This study presents the Chebyshev polynomials-based Ritz method to examine the thermal buckling and free vibration characteristics of metal foam beams.The analyses include three models for porosity distribution and tw...This study presents the Chebyshev polynomials-based Ritz method to examine the thermal buckling and free vibration characteristics of metal foam beams.The analyses include three models for porosity distribution and two scenarios for thermal distribution.The material properties are assessed under two conditions,i.e.,temperature dependence and temperature independence.The theoretical framework for the beams is based on the higher-order shear deformation theory,which incorporates shear deformations with higher-order polynomials.The governing equations are established from the Lagrange equations,and the beam displacement fields are approximated by the Chebyshev polynomials.Numerical simulations are performed to evaluate the effects of thermal load,slenderness,boundary condition(BC),and porosity distribution on the buckling and vibration behaviors of metal foam beams.The findings highlight the significant influence of temperature-dependent(TD)material properties on metal foam beams'buckling and vibration responses.展开更多
In this paper,a method for designing supermassive sparse phased arrays(SMSPAs)known as the unitary modified matrix enhancement and matrix pencil(UMMEMP)method is proposed.In this method,an eigenvalue pairing method,wh...In this paper,a method for designing supermassive sparse phased arrays(SMSPAs)known as the unitary modified matrix enhancement and matrix pencil(UMMEMP)method is proposed.In this method,an eigenvalue pairing method,which is inspired by the modified MEMP,effectively pairs the repeated eigenvalues intractable in the unitary matrix pencil method,and it is more effective in determining the locations of elements in the sparse array.Three numerical examples and a full-wave validation are presented to demonstrate the effectiveness of the method,implemented via SMSPA,in achieving low sidelobe level wide-angle scanning radiation patterns,circular flattop radiation patterns,and ultra wide-angle scanning radiation patterns.展开更多
This paper is devoted to investigate the accuracy of the Pseudo spectral scheme with the Chebyshev tau method and Chebyshev collocation method. The computational results of the nonlinear disturbance development in p...This paper is devoted to investigate the accuracy of the Pseudo spectral scheme with the Chebyshev tau method and Chebyshev collocation method. The computational results of the nonlinear disturbance development in plane Poiseuille flow for both methods are presented and compared in detail. It is acknowledged that the Chebyshev collocation method has higher precision than the other one, especially for near netural situation.展开更多
In view of generating optimal trajectories of Bolza problems, standard Chebyshev pseudospectral （PS） method makes the points＇ accumulation near the extremities and rarefaction of nodes close to the center of interv...In view of generating optimal trajectories of Bolza problems, standard Chebyshev pseudospectral （PS） method makes the points＇ accumulation near the extremities and rarefaction of nodes close to the center of interval, which causes an ill-condition of differentiation matrix and an oscillation of the optimal solution. For improvement upon the difficulties, a mapped Chebyshev pseudospectral method is proposed. A conformal map is applied to Chebyshev points to move the points closer to equidistant nodes. Condition number and spectral radius of differentiation matrices from both methods are presented to show the improvement. Furthermore, the modification keeps the Chebyshev pseudospectral method＇s advantage, the spectral convergence rate. Based on three numerical examples, a comparison of the execution time, convergence and accuracy is presented among the standard Chebyshev pseudospectral method, other collocation methods and the proposed one. In one example, the error of results from mapped Chebyshev pseudospectral method is reduced to 5% of that from standard Chebyshev pseudospectral method.展开更多
In this paper, we discuss local convergence of a family of Chebychev Halley type methods with a parameter θ∈[0,1] in Banach space using Smale type δ criterion under 2 th γ condition. We will see that the propertie...In this paper, we discuss local convergence of a family of Chebychev Halley type methods with a parameter θ∈[0,1] in Banach space using Smale type δ criterion under 2 th γ condition. We will see that the properties of the condition used for local convergence is much more different from that used in [6][15] for the semi-local convergence.展开更多
