The three-dimensional finite element method of lines is presented, and the basic processing description of 3D FEMOL in cracking questions is given in detail. Applications to 3D bodies with cracks indicate that good ac...The three-dimensional finite element method of lines is presented, and the basic processing description of 3D FEMOL in cracking questions is given in detail. Applications to 3D bodies with cracks indicate that good accuracy can be obtained with relatively coarse girds. In particular, application to the tension specimen shows very good agreement with the evaluation of stress intensity factors, which is better than the results of other methods. This implies a considerable potential for using this method in the 3D analysis of finite geometry solids and suggests a possible extension of this technique to nonlinear material behavior.展开更多
The Finite Element Method of Lines （FEMOL） is a semi-analytic approach and takes a position between FEM and analytic methods. First, FEMOL in Fracture Mechanics is presented in detail. Then, the method is applied to...The Finite Element Method of Lines （FEMOL） is a semi-analytic approach and takes a position between FEM and analytic methods. First, FEMOL in Fracture Mechanics is presented in detail. Then, the method is applied to a set of examples such as edge-crack plate, the central-crack plate, the plate with cracks emanating from a hole under tensile or under combination loads of tensile and bending. Their dimensionless stress distribution, the stress intensify factor （SIF） and crack opening displacement （COD） are obtained, and comparison with known solutions by other methods are reported. It is found that a good accuracy is achieved by FEMOL. The method is successfully modified to remarkably increase the accuracy and reduce convergence difficulties. So it is a very useful and new tool in studying fracture mechanics problems.展开更多
The method of lines based on Hu Hai-chang 's theory for the vibration and stability of moderate thick plates is developed. The standard nonlinear ordinary differential equation (ODE) system for natural frequencies...The method of lines based on Hu Hai-chang 's theory for the vibration and stability of moderate thick plates is developed. The standard nonlinear ordinary differential equation (ODE) system for natural frequencies and critical load is given by use of ODE techniques, and then any indicated eigenvalue could be obtained directly from ODE solver by employing the so-called initial eigenfunction technique instead of the mode orthogonality condition. Numerical examples show that the present method is very effective and reliable.展开更多
Based on the sub-region generalized variationM principle, a sub-region mixed version of the newly-developed semi-analytical ＇finite element method of lines＇ （FEMOL） is proposed in this paper for accurate and effic...Based on the sub-region generalized variationM principle, a sub-region mixed version of the newly-developed semi-analytical ＇finite element method of lines＇ （FEMOL） is proposed in this paper for accurate and efficient computation of stress intensity factors （SIFs） of two-dimensional notches/cracks. The circular regions surrounding notch/crack tips are taken as the complementary energy region in which a number of leading terms of singular solutions for stresses are used, with the sought SIFs being among the unknown coefficients. The rest of the arbitrary domain is taken as the potential energy region in which FEMOL is applied to obtain approximate displacements. A mixed system of ordinary differential equations （ODEs） and algebraic equations is derived via the sub-region generalized variational principle. A singularity removal technique that eliminates the stress parameters from the mixed equation system eventually yields a standard FEMOL ODE system, the solution of which is no longer singular and is simply and efficiently obtained using a standard general-purpose ODE solver. A number of numerical examples, including bi-material notches/cracks in anti-plane and plane elasticity, are given to show the generally excellent performance of the proposed method.展开更多
The method of lines(MOL) for solving the problems of functionally gradient materials(FGMs) was studied. Navier’s equations for FGMs were derived, and were semi-discretized into a system of ordinary differential (equa...The method of lines(MOL) for solving the problems of functionally gradient materials(FGMs) was studied. Navier’s equations for FGMs were derived, and were semi-discretized into a system of ordinary differential (equations(ODEs)) defined on discrete lines with the finite difference. By solving the system of ODEs, the solutions to the problem can be obtained. An example of three-point bending was given to demonstrate the application of MOL for a crack problem in the FGM. The computational results show that the more accurate results can be obtained with less computational time and resources. The obvious difficulties of numerical method for crack problems in FGMs, such as the effect of material nonhomogeneity and the existence of high gradient stress and strain near a crack tip, can be overcome without additional consideration if this method is adopted.展开更多
