This paper aims to investigate the tamed Euler method for the random periodic solution of semilinear SDEs with one-sided Lipschitz coefficient.We introduce a novel approach to analyze mean-square error bounds of the n...This paper aims to investigate the tamed Euler method for the random periodic solution of semilinear SDEs with one-sided Lipschitz coefficient.We introduce a novel approach to analyze mean-square error bounds of the novel schemes,without relying on a priori high-order moment bound of the numerical approximation.The expected order-one mean square convergence is attained for the proposed scheme.Moreover,a numerical example is presented to verify our theoretical analysis.展开更多
Mineral exploration is done by different methods. Geophysical and geochemical studies are two powerful tools in this field. In integrated studies, the results of each study are used to determine the location of the dr...Mineral exploration is done by different methods. Geophysical and geochemical studies are two powerful tools in this field. In integrated studies, the results of each study are used to determine the location of the drilling boreholes. The purpose of this study is to use field geophysics to calculate the depth of mineral reserve. The study area is located 38 km from Zarand city called Jalalabad iron mine. In this study, gravimetric data were measured and mineral depth was calculated using the Euler method. 1314 readings have been performed in this area. The rocks of the region include volcanic and sedimentary. The source of the mineralization in the area is hydrothermal processes. After gravity measuring in the region, the data were corrected, then various methods such as anomalous map remaining in levels one and two, upward expansion, first and second-degree vertical derivatives, analytical method, and analytical signal were drawn, and finally, the depth of the deposit was estimated by Euler method. As a result, the depth of the mineral deposit was calculated to be between 20 and 30 meters on average.展开更多
In this paper,we present the semi-implicit Euler（SIE）numerical solution for stochastic pantograph equations with jumps and prove that the SIE approximation solution converges to the exact solution in the mean-square...In this paper,we present the semi-implicit Euler（SIE）numerical solution for stochastic pantograph equations with jumps and prove that the SIE approximation solution converges to the exact solution in the mean-square sense under the Local Lipschitz condition.展开更多
The paper presents the comparative study on numerical methods of Euler method,Improved Euler method and fourth-order Runge-Kutta method for solving the engineering problems and applications.The three proposed methods ...The paper presents the comparative study on numerical methods of Euler method,Improved Euler method and fourth-order Runge-Kutta method for solving the engineering problems and applications.The three proposed methods are quite efficient and practically well suited for solving the unknown engineering problems.This paper aims to enhance the teaching and learning quality of teachers and students for various levels.At each point of the interval,the value of y is calculated and compared with its exact value at that point.The next interesting point is the observation of error from those methods.Error in the value of y is the difference between calculated and exact value.A mathematical equation which relates various functions with its derivatives is known as a differential equation.It is a popular field of mathematics because of its application to real-world problems.To calculate the exact values,the approximate values and the errors,the numerical tool such as MATLAB is appropriate for observing the results.This paper mainly concentrates on identifying the method which provides more accurate results.Then the analytical results and calculates their corresponding error were compared in details.The minimum error directly reflected to realize the best method from different numerical methods.According to the analyses from those three approaches,we observed that only the error is nominal for the fourth-order Runge-Kutta method.展开更多
In the literature （Tan and Wang, 2010）, Tan and Wang investigated the convergence of the split-step backward Euler （SSBE） method for linear stochastic delay integro-differential equations （SDIDEs） and proved the...In the literature （Tan and Wang, 2010）, Tan and Wang investigated the convergence of the split-step backward Euler （SSBE） method for linear stochastic delay integro-differential equations （SDIDEs） and proved the mean-square stability of SSBE method under some condition. Unfortu- nately, the main result of stability derived by the condition is somewhat restrictive to be applied for practical application. This paper improves the corresponding results. The authors not only prove the mean-square stability of the numerical method but also prove the general mean-square stability of the numerical method. Furthermore, an example is given to illustrate the theory.展开更多
