The compact implicit integration factor (cIIF) method is an efficient time discretization scheme for stiff nonlinear diffusion equations in two and three spatial dimensions. In the current work, we apply the cIIF me...The compact implicit integration factor (cIIF) method is an efficient time discretization scheme for stiff nonlinear diffusion equations in two and three spatial dimensions. In the current work, we apply the cIIF method to some complex-valued nonlinear evolutionary equations such as the nonlinear SchrSdinger (NLS) equation and the complex Ginzburg-Landau (GL) equation. Detailed algorithm formulation and practical implementation of cIIF method are performed. The numerical results indicate that this method is very accurate and efficient.展开更多
In this work,a method is put forward to obtain the dynamic solution efficiently and accurately for a large-scale train-track-substructure(TTS)system.It is called implicit-explicit integration and multi-time-step solut...In this work,a method is put forward to obtain the dynamic solution efficiently and accurately for a large-scale train-track-substructure(TTS)system.It is called implicit-explicit integration and multi-time-step solution method(abbreviated as mI-nE-MTS method).The TTS system is divided into train-track subsystem and substruc-ture subsystem.Considering that the root cause of low effi-ciency of obtaining TTS solution lies in solving the alge-braic equation of the substructures,the high-efficient Zhai method,an explicit integration scheme,can be introduced to avoid matrix inversion process.The train-track system is solved by implicitly Park method.Moreover,it is known that the requirement of time step size differs for different sub-systems,integration methods and structural frequency response characteristics.A multi-time-step solution is pro-posed,in which time step size for the train-track subsystem and the substructure subsystem can be arbitrarily chosen once satisfying stability and precision demand,namely the time spent for m implicit integral steps is equal to n explicit integral steps,i.e.,mI=nE as mentioned above.The numeri-cal examples show the accuracy,efficiency,and engineering practicality of the proposed method.展开更多
A high-order accuracy time discretization method is developed in this paper to solve the one-dimensional nonlinear Dirac(NLD)equation.Based on the implicit integration factor(IIF)method,two schemes are proposed.Centra...A high-order accuracy time discretization method is developed in this paper to solve the one-dimensional nonlinear Dirac(NLD)equation.Based on the implicit integration factor(IIF)method,two schemes are proposed.Central differences are applied to the spatial discretization.The semi-discrete scheme keeps the conservation of the charge and energy.For the temporal discretization,second-order IIF method and fourth-order IIF method are applied respectively to the nonlinear system arising from the spatial discretization.Numerical experiments are given to validate the accuracy of these schemes and to discuss the interaction dynamics of the NLD solitary waves.展开更多
This paper presents a new simple method of implicit time integration with two control parameters for solving initial-value problems of dynamics such that its accuracy is at least of order two along with the conditiona...This paper presents a new simple method of implicit time integration with two control parameters for solving initial-value problems of dynamics such that its accuracy is at least of order two along with the conditional and unconditional stability regions of the parameters. When the control parameters in the method are optimally taken in their regions, the accuracy may be improved to reach of order three. It is found that the new scheme can achieve lower numerical amplitude dissipation and period dispersion than some of the existing methods, e.g. the Newmark method and Zhai's approach, when the same time step size is used. The region of time step dependent on the parameters in the new scheme is explicitly obtained. Finally, some examples of dynamic problems are given to show the accuracy and efficiency of the proposed scheme applied in dynamic systems.展开更多
A balanced adaptive time-stepping strategy is implemented in an implicit discontinuous Galerkin solver to guarantee the temporal accuracy of unsteady simulations.A proper relation between the spatial,temporal and iter...A balanced adaptive time-stepping strategy is implemented in an implicit discontinuous Galerkin solver to guarantee the temporal accuracy of unsteady simulations.A proper relation between the spatial,temporal and iterative errors generated within one time step is constructed.With an estimate of temporal and spatial error using an embedded RungeKutta scheme and a higher order spatial discretization,an adaptive time-stepping strategy is proposed based on the idea that the time step should be the maximum without obviously infuencing the total error of the discretization.The designed adaptive time-stepping strategy is then tested in various types of problems including isentropic vortex convection,steady-state fow past a fat plate,Taylor-Green vortex and turbulent fow over a circular cylinder at Re=3900.The results indicate that the adaptive time-stepping strategy can maintain that the discretization error is dominated by the spatial error and relatively high efciency is obtained for unsteady and steady,well-resolved and under-resolved simulations.展开更多
A generalized solution scheme using implicit time integrators for piecewise linear and nonlinear systems is developed.The piecewise linear characteristic has been well‐discussed in previous studies,in which the origi...A generalized solution scheme using implicit time integrators for piecewise linear and nonlinear systems is developed.The piecewise linear characteristic has been well‐discussed in previous studies,in which the original problem has been transformed into linear complementarity problems(LCPs)and then solved via the Lemke algorithm for each time step.The proposed scheme,instead,uses the projection function to describe the discontinuity in the dynamics equations,and solves for each step the nonlinear equations obtained from the implicit integrator by the semismooth Newton iteration.Compared with the LCP‐based scheme,the new scheme offers a more general choice by allowing other nonlinearities in the governing equations.To assess its performances,several illustrative examples are solved.The numerical solutions demonstrate that the new scheme can not only predict satisfactory results for piecewise nonlinear systems,but also exhibits substantial efficiency advantages over the LCP‐based scheme when applied to piecewise linear systems.展开更多
