In this paper,the method of polynomial particular solutions is used to solve nonlinear Poisson-type partial differential equations in one,two,and three dimensions.The condition number of the coefficient matrix is redu...In this paper,the method of polynomial particular solutions is used to solve nonlinear Poisson-type partial differential equations in one,two,and three dimensions.The condition number of the coefficient matrix is reduced through the implementation of multiple scale technique,ultimately yielding a stable numerical solution.The methodological process can be divided into two main parts:first,identifying the corresponding polynomial particular solutions for the linear differential operator terms in the governing equations,and second,employing these polynomial particular solutions as basis function to iteratively solve the remaining nonlinear terms within the governing equations.Additionally,we investigate the potential improvement in numerical accuracy for equations with singularities in the analytical solution by shifting the computational domain a certain distance.Numerical experiments are conducted to assess both the accuracy and stability of the proposed method.A comparison of the obtained results with those produced by other numerical methods demonstrates the accuracy,stability,and efficiency of the proposed method in handling nonlinear Poisson-type partial differential equations.展开更多
In this study,the polynomial expansion method(PEM)and the polynomial method of particular solutions(PMPS)are applied to solve a class of linear elliptic partial differential equations(PDEs)in two dimensions with const...In this study,the polynomial expansion method(PEM)and the polynomial method of particular solutions(PMPS)are applied to solve a class of linear elliptic partial differential equations(PDEs)in two dimensions with constant coefficients.In the solution procedure,the sought solution is approximated by the Pascal polynomials and their particular solutions for the PEM and PMPS,respectively.The multiple-scale technique is applied to improve the conditioning of the resulted linear equations and the accuracy of numerical results for both of the PEM and PMPS.Some mathematical statements are provided to demonstrate the equivalence of the PEM and PMPS bases as they are both bases of a certain polynomial vector space.Then,some numerical experiments were conducted to validate the implementation of the PEM and PMPS.Numerical results demonstrated that the PEM is more accurate and well-conditioned than the PMPS and the multiple-scale technique is essential in these polynomial methods.展开更多
In this paper,we propose efficient algorithms for approximating particular solutions of second and fourth order elliptic equations.The approximation of the particular solution by a truncated series of Chebyshev polyno...In this paper,we propose efficient algorithms for approximating particular solutions of second and fourth order elliptic equations.The approximation of the particular solution by a truncated series of Chebyshev polynomials and the satisfaction of the differential equation lead to upper triangular block systems,each block being an upper triangular system.These systems can be solved efficiently by standard techniques.Several numerical examples are presented for each case.展开更多
In this paper we develop an efficient meshless method for solving inhomogeneous elliptic partial differential equations.We first approximate the source function by Chebyshev polynomials.We then focus on how to find a ...In this paper we develop an efficient meshless method for solving inhomogeneous elliptic partial differential equations.We first approximate the source function by Chebyshev polynomials.We then focus on how to find a polynomial particular solution when the source function is a polynomial.Through the principle of the method of undetermined coefficients and a proper arrangement of the terms for the polynomial particular solution to be determined,the coefficients of the particular solution satisfy a triangular system of linear algebraic equations.Explicit recursive formulas for the coefficients of the particular solutions are derived for different types of elliptic PDEs.The method is further incorporated into the method of fundamental solutions for solving inhomogeneous elliptic PDEs.Numerical results show that our approach is efficient and accurate.展开更多
The boundary element method for the modal analysis of freevibration for 3-D anisotropic structures using particular solutionshas been developed. The complete polynomials of order two are used toconstruct the particula...The boundary element method for the modal analysis of freevibration for 3-D anisotropic structures using particular solutionshas been developed. The complete polynomials of order two are used toconstruct the particular solutions for general anisotropic materials.The numerical results for 3-D free vibra- tion analysis of anisotropic cantilever beam by the method presented is in goodagreement with the results us- ing the Ritz technique. Foranisotropic materials, the numerical results calculated form theproposed method are in good agreement with the results from MSC.NASTRAN.展开更多
Based on the nonlocal theory and Mindlin plate theory,the governing equations(i.e.,a system of partial differential equations(PDEs)for bending problem)of magnetoelectroelastic(MEE)nanoplates resting on the Pasternak e...Based on the nonlocal theory and Mindlin plate theory,the governing equations(i.e.,a system of partial differential equations(PDEs)for bending problem)of magnetoelectroelastic(MEE)nanoplates resting on the Pasternak elastic foundation are first derived by the variational principle.The polynomial particular solutions corresponding to the established model are then obtained and further employed as basis functions with the method of particular solutions(MPS)to solve the governing equations numerically.It is confirmed that for the present bending model,the new solution strategy possesses more general applicability and superior flexibility in the selection of collocation points.The effects of different boundary conditions,applied loads,and geometrical shapes on the bending properties of MEE nanoplates are evaluated by using the developed method.Some important conclusions are drawn,which should be helpful for the design and applications of electromagnetic nanoplate structures.展开更多
