The energy norm convergence rate of the finite element solution of the heat equation is reduced by the time-regularity of the exact solution. This paper presents an adaptive finite element treatment of time-dependent ...The energy norm convergence rate of the finite element solution of the heat equation is reduced by the time-regularity of the exact solution. This paper presents an adaptive finite element treatment of time-dependent singularities on the one-dimensional heat equation. The method is based on a Fourier decomposition of the solution and an extraction formula of the coefficients of the singularities coupled with a predictor-corrector algorithm. The method recovers the optimal convergence rate of the finite element method on a quasi-uniform mesh refinement. Numerical results are carried out to show the efficiency of the method.展开更多
The multi-variable finite element algorithm based on the generalized Gulerkin's method is more flexible to establish a finite element model in the continuum mechanics. By using this algorithm and numerical tests a...The multi-variable finite element algorithm based on the generalized Gulerkin's method is more flexible to establish a finite element model in the continuum mechanics. By using this algorithm and numerical tests a new singular finite element for elasto-plastic fracture analysis has been formulated. The results of numerical tests show that the new element possesses high accuracy and good performance. Some rules for formulating a multi-variable singular finite element are also discussed in this paper.展开更多
Using the method of the boundary integral equation, a set of singular integral equations of the hear transfer problems and the thermo-elastic problems of a crack embedded in a two-dimensional finite body is derived, a...Using the method of the boundary integral equation, a set of singular integral equations of the hear transfer problems and the thermo-elastic problems of a crack embedded in a two-dimensional finite body is derived, and then,its numerical method is proposed by the numerical method of the singular integral equations combined with boundary element method. Moreover, the singular nature of temperature gradient field near the crack front is proved by the main-part analysis method of the singular integral equation, and the singular temperature gradients are exactly obtained. Finally, several typical examples calculated.展开更多
Quadrature rules for evaluating singular integrals that typically occur in the boundary element method (BEM) for two-dimensional and axisymmetric three-dimensional problems are considered. This paper focuses on the nu...Quadrature rules for evaluating singular integrals that typically occur in the boundary element method (BEM) for two-dimensional and axisymmetric three-dimensional problems are considered. This paper focuses on the numerical integration of the functions on the standard domain [-1, 1], with a logarithmic singularity at the centre. The substitution x = tp, where p (≥ 3) is an odd integer is given particular attention, as this returns a regular integral and the domain unchanged. Gauss-Legendre quadrature rules are applied to the transformed integrals for a number of values of p. It is shown that a high value for p typically gives more accurate results.展开更多
A new method named the state space boundary element method （SSBEM） is estab- lished, in which the problem domain is divided into two parts. One is the boundary element domain which includes the interested inner poin...A new method named the state space boundary element method （SSBEM） is estab- lished, in which the problem domain is divided into two parts. One is the boundary element domain which includes the interested inner point, and the other is the state space domain. The boundary integral equation and the state space equation are combined together based on the interfacial continuity condition to form the system equation of the SSBEM. The SSBEM synthe- sizes both advantages of the boundary element method and the state space method. However, it can give inaccurate results when being used to evaluate the mechanical quantity of a point very close to the boundary element, because the Gaussian quadrature fails to calculate the nearly singular integrals. The analytical formulas were developed by part of the authors before intro- duced to deal with the nearly singular integrals. Thus, the SSBEM can yield accurate physical quantities for the points very close to the boundary element. The SSBEM results agree well with those of the finite element method （FEM）, while the discretized elements are far fewer than those of the FEM. Meanwhile, the SSBEM can analyze very thin coating, while the FEM fails due to the limitation of tolerance for Boolean operations.展开更多
A general algorithm is applied to the regularization of nearly singular integrals in the boundary element method of planar potential problems. For linear elements, the strongly singular and hypersingular integrals of ...A general algorithm is applied to the regularization of nearly singular integrals in the boundary element method of planar potential problems. For linear elements, the strongly singular and hypersingular integrals of the interior points very close to boundary were categorized into two forms. The factor leading to the singularity was transformed out of the integral representations with integration by parts, so non-singular regularized formulas were presented for the two forms of integrals. Furthermore, quadratic elements are used in addition to linear ones. The quadratic element very close to the internal point can be divided into two linear ones, so that the algorithm is still valid. Numerical examples demonstrate the effectiveness and accuracy of this algorithm. Especially for problems with curved boundaries, the combination of quadratic elements and linear elements can give more accurate results.展开更多
A singularly perturbed advection-diffusion two-point Robin boundary value problem whose solution has a single boundary layer is considered. Based on the piecewise linear polynomial approximation, the finite element me...A singularly perturbed advection-diffusion two-point Robin boundary value problem whose solution has a single boundary layer is considered. Based on the piecewise linear polynomial approximation, the finite element method is applied to the problem. Estimation of the error between solution and the finite element approximation is given in energy norm on shishkin-type mesh.展开更多