This paper introduces two new types of precise integration methods based on Chebyshev polynomial of the first kind for dynamic response analysis of structures, namely the integral formula method （IFM） and the homoge...This paper introduces two new types of precise integration methods based on Chebyshev polynomial of the first kind for dynamic response analysis of structures, namely the integral formula method （IFM） and the homogenized initial system method （HISM）. In both methods, nonlinear variable loadings within time intervals are simulated using Chebyshev polynomials of the first kind before a direct integration is performed. Developed on the basis of the integral formula, the recurrence relationship of the integral computation suggested in this paper is combined with the Crout decomposed method to solve linear algebraic equations. In this way, the IFM based on Chebyshev polynomial of the first kind is constructed. Transforming the non-homogenous initial system to the homogeneous dynamic system, and developing a special scheme without dimensional expansion, the HISM based on Chebyshev polynomial of the first kind is able to avoid the matrix inversion operation. The accuracy of the time integration schemes is examined and compared with other commonly used schemes, and it is shown that a greater accuracy as well as less time consuming can be achieved. Two numerical examples are presented to demonstrate the applicability of these new methods.展开更多
The dynamic characteristics of a beam-cable coupled system are investigated using an improved Chebyshev spectral element method in order to observe the effects of adding cables on the beam. The system is modeled as a ...The dynamic characteristics of a beam-cable coupled system are investigated using an improved Chebyshev spectral element method in order to observe the effects of adding cables on the beam. The system is modeled as a double Timoshenko beam system interconnected by discrete springs. Utilizing Chebyshev series expansion and meshing the system according to the locations of its connections, numerical results of the natural frequencies and mode shapes are obtained using only a few elements, and the results are validated by comparing them with the results of a finite-element method. Then the effects of the cable parameters and layout of connections on the natural frequencies and mode shapes of a fixed-pinned beam are studied. The results show that the modes of a beam-cable coupled system can be classified into two types, beam mode and cable mode, according to the dominant deformation. To avoid undesirable vibrations of the cable, its parameters should be controlled in a reasonable range, or the layout of the connections should be optimized.展开更多
Fractional differential equations have recently been applied in various areas of engineering, science, finance, applied mathematics, bio-engineering and others. However, many researchers remain unaware of this field. ...Fractional differential equations have recently been applied in various areas of engineering, science, finance, applied mathematics, bio-engineering and others. However, many researchers remain unaware of this field. In this paper, an efficient numerical method for solving the fractional Advection-dispersion equation (ADE) is considered. The fractional derivative is described in the Caputo sense. The method is based on Chebyshev approximations. The properties of Chebyshev polynomials are used to reduce ADE to a system of ordinary differential equations, which are solved using the finite difference method (FDM). Moreover, the convergence analysis and an upper bound of the error for the derived formula are given. Numerical solutions of ADE are presented and the results are compared with the exact solution.展开更多
In this paper, we suggest a method for solving Fredholm integral equation of the first kind based on wavelet basis. The continuous Legendre and Chebyshev wavelets of the first, second, third and fourth kind on [0,1] a...In this paper, we suggest a method for solving Fredholm integral equation of the first kind based on wavelet basis. The continuous Legendre and Chebyshev wavelets of the first, second, third and fourth kind on [0,1] are used and are utilized as a basis in Galerkin method to approximate the solution of integral equations. Then, in some examples the mentioned wavelets are compared with each other.展开更多