The stress rate integral equations of elastoplasticity are deduced based on Ref. [1] by consistent methods. The point at which the stresses and/or displacements are calculated can be in the body or on the boundary, an...The stress rate integral equations of elastoplasticity are deduced based on Ref. [1] by consistent methods. The point at which the stresses and/or displacements are calculated can be in the body or on the boundary, and in the plastic region or elastic one. The existence of the principal value integral in the plastic region is demonstrated strictly, and the theoretical basis is presented for the paticular solution method by unit initial stress fields. In the present method, programming is easy and general, and the numerical results are excellent.展开更多
The boundary element method is used for he modal analysis of freevibration of 2-D composite structures in this paper. Since theparticular solution method is used to treat the terms of body forces(inertial forces) in t...The boundary element method is used for he modal analysis of freevibration of 2-D composite structures in this paper. Since theparticular solution method is used to treat the terms of body forces(inertial forces) in the equation of motion, only static fundamentalsolutions are needed in solving the problem. For an isotropiccantilever beam, the numerical results obtained by using the BEMpresented in this paper are in good agreement, with those of usingFEM or other BEM, but this BEM can also be used to analyze problemsfor anisotropic materials.展开更多
In this paper,we applied the polyharmonic splines as the basis functions to derive particular solutions for the differential operator ∆^(2) ± λ^(2).Similar to the derivation of fundamental solutions,it is non-tr...In this paper,we applied the polyharmonic splines as the basis functions to derive particular solutions for the differential operator ∆^(2) ± λ^(2).Similar to the derivation of fundamental solutions,it is non-trivial to derive particular solutions for higher order differential operators.In this paper,we provide a simple algebraic factorization approach to derive particular solutions for these types of differential operators in 2D and 3D.The main focus of this paper is its simplicity in the sense that minimal mathematical background is required for numerically solving higher order partial differential equations such as thin plate vibration.Three numerical examples in both 2D and 3D are given to validate particular solutions we derived.展开更多
Nonlinear solution of reinforced concrete structures, particularly complete load-deflection response, requires tracing of the equilibrium path and proper treatment of the limit and bifurcation points. In this regard, ...Nonlinear solution of reinforced concrete structures, particularly complete load-deflection response, requires tracing of the equilibrium path and proper treatment of the limit and bifurcation points. In this regard, ordinary solution techniques lead to instability near the limit points and also have problems in case of snap-through and snap-back. Thus they fail to predict the complete load-displacement response. The arc-length method serves the purpose well in principle, received wide acceptance in finite element analysis, and has been used extensively. However modifications to the basic idea are vital to meet the particular needs of the analysis. This paper reviews some of the recent developments of the method in the last two decades, with particular emphasis on nonlinear finite element analysis of reinforced concrete structures.展开更多
The aim of this paper is to propose a fast meshless numerical scheme for the simulation of non-linear Schrodinger equations.In the proposed scheme,the implicit-Euler scheme is used for the temporal discretization and ...The aim of this paper is to propose a fast meshless numerical scheme for the simulation of non-linear Schrodinger equations.In the proposed scheme,the implicit-Euler scheme is used for the temporal discretization and the localized method of approximate particular solution(LMAPS)is utilized for the spatial discretization.The multiple-scale technique is introduced to obtain the shape parameters of the multiquadric radial basis function for 2D problems and the Gaussian radial basis function for 3D problems.Six numerical examples are carried out to verify the accuracy and efficiency of the proposed scheme.Compared with well-known techniques,numerical results illustrate that the proposed scheme is of merits being easy-to-program,high accuracy,and rapid convergence even for long-term problems.These results also indicate that the proposed scheme has great potential in large scale problems and real-world applications.展开更多
A meshless method based on the method of fundamental solutions(MFS)is proposed to solve the time-dependent partial differential equations with variable coefficients.The proposed method combines the time discretization...A meshless method based on the method of fundamental solutions(MFS)is proposed to solve the time-dependent partial differential equations with variable coefficients.The proposed method combines the time discretization and the onestage MFS for spatial discretization.In contrast to the traditional two-stage process,the one-stage MFS approach is capable of solving a broad spectrum of partial differential equations.The numerical implementation is simple since both closed-form approximate particular solution and fundamental solution are easy to find than the traditional approach.The numerical results show that the one-stage approach is robust and stable.展开更多