提出一种改进的边界元方法(boundary element method,BEM),采用常数四边形单元离散结构边界以方便计算奇异积分,运用CHIEF方法以保证常规边界积分方程(conventional boundary integral equation,CBIE)在低频振动条件下解的唯一性,并利用...提出一种改进的边界元方法(boundary element method,BEM),采用常数四边形单元离散结构边界以方便计算奇异积分,运用CHIEF方法以保证常规边界积分方程(conventional boundary integral equation,CBIE)在低频振动条件下解的唯一性,并利用C++程序设计语言开发该边界元算法的求解器及其图形用户界面。以典型模型的声辐射问题计算为例验证该求解器的正确性,计算结果与参考结果吻合良好,说明改进方法适用于结构振动噪声的数值仿真。展开更多
The basic principle and numerical technique for simulating two three-dimensional bubbles near a free surface are studied in detail by using boundary element method. The singularities of influence coefficient matrix ar...The basic principle and numerical technique for simulating two three-dimensional bubbles near a free surface are studied in detail by using boundary element method. The singularities of influence coefficient matrix are eliminated using coordinate transformation and so-called 4π rule. The solid angle for the open surface is treated in direct method based on its definition. Several kinds of configurations for the bubbles and free surface have been investigated. The pressure contours during the evolution of bubbles are obtained in our model and can better illuminate the mechanism underlying the motions of bubbles and free surface. The bubble dynamics and their interactions have close relation with the standoff distances, buoyancy parameters and initial sizes of bubbles. Completely different bubble shapes, free surface motions, jetting patterns and pressure distributions under different parameters can be observed in our model, as demon- strated in our calculation results.展开更多
In this paper, we shall give a group of new isoparametric elements suitable .for St.Wnant's torsion of a bar with vertical cracks. These elements are eight-pointisoparametric element, quarter eight-Point isoparame...In this paper, we shall give a group of new isoparametric elements suitable .for St.Wnant's torsion of a bar with vertical cracks. These elements are eight-pointisoparametric element, quarter eight-Point isoparametric element and eight-pointisoparametric transition element. Among these elements, the second and the thirdelements possess the singularity of order r-1/2 at crack tip. Using these elements, wehave completed the calculations of St. Venant's torsion for a cylinder with a radialvertical crack. The calculated results show that the isoparametric elements given bythis paper have ideal accuracy. good convergence. high speed of convergence, lowfreedom and little computational time, and so they can be widely applied to practice.展开更多
In this article, a direct stress approach based on finite element analysis to determine the stress intensity factor is improved. Firstly, by comparing the rigorous solution against the asymptotic solution for a proble...In this article, a direct stress approach based on finite element analysis to determine the stress intensity factor is improved. Firstly, by comparing the rigorous solution against the asymptotic solution for a problem of an infinite plate embedded a central crack, we found that the stresses in a restrictive interval near the crack tip given by the rigorous solution can be used to determine the stress intensity factor, which is nearly equal to the stress intensity factor given by the asymptotic solution. Secondly, the crack problem is solved numerically by the finite element method. Depending on the modeling capability of the software, we designed an adaptive mesh model to simulate the stress singularity. Thus, the stress result in an appropriate interval near the crack tip is fairly approximated to the rigorous solution of the corresponding crack problem. Therefore, the stress intensity factor may be calculated from the stress distribution in the appropriate interval, with a high accuracy.展开更多
Although boundary displacement and traction are independent field variables in boundary conditions of an elasticity problem at a non-singular boundary point, there exist definite relations of singularity intensities b...Although boundary displacement and traction are independent field variables in boundary conditions of an elasticity problem at a non-singular boundary point, there exist definite relations of singularity intensities between boundary displacement derivatives and tractions at a singular boundary point. The analytical forms of the relations at a singular smooth point for 2D isotropic elastic problems have been established in this work. By using the relations, positions of the singular boundary points and the corresponding singularity intensities of the unknown boundary field variables can be determined a priori. Therefore, more appropriate shape functions of the unknown boundary field variables in singular elements can be constructed. A numerical example shows that the accuracy of the BEM analysis using the developed theory is greatly increased.展开更多
The paper discusses how to reduce higher singularity order of a boundary integral equation. The approach will be discussed in some detail for plane elasticity.Numerical results for the meshes of unequal length boundar...The paper discusses how to reduce higher singularity order of a boundary integral equation. The approach will be discussed in some detail for plane elasticity.Numerical results for the meshes of unequal length boundary elements are reported.Higher precision for both deflection and force is obtained than that obtained with a general boundary element method.展开更多
文摘The energy norm convergence rate of the finite element solution of the heat equation is reduced by the time-regularity of the exact solution. This paper presents an adaptive finite element treatment of time-dependent singularities on the one-dimensional heat equation. The method is based on a Fourier decomposition of the solution and an extraction formula of the coefficients of the singularities coupled with a predictor-corrector algorithm. The method recovers the optimal convergence rate of the finite element method on a quasi-uniform mesh refinement. Numerical results are carried out to show the efficiency of the method.
文摘The multi-variable finite element algorithm based on the generalized Gulerkin's method is more flexible to establish a finite element model in the continuum mechanics. By using this algorithm and numerical tests a new singular finite element for elasto-plastic fracture analysis has been formulated. The results of numerical tests show that the new element possesses high accuracy and good performance. Some rules for formulating a multi-variable singular finite element are also discussed in this paper.
