A planar nonlinear weak form quadrature beam element of arbitrary number of axial nodes is proposed on the basis of the absolute nodal coordinate formulation (ANCF). Elastic forces of the element are established throu...A planar nonlinear weak form quadrature beam element of arbitrary number of axial nodes is proposed on the basis of the absolute nodal coordinate formulation (ANCF). Elastic forces of the element are established through geometrically exact beam theory, resulting in good consistency with classical beam theory. Two examples with strong geometrical nonlinearity are presented to verify the effec-tiveness of the formulation.展开更多
This paper, based on the hi sto rical background of CLT(communicative language teaching), mainly addresses the k ey issues concerning the weak form of CLT. This is an area few people have paid attention to in the past...This paper, based on the hi sto rical background of CLT(communicative language teaching), mainly addresses the k ey issues concerning the weak form of CLT. This is an area few people have paid attention to in the past few years in China. Therefore, its main concepts and f e atures are discussed with relevance to literature and an empirical study on four novice teachers of English. Finally, the research shows that the weak form of CLT is being conducted in Chinese secondary schools.展开更多
The recently proposed weak form quadrature element method(QEM) is applied to flexural and vibrational analysis of thin plates.The integrals involved in the variational description of a thin plate are evaluated by an e...The recently proposed weak form quadrature element method(QEM) is applied to flexural and vibrational analysis of thin plates.The integrals involved in the variational description of a thin plate are evaluated by an efficient numerical scheme and the partial derivatives at the integration sampling points are then approximated using differential quadrature analogs.Neither the grid pattern nor the number of nodes is fixed,being adjustable according to convergence need.The C1 continuity conditions characterizing the thin plate theory are discussed and the robustness of the weak form quadrature element for thin plates against shape distortion is examined.Examples are presented and comparisons with analytical solutions and the results of the finite element method are made to demonstrate the convergence and computational efficiency of the weak form quadrature element method.It is shown that the present formulation is applicable to thin plates with varying thickness as well as uniform plates.展开更多
In this paper, we give the definition of weak WT 2-class of differential forms, and then obtain its weak reverse Hlder inequality. As an application, we give an alternative proof of the higher integrability result o...In this paper, we give the definition of weak WT 2-class of differential forms, and then obtain its weak reverse Hlder inequality. As an application, we give an alternative proof of the higher integrability result of weakly A-harmonic tensors due to B. Stroffolini.展开更多
Strong and weak limits as well as strong and weak differentials of multi-variable indeterminate forms are discussed based on the thorough exploration of differentiation to solve the strong and weak limits and differen...Strong and weak limits as well as strong and weak differentials of multi-variable indeterminate forms are discussed based on the thorough exploration of differentiation to solve the strong and weak limits and differentials of unitary indeterminate forms. The fruit of this work is going to be reported in three parts. The first part presents the standard analysis on this subject which supplements, systematizes and advances L. Hospital抯 principles on differential calculus by applying special ,general, and limit guaranteeing theories together with K(t) and XhK0 theories. The combination of theoretical analysis and geometric signification makes the derivation intuitional, visual and easy to perceive.展开更多
The main aim of this paper is to analyze the numerical method based upon the spectral element technique for the numerical solution of the fractional advection-diffusion equa-tion.The time variable has been discretized...The main aim of this paper is to analyze the numerical method based upon the spectral element technique for the numerical solution of the fractional advection-diffusion equa-tion.The time variable has been discretized by a second-order finite difference procedure.The stability and the convergence of the semi-discrete formula have been proven.Then,the spatial variable of the main PDEs is approximated by the spectral element method.The convergence order of the fully discrete scheme is studied.The basis functions of the spectral element method are based upon a class of Legendre polynomials.The numerical experiments confirm the theoretical results.展开更多
The Meshless Local Petrov-Galerkin (MLPG) with Laplace transform is used for solving partial differential equation. Local weak form is developed using the weighted residual method locally from the dynamic partial diff...The Meshless Local Petrov-Galerkin (MLPG) with Laplace transform is used for solving partial differential equation. Local weak form is developed using the weighted residual method locally from the dynamic partial differential equation and using the moving least square (MLS) method to construct shape function. This method is a more effective alternative than the finite element method for computer modelling and simulation of problems in engineering;however, the accuracy of the present method depends on a number of parameters deriving from local weak form and different subdomains. In this paper, the meshless local Petrov-Galerkin (MLPG) formulation is proposed for forced vibration analysis. First, the results are presented for different values of as, and aq?with regular distribution of nodes nt=55. After, the results are presented with fixed values of?as?and aq?for different time-step.展开更多
Bell’s theorem determines the number of representations of a positive integer in terms of the ternary quadratic forms x2+by2+cz2 with b,c {1,2,4,8}. This number depends only on the number of representations of an int...Bell’s theorem determines the number of representations of a positive integer in terms of the ternary quadratic forms x2+by2+cz2 with b,c {1,2,4,8}. This number depends only on the number of representations of an integer as a sum of three squares. We present a modern elementary proof of Bell’s theorem that is based on three standard Ramanujan theta function identities and a set of five so-called three-square identities by Hurwitz. We use Bell’s theorem and a slight extension of it to find explicit and finite computable expressions for Tunnel’s congruent number criterion. It is known that this criterion settles the congruent number problem under the weak Birch-Swinnerton-Dyer conjecture. Moreover, we present for the first time an unconditional proof that a square-free number n 3(mod 8) is not congruent.展开更多
文摘A planar nonlinear weak form quadrature beam element of arbitrary number of axial nodes is proposed on the basis of the absolute nodal coordinate formulation (ANCF). Elastic forces of the element are established through geometrically exact beam theory, resulting in good consistency with classical beam theory. Two examples with strong geometrical nonlinearity are presented to verify the effec-tiveness of the formulation.