Choosing particular solution source and its position have great influence on accu- racy of sound field prediction in distributed source boundary point method. An optimization method for determining the position of par...Choosing particular solution source and its position have great influence on accu- racy of sound field prediction in distributed source boundary point method. An optimization method for determining the position of particular solution sources is proposed to get high accu- racy prediction result. In this method, tripole is chosen as the particular solution. The upper limit frequency of calculation is predicted by setting 1% volume velocity relative error limit using vibration velocity of structure surface. Then, the optimal position of particular solution sources, in which the relative error of volume velocity is minimum, is determined within the range of upper limit frequency by searching algorithm using volume velocity matching. The transfer matrix between pressure and surface volume velocity is constructed in the optimal position. After that, the sound radiation of structure is calculated by the matrix. The results of numerical simulation show that the calculation error is significantly reduced by the proposed method. When there are vibration velocity measurement errors, the calculation errors can be controlled within 5% by the method.展开更多
Kernel-basedmethods are popular in computer graphics,machine learning,and statistics,among other fields;because they do not require meshing of the domain under consideration,higher dimensions and complicated domains c...Kernel-basedmethods are popular in computer graphics,machine learning,and statistics,among other fields;because they do not require meshing of the domain under consideration,higher dimensions and complicated domains can be managed with reasonable effort.Traditionally,the high order of accuracy associated with these methods has been tempered by ill-conditioning,which ariseswhen highly smooth kernels are used to conduct the approximation.Recent advances in representing Gaussians using eigenfunctions have proven successful at avoiding this destabilization in scattered data approximation problems.This paper will extend these techniques to the solution of boundary value problems using collocation.The method of particular solutions will also be considered for elliptic problems,using Gaussian eigenfunctions to stably produce an approximate particular solution.展开更多
Fourier transform is applied to remove the time-dependent variable in the diffusion equation.Under non-harmonic initial conditions this gives rise to a non-homogeneous Helmholtz equation,which is solved by the method ...Fourier transform is applied to remove the time-dependent variable in the diffusion equation.Under non-harmonic initial conditions this gives rise to a non-homogeneous Helmholtz equation,which is solved by the method of fundamental solutions and the method of particular solutions.The particular solution of Helmholtz equation is available as shown in[4,15].The approximate solution in frequency domain is then inverted numerically using the inverse Fourier transform algorithm.Complex frequencies are used in order to avoid aliasing phenomena and to allow the computation of the static response.Two numerical examples are given to illustrate the effectiveness of the proposed approach for solving 2-D diffusion equations.展开更多
The author studies the structure of solutions to the interface problems for second order linear elliptic partial differential equations in three space dimension. The set of singular points consists of some singular li...The author studies the structure of solutions to the interface problems for second order linear elliptic partial differential equations in three space dimension. The set of singular points consists of some singular lines and some isolated singular points. It is proved that near a singular line or a singular point, each weak solution can be decomposed into two parts, a singular part and a regular part. The singular parts are some finite sum of particular solutions to some simpler equations, and the regular parts are bounded in some norms, which are slightly weaker than that in the Sobolev space H^2.展开更多
This paper presents the contour integral method for solving the linear constant coefficient ordinary differential equations in complex plane,and obtains the uniform expressions of the general solutions.Firstly,by usin...This paper presents the contour integral method for solving the linear constant coefficient ordinary differential equations in complex plane,and obtains the uniform expressions of the general solutions.Firstly,by using Residue Theorem,the general form of the contour integral representation for the homogeneous complex differential equation is obtained,which can be degenerated to classical results in real line.As for inhomogeneous complex differential equations with constant coefficients,we construct the integral expression of the particular solution for any continuous forcing term,and give rigorous proof via Residue Theorem.Thus the general solutions of inhomogeneous complex differential equations are also given.The main purpose of this paper is to give a foundation for a complete theory of linear complex differential equations with constant coefficients by a contour integral method.The results can not only solve the inhomogeneous complex differential equation well,but also explain the forms that are difficult to be understood in the classical solutions.展开更多
基金supported by the Science and Technology Research Project of Education Department of Jiangxi Province,Philippine Council for Industry,Energy,and Emerging Technology Research and Development(Grant Nos.GJJ2200605)the National Natural Science Foundation of China(11461026)Jiangxi Provincial Natural Science Foundation(20282BA8201016).
