In order to solve the problem of the variable coefficient ordinary differen-tial equation on the bounded domain,the Lagrange interpolation method is used to approximate the exact solution of the equation,and the error...In order to solve the problem of the variable coefficient ordinary differen-tial equation on the bounded domain,the Lagrange interpolation method is used to approximate the exact solution of the equation,and the error between the numerical solution and the exact solution is obtained,and then compared with the error formed by the difference method,it is concluded that the Lagrange interpolation method is more effective in solving the variable coefficient ordinary differential equation.展开更多
This paper presents a novel element differential method for modeling cracks in piezoelectric materials,aiming to simulate fracture behaviors and predict the fracture parameter known as the J-integral accurately.The me...This paper presents a novel element differential method for modeling cracks in piezoelectric materials,aiming to simulate fracture behaviors and predict the fracture parameter known as the J-integral accurately.The method leverages an efficient collocation technique to satisfy traction and electric charge equilibrium on the crack surface,aligning internal nodes with piezoelectric governing equations without needing integration or variational principles.It combines the strengths of the strong form collocation and finite element methods.The J-integral is derived analytically using the equivalent domain integral method,employing Green's formula and Gauss's divergence theorem to transform line integrals into area integrals for solving two-dimensional piezoelectric material problems.The accuracy of the method is validated through comparison with three typical examples,and it offers fracture prevention strategies for engineering piezoelectric structures under different electrical loading patterns.展开更多
In this paper,the convergence of the split-step theta method for stochastic differential equations is analyzed using stochastic C-stability and stochastic B-consistency.The fact that the numerical scheme,which is both...In this paper,the convergence of the split-step theta method for stochastic differential equations is analyzed using stochastic C-stability and stochastic B-consistency.The fact that the numerical scheme,which is both stochastically C-stable and stochastically B-consistent,is convergent has been proved in a previous paper.In order to analyze the convergence of the split-step theta method(θ∈[1/2,1]),the stochastic C-stability and stochastic B-consistency under the condition of global monotonicity have been researched,and the rate of convergence 1/2 has been explored in this paper.It can be seen that the convergence does not require the drift function should satisfy the linear growth condition whenθ=1/2 Furthermore,the rate of the convergence of the split-step scheme for stochastic differential equations with additive noise has been researched and found to be 1.Finally,an example is given to illustrate the convergence with the theoretical results.展开更多
Aimed at the demand of contingency return at any time during the near-moon phase in the manned lunar landing missions,a fast calculation method for three-impulse contingency return trajectories is proposed.Firstly,a t...Aimed at the demand of contingency return at any time during the near-moon phase in the manned lunar landing missions,a fast calculation method for three-impulse contingency return trajectories is proposed.Firstly,a three-impulse contingency return trajectory scheme is presented by combining the Lambert transfer and maneuver at the special point.Secondly,a calculation model of three-impulse contingency return trajectories is established.Then,fast calculation methods are proposed by adopting the high-order Taylor expansion of differential algebra in the twobody trajectory dynamics model and perturbed trajectory dynamics model.Finally,the performance of the proposed methods is verified by numerical simulation.The results indicate that the fast calculation method of two-body trajectory has higher calculation efficiency compared to the semi-analytical calculation method under a certain accuracy condition.Due to its high efficiency,the characteristics of the three-impulse contingency return trajectories under different contingency scenarios are further analyzed expeditiously.These findings can be used for the design of contingency return trajectories in future manned lunar landing missions.展开更多
This paper investigates the active traveling wave vibration control of an elastic supported rotating porous aluminium conical shell(CS)under impact loading.Piezoelectric smart materials in the form of micro fiber comp...This paper investigates the active traveling wave vibration control of an elastic supported rotating porous aluminium conical shell(CS)under impact loading.Piezoelectric smart materials in the form of micro fiber composites(MFCs)are used as actuators and sensors.To this end,a metal pore truncated CS with MFCs attached to its surface is considered.Adding artificial virtual springs at two edges of the truncated CS achieves various elastic supported boundaries by changing the spring stiffness.Based on the first-order shear deformation theory(FSDT),minimum energy principle,and artificial virtual spring technology,the theoretical formulations considering the electromechanical coupling are derived.The comparison of the natural frequency of the present results with the natural frequencies reported in previous literature evaluates the accuracy of the present approach.To study the vibration control,the integral quadrature method in conjunction with the differential quadrature approximation in the length direction is used to discretize the partial differential dynamical system to form a set of ordinary differential equations.With the aid of the velocity negative feedback method,both the time history and the input control voltage on the actuator are demonstrated to present the effects of velocity feedback gain,pore distribution type,semi-vertex angle,impact loading,and rotational angular velocity on the traveling wave vibration control.展开更多