In this paper, we solve chiral nonlinear Schrodinger equation (CNSE) numerically. Two numerical methods are derived using the explicit Runge-Kutta method of order four and the linear multistep method (Predictor-Correc...In this paper, we solve chiral nonlinear Schrodinger equation (CNSE) numerically. Two numerical methods are derived using the explicit Runge-Kutta method of order four and the linear multistep method (Predictor-Corrector method of fourth order). The resulting schemes of fourth order accuracy in spatial and temporal directions. The CNSE is non-integrable and has two kinds of soliton solutions: bright and dark soliton. The exact solutions and the conserved quantities of CNSE are used to display the efficiency and robustness of the numerical methods we derived. Interaction of two bright solitons for different parameters is also displayed.展开更多
This work deals with the determination of the temperature profile within a direct heating<span style="font-family:;" "=""> </span><span style="font-family:;" "=&q...This work deals with the determination of the temperature profile within a direct heating<span style="font-family:;" "=""> </span><span style="font-family:;" "=""><span style="font-family:Verdana;">moving bed </span><span style="font-family:Verdana;">torrefier</span> <span style="font-family:Verdana;">in order to</span><span style="font-family:Verdana;"> determine its minimum column height. A thermal model based on </span><span style="font-family:Verdana;">eulerian-eulerian</span> <span style="font-family:Verdana;">two</span></span><span style="font-family:Verdana;">-</span><span style="font-family:Verdana;">phase</span><span style="font-family:;" "=""><span style="font-family:Verdana;"> solid-gas theory was performed and solved with the method of lines. </span><span style="font-family:Verdana;">In addition</span><span style="font-family:Verdana;">, this study allows </span><span style="font-family:Verdana;">to investigate</span><span style="font-family:Verdana;"> the effect of the biomass particle size on the minimum </span><span style="font-family:Verdana;">torrefier</span><span style="font-family:Verdana;"> column height. This</span></span><span style="font-family:Verdana;"> investigation </span><span style="font-family:Verdana;">was performed by changing, simultaneously, the diameter of particles and the minimum fluidization velocity of the bed. Then, the calculations were made for a counter-current torrefaction reactor of 30</span><span style="font-family:;" "=""> </span><span style="font-family:;" "=""><span style="font-family:Verdana;">cm in diameter and </span><span style="font-family:Verdana;">for</span><span style="font-family:Verdana;"> 5</span></span><span style="font-family:;" "=""> </span><span style="font-family:Verdana;">kg/h of the feed rate of raw sugarcane bagasse.</span><span style="font-family:Verdana;"> Results show that the height of the reactor column must be at least 30 cm for that are 1 mm in diameter and 108 cm for particles that are 2 mm in diameter.</span>展开更多
This paper presents a method of lines solution based on the reproducing kernel Hilbert space method to the nonlinear one-dimensional Klein-Gordon equation that arises in many scientific fields areas.Our method uses di...This paper presents a method of lines solution based on the reproducing kernel Hilbert space method to the nonlinear one-dimensional Klein-Gordon equation that arises in many scientific fields areas.Our method uses discretization of the partial derivatives of the space variable to get a system of ODEs in the time variable and then solve the system of ODEs using reproducing kernel Hilbert space method.Consider two examples to validate the proposed method.Compare the results with the exact solution by calculating the error norms L_(2) and L_(∞) at various time levels.The results show that the presented scheme is a systematic,effective and powerful technique for the solution of Klein-Gordon equation.展开更多
This article extends the finite element method of lines to a parabolic initial boundary value problem whose diffusion coefficient is discontinuous across an interface that changes with respect to time.The method prese...This article extends the finite element method of lines to a parabolic initial boundary value problem whose diffusion coefficient is discontinuous across an interface that changes with respect to time.The method presented here uses immersed finite element(IFE)functions for the discretization in spatial variables that can be carried out over a fixedmesh(such as a Cartesianmesh if desired),and this featuremakes it possible to reduce the parabolic equation to a system of ordinary differential equations(ODE)through the usual semi-discretization procedure.Therefore,with a suitable choice of the ODE solver,this method can reliably and efficiently solve a parabolic moving interface problem over a fixed structured(Cartesian)mesh.Numerical examples are presented to demonstrate features of this new method.展开更多