文摘In this paper,the method of polynomial particular solutions is used to solve nonlinear Poisson-type partial differential equations in one,two,and three dimensions.The condition number of the coefficient matrix is reduced through the implementation of multiple scale technique,ultimately yielding a stable numerical solution.The methodological process can be divided into two main parts:first,identifying the corresponding polynomial particular solutions for the linear differential operator terms in the governing equations,and second,employing these polynomial particular solutions as basis function to iteratively solve the remaining nonlinear terms within the governing equations.Additionally,we investigate the potential improvement in numerical accuracy for equations with singularities in the analytical solution by shifting the computational domain a certain distance.Numerical experiments are conducted to assess both the accuracy and stability of the proposed method.A comparison of the obtained results with those produced by other numerical methods demonstrates the accuracy,stability,and efficiency of the proposed method in handling nonlinear Poisson-type partial differential equations.
基金The Ministry of Science and Technology of Taiwan is gratefully acknowledged for providing financial support to carry out the present work under the Grant No.MOST 109-2221-E-992-046-MY3.
文摘In this study,the polynomial expansion method(PEM)and the polynomial method of particular solutions(PMPS)are applied to solve a class of linear elliptic partial differential equations(PDEs)in two dimensions with constant coefficients.In the solution procedure,the sought solution is approximated by the Pascal polynomials and their particular solutions for the PEM and PMPS,respectively.The multiple-scale technique is applied to improve the conditioning of the resulted linear equations and the accuracy of numerical results for both of the PEM and PMPS.Some mathematical statements are provided to demonstrate the equivalence of the PEM and PMPS bases as they are both bases of a certain polynomial vector space.Then,some numerical experiments were conducted to validate the implementation of the PEM and PMPS.Numerical results demonstrated that the PEM is more accurate and well-conditioned than the PMPS and the multiple-scale technique is essential in these polynomial methods.
文摘In this paper,we propose efficient algorithms for approximating particular solutions of second and fourth order elliptic equations.The approximation of the particular solution by a truncated series of Chebyshev polynomials and the satisfaction of the differential equation lead to upper triangular block systems,each block being an upper triangular system.These systems can be solved efficiently by standard techniques.Several numerical examples are presented for each case.
文摘In this paper we develop an efficient meshless method for solving inhomogeneous elliptic partial differential equations.We first approximate the source function by Chebyshev polynomials.We then focus on how to find a polynomial particular solution when the source function is a polynomial.Through the principle of the method of undetermined coefficients and a proper arrangement of the terms for the polynomial particular solution to be determined,the coefficients of the particular solution satisfy a triangular system of linear algebraic equations.Explicit recursive formulas for the coefficients of the particular solutions are derived for different types of elliptic PDEs.The method is further incorporated into the method of fundamental solutions for solving inhomogeneous elliptic PDEs.Numerical results show that our approach is efficient and accurate.
基金the National Natural Science Foundation of China (No.19872011)
文摘The boundary element method for the modal analysis of freevibration for 3-D anisotropic structures using particular solutionshas been developed. The complete polynomials of order two are used toconstruct the particular solutions for general anisotropic materials.The numerical results for 3-D free vibra- tion analysis of anisotropic cantilever beam by the method presented is in goodagreement with the results us- ing the Ritz technique. Foranisotropic materials, the numerical results calculated form theproposed method are in good agreement with the results from MSC.NASTRAN.