The static behavior of piezoelectric circular spherical shallow shells under both electrical and mechanical loads is studied by using the differential quadrature element method (DQEM). Geometrical nonlinearity effect ...The static behavior of piezoelectric circular spherical shallow shells under both electrical and mechanical loads is studied by using the differential quadrature element method (DQEM). Geometrical nonlinearity effect is considered. Detailed formulations and procedures are given for the first time. Several examples are analyzed and accurate results are obtained by the DQEM. Based on the results in this paper, one may conclude that the DQEM is a useful tool for obtaining solutions of structural elements. It can be seen that the shell shape may be theore tically controlled and snap through may occur when the applied voltage reaches a critical value even without mechanical load for certain geometric configurations.展开更多
It is a new attempt to extend the differential quadrature method(DQM) to stability analysis of the straight and curved centerlinepipes conveying fluid. Emphasis is placed on the study of theinfluences of several param...It is a new attempt to extend the differential quadrature method(DQM) to stability analysis of the straight and curved centerlinepipes conveying fluid. Emphasis is placed on the study of theinfluences of several parameters on the critical flow velocity.Compared to other methods, this method can more easily deal with thepipe with spring support at its boundaries and asks for much lesscomputing effort while giving ac- ceptable precision in the numericalresults.展开更多
To enhance the integrity, an analytic method （AM） which has less execution time is proposed to calculate the user differential range error （UDRE） used by the user to detect the potential risk. An ephemeris and clo...To enhance the integrity, an analytic method （AM） which has less execution time is proposed to calculate the user differential range error （UDRE） used by the user to detect the potential risk. An ephemeris and clock correction calculation method is introduced first. It shows that the most important thing of computing UDRE is to find the worst user location （WUL） in the service volume. Then, a UDRE algorithm using AM is described to solve this problem. By using the covariance matrix of the error vector, the searching of WUL is converted to an analytic geometry problem. The location of WUL can be obtained directly by mathematical derivation. Experiments are conducted to compare the performance between the proposed AM algorithm and the exhaustive grid search （EGS） method used in the master station. The results show that the correctness of the AM algorithm can be proved by the EGS method and the AM algorithm can reduce the calculation time by more than 90%. The computational complexity of this proposed algorithm is better than that of EGS. Thereby this algorithm is more suitable for computing UDRE at the master station.展开更多
The Hamilton-Jacobi method for solving ordinary differential equations is presented in this paper. A system of ordinary differential equations of first order or second order can be expressed as a Hamilton system under...The Hamilton-Jacobi method for solving ordinary differential equations is presented in this paper. A system of ordinary differential equations of first order or second order can be expressed as a Hamilton system under certain conditions. Then the Hamilton-Jacobi method is used in the integration of the Hamilton system and the solution of the original ordinary differential equations can be found. Finally, an example is given to illustrate the application of the result.展开更多
A differential equation of first order can be expressed by the equation of motion of a mechanical system. In this paper, three methods of analytical mechanics, i.e. the Hamilton-Noether method, the Lagrange-Noether me...A differential equation of first order can be expressed by the equation of motion of a mechanical system. In this paper, three methods of analytical mechanics, i.e. the Hamilton-Noether method, the Lagrange-Noether method and the Poisson method, are given to solve a differential equation of first order, of which the way may be called the mechanical methodology in mathematics.展开更多
The quasi-static and dynamic responses of a thermoviscoelastic Timoshenko beam subject to thermal loads are analyzed. First, based on the small geometric deformation assumption and Boltzmann constitutive relation, the...The quasi-static and dynamic responses of a thermoviscoelastic Timoshenko beam subject to thermal loads are analyzed. First, based on the small geometric deformation assumption and Boltzmann constitutive relation, the governing equations for the beam are presented. Second, an extended differential quadrature method(DQM)in the spatial domain and a differential method in the temporal domain are combined to transform the integro-partial-differential governing equations into the ordinary differential equations. Third, the accuracy of the present discrete method is verified by elastic/viscoelastic examples, and the effects of thermal load parameters, material and geometrical parameters on the quasi-static and dynamic responses of the beam are discussed. Numerical results show that the thermal function parameter has a great effect on quasi-static and dynamic responses of the beam. Compared with the thermal relaxation time, the initial vibrational responses of the beam are more sensitive to the mechanical relaxation time of the thermoviscoelastic material.展开更多