In this paper,a nonlinear shallow-water model of tsunami wave propagation at different points along a coastline of an ocean has been numerically simulated using method of lines.The simulation is carried out for variou...In this paper,a nonlinear shallow-water model of tsunami wave propagation at different points along a coastline of an ocean has been numerically simulated using method of lines.The simulation is carried out for various coastal slopes and the ocean depths.The effects of the coast slope and sea depth on the tsunami wave run-up height and velocity are illustrated.The accuracy of the mathematical model is verified by solving a classical test problem with known analytic solution.The computed run-up height and velocity show satisfactory agreement with the tsunami wave physics.展开更多
The model of electrically driven jet is governed by a series of quasi 1D dimensionless partial differential equations(PDEs).Following the method of lines,the Chebyshev collocation method is employed to discretize the ...The model of electrically driven jet is governed by a series of quasi 1D dimensionless partial differential equations(PDEs).Following the method of lines,the Chebyshev collocation method is employed to discretize the PDEs and obtain a system of differential-algebraic equations(DAEs).By differentiating constrains in DAEs twice,the system is transformed into a set of ordinary differential equations(ODEs) with invariants.Then the implicit differential equations solver 'ddaskr' is used to solve the ODEs and post-stabilization is executed at the end of each step.Results show the distributions of radius,linear charge density,stretching ratio and also the horizontal velocity at a time point.Meanwhile,the spiral and expanding projections to X-Y plane of the jet centerline suggest the occurring of bending instability.展开更多
A first principles-based dynamic model for a continuous catalyst regeneration (CCR) platforming process, the UOP commercial naphtha catalytic reforming process, is developed in this paper. The lumping details of the n...A first principles-based dynamic model for a continuous catalyst regeneration (CCR) platforming process, the UOP commercial naphtha catalytic reforming process, is developed in this paper. The lumping details of the naphtha feed and reaction scheme of the reaction model are given. The process model is composed of the reforming reaction model with catalyst deactivation, the furnace model and the separator model, which is capable of capturing the major dynamics that occurs in this process system. Dynamic simulations are performed based on Gear numerical algorithm and method of lines (MOL), a numerical technique dealing with partial differential equations (PDEs). The results of simulation are also presented. Dynamic responses caused by disturbances in the process system can be correctly predicted through simulations.展开更多
The ultimate goal and highlight of this paper are to explore water levels along the coast of Bangladesh efficiently due to the nonlinear interaction of tide and surge by employing the method of lines(MOLs)with the aid...The ultimate goal and highlight of this paper are to explore water levels along the coast of Bangladesh efficiently due to the nonlinear interaction of tide and surge by employing the method of lines(MOLs)with the aid of newly proposed RKAHeM(4,4)technique.In this regard,the spatial derivatives of shallow water equations(SWEs)were discretized by means of a finite difference method to obtain a system of ordinary differential equations(ODEs)of initial valued with time as an independent variable.The obtained system of ODEs was solved by the RKAHeM(4,4)technique.One-way nested grid technique was exercised to incorporate coastal complexities closely with minimum computational cost.A stable tidal oscillation was produced over the region of interest by applying the most influential tidal constituent M2 along the southern open boundary of the outer scheme.The newly developed model was applied to estimate water levels due to the non-linear interaction of tide and surge associated with the catastrophic cyclone April 1991 along the coast of Bangladesh.The approach employed in the study was found to perform well and ensure conformity with real-time observations.展开更多