In structural design optimization involving transient responses,time integration scheme plays a crucial role in sensitivity analysis because it affects the accuracy and stability of transient analysis.In this work,the...In structural design optimization involving transient responses,time integration scheme plays a crucial role in sensitivity analysis because it affects the accuracy and stability of transient analysis.In this work,the influence of time integration scheme is studied numerically for the adjoint shape sensitivity analysis of two benchmark transient heat conduction problems within the framework of isogeometric analysis.It is found that(i)the explicit approach(β=0)and semi-implicit approach withβ<0.5 impose a strict stability condition of the transient analysis;(ii)the implicit approach(β=1)and semi-implicit approach withβ>0.5 are generally preferred for their unconditional stability;and(iii)Crank–Nicolson type approach(β=0.5)may induce a large error for large time-step sizes due to the oscillatory solutions.The numerical results also show that the time-step size does not have to be chosen to satisfy the critical conditions for all of the eigen-frequencies.It is recommended to useβ≈0.75 for unconditional stability,such that the oscillation condition is much less critical than the Crank–Nicolson scheme,and the accuracy is higher than a fully implicit approach.展开更多
In this study, we present a conservative local discontinuous Galerkin(LDG) method for numerically solving the two-dimensional nonlinear Schrdinger(NLS) equation. The NLS equation is rewritten as a firstorder system an...In this study, we present a conservative local discontinuous Galerkin(LDG) method for numerically solving the two-dimensional nonlinear Schrdinger(NLS) equation. The NLS equation is rewritten as a firstorder system and then we construct the LDG formulation with appropriate numerical flux. The mass and energy conserving laws for the semi-discrete formulation can be proved based on different choices of numerical fluxes such as the central, alternative and upwind-based flux. We will propose two kinds of time discretization methods for the semi-discrete formulation. One is based on Crank-Nicolson method and can be proved to preserve the discrete mass and energy conservation. The other one is Krylov implicit integration factor(IIF) method which demands much less computational effort. Various numerical experiments are presented to demonstrate the conservation law of mass and energy, the optimal rates of convergence, and the blow-up phenomenon.展开更多
文摘The compact implicit integration factor (cIIF) method is an efficient time discretization scheme for stiff nonlinear diffusion equations in two and three spatial dimensions. In the current work, we apply the cIIF method to some complex-valued nonlinear evolutionary equations such as the nonlinear SchrSdinger (NLS) equation and the complex Ginzburg-Landau (GL) equation. Detailed algorithm formulation and practical implementation of cIIF method are performed. The numerical results indicate that this method is very accurate and efficient.
基金This work was supported by the National Natural Science Foundation of China(Grant Nos.52008404,U1934217 and 11790283)Science and Technology Research and Development Program Project of China Railway Group Limited(Major Special Project,No.2020-Special-02)the National Natural Science Foundation of Hunan Province(Grant No.2021JJ30850).
文摘In this work,a method is put forward to obtain the dynamic solution efficiently and accurately for a large-scale train-track-substructure(TTS)system.It is called implicit-explicit integration and multi-time-step solution method(abbreviated as mI-nE-MTS method).The TTS system is divided into train-track subsystem and substruc-ture subsystem.Considering that the root cause of low effi-ciency of obtaining TTS solution lies in solving the alge-braic equation of the substructures,the high-efficient Zhai method,an explicit integration scheme,can be introduced to avoid matrix inversion process.The train-track system is solved by implicitly Park method.Moreover,it is known that the requirement of time step size differs for different sub-systems,integration methods and structural frequency response characteristics.A multi-time-step solution is pro-posed,in which time step size for the train-track subsystem and the substructure subsystem can be arbitrarily chosen once satisfying stability and precision demand,namely the time spent for m implicit integral steps is equal to n explicit integral steps,i.e.,mI=nE as mentioned above.The numeri-cal examples show the accuracy,efficiency,and engineering practicality of the proposed method.
基金the National Natural Science Foundation of China(No.11671044)the Science Challenge Project(No.TZ2016001)the Beijing Municipal Education Commission(No.PXM2017014224000020).
文摘A high-order accuracy time discretization method is developed in this paper to solve the one-dimensional nonlinear Dirac(NLD)equation.Based on the implicit integration factor(IIF)method,two schemes are proposed.Central differences are applied to the spatial discretization.The semi-discrete scheme keeps the conservation of the charge and energy.For the temporal discretization,second-order IIF method and fourth-order IIF method are applied respectively to the nonlinear system arising from the spatial discretization.Numerical experiments are given to validate the accuracy of these schemes and to discuss the interaction dynamics of the NLD solitary waves.