This paper presents a combined application of the finite element method （FEM） and the differential quadrature method （DQM） to vibration and buckling problems of rectangular plates. The proposed scheme combines the...This paper presents a combined application of the finite element method （FEM） and the differential quadrature method （DQM） to vibration and buckling problems of rectangular plates. The proposed scheme combines the geometry flexibility of the FEM and the high accuracy and efficiency of the DQM. The accuracy of the present method is demonstrated by comparing the obtained results with those available in the literature. It is shown that highly accurate results can be obtained by using a small number of finite elements and DQM sample points. The proposed method is suitable for the problems considered due to its simplicity and potential for further development.展开更多
This article presents a numerical solution for the flow of a Newtonian fluid over an impermeable stretching sheet embedded in a porous medium with the power law surface velocity and variable thickness in the presence ...This article presents a numerical solution for the flow of a Newtonian fluid over an impermeable stretching sheet embedded in a porous medium with the power law surface velocity and variable thickness in the presence of thermal radiation. The flow is caused by non-linear stretching of a sheet. Thermal conductivity of the fluid is assumed to vary linearly with temperature. The governing partial differential equations （PDEs） are transformed into a system of coupled non-linear ordinary differential equations （ODEs） with appropriate boundary conditions for various physical parameters. The remaining system of ODEs is solved numerically using a differential transformation method （DTM）. The effects of the porous parameter, the wall thickness parameter, the radiation parameter, the thermal conductivity parameter, and the Prandtl number on the flow and temperature profiles are presented. Moreover, the local skin-friction and the Nusselt numbers are presented. Comparison of the obtained numerical results is made with previously published results in some special cases, with good agreement. The results obtained in this paper confirm the idea that DTM is a powerful mathematical tool and can be applied to a large class of linear and non-linear problems in different fields of science and engineering.展开更多
In this paper, a high accuracy finite volume element method is presented for two-point boundary value problem of second order ordinary differential equation, which differs from the high order generalized difference me...In this paper, a high accuracy finite volume element method is presented for two-point boundary value problem of second order ordinary differential equation, which differs from the high order generalized difference methods. It is proved that the method has optimal order error estimate O(h3) in H1 norm. Finally, two examples show that the method is effective.展开更多
In this paper, a new fractional projective Riccati equation method is proposed to establish exact solutions for fractional partial differential equations in the sense of modified Riemann–Liouville derivative. This me...In this paper, a new fractional projective Riccati equation method is proposed to establish exact solutions for fractional partial differential equations in the sense of modified Riemann–Liouville derivative. This method can be seen as the fractional version of the known projective Riccati equation method. For illustrating the validity of this method,we apply this method to solve the space-time fractional Whitham–Broer–Kaup(WBK) equations and the nonlinear fractional Sharma–Tasso–Olever(STO) equation, and as a result, some new exact solutions for them are obtained.展开更多
A differential quadrature (DQ) method for orthotropic plates was proposed based on Reddy' s theory of plates with the effects of the higher-order transverse shear deformations. Wang-Bert's DQ approach was also...A differential quadrature (DQ) method for orthotropic plates was proposed based on Reddy' s theory of plates with the effects of the higher-order transverse shear deformations. Wang-Bert's DQ approach was also further extended to handle the boundary conditions of plates. The computational convergence was studied, and the numerical results were obtained for different grid spacings and compared with the existing results. The results show that the DQ method is fairly reliable and effective.展开更多
This paper deals with the numerical solution of initial value problems for systems of differential equations with a delay argument. The numerical stability of a linear multistep method is investigated by analysing the...This paper deals with the numerical solution of initial value problems for systems of differential equations with a delay argument. The numerical stability of a linear multistep method is investigated by analysing the solution of the lest equation y’(t)=Ay(t) + By(1-t),where A,B denote constant complex N×N-matrices,and t】0.We investigate carefully the characterization of the stability region.展开更多
文摘In order to solve the problem of the variable coefficient ordinary differen-tial equation on the bounded domain,the Lagrange interpolation method is used to approximate the exact solution of the equation,and the error between the numerical solution and the exact solution is obtained,and then compared with the error formed by the difference method,it is concluded that the Lagrange interpolation method is more effective in solving the variable coefficient ordinary differential equation.