The numerical method of lines(MOLs)in coordination with the classical fourth-order Runge Kutta(RK(4,4))method is used to solve shallow water equations(SWEs)for foreseeing water levels owing to the nonlinear interactio...The numerical method of lines(MOLs)in coordination with the classical fourth-order Runge Kutta(RK(4,4))method is used to solve shallow water equations(SWEs)for foreseeing water levels owing to the nonlinear interaction of tide and surge accompanying with a storm along the coast of Bangladesh.The SWEs are developed by extending the body forces with tide generating forces(TGFs).Spatial variables of the SWEs along with the boundary conditions are approximated by means of finite difference technique on an Arakawa C-grid to attain a system of ordinary differential equations(ODEs)of initial valued in time,which are being solved with the aid of the RK(4,4)method.Nested grid technique is adopted to solve coastal complexities closely with least computational cost.A stable tidal solution in the region of our choice is produced by applying the tidal forcing with the major tidal constituent M2(lunar semi-diurnal)along the southern open-sea boundary of the outer scheme.Numerical experimentations are carried out to simulate water levels generated by the cyclonic storm AILA along the coast of Bangladesh.The model simulated results are found to be in a reasonable agreement with the limited available reported data and observations.展开更多
For an anti-plane problem, the differential operator is self-adjoint and the corresponding eigenfunctions belong to the Hilbert space. The orthogonal property between eigenfunctions （or between the derivatives of eig...For an anti-plane problem, the differential operator is self-adjoint and the corresponding eigenfunctions belong to the Hilbert space. The orthogonal property between eigenfunctions （or between the derivatives of eigenfunctions） of anti-plane problem is exploited. We developed for the first time two sets of radius-independent orthogonal integrals for extraction of stress intensity factors （SIFs）, so any order SIF can be extracted based on a certain known solution of displacement （an analytic result or a numerical result）. Many numerical examples based on the finite element method of lines （FEMOL） show that the present method is very powerful and efficient.展开更多
Usually, it is very difficult to find out an analytical solution to thermal conduction problems during high temperature welding. Therefore, as an important numerical approach, the method of lines （MOLs） is introduce...Usually, it is very difficult to find out an analytical solution to thermal conduction problems during high temperature welding. Therefore, as an important numerical approach, the method of lines （MOLs） is introduced to solve the temperature field characterized by high gradients. The basic idea of the method is to semi-discretize the governing equation of the problem into a system of ordinary differential equations （ODEs） defined on discrete lines by means of the finite difference method, by which the thermal boundary condition with high gradients are directly embodied in formulation. Thus the temperature field can be obtained by solving the ODEs. As a numerical example, the variation of an axisymmetrical temperature field along the plate thickness can be obtained.展开更多
In this paper, the one-dimensional time dependent Schr?dinger equation is discretized by the method of lines using a second order finite difference approximation to replace the second order spatial derivative. The evo...In this paper, the one-dimensional time dependent Schr?dinger equation is discretized by the method of lines using a second order finite difference approximation to replace the second order spatial derivative. The evolving system of stiff Ordinary Differential Equation (ODE) in time is solved numerically by an L-stable trapezoidal-like integrator. Results show accuracy of relative maximum error of order 10?4 in the interval of consideration. The performance of the method as compared to an existing scheme is considered favorable.展开更多
Microbiological experiments show that the colonies of the bacterium bacillus subtilis placed on a dish filled with an agar medium and nutrient form varied spatial patterns while the individual cells grow, reproduce an...Microbiological experiments show that the colonies of the bacterium bacillus subtilis placed on a dish filled with an agar medium and nutrient form varied spatial patterns while the individual cells grow, reproduce and migrate on the dish in clumps. In this paper, we discuss a system of reaction-diffusion equations that can be used with a view to modelling this phenomenon and we solve it numerically by means of the method of lines. For the spatial discretization, we use the finite difference method and Galerkin finite element method. We present how the spatial patterns obtained depend on the spatial discretization employed and we measure the experimental order of convergence of the numerical schemes used. Further, we present the numerical results obtained by solving the model in a cubic domain.展开更多