A class of stochastic Besov spaces BpL^(2)(Ω;˙H^(α)(O)),1≤p≤∞andα∈[−2,2],is introduced to characterize the regularity of the noise in the semilinear stochastic heat equation du−Δudt=f(u)dt+dW(t),under the fol...A class of stochastic Besov spaces BpL^(2)(Ω;˙H^(α)(O)),1≤p≤∞andα∈[−2,2],is introduced to characterize the regularity of the noise in the semilinear stochastic heat equation du−Δudt=f(u)dt+dW(t),under the following conditions for someα∈(0,1]:||∫_(0)^(t)e−(t−s)^(A)dW(s)||L^(2)(Ω;L^(2)(O))≤C^(t^(α/2))and||∫_(0)^(t)e−(t−s)^(A)dW(s)||_B^(∞)L^(2)(Ω:H^(α)(O))≤C..The conditions above are shown to be satisfied by both trace-class noises(withα=1)and one-dimensional space-time white noises(withα=1/2).The latter would fail to satisfy the conditions withα=1/2 if the stochastic Besov norm||·||B∞L^(2)(Ω;˙H^(α)(O))is replaced by the classical Sobolev norm||·||L^(2)(Ω;˙Hα(O)),and this often causes reduction of the convergence order in the numerical analysis of the semilinear stochastic heat equation.In this paper,the convergence of a modified exponential Euler method,with a spectral method for spatial discretization,is proved to have orderαin both the time and space for possibly nonsmooth initial data in L^(4)(Ω;˙H^(β)(O))withβ>−1,by utilizing the real interpolation properties of the stochastic Besov spaces and a class of locally refined stepsizes to resolve the singularity of the solution at t=0.展开更多
In this paper,the numerical methods for semi-linear stochastic delay integro-difFerential equations are studied.The uniqueness,existence and stability of analytic solutions of semi-linear stochastic delay integro-diff...In this paper,the numerical methods for semi-linear stochastic delay integro-difFerential equations are studied.The uniqueness,existence and stability of analytic solutions of semi-linear stochastic delay integro-differential equations are studied and some suitable conditions for the mean-square stability of the analytic solutions are also obtained.Then the numerical approximation of exponential Euler method for semi-linear stochastic delay integro-differential equations is constructed and the convergence and the stability of the numerical method are studied.It is proved that the exponential Euler method is convergent with strong order 1/2 and can keep the mean-square exponential stability of the analytical solutions under some restrictions on the step size.In addition,numerical experiments are presented to confirm the theoretical results.展开更多
Many problems in applied mathematics lead to ordinary differential equation. In this paper, a considerable refinement and improvement of the Euler＇s method obtained using PSO （particle swarm optimization） was prese...Many problems in applied mathematics lead to ordinary differential equation. In this paper, a considerable refinement and improvement of the Euler＇s method obtained using PSO （particle swarm optimization） was presented. PSO is a technique based on the cooperation between particles. The exchange of information between these particles allows to resolve difficult problems. This approach is carefully handled and tested with an illustrated example.展开更多
In this paper,we propose a multiphysics finite element method for a nonlinear poroelasticity model with nonlinear stress-strain relation.Firstly,we reformulate the original problem into a new coupled fluid system-a ge...In this paper,we propose a multiphysics finite element method for a nonlinear poroelasticity model with nonlinear stress-strain relation.Firstly,we reformulate the original problem into a new coupled fluid system-a generalized nonlinear Stokes problem of displacement vector field related to pseudo pressure and a diffusion problem of other pseudo pressure fields.Secondly,a fully discrete multiphysics finite element method is performed to solve the reformulated system numerically.Thirdly,existence and uniqueness of the weak solution of the reformulated model and stability analysis and optimal convergence order for the multiphysics finite element method are proven theoretically.Lastly,numerical tests are given to verify the theoretical results.展开更多
Level Set interface treatment method is introduced into Euler method,which is employed for interface treatment method for multi-materials. Combined with the ghost fluid method,the moving interface is tracked. Fifth-or...Level Set interface treatment method is introduced into Euler method,which is employed for interface treatment method for multi-materials. Combined with the ghost fluid method,the moving interface is tracked. Fifth-order WENO spatial discretization and third-order TVD Runge-Kutta time discretization methods are used. Shock-wave action on bubble,implosion and velocity field Shock effect bubbles; implosion and velocity field are simulated by means of LS-MMIC3D programmed by C++. Nu-merical results show that the Level Set interface treatment method is effective and feasible for multi-material interface treatment in comparison with the WENO method.展开更多