基金Project supported by the National Natural Science Foundation of China(Nos.11872257 and 11572358)the German Research Foundation(No.ZH 15/14-1)。
文摘Based on the nonlocal theory and Mindlin plate theory,the governing equations(i.e.,a system of partial differential equations(PDEs)for bending problem)of magnetoelectroelastic(MEE)nanoplates resting on the Pasternak elastic foundation are first derived by the variational principle.The polynomial particular solutions corresponding to the established model are then obtained and further employed as basis functions with the method of particular solutions(MPS)to solve the governing equations numerically.It is confirmed that for the present bending model,the new solution strategy possesses more general applicability and superior flexibility in the selection of collocation points.The effects of different boundary conditions,applied loads,and geometrical shapes on the bending properties of MEE nanoplates are evaluated by using the developed method.Some important conclusions are drawn,which should be helpful for the design and applications of electromagnetic nanoplate structures.
基金The project supported by the National Natural Science Foundation of China
文摘The stress rate integral equations of elastoplasticity are deduced based on Ref. [1] by consistent methods. The point at which the stresses and/or displacements are calculated can be in the body or on the boundary, and in the plastic region or elastic one. The existence of the principal value integral in the plastic region is demonstrated strictly, and the theoretical basis is presented for the paticular solution method by unit initial stress fields. In the present method, programming is easy and general, and the numerical results are excellent.
文摘The boundary element method is used for he modal analysis of freevibration of 2-D composite structures in this paper. Since theparticular solution method is used to treat the terms of body forces(inertial forces) in the equation of motion, only static fundamentalsolutions are needed in solving the problem. For an isotropiccantilever beam, the numerical results obtained by using the BEMpresented in this paper are in good agreement, with those of usingFEM or other BEM, but this BEM can also be used to analyze problemsfor anisotropic materials.
文摘In this paper,we applied the polyharmonic splines as the basis functions to derive particular solutions for the differential operator ∆^(2) ± λ^(2).Similar to the derivation of fundamental solutions,it is non-trivial to derive particular solutions for higher order differential operators.In this paper,we provide a simple algebraic factorization approach to derive particular solutions for these types of differential operators in 2D and 3D.The main focus of this paper is its simplicity in the sense that minimal mathematical background is required for numerically solving higher order partial differential equations such as thin plate vibration.Three numerical examples in both 2D and 3D are given to validate particular solutions we derived.
文摘Nonlinear solution of reinforced concrete structures, particularly complete load-deflection response, requires tracing of the equilibrium path and proper treatment of the limit and bifurcation points. In this regard, ordinary solution techniques lead to instability near the limit points and also have problems in case of snap-through and snap-back. Thus they fail to predict the complete load-displacement response. The arc-length method serves the purpose well in principle, received wide acceptance in finite element analysis, and has been used extensively. However modifications to the basic idea are vital to meet the particular needs of the analysis. This paper reviews some of the recent developments of the method in the last two decades, with particular emphasis on nonlinear finite element analysis of reinforced concrete structures.
基金The authors thank the editor and anonymous reviewers for their constructive comments on the manuscript.The research of the authors was supported by the Natural Science Foundation of Jiangsu Province(No.BK20150795)the Fundamental Research Funds for the Central Universities(No.2018B16714)+3 种基金the National Natural Science Foundation of China(Nos.11702083,51679150,51579153,51739008,51527811)the State Key Laboratory of Mechanics and Control of Mechanical Structures(Nanjing University of Aeronautics and Astronautics)(No.MCMS-0218G01)the National Key R&D Program of China(No.2016YFC0401902)the Fund Project of NHRI(Nos.Y417002,Y417015).
文摘The aim of this paper is to propose a fast meshless numerical scheme for the simulation of non-linear Schrodinger equations.In the proposed scheme,the implicit-Euler scheme is used for the temporal discretization and the localized method of approximate particular solution(LMAPS)is utilized for the spatial discretization.The multiple-scale technique is introduced to obtain the shape parameters of the multiquadric radial basis function for 2D problems and the Gaussian radial basis function for 3D problems.Six numerical examples are carried out to verify the accuracy and efficiency of the proposed scheme.Compared with well-known techniques,numerical results illustrate that the proposed scheme is of merits being easy-to-program,high accuracy,and rapid convergence even for long-term problems.These results also indicate that the proposed scheme has great potential in large scale problems and real-world applications.