A diagonal or lumped mass matrix is of great value for time-domain analysis of structural dynamic and wave propagation problems,as the computational efforts can be greatly reduced in the process of mass matrix inversi...A diagonal or lumped mass matrix is of great value for time-domain analysis of structural dynamic and wave propagation problems,as the computational efforts can be greatly reduced in the process of mass matrix inversion.In this study,the nodal quadrature method is employed to construct a lumped mass matrix for the Chebyshev spectral element method(CSEM).A Gauss-Lobatto type quadrature,based on Gauss-Lobatto-Chebyshev points with a weighting function of unity,is thus derived.With the aid of this quadrature,the CSEM can take advantage of explicit time-marching schemes and provide an efficient new tool for solving structural dynamic problems.Several types of lumped mass Chebyshev spectral elements are designed,including rod,beam and plate elements.The performance of the developed method is examined via some numerical examples of natural vibration and elastic wave propagation,accompanied by their comparison to that of traditional consistent-mass CSEM or the classical finite element method(FEM).Numerical results indicate that the proposed method displays comparable accuracy as its consistent-mass counterpart,and is more accurate than classical FEM.For the simulation of elastic wave propagation in structures induced by high-frequency loading,this method achieves satisfactory performance in accuracy and efficiency.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
The magnetohydrodynamics （MHD） Falkner-Skan flow of the Maxwell fluid is studied. Suitable transform reduces the partial differential equation into a nonlinear three order boundary value problem over a semi-infinite...The magnetohydrodynamics （MHD） Falkner-Skan flow of the Maxwell fluid is studied. Suitable transform reduces the partial differential equation into a nonlinear three order boundary value problem over a semi-infinite interval. An efficient approach based on the rational Chebyshev collocation method is performed to find the solution to the proposed boundary value problem. The rational Chebyshev collocation method is equipped with the orthogonal rational Chebyshev function which solves the problem on the semi-infinite domain without truncating it to a finite domain. The obtained results are presented through the illustrative graphs and tables which demonstrate the affectivity, stability, and convergence of the rational Chebyshev collocation method. To check the accuracy of the obtained results, a numerical method is applied for solving the problem. The variations of various embedded parameters into the problem are examined.展开更多
We study approximate solutions of a nonlinear integral equation of Hammerstein type. We describe the principle of discrete Adomian decomposition method (DADM). DADM is considered in the case we evaluate numerical inte...We study approximate solutions of a nonlinear integral equation of Hammerstein type. We describe the principle of discrete Adomian decomposition method (DADM). DADM is considered in the case we evaluate numerical integration by using Chebyshev roots. This technique gives an accurate solutions as will shown by illustrate examples.展开更多
In this paper, we propose to replace the Chebyshev series used in pseudospectral methods with the equivalent Chebyshev economized power series that can be evaluated more rapidly. We keep the rest of the implementation...In this paper, we propose to replace the Chebyshev series used in pseudospectral methods with the equivalent Chebyshev economized power series that can be evaluated more rapidly. We keep the rest of the implementation the same as the spectral method so that there is no new mathematical principle involved. We show by numerical examples that the new approach works well and there is indeed no significant loss of solution accuracy. The advantages of using power series also include simplicity in its formulation and implementation such that it could be used for complex systems. We investigate the important issue of collocation point selection. Our numerical results indicate that there is a clear accuracy advantage of using collocation points corresponding to roots of the Chebyshev polynomial.展开更多
The attitude optimal control problem （OCP） of a two-rigid-body space- craft with two rigid bodies coupled by a ball-in-socket joint is considered. Based on conservation of angular momentum of the system without the ...The attitude optimal control problem （OCP） of a two-rigid-body space- craft with two rigid bodies coupled by a ball-in-socket joint is considered. Based on conservation of angular momentum of the system without the external torque, a dynamic equation of three-dimensional attitude motion of the system is formulated. The attitude motion planning problem of the coupled-rigid-body spacecraft can be converted to a dis- crete nonlinear programming （NLP） problem using the Chebyshev-Gauss pseudospectral method （CGPM）. Solutions of the NLP problem can be obtained using the sequential quadratic programming （SQP） algorithm. Since the collocation points of the CGPM are Chebyshev-Gauss （CG） points, the integration of cost function can be approximated by the Clenshaw-Curtis quadrature, and the corresponding quadrature weights can be calculated efficiently using the fast Fourier transform （FFT）. To improve computational efficiency and numerical stability, the barycentric Lagrange interpolation is presented to substitute for the classic Lagrange interpolation in the approximation of state and con- trol variables. Furthermore, numerical float errors of the state differential matrix and barycentric weights can be alleviated using trigonometric identity especially when the number of CG points is large. A simple yet efficient method is used to avoid sensitivity to the initial values for the SQP algorithm using a layered optimization strategy from a feasible solution to an optimal solution. Effectiveness of the proposed algorithm is perfect for attitude motion planning of a two-rigid-body spacecraft coupled by a ball-in-socket joint through numerical simulation.展开更多