基金Financial support of this work by the Technology Development program of China(Grant No.2022204B003)National Natural Science Foundation of China(12272083 and 12172078)the Fundamental Research Funds for the Central Universities(DUT24YJ136)is gratefully acknowledged.
文摘This paper presents a novel element differential method for modeling cracks in piezoelectric materials,aiming to simulate fracture behaviors and predict the fracture parameter known as the J-integral accurately.The method leverages an efficient collocation technique to satisfy traction and electric charge equilibrium on the crack surface,aligning internal nodes with piezoelectric governing equations without needing integration or variational principles.It combines the strengths of the strong form collocation and finite element methods.The J-integral is derived analytically using the equivalent domain integral method,employing Green's formula and Gauss's divergence theorem to transform line integrals into area integrals for solving two-dimensional piezoelectric material problems.The accuracy of the method is validated through comparison with three typical examples,and it offers fracture prevention strategies for engineering piezoelectric structures under different electrical loading patterns.
基金Supported by the National Natural Science Foundation of China (Grant No. 12301521)the Natural Science Foundation of Shanxi Province (Grant No. 20210302124081)。
文摘In this paper,the convergence of the split-step theta method for stochastic differential equations is analyzed using stochastic C-stability and stochastic B-consistency.The fact that the numerical scheme,which is both stochastically C-stable and stochastically B-consistent,is convergent has been proved in a previous paper.In order to analyze the convergence of the split-step theta method(θ∈[1/2,1]),the stochastic C-stability and stochastic B-consistency under the condition of global monotonicity have been researched,and the rate of convergence 1/2 has been explored in this paper.It can be seen that the convergence does not require the drift function should satisfy the linear growth condition whenθ=1/2 Furthermore,the rate of the convergence of the split-step scheme for stochastic differential equations with additive noise has been researched and found to be 1.Finally,an example is given to illustrate the convergence with the theoretical results.
基金co-supported by the National Natural Science Foundation of China(No.12072365)the Technology Innovation Team of Manned Space Engineering,China。
文摘Aimed at the demand of contingency return at any time during the near-moon phase in the manned lunar landing missions,a fast calculation method for three-impulse contingency return trajectories is proposed.Firstly,a three-impulse contingency return trajectory scheme is presented by combining the Lambert transfer and maneuver at the special point.Secondly,a calculation model of three-impulse contingency return trajectories is established.Then,fast calculation methods are proposed by adopting the high-order Taylor expansion of differential algebra in the twobody trajectory dynamics model and perturbed trajectory dynamics model.Finally,the performance of the proposed methods is verified by numerical simulation.The results indicate that the fast calculation method of two-body trajectory has higher calculation efficiency compared to the semi-analytical calculation method under a certain accuracy condition.Due to its high efficiency,the characteristics of the three-impulse contingency return trajectories under different contingency scenarios are further analyzed expeditiously.These findings can be used for the design of contingency return trajectories in future manned lunar landing missions.