This paper develops the mean-square exponential input-to-state stability（exp-ISS） of the Euler-Maruyama（EM） method for stochastic delay control systems（SDCSs）.The definition of mean-square exp-ISS of numerical m...This paper develops the mean-square exponential input-to-state stability（exp-ISS） of the Euler-Maruyama（EM） method for stochastic delay control systems（SDCSs）.The definition of mean-square exp-ISS of numerical methods is established.The conditions of the exact and EM method for an SDCS with the property of mean-square exp-ISS are obtained without involving control Lyapunov functions or functional.Under the global Lipschitz coefficients and mean-square continuous measurable inputs,it is proved that the mean-square exp-ISS of an SDCS holds if and only if that of the EM method is preserved for a sufficiently small step size.The proposed results are evaluated by using numerical experiments to show their effectiveness.展开更多
This paper mainly presents Euler method and fourth-order Runge Kutta Method (RK4) for solving initial value problems (IVP) for ordinary differential equations (ODE). The two proposed methods are quite efficient and pr...This paper mainly presents Euler method and fourth-order Runge Kutta Method (RK4) for solving initial value problems (IVP) for ordinary differential equations (ODE). The two proposed methods are quite efficient and practically well suited for solving these problems. In order to verify the ac-curacy, we compare numerical solutions with the exact solutions. The numerical solutions are in good agreement with the exact solutions. Numerical comparisons between Euler method and Runge Kutta method have been presented. Also we compare the performance and the computational effort of such methods. In order to achieve higher accuracy in the solution, the step size needs to be very small. Finally we investigate and compute the errors of the two proposed methods for different step sizes to examine superiority. Several numerical examples are given to demonstrate the reliability and efficiency.展开更多
Using Euler’s first-order explicit(EE)method and the peridynamic differential operator(PDDO)to discretize the time and internal crystal-size derivatives,respectively,the Euler’s first-order explicit method–peridyna...Using Euler’s first-order explicit(EE)method and the peridynamic differential operator(PDDO)to discretize the time and internal crystal-size derivatives,respectively,the Euler’s first-order explicit method–peridynamic differential operator(EE–PDDO)was obtained for solving the one-dimensional population balance equation in crystallization.Four different conditions during crystallization were studied:size-independent growth,sizedependent growth in a batch process,nucleation and size-independent growth,and nucleation and size-dependent growth in a continuous process.The high accuracy of the EE–PDDO method was confirmed by comparing it with the numerical results obtained using the second-order upwind and HR-van methods.The method is characterized by non-oscillation and high accuracy,especially in the discontinuous and sharp crystal size distribution.The stability of the EE–PDDO method,choice of weight function in the PDDO method,and optimal time step are also discussed.展开更多
A preconditioned gridless method is developed for solving the Euler equations at low Mach numbers.The preconditioned system in a conservation form is obtained by multiplying apreconditioning matrix of the type of Weis...A preconditioned gridless method is developed for solving the Euler equations at low Mach numbers.The preconditioned system in a conservation form is obtained by multiplying apreconditioning matrix of the type of Weiss and Smith to the time derivative of the Euler equations,which are discretized using agridless technique wherein the physical domain is distributed by clouds of points.The implementation of the preconditioned gridless method is mainly based on the frame of the traditional gridless method without preconditioning,which may fail to converge for low Mach number simulations.Therefore,the modifications corresponding to the affected terms of preconditioning are mainly addressed.The numerical results show that the preconditioned gridless method still functions for compressible transonic flow simulations and additionally,for nearly incompressible flow simulations at low Mach numbers as well.The paper ends with the nearly incompressible flow over a multi-element airfoil,which demonstrates the ability of the method presented for treating flows over complicated geometries.展开更多
Chaotic vibrations of flexible non-linear Euler-Bernoulli beams subjected to harmonic load and with various boundary conditions（symmetric and non-symmetric）are studied in this work.Reliability of the obtained result...Chaotic vibrations of flexible non-linear Euler-Bernoulli beams subjected to harmonic load and with various boundary conditions（symmetric and non-symmetric）are studied in this work.Reliability of the obtained results is verified by the finite difference method（FDM）and the finite element method（FEM）with the Bubnov-Galerkin approximation for various boundary conditions and various dynamic regimes（regular and non-regular）.The influence of boundary conditions on the Euler-Bernoulli beams dynamics is studied mainly,dynamic behavior vs.control parameters { ωp,q0 } is reported,and scenarios of the system transition into chaos are illustrated.展开更多
