A type of 3 node triangular element is constructed by the Quasi-conforming method, which may be used to solve the equation of a type of inverse problem of wave propagation after Laplace transformation △u-A^2u=0. The ...A type of 3 node triangular element is constructed by the Quasi-conforming method, which may be used to solve the equation of a type of inverse problem of wave propagation after Laplace transformation △u-A^2u=0. The strains in the element are approximated by an expohential function and the string-net function between neigh- bouring elements is apporoximated by one dimensional general solution of the equation. Furthermore the strain field satisfies the equation, and therefore in the derivation of the element formulation, no shape function is needed. In this sense, it is a kind of hybrid element. Compared with the ordinary linear triangular element, the new one features higher precision with coarse meshes. Some numerical tests are presented.展开更多
The superconvergence in the finite element method is a phenomenon in which the finite element approximation converges to the exact solution at a rate higher than the optimal order error estimate. Wang proposed and ana...The superconvergence in the finite element method is a phenomenon in which the finite element approximation converges to the exact solution at a rate higher than the optimal order error estimate. Wang proposed and analyzed superconvergence of the conforming finite element method by L2-projections. The goal of this paper is to perform numerical experiments using MATLAB to support and to verify the theoretical results in Wang for the superconvergence of the conforming finite element method (CFEM) for the second order elliptic problems by L2-projection methods. MATLAB codes are published at https://github.com/annaleeharris/Superconvergence-CFEM for anyone to use and to study.展开更多
The nonlinear quasi-conforming FEM is presented based on the basic concept of the quasi- -conforming finite element. First, the incremental principle of stationary potential energy is discussed, Then, the formulation ...The nonlinear quasi-conforming FEM is presented based on the basic concept of the quasi- -conforming finite element. First, the incremental principle of stationary potential energy is discussed, Then, the formulation process of the nonlinear quasi-conforming FEM is given. Lastly, two computational examples of shells are given.展开更多
In this paper, we establish the maximum norm estimates of the solutions of the finite volume element method （FVE） based on the P1 conforming element for the non-selfadjoint and indefinite elliptic problems.
In this paper, we prove the existence, uniqueness and uniform convergence of the solution of finite volume element method based on the P1 conforming element for non-selfadjoint and indefinite elliptic problems under m...In this paper, we prove the existence, uniqueness and uniform convergence of the solution of finite volume element method based on the P1 conforming element for non-selfadjoint and indefinite elliptic problems under minimal elliptic regularity assumption.展开更多
With a generalized conforming element as a typical example, the spectral equivalence of unconventional finite elements and their conventional relatives is proved. This result is very important for the construction of ...With a generalized conforming element as a typical example, the spectral equivalence of unconventional finite elements and their conventional relatives is proved. This result is very important for the construction of domain decomposition parallel algorithms for unconventional finite elements.展开更多
Based on the strain formulation of the quasi-conforming finite element, displacement functions are constructed which have definite physical meaning, and a conclusion can be obtained that the coefficients of the consta...Based on the strain formulation of the quasi-conforming finite element, displacement functions are constructed which have definite physical meaning, and a conclusion can be obtained that the coefficients of the constant and the linear strain are uniquely determined, and the quasi-conforming finite element method is convergent to constant strain. There are different methods for constructing the rigid displacement items, and different methods correspond to different order node errors, and this is different from ordinary displacement method finite element.展开更多
Numerical quadrature schemes of a non-conforming finite element method for general second order elliptic problems in two dimensional （2-D） and three dimensional （3-D） space are discussed in this paper. We present ...Numerical quadrature schemes of a non-conforming finite element method for general second order elliptic problems in two dimensional （2-D） and three dimensional （3-D） space are discussed in this paper. We present and analyze some optimal numerical quadrature schemes. One of the schemes contains only three sampling points, which greatly improves the efficiency of numerical computations. The optimal error estimates are derived by using some traditional approaches and techniques. Lastly, some numerical results are provided to verify our theoretical analysis.展开更多
Based on the work of Xu and Zhou(2000),this paper makes a further discussion on conforming finite elements approximation for Steklov eigenvalue problems,and proves a local a priori error estimate and a new local a pos...Based on the work of Xu and Zhou(2000),this paper makes a further discussion on conforming finite elements approximation for Steklov eigenvalue problems,and proves a local a priori error estimate and a new local a posteriori error estimate in ||·||1,Ω0 norm for conforming elements eigenfunction,which has not been studied in existing literatures.展开更多
In this paper, the method of non-conforming mixed finite element for second order elliptic problems is discussed and a format of real optimal order for the lowest order error estimate.