This paper ix devoted to establishment of the Chebyshev pseudospectral domain de-composition scheme for solving two-dimensional elliptic equation. By the generalized equivalent variatiunal form, we can get the stabili...This paper ix devoted to establishment of the Chebyshev pseudospectral domain de-composition scheme for solving two-dimensional elliptic equation. By the generalized equivalent variatiunal form, we can get the stability and convergence of this new scheme.展开更多
文摘This study presents the Chebyshev polynomials-based Ritz method to examine the thermal buckling and free vibration characteristics of metal foam beams.The analyses include three models for porosity distribution and two scenarios for thermal distribution.The material properties are assessed under two conditions,i.e.,temperature dependence and temperature independence.The theoretical framework for the beams is based on the higher-order shear deformation theory,which incorporates shear deformations with higher-order polynomials.The governing equations are established from the Lagrange equations,and the beam displacement fields are approximated by the Chebyshev polynomials.Numerical simulations are performed to evaluate the effects of thermal load,slenderness,boundary condition(BC),and porosity distribution on the buckling and vibration behaviors of metal foam beams.The findings highlight the significant influence of temperature-dependent(TD)material properties on metal foam beams'buckling and vibration responses.
文摘In this paper,a method for designing supermassive sparse phased arrays(SMSPAs)known as the unitary modified matrix enhancement and matrix pencil(UMMEMP)method is proposed.In this method,an eigenvalue pairing method,which is inspired by the modified MEMP,effectively pairs the repeated eigenvalues intractable in the unitary matrix pencil method,and it is more effective in determining the locations of elements in the sparse array.Three numerical examples and a full-wave validation are presented to demonstrate the effectiveness of the method,implemented via SMSPA,in achieving low sidelobe level wide-angle scanning radiation patterns,circular flattop radiation patterns,and ultra wide-angle scanning radiation patterns.
文摘This paper is devoted to investigate the accuracy of the Pseudo spectral scheme with the Chebyshev tau method and Chebyshev collocation method. The computational results of the nonlinear disturbance development in plane Poiseuille flow for both methods are presented and compared in detail. It is acknowledged that the Chebyshev collocation method has higher precision than the other one, especially for near netural situation.
基金supported by the National Natural Science Foundation of China (No.61203022)the Aeronautical Science Foundation of China (2012CZ51029)
文摘In view of generating optimal trajectories of Bolza problems, standard Chebyshev pseudospectral （PS） method makes the points＇ accumulation near the extremities and rarefaction of nodes close to the center of interval, which causes an ill-condition of differentiation matrix and an oscillation of the optimal solution. For improvement upon the difficulties, a mapped Chebyshev pseudospectral method is proposed. A conformal map is applied to Chebyshev points to move the points closer to equidistant nodes. Condition number and spectral radius of differentiation matrices from both methods are presented to show the improvement. Furthermore, the modification keeps the Chebyshev pseudospectral method＇s advantage, the spectral convergence rate. Based on three numerical examples, a comparison of the execution time, convergence and accuracy is presented among the standard Chebyshev pseudospectral method, other collocation methods and the proposed one. In one example, the error of results from mapped Chebyshev pseudospectral method is reduced to 5% of that from standard Chebyshev pseudospectral method.
文摘In this paper, we discuss local convergence of a family of Chebychev Halley type methods with a parameter θ∈[0,1] in Banach space using Smale type δ criterion under 2 th γ condition. We will see that the properties of the condition used for local convergence is much more different from that used in [6][15] for the semi-local convergence.