基金Supported by the National Natural Science Foundation of China(Nos.12272056 and 11832002)。
文摘This paper investigates the active traveling wave vibration control of an elastic supported rotating porous aluminium conical shell(CS)under impact loading.Piezoelectric smart materials in the form of micro fiber composites(MFCs)are used as actuators and sensors.To this end,a metal pore truncated CS with MFCs attached to its surface is considered.Adding artificial virtual springs at two edges of the truncated CS achieves various elastic supported boundaries by changing the spring stiffness.Based on the first-order shear deformation theory(FSDT),minimum energy principle,and artificial virtual spring technology,the theoretical formulations considering the electromechanical coupling are derived.The comparison of the natural frequency of the present results with the natural frequencies reported in previous literature evaluates the accuracy of the present approach.To study the vibration control,the integral quadrature method in conjunction with the differential quadrature approximation in the length direction is used to discretize the partial differential dynamical system to form a set of ordinary differential equations.With the aid of the velocity negative feedback method,both the time history and the input control voltage on the actuator are demonstrated to present the effects of velocity feedback gain,pore distribution type,semi-vertex angle,impact loading,and rotational angular velocity on the traveling wave vibration control.
文摘The static behavior of piezoelectric circular spherical shallow shells under both electrical and mechanical loads is studied by using the differential quadrature element method (DQEM). Geometrical nonlinearity effect is considered. Detailed formulations and procedures are given for the first time. Several examples are analyzed and accurate results are obtained by the DQEM. Based on the results in this paper, one may conclude that the DQEM is a useful tool for obtaining solutions of structural elements. It can be seen that the shell shape may be theore tically controlled and snap through may occur when the applied voltage reaches a critical value even without mechanical load for certain geometric configurations.
基金National Key Project of China (No.PD9521907)the National Science Foundation of China (No.19872025).
文摘It is a new attempt to extend the differential quadrature method(DQM) to stability analysis of the straight and curved centerlinepipes conveying fluid. Emphasis is placed on the study of theinfluences of several parameters on the critical flow velocity.Compared to other methods, this method can more easily deal with thepipe with spring support at its boundaries and asks for much lesscomputing effort while giving ac- ceptable precision in the numericalresults.
文摘To enhance the integrity, an analytic method （AM） which has less execution time is proposed to calculate the user differential range error （UDRE） used by the user to detect the potential risk. An ephemeris and clock correction calculation method is introduced first. It shows that the most important thing of computing UDRE is to find the worst user location （WUL） in the service volume. Then, a UDRE algorithm using AM is described to solve this problem. By using the covariance matrix of the error vector, the searching of WUL is converted to an analytic geometry problem. The location of WUL can be obtained directly by mathematical derivation. Experiments are conducted to compare the performance between the proposed AM algorithm and the exhaustive grid search （EGS） method used in the master station. The results show that the correctness of the AM algorithm can be proved by the EGS method and the AM algorithm can reduce the calculation time by more than 90%. The computational complexity of this proposed algorithm is better than that of EGS. Thereby this algorithm is more suitable for computing UDRE at the master station.
基金Project supported by the National Natural Science Foundation of China （Grant Nos 10272021, 10572021） and the Doctoral Program Foundation of Institution of Higher Education of China （Grant No 20040007022）.
文摘The Hamilton-Jacobi method for solving ordinary differential equations is presented in this paper. A system of ordinary differential equations of first order or second order can be expressed as a Hamilton system under certain conditions. Then the Hamilton-Jacobi method is used in the integration of the Hamilton system and the solution of the original ordinary differential equations can be found. Finally, an example is given to illustrate the application of the result.
基金Project supported by the National Natural Science Foundation of China （Grant No 10272021） and the Doctorate Foundation of the State Education Ministry of China （Grant No 20040007022）.
文摘A differential equation of first order can be expressed by the equation of motion of a mechanical system. In this paper, three methods of analytical mechanics, i.e. the Hamilton-Noether method, the Lagrange-Noether method and the Poisson method, are given to solve a differential equation of first order, of which the way may be called the mechanical methodology in mathematics.