The classical problem of the construction of C^1 conforming single-patch quadrilateral finite elements has been solved in this investigation by using the blending function interpolation method.In order to achieve the ...The classical problem of the construction of C^1 conforming single-patch quadrilateral finite elements has been solved in this investigation by using the blending function interpolation method.In order to achieve the C^1 conformity on the interfaces of quadrilateral elements,complete second-order derivatives are used at the element vertices,and the information of geometrical mapping is also considered into the construction of shape functions.It is found that the shape functions and the polynomial spaces of the present elements vary with element shapes.However,the developed quadrilateral elements are at least third order for general quadrilateral shapes and fifth order for rectangular shapes.Therefore,very fast convergence can be achieved.A promising feature of the present elements is that they can be used in cooperation with those high-precision rectangular and triangular elements.Since the present elements are over conforming on element vertices,an approach for handling problems of material discontinuity is also proposed.Numerical examples of Kirchhoff plates are employed to demonstrate the computational performance of the present elements.展开更多
Optimal convergence rates of adaptive finite element methods are well understood in terms of the axioms of adaptivity.One key ingredient is the discrete reliability of a residualbased a posteriori error estimator,whic...Optimal convergence rates of adaptive finite element methods are well understood in terms of the axioms of adaptivity.One key ingredient is the discrete reliability of a residualbased a posteriori error estimator,which controls the error of two discrete finite element solutions based on two nested triangulations.In the error analysis of nonconforming finite element methods,like the Crouzeix-Raviart or Morley finite element schemes,the difference of the piecewise derivatives of discontinuous approximations to the distributional gradients of global Sobolev functions plays a dominant role and is the object of this paper.The nonconforming interpolation operator,which comes natural with the definition of the aforementioned nonconforming finite element in the sense of Ciarlet,allows for stability and approximation properties that enable direct proofs of the reliability for the residual that monitors the equilibrium condition.The novel approach of this paper is the suggestion of a right-inverse of this interpolation operator in conforming piecewise polynomials to design a nonconforming approximation of a given coarse-grid approximation on a refined triangulation.The results of this paper allow for simple proofs of the discrete reliability in any space dimension and multiply connected domains on general shape-regular triangulations beyond newest-vertex bisection of simplices.Particular attention is on optimal constants in some standard discrete estimates listed in the appendices.展开更多
Nanoplates have been widely used as elementary components for ultrasensitive and ultrafine resolution applications in the field of nano-electro-mechanical systems because of their potentially remarkable mechanical pro...Nanoplates have been widely used as elementary components for ultrasensitive and ultrafine resolution applications in the field of nano-electro-mechanical systems because of their potentially remarkable mechanical properties.The accurate analysis of their mechanical behavior is currently of particular interest in the function design and reliability analysis of nano-scaled devices.To examine the size-dependent bending and vibration behavior of nanoplates with curvilinear and irregular shapes,a new p-version curved C^(1)finite element is formulated in the framework of the nonlocal Kirchhoff plate model.This newly developed element not only enables an accurate geometry representation and easy mesh generation of curvilinear domains but also overcomes the difficulty of imposing C^(1)conformity required by the nonlocal Kirchhoff plate model,particularly on the curvilinear inter-element boundaries.Numerical examples show that this element can produce an exponential rate of convergence even when curved elements are used in the domain discretization.Vast numerical results are presented for nanoplates with various geometric shapes,including rectangular,circular,elliptic,annular,and sectorial.The high accuracy of the present element is verified by comparing the obtained results with analytical and numerical results in the literature.Additionally,a comprehensive parametric analysis is conducted to investigate the influences of nonlocal parameters,plate dimensions,and boundary conditions on the nonlocal behavior of nanoplates.The present element can be envisaged to allow large-scale mechanical simulations of nanoplates,with a guarantee of accuracy and efficiency.展开更多
基金The project is supported by the National Natural Science Foundation of China
文摘A type of 3 node triangular element is constructed by the Quasi-conforming method, which may be used to solve the equation of a type of inverse problem of wave propagation after Laplace transformation △u-A^2u=0. The strains in the element are approximated by an expohential function and the string-net function between neigh- bouring elements is apporoximated by one dimensional general solution of the equation. Furthermore the strain field satisfies the equation, and therefore in the derivation of the element formulation, no shape function is needed. In this sense, it is a kind of hybrid element. Compared with the ordinary linear triangular element, the new one features higher precision with coarse meshes. Some numerical tests are presented.