基金the support of NATO Collaborative Linkage Grant under reference ESP.CLG.982891。
文摘A meshless method based on the method of fundamental solutions(MFS)is proposed to solve the time-dependent partial differential equations with variable coefficients.The proposed method combines the time discretization and the onestage MFS for spatial discretization.In contrast to the traditional two-stage process,the one-stage MFS approach is capable of solving a broad spectrum of partial differential equations.The numerical implementation is simple since both closed-form approximate particular solution and fundamental solution are easy to find than the traditional approach.The numerical results show that the one-stage approach is robust and stable.
文摘Choosing particular solution source and its position have great influence on accu- racy of sound field prediction in distributed source boundary point method. An optimization method for determining the position of particular solution sources is proposed to get high accu- racy prediction result. In this method, tripole is chosen as the particular solution. The upper limit frequency of calculation is predicted by setting 1% volume velocity relative error limit using vibration velocity of structure surface. Then, the optimal position of particular solution sources, in which the relative error of volume velocity is minimum, is determined within the range of upper limit frequency by searching algorithm using volume velocity matching. The transfer matrix between pressure and surface volume velocity is constructed in the optimal position. After that, the sound radiation of structure is calculated by the matrix. The results of numerical simulation show that the calculation error is significantly reduced by the proposed method. When there are vibration velocity measurement errors, the calculation errors can be controlled within 5% by the method.
文摘Kernel-basedmethods are popular in computer graphics,machine learning,and statistics,among other fields;because they do not require meshing of the domain under consideration,higher dimensions and complicated domains can be managed with reasonable effort.Traditionally,the high order of accuracy associated with these methods has been tempered by ill-conditioning,which ariseswhen highly smooth kernels are used to conduct the approximation.Recent advances in representing Gaussians using eigenfunctions have proven successful at avoiding this destabilization in scattered data approximation problems.This paper will extend these techniques to the solution of boundary value problems using collocation.The method of particular solutions will also be considered for elliptic problems,using Gaussian eigenfunctions to stably produce an approximate particular solution.
基金The second author acknowledges the support of the Distinguished Overseas Visiting Scholar Fellowship funded by the Minister of Education in China and the support of NATO-RTA project under reference AVT-08/1.
文摘Fourier transform is applied to remove the time-dependent variable in the diffusion equation.Under non-harmonic initial conditions this gives rise to a non-homogeneous Helmholtz equation,which is solved by the method of fundamental solutions and the method of particular solutions.The particular solution of Helmholtz equation is available as shown in[4,15].The approximate solution in frequency domain is then inverted numerically using the inverse Fourier transform algorithm.Complex frequencies are used in order to avoid aliasing phenomena and to allow the computation of the static response.Two numerical examples are given to illustrate the effectiveness of the proposed approach for solving 2-D diffusion equations.
文摘The author studies the structure of solutions to the interface problems for second order linear elliptic partial differential equations in three space dimension. The set of singular points consists of some singular lines and some isolated singular points. It is proved that near a singular line or a singular point, each weak solution can be decomposed into two parts, a singular part and a regular part. The singular parts are some finite sum of particular solutions to some simpler equations, and the regular parts are bounded in some norms, which are slightly weaker than that in the Sobolev space H^2.
基金Supported by the National Natural Science Foundation of China(11561055)the Natural Science Foundation of Ningxia(2018AAC03057)。
文摘This paper presents the contour integral method for solving the linear constant coefficient ordinary differential equations in complex plane,and obtains the uniform expressions of the general solutions.Firstly,by using Residue Theorem,the general form of the contour integral representation for the homogeneous complex differential equation is obtained,which can be degenerated to classical results in real line.As for inhomogeneous complex differential equations with constant coefficients,we construct the integral expression of the particular solution for any continuous forcing term,and give rigorous proof via Residue Theorem.Thus the general solutions of inhomogeneous complex differential equations are also given.The main purpose of this paper is to give a foundation for a complete theory of linear complex differential equations with constant coefficients by a contour integral method.The results can not only solve the inhomogeneous complex differential equation well,but also explain the forms that are difficult to be understood in the classical solutions.