Through transformations, the time-dependent boundary condition on the airfoil contour and the boundary condition at infinity are brought fixed to the boundaries of a finite domain. The boundary conditions can thus be ...Through transformations, the time-dependent boundary condition on the airfoil contour and the boundary condition at infinity are brought fixed to the boundaries of a finite domain. The boundary conditions can thus be satisfied exactly without increasing the computational time. The novel scheme is useful for computing transonic, strong disturbance, unsteady flows with high reduced frequencies. The scheme makes use of curvefitted orthogonal meshes and the lattice control technique to obtain the optimal grid distribution. The numerical results are satisfactory.展开更多
The numerical solutions of standing waves for Euler equations with the nonlinear free surface boundary condition in a two-dimensional （2D） tank are studied. The irregular tank is mapped onto a fixed square domain th...The numerical solutions of standing waves for Euler equations with the nonlinear free surface boundary condition in a two-dimensional （2D） tank are studied. The irregular tank is mapped onto a fixed square domain through proper mapping functions. A staggered mesh system is employed in a 2D tank to calculate the elevation of the transient fluid. A time-independent finite difference method, which is developed by Bang- fuh Chen, is used to solve the Euler equations for incompressible and inviscid fluids. The numerical results agree well with the analytic solutions and previously published results. The sloshing profiles of surge and heave motion with initial standing waves are presented. The results show very clear nonlinear and beating phenomena.展开更多
Euler-Bernoulli beam equation is very important that can be applied in the field of mechanics, science and technology. Some authors have put forward many different numerical methods, but the precision is not enough hi...Euler-Bernoulli beam equation is very important that can be applied in the field of mechanics, science and technology. Some authors have put forward many different numerical methods, but the precision is not enough high. In this paper, we will illustrate the high-precision numerical method to solve Euler-Bernoulli beam equation. Three numerical examples are studied to demonstrate the accuracy of the present method. Results obtained by our method indicate new algorithm has the following advantages: small computational work, fast convergence speed and high precision.展开更多
Several efficient analytical methods have been developed to solve the solid-state diffusion problem, for constant diffusion coefficient problems. However, these methods cannot be applied for concentration-dependent di...Several efficient analytical methods have been developed to solve the solid-state diffusion problem, for constant diffusion coefficient problems. However, these methods cannot be applied for concentration-dependent diffusion coefficient problems and numerical methods are used instead. Herein, grid-based numerical methods derived from the control volume discretization are presented to resolve the characteristic nonlinear system of partial differential equations. A novel hybrid backward Euler control volume (HBECV) method is presented which requires only one iteration to reach an implicit solution. The HBECV results are shown to be stable and accurate for a moderate number of grid points. The computational speed and accuracy of the HBECV, justify its use in battery simulations, in which the solid-state diffusion coefficient is a strong function of the concentration.展开更多
基金supported by the National Natural Science Foundation of China(Nos.12471394,12371417)Natural Science Foundation of Changsha(No.kq2502101)。
文摘This paper aims to investigate the tamed Euler method for the random periodic solution of semilinear SDEs with one-sided Lipschitz coefficient.We introduce a novel approach to analyze mean-square error bounds of the novel schemes,without relying on a priori high-order moment bound of the numerical approximation.The expected order-one mean square convergence is attained for the proposed scheme.Moreover,a numerical example is presented to verify our theoretical analysis.
文摘Mineral exploration is done by different methods. Geophysical and geochemical studies are two powerful tools in this field. In integrated studies, the results of each study are used to determine the location of the drilling boreholes. The purpose of this study is to use field geophysics to calculate the depth of mineral reserve. The study area is located 38 km from Zarand city called Jalalabad iron mine. In this study, gravimetric data were measured and mineral depth was calculated using the Euler method. 1314 readings have been performed in this area. The rocks of the region include volcanic and sedimentary. The source of the mineralization in the area is hydrothermal processes. After gravity measuring in the region, the data were corrected, then various methods such as anomalous map remaining in levels one and two, upward expansion, first and second-degree vertical derivatives, analytical method, and analytical signal were drawn, and finally, the depth of the deposit was estimated by Euler method. As a result, the depth of the mineral deposit was calculated to be between 20 and 30 meters on average.