文摘The superconvergence in the finite element method is a phenomenon in which the finite element approximation converges to the exact solution at a rate higher than the optimal order error estimate. Wang proposed and analyzed superconvergence of the conforming finite element method by L2-projections. The goal of this paper is to perform numerical experiments using MATLAB to support and to verify the theoretical results in Wang for the superconvergence of the conforming finite element method (CFEM) for the second order elliptic problems by L2-projection methods. MATLAB codes are published at https://github.com/annaleeharris/Superconvergence-CFEM for anyone to use and to study.
文摘The nonlinear quasi-conforming FEM is presented based on the basic concept of the quasi- -conforming finite element. First, the incremental principle of stationary potential energy is discussed, Then, the formulation process of the nonlinear quasi-conforming FEM is given. Lastly, two computational examples of shells are given.
基金The Major State Basic Research Program (19871051) of China and the NNSP (19972039) of China.
文摘In this paper, we establish the maximum norm estimates of the solutions of the finite volume element method （FVE） based on the P1 conforming element for the non-selfadjoint and indefinite elliptic problems.
基金The Major State Basic Research Program (19871051) of China the NNSF (19972039) of China and Yantai University Doctor Foundation (SX03B20).
文摘In this paper, we prove the existence, uniqueness and uniform convergence of the solution of finite volume element method based on the P1 conforming element for non-selfadjoint and indefinite elliptic problems under minimal elliptic regularity assumption.
文摘With a generalized conforming element as a typical example, the spectral equivalence of unconventional finite elements and their conventional relatives is proved. This result is very important for the construction of domain decomposition parallel algorithms for unconventional finite elements.
文摘Based on the strain formulation of the quasi-conforming finite element, displacement functions are constructed which have definite physical meaning, and a conclusion can be obtained that the coefficients of the constant and the linear strain are uniquely determined, and the quasi-conforming finite element method is convergent to constant strain. There are different methods for constructing the rigid displacement items, and different methods correspond to different order node errors, and this is different from ordinary displacement method finite element.
基金Supported by the National Natural Science Foundation of China (No. 50838004, 50908167)supported by the Fundamental Research Funds for the Central Universities of China (No. 2011YYL078)the National Natural Science Foundation of China (No. 11101386)
文摘Numerical quadrature schemes of a non-conforming finite element method for general second order elliptic problems in two dimensional （2-D） and three dimensional （3-D） space are discussed in this paper. We present and analyze some optimal numerical quadrature schemes. One of the schemes contains only three sampling points, which greatly improves the efficiency of numerical computations. The optimal error estimates are derived by using some traditional approaches and techniques. Lastly, some numerical results are provided to verify our theoretical analysis.
基金supported by National Natural Science Foundation of China(Grant Nos.11201093 and 11161012)
文摘Based on the work of Xu and Zhou(2000),this paper makes a further discussion on conforming finite elements approximation for Steklov eigenvalue problems,and proves a local a priori error estimate and a new local a posteriori error estimate in ||·||1,Ω0 norm for conforming elements eigenfunction,which has not been studied in existing literatures.