In this paper, we describe a comparison of two spatial discretization schemes for the advection equation, namely the first finite difference method and the method of lines. The stability of the methods has been studie...In this paper, we describe a comparison of two spatial discretization schemes for the advection equation, namely the first finite difference method and the method of lines. The stability of the methods has been studied by Von Neumann method and with the matrix analysis. The methods are applied to a number of test problems to compare the accuracy and computational efficiency. We show that both discretization techniques approximate correctly solution of advection equation and compare their accuracy and performance.展开更多
A mathematical model is presented for the charging-up process in an air-entrapped pipeline with moving boundary conditions. A coordinate transformation technique is employed to reduce fluid motion in time-dependent do...A mathematical model is presented for the charging-up process in an air-entrapped pipeline with moving boundary conditions. A coordinate transformation technique is employed to reduce fluid motion in time-dependent domains to ones in time-independent domains. The nonlinear hyperbolic partial differential equations governing the unsteady motion of fluid combined with an equation for transient shear stress between the pipe wall and the flowing fluid are solved by the method of lines. Results show that ignoring elastic effects overestimates the maximum pressure and underestimates the maximum front velocity of filling fluid. The peak pressure of the entrapped air is sensitive to the length of the initial entrapped air pocket.展开更多
文摘The three-dimensional finite element method of lines is presented, and the basic processing description of 3D FEMOL in cracking questions is given in detail. Applications to 3D bodies with cracks indicate that good accuracy can be obtained with relatively coarse girds. In particular, application to the tension specimen shows very good agreement with the evaluation of stress intensity factors, which is better than the results of other methods. This implies a considerable potential for using this method in the 3D analysis of finite geometry solids and suggests a possible extension of this technique to nonlinear material behavior.
文摘The Finite Element Method of Lines （FEMOL） is a semi-analytic approach and takes a position between FEM and analytic methods. First, FEMOL in Fracture Mechanics is presented in detail. Then, the method is applied to a set of examples such as edge-crack plate, the central-crack plate, the plate with cracks emanating from a hole under tensile or under combination loads of tensile and bending. Their dimensionless stress distribution, the stress intensify factor （SIF） and crack opening displacement （COD） are obtained, and comparison with known solutions by other methods are reported. It is found that a good accuracy is achieved by FEMOL. The method is successfully modified to remarkably increase the accuracy and reduce convergence difficulties. So it is a very useful and new tool in studying fracture mechanics problems.
基金The project supported by the Pioneer Fundation of Tongji University
文摘The method of lines based on Hu Hai-chang 's theory for the vibration and stability of moderate thick plates is developed. The standard nonlinear ordinary differential equation (ODE) system for natural frequencies and critical load is given by use of ODE techniques, and then any indicated eigenvalue could be obtained directly from ODE solver by employing the so-called initial eigenfunction technique instead of the mode orthogonality condition. Numerical examples show that the present method is very effective and reliable.
基金Project supported by the National Natural Sciences Foundation of China(Nos.59525813 and 19872066)the Cardiff Advanced Chinese Engineering Centre of Cardiff University.
文摘Based on the sub-region generalized variationM principle, a sub-region mixed version of the newly-developed semi-analytical ＇finite element method of lines＇ （FEMOL） is proposed in this paper for accurate and efficient computation of stress intensity factors （SIFs） of two-dimensional notches/cracks. The circular regions surrounding notch/crack tips are taken as the complementary energy region in which a number of leading terms of singular solutions for stresses are used, with the sought SIFs being among the unknown coefficients. The rest of the arbitrary domain is taken as the potential energy region in which FEMOL is applied to obtain approximate displacements. A mixed system of ordinary differential equations （ODEs） and algebraic equations is derived via the sub-region generalized variational principle. A singularity removal technique that eliminates the stress parameters from the mixed equation system eventually yields a standard FEMOL ODE system, the solution of which is no longer singular and is simply and efficiently obtained using a standard general-purpose ODE solver. A number of numerical examples, including bi-material notches/cracks in anti-plane and plane elasticity, are given to show the generally excellent performance of the proposed method.