文摘This paper, based on the hi sto rical background of CLT(communicative language teaching), mainly addresses the k ey issues concerning the weak form of CLT. This is an area few people have paid attention to in the past few years in China. Therefore, its main concepts and f e atures are discussed with relevance to literature and an empirical study on four novice teachers of English. Finally, the research shows that the weak form of CLT is being conducted in Chinese secondary schools.
基金supported by the National Natural Science Foundation of China (Grant Nos.51178247 and 50778104)the National High Technology Research and Development Program of China (Grant No.2009AA04Z401)
文摘The recently proposed weak form quadrature element method(QEM) is applied to flexural and vibrational analysis of thin plates.The integrals involved in the variational description of a thin plate are evaluated by an efficient numerical scheme and the partial derivatives at the integration sampling points are then approximated using differential quadrature analogs.Neither the grid pattern nor the number of nodes is fixed,being adjustable according to convergence need.The C1 continuity conditions characterizing the thin plate theory are discussed and the robustness of the weak form quadrature element for thin plates against shape distortion is examined.Examples are presented and comparisons with analytical solutions and the results of the finite element method are made to demonstrate the convergence and computational efficiency of the weak form quadrature element method.It is shown that the present formulation is applicable to thin plates with varying thickness as well as uniform plates.
基金Supported by the National Natural Science Foundation of China (10971224)the Hebei Natural ScienceFoundation (07M003)
文摘In this paper, we give the definition of weak WT 2-class of differential forms, and then obtain its weak reverse Hlder inequality. As an application, we give an alternative proof of the higher integrability result of weakly A-harmonic tensors due to B. Stroffolini.
文摘Strong and weak limits as well as strong and weak differentials of multi-variable indeterminate forms are discussed based on the thorough exploration of differentiation to solve the strong and weak limits and differentials of unitary indeterminate forms. The fruit of this work is going to be reported in three parts. The first part presents the standard analysis on this subject which supplements, systematizes and advances L. Hospital抯 principles on differential calculus by applying special ,general, and limit guaranteeing theories together with K(t) and XhK0 theories. The combination of theoretical analysis and geometric signification makes the derivation intuitional, visual and easy to perceive.
基金The authors are grateful to the two reviewers for carefully reading this paper and for their comments and suggestions which have highly improved the paper.
文摘The main aim of this paper is to analyze the numerical method based upon the spectral element technique for the numerical solution of the fractional advection-diffusion equa-tion.The time variable has been discretized by a second-order finite difference procedure.The stability and the convergence of the semi-discrete formula have been proven.Then,the spatial variable of the main PDEs is approximated by the spectral element method.The convergence order of the fully discrete scheme is studied.The basis functions of the spectral element method are based upon a class of Legendre polynomials.The numerical experiments confirm the theoretical results.
文摘The Meshless Local Petrov-Galerkin (MLPG) with Laplace transform is used for solving partial differential equation. Local weak form is developed using the weighted residual method locally from the dynamic partial differential equation and using the moving least square (MLS) method to construct shape function. This method is a more effective alternative than the finite element method for computer modelling and simulation of problems in engineering;however, the accuracy of the present method depends on a number of parameters deriving from local weak form and different subdomains. In this paper, the meshless local Petrov-Galerkin (MLPG) formulation is proposed for forced vibration analysis. First, the results are presented for different values of as, and aq?with regular distribution of nodes nt=55. After, the results are presented with fixed values of?as?and aq?for different time-step.
文摘Bell’s theorem determines the number of representations of a positive integer in terms of the ternary quadratic forms x2+by2+cz2 with b,c {1,2,4,8}. This number depends only on the number of representations of an integer as a sum of three squares. We present a modern elementary proof of Bell’s theorem that is based on three standard Ramanujan theta function identities and a set of five so-called three-square identities by Hurwitz. We use Bell’s theorem and a slight extension of it to find explicit and finite computable expressions for Tunnel’s congruent number criterion. It is known that this criterion settles the congruent number problem under the weak Birch-Swinnerton-Dyer conjecture. Moreover, we present for the first time an unconditional proof that a square-free number n 3(mod 8) is not congruent.