基金Hunan Provincial Natural Science Foundation Under Grant No.02JJY2085
文摘This paper introduces two new types of precise integration methods based on Chebyshev polynomial of the first kind for dynamic response analysis of structures, namely the integral formula method （IFM） and the homogenized initial system method （HISM）. In both methods, nonlinear variable loadings within time intervals are simulated using Chebyshev polynomials of the first kind before a direct integration is performed. Developed on the basis of the integral formula, the recurrence relationship of the integral computation suggested in this paper is combined with the Crout decomposed method to solve linear algebraic equations. In this way, the IFM based on Chebyshev polynomial of the first kind is constructed. Transforming the non-homogenous initial system to the homogeneous dynamic system, and developing a special scheme without dimensional expansion, the HISM based on Chebyshev polynomial of the first kind is able to avoid the matrix inversion operation. The accuracy of the time integration schemes is examined and compared with other commonly used schemes, and it is shown that a greater accuracy as well as less time consuming can be achieved. Two numerical examples are presented to demonstrate the applicability of these new methods.
基金supported by the National Basic Research Program of China (Grant 2013CB733004)
文摘The dynamic characteristics of a beam-cable coupled system are investigated using an improved Chebyshev spectral element method in order to observe the effects of adding cables on the beam. The system is modeled as a double Timoshenko beam system interconnected by discrete springs. Utilizing Chebyshev series expansion and meshing the system according to the locations of its connections, numerical results of the natural frequencies and mode shapes are obtained using only a few elements, and the results are validated by comparing them with the results of a finite-element method. Then the effects of the cable parameters and layout of connections on the natural frequencies and mode shapes of a fixed-pinned beam are studied. The results show that the modes of a beam-cable coupled system can be classified into two types, beam mode and cable mode, according to the dominant deformation. To avoid undesirable vibrations of the cable, its parameters should be controlled in a reasonable range, or the layout of the connections should be optimized.
文摘Fractional differential equations have recently been applied in various areas of engineering, science, finance, applied mathematics, bio-engineering and others. However, many researchers remain unaware of this field. In this paper, an efficient numerical method for solving the fractional Advection-dispersion equation (ADE) is considered. The fractional derivative is described in the Caputo sense. The method is based on Chebyshev approximations. The properties of Chebyshev polynomials are used to reduce ADE to a system of ordinary differential equations, which are solved using the finite difference method (FDM). Moreover, the convergence analysis and an upper bound of the error for the derived formula are given. Numerical solutions of ADE are presented and the results are compared with the exact solution.
文摘In this paper, we suggest a method for solving Fredholm integral equation of the first kind based on wavelet basis. The continuous Legendre and Chebyshev wavelets of the first, second, third and fourth kind on [0,1] are used and are utilized as a basis in Galerkin method to approximate the solution of integral equations. Then, in some examples the mentioned wavelets are compared with each other.
基金Supported by:Joint Research Fund for Earthquake Science,launched by the National Natural Science Foundation of China and the China Earthquake Administration under Grant No.U2039208。
文摘A diagonal or lumped mass matrix is of great value for time-domain analysis of structural dynamic and wave propagation problems,as the computational efforts can be greatly reduced in the process of mass matrix inversion.In this study,the nodal quadrature method is employed to construct a lumped mass matrix for the Chebyshev spectral element method(CSEM).A Gauss-Lobatto type quadrature,based on Gauss-Lobatto-Chebyshev points with a weighting function of unity,is thus derived.With the aid of this quadrature,the CSEM can take advantage of explicit time-marching schemes and provide an efficient new tool for solving structural dynamic problems.Several types of lumped mass Chebyshev spectral elements are designed,including rod,beam and plate elements.The performance of the developed method is examined via some numerical examples of natural vibration and elastic wave propagation,accompanied by their comparison to that of traditional consistent-mass CSEM or the classical finite element method(FEM).Numerical results indicate that the proposed method displays comparable accuracy as its consistent-mass counterpart,and is more accurate than classical FEM.For the simulation of elastic wave propagation in structures induced by high-frequency loading,this method achieves satisfactory performance in accuracy and efficiency.