基金Projects(90305023 59731020) supported by the National Natural Science Foundation of China
文摘The method of lines(MOL) for solving the problems of functionally gradient materials(FGMs) was studied. Navier’s equations for FGMs were derived, and were semi-discretized into a system of ordinary differential (equations(ODEs)) defined on discrete lines with the finite difference. By solving the system of ODEs, the solutions to the problem can be obtained. An example of three-point bending was given to demonstrate the application of MOL for a crack problem in the FGM. The computational results show that the more accurate results can be obtained with less computational time and resources. The obvious difficulties of numerical method for crack problems in FGMs, such as the effect of material nonhomogeneity and the existence of high gradient stress and strain near a crack tip, can be overcome without additional consideration if this method is adopted.
文摘In this paper, we solve chiral nonlinear Schrodinger equation (CNSE) numerically. Two numerical methods are derived using the explicit Runge-Kutta method of order four and the linear multistep method (Predictor-Corrector method of fourth order). The resulting schemes of fourth order accuracy in spatial and temporal directions. The CNSE is non-integrable and has two kinds of soliton solutions: bright and dark soliton. The exact solutions and the conserved quantities of CNSE are used to display the efficiency and robustness of the numerical methods we derived. Interaction of two bright solitons for different parameters is also displayed.
文摘This work deals with the determination of the temperature profile within a direct heating<span style="font-family:;" "=""> </span><span style="font-family:;" "=""><span style="font-family:Verdana;">moving bed </span><span style="font-family:Verdana;">torrefier</span> <span style="font-family:Verdana;">in order to</span><span style="font-family:Verdana;"> determine its minimum column height. A thermal model based on </span><span style="font-family:Verdana;">eulerian-eulerian</span> <span style="font-family:Verdana;">two</span></span><span style="font-family:Verdana;">-</span><span style="font-family:Verdana;">phase</span><span style="font-family:;" "=""><span style="font-family:Verdana;"> solid-gas theory was performed and solved with the method of lines. </span><span style="font-family:Verdana;">In addition</span><span style="font-family:Verdana;">, this study allows </span><span style="font-family:Verdana;">to investigate</span><span style="font-family:Verdana;"> the effect of the biomass particle size on the minimum </span><span style="font-family:Verdana;">torrefier</span><span style="font-family:Verdana;"> column height. This</span></span><span style="font-family:Verdana;"> investigation </span><span style="font-family:Verdana;">was performed by changing, simultaneously, the diameter of particles and the minimum fluidization velocity of the bed. Then, the calculations were made for a counter-current torrefaction reactor of 30</span><span style="font-family:;" "=""> </span><span style="font-family:;" "=""><span style="font-family:Verdana;">cm in diameter and </span><span style="font-family:Verdana;">for</span><span style="font-family:Verdana;"> 5</span></span><span style="font-family:;" "=""> </span><span style="font-family:Verdana;">kg/h of the feed rate of raw sugarcane bagasse.</span><span style="font-family:Verdana;"> Results show that the height of the reactor column must be at least 30 cm for that are 1 mm in diameter and 108 cm for particles that are 2 mm in diameter.</span>
文摘This paper presents a method of lines solution based on the reproducing kernel Hilbert space method to the nonlinear one-dimensional Klein-Gordon equation that arises in many scientific fields areas.Our method uses discretization of the partial derivatives of the space variable to get a system of ODEs in the time variable and then solve the system of ODEs using reproducing kernel Hilbert space method.Consider two examples to validate the proposed method.Compare the results with the exact solution by calculating the error norms L_(2) and L_(∞) at various time levels.The results show that the presented scheme is a systematic,effective and powerful technique for the solution of Klein-Gordon equation.
基金This work is partially supported by NSF grant DMS-1016313,GRF grant of Hong Kong(Project No.PolyU 501709),AMA-JRI of PolyU,Polyu grant No.5020/10P and NSERC(Canada).
