional compressible lattice Boltzmann(LB)model along the normal direction to the cell interface for simulation of compressible inviscid flows with shock waves.The other is to locally apply multi-dimensional LB model at cell interface for simulation of incompressible viscous and inviscid flows.The present solver removes the drawbacks of conventional lattice Boltzmann method(LBM)such as limitation to uniform mesh,tie-up of mesh spacing and time interval,limitation to viscous flows.Numerical examples show that the present solver can be well applied to simulate fluid flows with non-uniform mesh and curved boundary.
基金supported by the National Natural Science Foundation of China (Grants No. 50909065 and 51109039)the National Basic Research Program of China (973 Program, Grant No. 2012CB417002)
文摘In this study, porosity was introduced into two-dimensional shallow water equations to reflect the effects of obstructions, leading to the modification of the expressions for the flux and source terms. An extra porosity source term appears in the momentum equation. The numerical model of the shallow water equations with porosity is presented with the finite volume method on unstructured grids and the modified Roe-type approximate Riemann solver. The source terms of the bed slope and porosity are both decomposed in the characteristic direction so that the numerical scheme can exactly satisfy the conservative property. The present model was tested with a dam break with discontinuous porosity and a flash flood in the Toce River Valley. The results show that the model can simulate the influence of obstructions, and the numerical scheme can maintain the flux balance at the interface with high efficiency and resolution.
基金Project supported by the National Natural Science Foundation of China (Grant No 10672053)
文摘By introducing a more general auxiliary ordinary differential equation （ODE）, a modified variable separated ODE method is developed for solving the mKdV-sinh-Gordon equation. As a result, many explicit and exact solutions including some new formal solutions are successfully picked up for the mKdV-sinh-Gordon equation by this approach.
文摘The emerging push of the differentiable programming paradigm in scientific computing is conducive to training deep learning turbulence models using indirect observations.This paper demonstrates the viability of this approach and presents an end-to-end differentiable framework for training deep neural networks to learn eddy viscosity models from indirect observations derived from the velocity and pressure fields.The framework consists of a Reynolds-averaged Navier–Stokes(RANS)solver and a neuralnetwork-represented turbulence model,each accompanied by its derivative computations.For computing the sensitivities of the indirect observations to the Reynolds stress field,we use the continuous adjoint equations for the RANS equations,while the gradient of the neural network is obtained via its built-in automatic differentiation capability.We demonstrate the ability of this approach to learn the true underlying turbulence closure when one exists by training models using synthetic velocity data from linear and nonlinear closures.We also train a linear eddy viscosity model using synthetic velocity measurements from direct numerical simulations of the Navier–Stokes equations for which no true underlying linear closure exists.The trained deep-neural-network turbulence model showed predictive capability on similar flows.
基金supported by National Key Research and Development Program of China(2019YFC0605300)the National Natural Science Foundation of China(61873299,61902022,61972028)+2 种基金Scientific and Technological Innovation Foundation of Shunde Graduate School,University of Science and Technology Beijing(BK21BF002)Macao Science and Technology Development Fund under Macao Funding Scheme for Key R&D Projects(0025/2019/AKP)Macao Science and Technology Development Fund(0015/2020/AMJ)。
文摘It is crucial to predict the outputs of a thickening system,including the underflow concentration(UC)and mud pressure,for optimal control of the process.The proliferation of industrial sensors and the availability of thickening-system data make this possible.However,the unique properties of thickening systems,such as the non-linearities,long-time delays,partially observed data,and continuous time evolution pose challenges on building data-driven predictive models.To address the above challenges,we establish an integrated,deep-learning,continuous time network structure that consists of a sequential encoder,a state decoder,and a derivative module to learn the deterministic state space model from thickening systems.Using a case study,we examine our methods with a tailing thickener manufactured by the FLSmidth installed with massive sensors and obtain extensive experimental results.The results demonstrate that the proposed continuous-time model with the sequential encoder achieves better prediction performances than the existing discrete-time models and reduces the negative effects from long time delays by extracting features from historical system trajectories.The proposed method also demonstrates outstanding performances for both short and long term prediction tasks with the two proposed derivative types.