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.
基金supported by the Imam Khomeini International University of Iran(No.751166-1392)the Deanship of Scientific Research(DSR)in King Abdulaziz University of Saudi Arabia
文摘The magnetohydrodynamics （MHD） Falkner-Skan flow of the Maxwell fluid is studied. Suitable transform reduces the partial differential equation into a nonlinear three order boundary value problem over a semi-infinite interval. An efficient approach based on the rational Chebyshev collocation method is performed to find the solution to the proposed boundary value problem. The rational Chebyshev collocation method is equipped with the orthogonal rational Chebyshev function which solves the problem on the semi-infinite domain without truncating it to a finite domain. The obtained results are presented through the illustrative graphs and tables which demonstrate the affectivity, stability, and convergence of the rational Chebyshev collocation method. To check the accuracy of the obtained results, a numerical method is applied for solving the problem. The variations of various embedded parameters into the problem are examined.
基金Supported by Natural Science Basic Research Plan in Shaanxi Province of China(2014JQ8366)Fundamental Research Foundation of Northwestern Polytechnical University(JC20120210,JC20110238)Aeronautical Science Foundation of China(20120853007)
文摘We study approximate solutions of a nonlinear integral equation of Hammerstein type. We describe the principle of discrete Adomian decomposition method (DADM). DADM is considered in the case we evaluate numerical integration by using Chebyshev roots. This technique gives an accurate solutions as will shown by illustrate examples.
文摘In this paper, we propose to replace the Chebyshev series used in pseudospectral methods with the equivalent Chebyshev economized power series that can be evaluated more rapidly. We keep the rest of the implementation the same as the spectral method so that there is no new mathematical principle involved. We show by numerical examples that the new approach works well and there is indeed no significant loss of solution accuracy. The advantages of using power series also include simplicity in its formulation and implementation such that it could be used for complex systems. We investigate the important issue of collocation point selection. Our numerical results indicate that there is a clear accuracy advantage of using collocation points corresponding to roots of the Chebyshev polynomial.
基金supported by the National Natural Science Foundation of China(No.11472058)
文摘The attitude optimal control problem （OCP） of a two-rigid-body space- craft with two rigid bodies coupled by a ball-in-socket joint is considered. Based on conservation of angular momentum of the system without the external torque, a dynamic equation of three-dimensional attitude motion of the system is formulated. The attitude motion planning problem of the coupled-rigid-body spacecraft can be converted to a dis- crete nonlinear programming （NLP） problem using the Chebyshev-Gauss pseudospectral method （CGPM）. Solutions of the NLP problem can be obtained using the sequential quadratic programming （SQP） algorithm. Since the collocation points of the CGPM are Chebyshev-Gauss （CG） points, the integration of cost function can be approximated by the Clenshaw-Curtis quadrature, and the corresponding quadrature weights can be calculated efficiently using the fast Fourier transform （FFT）. To improve computational efficiency and numerical stability, the barycentric Lagrange interpolation is presented to substitute for the classic Lagrange interpolation in the approximation of state and con- trol variables. Furthermore, numerical float errors of the state differential matrix and barycentric weights can be alleviated using trigonometric identity especially when the number of CG points is large. A simple yet efficient method is used to avoid sensitivity to the initial values for the SQP algorithm using a layered optimization strategy from a feasible solution to an optimal solution. Effectiveness of the proposed algorithm is perfect for attitude motion planning of a two-rigid-body spacecraft coupled by a ball-in-socket joint through numerical simulation.
文摘This paper ix devoted to establishment of the Chebyshev pseudospectral domain de-composition scheme for solving two-dimensional elliptic equation. By the generalized equivalent variatiunal form, we can get the stability and convergence of this new scheme.