基金Project supported by the Cultivating Foundation of Youthful Backbone of Science and Technologyof Beijing, the National Science
文摘In this paper, the method of non-conforming mixed finite element for second order elliptic problems is discussed and a format of real optimal order for the lowest order error estimate.
基金supported by the National Natural Science Foundation of China(Grant Nos.11402015,11872090&11672019)。
文摘The classical problem of the construction of C^1 conforming single-patch quadrilateral finite elements has been solved in this investigation by using the blending function interpolation method.In order to achieve the C^1 conformity on the interfaces of quadrilateral elements,complete second-order derivatives are used at the element vertices,and the information of geometrical mapping is also considered into the construction of shape functions.It is found that the shape functions and the polynomial spaces of the present elements vary with element shapes.However,the developed quadrilateral elements are at least third order for general quadrilateral shapes and fifth order for rectangular shapes.Therefore,very fast convergence can be achieved.A promising feature of the present elements is that they can be used in cooperation with those high-precision rectangular and triangular elements.Since the present elements are over conforming on element vertices,an approach for handling problems of material discontinuity is also proposed.Numerical examples of Kirchhoff plates are employed to demonstrate the computational performance of the present elements.
文摘Optimal convergence rates of adaptive finite element methods are well understood in terms of the axioms of adaptivity.One key ingredient is the discrete reliability of a residualbased a posteriori error estimator,which controls the error of two discrete finite element solutions based on two nested triangulations.In the error analysis of nonconforming finite element methods,like the Crouzeix-Raviart or Morley finite element schemes,the difference of the piecewise derivatives of discontinuous approximations to the distributional gradients of global Sobolev functions plays a dominant role and is the object of this paper.The nonconforming interpolation operator,which comes natural with the definition of the aforementioned nonconforming finite element in the sense of Ciarlet,allows for stability and approximation properties that enable direct proofs of the reliability for the residual that monitors the equilibrium condition.The novel approach of this paper is the suggestion of a right-inverse of this interpolation operator in conforming piecewise polynomials to design a nonconforming approximation of a given coarse-grid approximation on a refined triangulation.The results of this paper allow for simple proofs of the discrete reliability in any space dimension and multiply connected domains on general shape-regular triangulations beyond newest-vertex bisection of simplices.Particular attention is on optimal constants in some standard discrete estimates listed in the appendices.
基金the National Major Science and Technology Projects of China(Grant No.J2019-VI-0001-0114)the National Natural Science Foundation of China(Grant Nos.11972004,11772031,11402015)。
文摘Nanoplates have been widely used as elementary components for ultrasensitive and ultrafine resolution applications in the field of nano-electro-mechanical systems because of their potentially remarkable mechanical properties.The accurate analysis of their mechanical behavior is currently of particular interest in the function design and reliability analysis of nano-scaled devices.To examine the size-dependent bending and vibration behavior of nanoplates with curvilinear and irregular shapes,a new p-version curved C^(1)finite element is formulated in the framework of the nonlocal Kirchhoff plate model.This newly developed element not only enables an accurate geometry representation and easy mesh generation of curvilinear domains but also overcomes the difficulty of imposing C^(1)conformity required by the nonlocal Kirchhoff plate model,particularly on the curvilinear inter-element boundaries.Numerical examples show that this element can produce an exponential rate of convergence even when curved elements are used in the domain discretization.Vast numerical results are presented for nanoplates with various geometric shapes,including rectangular,circular,elliptic,annular,and sectorial.The high accuracy of the present element is verified by comparing the obtained results with analytical and numerical results in the literature.Additionally,a comprehensive parametric analysis is conducted to investigate the influences of nonlocal parameters,plate dimensions,and boundary conditions on the nonlocal behavior of nanoplates.The present element can be envisaged to allow large-scale mechanical simulations of nanoplates,with a guarantee of accuracy and efficiency.