基金Supported by the NSF of the Higher Education Institutions of Jiangsu Province(10KJD110006)Supported by the grant of Jiangsu Institute of Education(Jsjy2009zd03)Supported by the Qing Lan Project of Jiangsu Province(2010)
文摘In this paper,we present the semi-implicit Euler（SIE）numerical solution for stochastic pantograph equations with jumps and prove that the SIE approximation solution converges to the exact solution in the mean-square sense under the Local Lipschitz condition.
文摘The paper presents the comparative study on numerical methods of Euler method,Improved Euler method and fourth-order Runge-Kutta method for solving the engineering problems and applications.The three proposed methods are quite efficient and practically well suited for solving the unknown engineering problems.This paper aims to enhance the teaching and learning quality of teachers and students for various levels.At each point of the interval,the value of y is calculated and compared with its exact value at that point.The next interesting point is the observation of error from those methods.Error in the value of y is the difference between calculated and exact value.A mathematical equation which relates various functions with its derivatives is known as a differential equation.It is a popular field of mathematics because of its application to real-world problems.To calculate the exact values,the approximate values and the errors,the numerical tool such as MATLAB is appropriate for observing the results.This paper mainly concentrates on identifying the method which provides more accurate results.Then the analytical results and calculates their corresponding error were compared in details.The minimum error directly reflected to realize the best method from different numerical methods.According to the analyses from those three approaches,we observed that only the error is nominal for the fourth-order Runge-Kutta method.
基金supported by the Fundamental Research Funds for the Central Universities under Grant No. 2012089:31541111213China Postdoctoral Science Foundation Funded Project under Grant No.2012M511615the State Key Program of National Natural Science of China under Grant No.61134012
文摘In the literature （Tan and Wang, 2010）, Tan and Wang investigated the convergence of the split-step backward Euler （SSBE） method for linear stochastic delay integro-differential equations （SDIDEs） and proved the mean-square stability of SSBE method under some condition. Unfortu- nately, the main result of stability derived by the condition is somewhat restrictive to be applied for practical application. This paper improves the corresponding results. The authors not only prove the mean-square stability of the numerical method but also prove the general mean-square stability of the numerical method. Furthermore, an example is given to illustrate the theory.
基金supported by National Natural Science Foundation of China(Grant Nos.12071020,12131005 and U2230402)the Research Grants Council of Hong Kong(Grant No.Poly U15300519)an Internal Grant of The Hong Kong Polytechnic University(Grant No.P0038843,Work Programme:ZVX7)。
文摘A class of stochastic Besov spaces BpL^(2)(Ω;˙H^(α)(O)),1≤p≤∞andα∈[−2,2],is introduced to characterize the regularity of the noise in the semilinear stochastic heat equation du−Δudt=f(u)dt+dW(t),under the following conditions for someα∈(0,1]:||∫_(0)^(t)e−(t−s)^(A)dW(s)||L^(2)(Ω;L^(2)(O))≤C^(t^(α/2))and||∫_(0)^(t)e−(t−s)^(A)dW(s)||_B^(∞)L^(2)(Ω:H^(α)(O))≤C..The conditions above are shown to be satisfied by both trace-class noises(withα=1)and one-dimensional space-time white noises(withα=1/2).The latter would fail to satisfy the conditions withα=1/2 if the stochastic Besov norm||·||B∞L^(2)(Ω;˙H^(α)(O))is replaced by the classical Sobolev norm||·||L^(2)(Ω;˙Hα(O)),and this often causes reduction of the convergence order in the numerical analysis of the semilinear stochastic heat equation.In this paper,the convergence of a modified exponential Euler method,with a spectral method for spatial discretization,is proved to have orderαin both the time and space for possibly nonsmooth initial data in L^(4)(Ω;˙H^(β)(O))withβ>−1,by utilizing the real interpolation properties of the stochastic Besov spaces and a class of locally refined stepsizes to resolve the singularity of the solution at t=0.
基金This research is supported by National Natural Science Foundation of China(Project No.11901173)by the Heilongjiang province Natural Science Foundation(LH2019A030)by the Heilongjiang province Innovation Talent Foundation(2018CX17).