基金The project supported by the National Key Basic Research and Development Foundation of the Ministry of Science and Technology of China (G2000048702, 2003CB716707)the National Science Fund for Distinguished Young Scholars (10025208)+1 种基金 the National Natural Science Foundation of China (Key Program) (10532040) the Research Fund for 0versea Chinese (10228028).
文摘This paper presents a new simple method of implicit time integration with two control parameters for solving initial-value problems of dynamics such that its accuracy is at least of order two along with the conditional and unconditional stability regions of the parameters. When the control parameters in the method are optimally taken in their regions, the accuracy may be improved to reach of order three. It is found that the new scheme can achieve lower numerical amplitude dissipation and period dispersion than some of the existing methods, e.g. the Newmark method and Zhai's approach, when the same time step size is used. The region of time step dependent on the parameters in the new scheme is explicitly obtained. Finally, some examples of dynamic problems are given to show the accuracy and efficiency of the proposed scheme applied in dynamic systems.
基金Zhen-Guo Yan acknowledges supports from the National Natural Science Foundation of China(Grant no.11902344)National Numerical Windtunnel Project.The development of the implicit solver in Nektar++has been supported by EPSRC grant(EP/R029423/1)UK Turbulence Consortium grant(EP/R029326/1).
文摘A balanced adaptive time-stepping strategy is implemented in an implicit discontinuous Galerkin solver to guarantee the temporal accuracy of unsteady simulations.A proper relation between the spatial,temporal and iterative errors generated within one time step is constructed.With an estimate of temporal and spatial error using an embedded RungeKutta scheme and a higher order spatial discretization,an adaptive time-stepping strategy is proposed based on the idea that the time step should be the maximum without obviously infuencing the total error of the discretization.The designed adaptive time-stepping strategy is then tested in various types of problems including isentropic vortex convection,steady-state fow past a fat plate,Taylor-Green vortex and turbulent fow over a circular cylinder at Re=3900.The results indicate that the adaptive time-stepping strategy can maintain that the discretization error is dominated by the spatial error and relatively high efciency is obtained for unsteady and steady,well-resolved and under-resolved simulations.
文摘A generalized solution scheme using implicit time integrators for piecewise linear and nonlinear systems is developed.The piecewise linear characteristic has been well‐discussed in previous studies,in which the original problem has been transformed into linear complementarity problems(LCPs)and then solved via the Lemke algorithm for each time step.The proposed scheme,instead,uses the projection function to describe the discontinuity in the dynamics equations,and solves for each step the nonlinear equations obtained from the implicit integrator by the semismooth Newton iteration.Compared with the LCP‐based scheme,the new scheme offers a more general choice by allowing other nonlinearities in the governing equations.To assess its performances,several illustrative examples are solved.The numerical solutions demonstrate that the new scheme can not only predict satisfactory results for piecewise nonlinear systems,but also exhibits substantial efficiency advantages over the LCP‐based scheme when applied to piecewise linear systems.
基金The authors would like to thank Dr.Dan Wang from Institute of High Performance Computing(IHPC),A*STAR for the communications related to this work.
文摘In structural design optimization involving transient responses,time integration scheme plays a crucial role in sensitivity analysis because it affects the accuracy and stability of transient analysis.In this work,the influence of time integration scheme is studied numerically for the adjoint shape sensitivity analysis of two benchmark transient heat conduction problems within the framework of isogeometric analysis.It is found that(i)the explicit approach(β=0)and semi-implicit approach withβ<0.5 impose a strict stability condition of the transient analysis;(ii)the implicit approach(β=1)and semi-implicit approach withβ>0.5 are generally preferred for their unconditional stability;and(iii)Crank–Nicolson type approach(β=0.5)may induce a large error for large time-step sizes due to the oscillatory solutions.The numerical results also show that the time-step size does not have to be chosen to satisfy the critical conditions for all of the eigen-frequencies.It is recommended to useβ≈0.75 for unconditional stability,such that the oscillation condition is much less critical than the Crank–Nicolson scheme,and the accuracy is higher than a fully implicit approach.
基金supported by the Foundation of Liaoning Educational Committee (Grant No. L201604)China Scholarship Council, National Natural Science Foundation of China (Grant Nos. 11571002, 11171281 and 11671044)+1 种基金the Science Foundation of China Academy of Engineering Physics (Grant No. 2015B0101021)the Defense Industrial Technology Development Program (Grant No. B1520133015)
文摘In this study, we present a conservative local discontinuous Galerkin(LDG) method for numerically solving the two-dimensional nonlinear Schrdinger(NLS) equation. The NLS equation is rewritten as a firstorder system and then we construct the LDG formulation with appropriate numerical flux. The mass and energy conserving laws for the semi-discrete formulation can be proved based on different choices of numerical fluxes such as the central, alternative and upwind-based flux. We will propose two kinds of time discretization methods for the semi-discrete formulation. One is based on Crank-Nicolson method and can be proved to preserve the discrete mass and energy conservation. The other one is Krylov implicit integration factor(IIF) method which demands much less computational effort. Various numerical experiments are presented to demonstrate the conservation law of mass and energy, the optimal rates of convergence, and the blow-up phenomenon.