文摘This article extends the finite element method of lines to a parabolic initial boundary value problem whose diffusion coefficient is discontinuous across an interface that changes with respect to time.The method presented here uses immersed finite element(IFE)functions for the discretization in spatial variables that can be carried out over a fixedmesh(such as a Cartesianmesh if desired),and this featuremakes it possible to reduce the parabolic equation to a system of ordinary differential equations(ODE)through the usual semi-discretization procedure.Therefore,with a suitable choice of the ODE solver,this method can reliably and efficiently solve a parabolic moving interface problem over a fixed structured(Cartesian)mesh.Numerical examples are presented to demonstrate features of this new method.
基金supported by PSRC(A Project Funded by the Basic Science Research Center of Majmaah University,KSA)and Project No.60/38.
文摘In this paper,a nonlinear shallow-water model of tsunami wave propagation at different points along a coastline of an ocean has been numerically simulated using method of lines.The simulation is carried out for various coastal slopes and the ocean depths.The effects of the coast slope and sea depth on the tsunami wave run-up height and velocity are illustrated.The accuracy of the mathematical model is verified by solving a classical test problem with known analytic solution.The computed run-up height and velocity show satisfactory agreement with the tsunami wave physics.
基金supported by the National Natural Science Foundation of China(10772136)Shanghai Leading Academic Discipline Project(B302)The authors wish to thank Dr.Guyue Jiao for the literary suggestions on the manuscript
文摘The model of electrically driven jet is governed by a series of quasi 1D dimensionless partial differential equations(PDEs).Following the method of lines,the Chebyshev collocation method is employed to discretize the PDEs and obtain a system of differential-algebraic equations(DAEs).By differentiating constrains in DAEs twice,the system is transformed into a set of ordinary differential equations(ODEs) with invariants.Then the implicit differential equations solver 'ddaskr' is used to solve the ODEs and post-stabilization is executed at the end of each step.Results show the distributions of radius,linear charge density,stretching ratio and also the horizontal velocity at a time point.Meanwhile,the spiral and expanding projections to X-Y plane of the jet centerline suggest the occurring of bending instability.
文摘A first principles-based dynamic model for a continuous catalyst regeneration (CCR) platforming process, the UOP commercial naphtha catalytic reforming process, is developed in this paper. The lumping details of the naphtha feed and reaction scheme of the reaction model are given. The process model is composed of the reforming reaction model with catalyst deactivation, the furnace model and the separator model, which is capable of capturing the major dynamics that occurs in this process system. Dynamic simulations are performed based on Gear numerical algorithm and method of lines (MOL), a numerical technique dealing with partial differential equations (PDEs). The results of simulation are also presented. Dynamic responses caused by disturbances in the process system can be correctly predicted through simulations.
文摘The ultimate goal and highlight of this paper are to explore water levels along the coast of Bangladesh efficiently due to the nonlinear interaction of tide and surge by employing the method of lines(MOLs)with the aid of newly proposed RKAHeM(4,4)technique.In this regard,the spatial derivatives of shallow water equations(SWEs)were discretized by means of a finite difference method to obtain a system of ordinary differential equations(ODEs)of initial valued with time as an independent variable.The obtained system of ODEs was solved by the RKAHeM(4,4)technique.One-way nested grid technique was exercised to incorporate coastal complexities closely with minimum computational cost.A stable tidal oscillation was produced over the region of interest by applying the most influential tidal constituent M2 along the southern open boundary of the outer scheme.The newly developed model was applied to estimate water levels due to the non-linear interaction of tide and surge associated with the catastrophic cyclone April 1991 along the coast of Bangladesh.The approach employed in the study was found to perform well and ensure conformity with real-time observations.
文摘The numerical method of lines(MOLs)in coordination with the classical fourth-order Runge Kutta(RK(4,4))method is used to solve shallow water equations(SWEs)for foreseeing water levels owing to the nonlinear interaction of tide and surge accompanying with a storm along the coast of Bangladesh.The SWEs are developed by extending the body forces with tide generating forces(TGFs).Spatial variables of the SWEs along with the boundary conditions are approximated by means of finite difference technique on an Arakawa C-grid to attain a system of ordinary differential equations(ODEs)of initial valued in time,which are being solved with the aid of the RK(4,4)method.Nested grid technique is adopted to solve coastal complexities closely with least computational cost.A stable tidal solution in the region of our choice is produced by applying the tidal forcing with the major tidal constituent M2(lunar semi-diurnal)along the southern open-sea boundary of the outer scheme.Numerical experimentations are carried out to simulate water levels generated by the cyclonic storm AILA along the coast of Bangladesh.The model simulated results are found to be in a reasonable agreement with the limited available reported data and observations.