文摘In this paper,the numerical methods for semi-linear stochastic delay integro-difFerential equations are studied.The uniqueness,existence and stability of analytic solutions of semi-linear stochastic delay integro-differential equations are studied and some suitable conditions for the mean-square stability of the analytic solutions are also obtained.Then the numerical approximation of exponential Euler method for semi-linear stochastic delay integro-differential equations is constructed and the convergence and the stability of the numerical method are studied.It is proved that the exponential Euler method is convergent with strong order 1/2 and can keep the mean-square exponential stability of the analytical solutions under some restrictions on the step size.In addition,numerical experiments are presented to confirm the theoretical results.
文摘Many problems in applied mathematics lead to ordinary differential equation. In this paper, a considerable refinement and improvement of the Euler＇s method obtained using PSO （particle swarm optimization） was presented. PSO is a technique based on the cooperation between particles. The exchange of information between these particles allows to resolve difficult problems. This approach is carefully handled and tested with an illustrated example.
基金Supported by the National Natural Science Foundation of China(Grant Nos.12371393,11971150 and 11801143)Natural Science Foundation of Henan Province(Grant No.242300421047).
文摘In this paper,we propose a multiphysics finite element method for a nonlinear poroelasticity model with nonlinear stress-strain relation.Firstly,we reformulate the original problem into a new coupled fluid system-a generalized nonlinear Stokes problem of displacement vector field related to pseudo pressure and a diffusion problem of other pseudo pressure fields.Secondly,a fully discrete multiphysics finite element method is performed to solve the reformulated system numerically.Thirdly,existence and uniqueness of the weak solution of the reformulated model and stability analysis and optimal convergence order for the multiphysics finite element method are proven theoretically.Lastly,numerical tests are given to verify the theoretical results.
基金supported by the National Natural Science Foundation of China (Grant Nos. 10872085 and 10472042)Program for New Century Excellent Talents of Ministry of Education (Grant No. NCET-08-0043)+1 种基金National Basic Research Program of China (Grant No. 2010CB832700 6)Key Program of Numerical Simulation Key Laboratory of Sichuan Higher School (Grant No. 07NJZZ001)
文摘Level Set interface treatment method is introduced into Euler method,which is employed for interface treatment method for multi-materials. Combined with the ghost fluid method,the moving interface is tracked. Fifth-order WENO spatial discretization and third-order TVD Runge-Kutta time discretization methods are used. Shock-wave action on bubble,implosion and velocity field Shock effect bubbles; implosion and velocity field are simulated by means of LS-MMIC3D programmed by C++. Nu-merical results show that the Level Set interface treatment method is effective and feasible for multi-material interface treatment in comparison with the WENO method.
基金Supported by National Natural Science Foundation of China(10571036)the Key Discipline Development Program of Beijing Municipal Commission (XK100080537)
基金supported by the National Natural Science Foundation of China(6127312660904032)the Natural Science Foundation of Guangdong Province(10251064101000008)
文摘This paper develops the mean-square exponential input-to-state stability（exp-ISS） of the Euler-Maruyama（EM） method for stochastic delay control systems（SDCSs）.The definition of mean-square exp-ISS of numerical methods is established.The conditions of the exact and EM method for an SDCS with the property of mean-square exp-ISS are obtained without involving control Lyapunov functions or functional.Under the global Lipschitz coefficients and mean-square continuous measurable inputs,it is proved that the mean-square exp-ISS of an SDCS holds if and only if that of the EM method is preserved for a sufficiently small step size.The proposed results are evaluated by using numerical experiments to show their effectiveness.
文摘This paper mainly presents Euler method and fourth-order Runge Kutta Method (RK4) for solving initial value problems (IVP) for ordinary differential equations (ODE). The two proposed methods are quite efficient and practically well suited for solving these problems. In order to verify the ac-curacy, we compare numerical solutions with the exact solutions. The numerical solutions are in good agreement with the exact solutions. Numerical comparisons between Euler method and Runge Kutta method have been presented. Also we compare the performance and the computational effort of such methods. In order to achieve higher accuracy in the solution, the step size needs to be very small. Finally we investigate and compute the errors of the two proposed methods for different step sizes to examine superiority. Several numerical examples are given to demonstrate the reliability and efficiency.