基金supported by the National Natural Science Foundation of China(Nos.11772182 and90816001)
文摘The quasi-static and dynamic responses of a thermoviscoelastic Timoshenko beam subject to thermal loads are analyzed. First, based on the small geometric deformation assumption and Boltzmann constitutive relation, the governing equations for the beam are presented. Second, an extended differential quadrature method(DQM)in the spatial domain and a differential method in the temporal domain are combined to transform the integro-partial-differential governing equations into the ordinary differential equations. Third, the accuracy of the present discrete method is verified by elastic/viscoelastic examples, and the effects of thermal load parameters, material and geometrical parameters on the quasi-static and dynamic responses of the beam are discussed. Numerical results show that the thermal function parameter has a great effect on quasi-static and dynamic responses of the beam. Compared with the thermal relaxation time, the initial vibrational responses of the beam are more sensitive to the mechanical relaxation time of the thermoviscoelastic material.
文摘This paper presents a combined application of the finite element method （FEM） and the differential quadrature method （DQM） to vibration and buckling problems of rectangular plates. The proposed scheme combines the geometry flexibility of the FEM and the high accuracy and efficiency of the DQM. The accuracy of the present method is demonstrated by comparing the obtained results with those available in the literature. It is shown that highly accurate results can be obtained by using a small number of finite elements and DQM sample points. The proposed method is suitable for the problems considered due to its simplicity and potential for further development.
文摘This article presents a numerical solution for the flow of a Newtonian fluid over an impermeable stretching sheet embedded in a porous medium with the power law surface velocity and variable thickness in the presence of thermal radiation. The flow is caused by non-linear stretching of a sheet. Thermal conductivity of the fluid is assumed to vary linearly with temperature. The governing partial differential equations （PDEs） are transformed into a system of coupled non-linear ordinary differential equations （ODEs） with appropriate boundary conditions for various physical parameters. The remaining system of ODEs is solved numerically using a differential transformation method （DTM）. The effects of the porous parameter, the wall thickness parameter, the radiation parameter, the thermal conductivity parameter, and the Prandtl number on the flow and temperature profiles are presented. Moreover, the local skin-friction and the Nusselt numbers are presented. Comparison of the obtained numerical results is made with previously published results in some special cases, with good agreement. The results obtained in this paper confirm the idea that DTM is a powerful mathematical tool and can be applied to a large class of linear and non-linear problems in different fields of science and engineering.
基金heprojectissupportedbyNNSFofChina (No .1 9972 0 39) .
文摘In this paper, a high accuracy finite volume element method is presented for two-point boundary value problem of second order ordinary differential equation, which differs from the high order generalized difference methods. It is proved that the method has optimal order error estimate O(h3) in H1 norm. Finally, two examples show that the method is effective.
基金Supported by Natural Science Foundation of Shandong Province of China under Grant No.ZR2013AQ009National Training Programs of Innovation and Entrepreneurship for Undergraduates under Grant No.201310433031Doctoral initializing Foundation of Shandong University of Technology of China under Grant No.4041-413030
文摘In this paper, a new fractional projective Riccati equation method is proposed to establish exact solutions for fractional partial differential equations in the sense of modified Riemann–Liouville derivative. This method can be seen as the fractional version of the known projective Riccati equation method. For illustrating the validity of this method,we apply this method to solve the space-time fractional Whitham–Broer–Kaup(WBK) equations and the nonlinear fractional Sharma–Tasso–Olever(STO) equation, and as a result, some new exact solutions for them are obtained.
基金key Project of the Municipal Commission of Science and Technology of Shanghai
文摘A differential quadrature (DQ) method for orthotropic plates was proposed based on Reddy' s theory of plates with the effects of the higher-order transverse shear deformations. Wang-Bert's DQ approach was also further extended to handle the boundary conditions of plates. The computational convergence was studied, and the numerical results were obtained for different grid spacings and compared with the existing results. The results show that the DQ method is fairly reliable and effective.
文摘This paper deals with the numerical solution of initial value problems for systems of differential equations with a delay argument. The numerical stability of a linear multistep method is investigated by analysing the solution of the lest equation y’(t)=Ay(t) + By(1-t),where A,B denote constant complex N×N-matrices,and t】0.We investigate carefully the characterization of the stability region.