基金Project supported by the National Natural Science Foundation of China (Nos. 59525813 and 19872066).
文摘For an anti-plane problem, the differential operator is self-adjoint and the corresponding eigenfunctions belong to the Hilbert space. The orthogonal property between eigenfunctions （or between the derivatives of eigenfunctions） of anti-plane problem is exploited. We developed for the first time two sets of radius-independent orthogonal integrals for extraction of stress intensity factors （SIFs）, so any order SIF can be extracted based on a certain known solution of displacement （an analytic result or a numerical result）. Many numerical examples based on the finite element method of lines （FEMOL） show that the present method is very powerful and efficient.
基金National Natural Science Foundation of China (50574097 90305023)
文摘Usually, it is very difficult to find out an analytical solution to thermal conduction problems during high temperature welding. Therefore, as an important numerical approach, the method of lines （MOLs） is introduced to solve the temperature field characterized by high gradients. The basic idea of the method is to semi-discretize the governing equation of the problem into a system of ordinary differential equations （ODEs） defined on discrete lines by means of the finite difference method, by which the thermal boundary condition with high gradients are directly embodied in formulation. Thus the temperature field can be obtained by solving the ODEs. As a numerical example, the variation of an axisymmetrical temperature field along the plate thickness can be obtained.
文摘In this paper, the one-dimensional time dependent Schr?dinger equation is discretized by the method of lines using a second order finite difference approximation to replace the second order spatial derivative. The evolving system of stiff Ordinary Differential Equation (ODE) in time is solved numerically by an L-stable trapezoidal-like integrator. Results show accuracy of relative maximum error of order 10?4 in the interval of consideration. The performance of the method as compared to an existing scheme is considered favorable.
基金The projects “Applied Mathematics in Physics and Technical Sciences” number MSM684077 0010 of the Ministry of Education Youth and Sports of the Czech Republic and “Advanced Supercomputing Methods for Implementation of Mathematical Models” number SGS11/161/OHK4/3T/14
文摘Microbiological experiments show that the colonies of the bacterium bacillus subtilis placed on a dish filled with an agar medium and nutrient form varied spatial patterns while the individual cells grow, reproduce and migrate on the dish in clumps. In this paper, we discuss a system of reaction-diffusion equations that can be used with a view to modelling this phenomenon and we solve it numerically by means of the method of lines. For the spatial discretization, we use the finite difference method and Galerkin finite element method. We present how the spatial patterns obtained depend on the spatial discretization employed and we measure the experimental order of convergence of the numerical schemes used. Further, we present the numerical results obtained by solving the model in a cubic domain.
文摘In this paper, we describe a comparison of two spatial discretization schemes for the advection equation, namely the first finite difference method and the method of lines. The stability of the methods has been studied by Von Neumann method and with the matrix analysis. The methods are applied to a number of test problems to compare the accuracy and computational efficiency. We show that both discretization techniques approximate correctly solution of advection equation and compare their accuracy and performance.
文摘A mathematical model is presented for the charging-up process in an air-entrapped pipeline with moving boundary conditions. A coordinate transformation technique is employed to reduce fluid motion in time-dependent domains to ones in time-independent domains. The nonlinear hyperbolic partial differential equations governing the unsteady motion of fluid combined with an equation for transient shear stress between the pipe wall and the flowing fluid are solved by the method of lines. Results show that ignoring elastic effects overestimates the maximum pressure and underestimates the maximum front velocity of filling fluid. The peak pressure of the entrapped air is sensitive to the length of the initial entrapped air pocket.