基金Supported by the Scientific Research Foundation for Advisor Program of Higher Education of Gansu Province(1009-6)Supported by the Scientific Research Foundation for Youth Scholars of Hexi University(qn201015)
文摘The preconditioned generalized minimal residual(GMRES) method is a common method for solving non-symmetric,large and sparse linear systems which originated in discrete ordinary differential equations by Boundary value methods.In this paper,we propose a new circulant preconditioner to speed up the convergence rate of the GMRES method, which is a convex linear combination of P-circulant and Strang-type circulant preconditioners. Theoretical and practical arguments are given to show that this preconditioner is feasible and effective in some cases.
文摘A heuristic technique is developed for a nonlinear magnetohydrodynamics （MHD） Jeffery-Hamel problem with the help of the feed-forward artificial neural net- work （ANN） optimized with the genetic algorithm （GA） and the sequential quadratic programming （SQP） method. The twodimensional （2D） MHD Jeffery-Hamel problem is transformed into a higher order boundary value problem （BVP） of ordinary differential equations （ODEs）. The mathematical model of the transformed BVP is formulated with the ANN in an unsupervised manner. The training of the weights of the ANN is carried out with the evolutionary calculation based on the GA hybridized with the SQP method for the rapid local convergence. The proposed scheme is evaluated on the variants of the Jeffery-Hamel flow by varying the Reynold number, the Hartmann number, and the an- gles of the walls. A large number of simulations are performed with an extensive analysis to validate the accuracy, convergence, and effectiveness of the scheme. The comparison of the standard numerical solution and the analytic solution establishes the correctness of the proposed designed methodologies.
基金This research has been supported in part by the Natural Sciences and Engineering Research Council of Canada(Grant OGPIN-336)and by the"Ministere de l'Education du Quebec"(FCAR Grant-ER-0725)
文摘A formulation of a differential equation as projection and fixed point pi-Mem alloivs approximations using general piecnvise functions. We prone existence and uniqueness of the up proximate solution* convergence in the L2 norm and nodal supercnnvergence. These results generalize those obtained earlier by Hulme for continuous piecevjise polynomials and by Delfour-Dubeau for discontinuous pieceuiise polynomials. A duality relationship for the two types of approximations is also given.
基金Project supported by the National Natural Science Foundation of China(Nos.52009104 and 52079106)the Shaanxi Provincial Department of Water Resources Project(No.2017slkj-14)the Shaanxi Provincial Department of Science and Technology Project(No.2017JQ3043),China。
文摘Solute transport simulations are important in water pollution events.This paper introduces a finite volume Godunovtype model for solving a 4×4 matrix form of the hyperbolic conservation laws consisting of 2D shallow water equations and transport equations.The model adopts the Harten-Lax-van Leer-contact(HLLC)-approximate Riemann solution to calculate the cell interface fluxes.It can deal well with the changes in the dry and wet interfaces in an actual complex terrain,and it has a strong shock-wave capturing ability.Using monotonic upstream-centred scheme for conservation laws(MUSCL)linear reconstruction with finite slope and the Runge-Kutta time integration method can achieve second-order accuracy.At the same time,the introduction of graphics processing unit(GPU)-accelerated computing technology greatly increases the computing speed.The model is validated against multiple benchmarks,and the results are in good agreement with analytical solutions and other published numerical predictions.The third test case uses the GPU and central processing unit(CPU)calculation models which take 3.865 s and 13.865 s,respectively,indicating that the GPU calculation model can increase the calculation speed by 3.6 times.In the fourth test case,comparing the numerical model calculated by GPU with the traditional numerical model calculated by CPU,the calculation efficiencies of the numerical model calculated by GPU under different resolution grids are 9.8–44.6 times higher than those by CPU.Therefore,it has better potential than previous models for large-scale simulation of solute transport in water pollution incidents.It can provide a reliable theoretical basis and strong data support in the rapid assessment and early warning of water pollution accidents.
文摘We present here asymptotic solutions of equations of the type , where is a large parameter. The Bessel differential equation is considered as a typical example of the above and the solutions are provided as . Furthermore, the behaviour of the solutions as well as the stability of the Bessel ode is investigated numerically as the parameter grows indefinitely.