文摘Using Euler’s first-order explicit(EE)method and the peridynamic differential operator(PDDO)to discretize the time and internal crystal-size derivatives,respectively,the Euler’s first-order explicit method–peridynamic differential operator(EE–PDDO)was obtained for solving the one-dimensional population balance equation in crystallization.Four different conditions during crystallization were studied:size-independent growth,sizedependent growth in a batch process,nucleation and size-independent growth,and nucleation and size-dependent growth in a continuous process.The high accuracy of the EE–PDDO method was confirmed by comparing it with the numerical results obtained using the second-order upwind and HR-van methods.The method is characterized by non-oscillation and high accuracy,especially in the discontinuous and sharp crystal size distribution.The stability of the EE–PDDO method,choice of weight function in the PDDO method,and optimal time step are also discussed.
基金supported by the National Natural Science Foundation of China(No.11172134)
文摘A preconditioned gridless method is developed for solving the Euler equations at low Mach numbers.The preconditioned system in a conservation form is obtained by multiplying apreconditioning matrix of the type of Weiss and Smith to the time derivative of the Euler equations,which are discretized using agridless technique wherein the physical domain is distributed by clouds of points.The implementation of the preconditioned gridless method is mainly based on the frame of the traditional gridless method without preconditioning,which may fail to converge for low Mach number simulations.Therefore,the modifications corresponding to the affected terms of preconditioning are mainly addressed.The numerical results show that the preconditioned gridless method still functions for compressible transonic flow simulations and additionally,for nearly incompressible flow simulations at low Mach numbers as well.The paper ends with the nearly incompressible flow over a multi-element airfoil,which demonstrates the ability of the method presented for treating flows over complicated geometries.
文摘Chaotic vibrations of flexible non-linear Euler-Bernoulli beams subjected to harmonic load and with various boundary conditions（symmetric and non-symmetric）are studied in this work.Reliability of the obtained results is verified by the finite difference method（FDM）and the finite element method（FEM）with the Bubnov-Galerkin approximation for various boundary conditions and various dynamic regimes（regular and non-regular）.The influence of boundary conditions on the Euler-Bernoulli beams dynamics is studied mainly,dynamic behavior vs.control parameters { ωp,q0 } is reported,and scenarios of the system transition into chaos are illustrated.
文摘Through transformations, the time-dependent boundary condition on the airfoil contour and the boundary condition at infinity are brought fixed to the boundaries of a finite domain. The boundary conditions can thus be satisfied exactly without increasing the computational time. The novel scheme is useful for computing transonic, strong disturbance, unsteady flows with high reduced frequencies. The scheme makes use of curvefitted orthogonal meshes and the lattice control technique to obtain the optimal grid distribution. The numerical results are satisfactory.
文摘The numerical solutions of standing waves for Euler equations with the nonlinear free surface boundary condition in a two-dimensional （2D） tank are studied. The irregular tank is mapped onto a fixed square domain through proper mapping functions. A staggered mesh system is employed in a 2D tank to calculate the elevation of the transient fluid. A time-independent finite difference method, which is developed by Bang- fuh Chen, is used to solve the Euler equations for incompressible and inviscid fluids. The numerical results agree well with the analytic solutions and previously published results. The sloshing profiles of surge and heave motion with initial standing waves are presented. The results show very clear nonlinear and beating phenomena.
文摘Euler-Bernoulli beam equation is very important that can be applied in the field of mechanics, science and technology. Some authors have put forward many different numerical methods, but the precision is not enough high. In this paper, we will illustrate the high-precision numerical method to solve Euler-Bernoulli beam equation. Three numerical examples are studied to demonstrate the accuracy of the present method. Results obtained by our method indicate new algorithm has the following advantages: small computational work, fast convergence speed and high precision.
文摘Several efficient analytical methods have been developed to solve the solid-state diffusion problem, for constant diffusion coefficient problems. However, these methods cannot be applied for concentration-dependent diffusion coefficient problems and numerical methods are used instead. Herein, grid-based numerical methods derived from the control volume discretization are presented to resolve the characteristic nonlinear system of partial differential equations. A novel hybrid backward Euler control volume (HBECV) method is presented which requires only one iteration to reach an implicit solution. The HBECV results are shown to be stable and accurate for a moderate number of grid points. The computational speed and accuracy of the HBECV, justify its use in battery simulations, in which the solid-state diffusion coefficient is a strong function of the concentration.