文摘Using the method of the boundary integral equation, a set of singular integral equations of the hear transfer problems and the thermo-elastic problems of a crack embedded in a two-dimensional finite body is derived, and then,its numerical method is proposed by the numerical method of the singular integral equations combined with boundary element method. Moreover, the singular nature of temperature gradient field near the crack front is proved by the main-part analysis method of the singular integral equation, and the singular temperature gradients are exactly obtained. Finally, several typical examples calculated.
文摘Quadrature rules for evaluating singular integrals that typically occur in the boundary element method (BEM) for two-dimensional and axisymmetric three-dimensional problems are considered. This paper focuses on the numerical integration of the functions on the standard domain [-1, 1], with a logarithmic singularity at the centre. The substitution x = tp, where p (≥ 3) is an odd integer is given particular attention, as this returns a regular integral and the domain unchanged. Gauss-Legendre quadrature rules are applied to the transformed integrals for a number of values of p. It is shown that a high value for p typically gives more accurate results.
基金This work was supported by National Natural Science Foundation of China （No.11772114） and Grants from China Scholarship Council （No. 201706690019）.
文摘A new method named the state space boundary element method （SSBEM） is estab- lished, in which the problem domain is divided into two parts. One is the boundary element domain which includes the interested inner point, and the other is the state space domain. The boundary integral equation and the state space equation are combined together based on the interfacial continuity condition to form the system equation of the SSBEM. The SSBEM synthe- sizes both advantages of the boundary element method and the state space method. However, it can give inaccurate results when being used to evaluate the mechanical quantity of a point very close to the boundary element, because the Gaussian quadrature fails to calculate the nearly singular integrals. The analytical formulas were developed by part of the authors before intro- duced to deal with the nearly singular integrals. Thus, the SSBEM can yield accurate physical quantities for the points very close to the boundary element. The SSBEM results agree well with those of the finite element method （FEM）, while the discretized elements are far fewer than those of the FEM. Meanwhile, the SSBEM can analyze very thin coating, while the FEM fails due to the limitation of tolerance for Boolean operations.
文摘A general algorithm is applied to the regularization of nearly singular integrals in the boundary element method of planar potential problems. For linear elements, the strongly singular and hypersingular integrals of the interior points very close to boundary were categorized into two forms. The factor leading to the singularity was transformed out of the integral representations with integration by parts, so non-singular regularized formulas were presented for the two forms of integrals. Furthermore, quadratic elements are used in addition to linear ones. The quadratic element very close to the internal point can be divided into two linear ones, so that the algorithm is still valid. Numerical examples demonstrate the effectiveness and accuracy of this algorithm. Especially for problems with curved boundaries, the combination of quadratic elements and linear elements can give more accurate results.
文摘A singularly perturbed advection-diffusion two-point Robin boundary value problem whose solution has a single boundary layer is considered. Based on the piecewise linear polynomial approximation, the finite element method is applied to the problem. Estimation of the error between solution and the finite element approximation is given in energy norm on shishkin-type mesh.
文摘提出一种改进的边界元方法(boundary element method,BEM),采用常数四边形单元离散结构边界以方便计算奇异积分,运用CHIEF方法以保证常规边界积分方程(conventional boundary integral equation,CBIE)在低频振动条件下解的唯一性,并利用C++程序设计语言开发该边界元算法的求解器及其图形用户界面。以典型模型的声辐射问题计算为例验证该求解器的正确性,计算结果与参考结果吻合良好,说明改进方法适用于结构振动噪声的数值仿真。
基金supported by the Funds for Creative Research Groups of China (50921001)the State Key Development Program for Basic Research of China (2010CB832704)
文摘The basic principle and numerical technique for simulating two three-dimensional bubbles near a free surface are studied in detail by using boundary element method. The singularities of influence coefficient matrix are eliminated using coordinate transformation and so-called 4π rule. The solid angle for the open surface is treated in direct method based on its definition. Several kinds of configurations for the bubbles and free surface have been investigated. The pressure contours during the evolution of bubbles are obtained in our model and can better illuminate the mechanism underlying the motions of bubbles and free surface. The bubble dynamics and their interactions have close relation with the standoff distances, buoyancy parameters and initial sizes of bubbles. Completely different bubble shapes, free surface motions, jetting patterns and pressure distributions under different parameters can be observed in our model, as demon- strated in our calculation results.
文摘In this paper, we shall give a group of new isoparametric elements suitable .for St.Wnant's torsion of a bar with vertical cracks. These elements are eight-pointisoparametric element, quarter eight-Point isoparametric element and eight-pointisoparametric transition element. Among these elements, the second and the thirdelements possess the singularity of order r-1/2 at crack tip. Using these elements, wehave completed the calculations of St. Venant's torsion for a cylinder with a radialvertical crack. The calculated results show that the isoparametric elements given bythis paper have ideal accuracy. good convergence. high speed of convergence, lowfreedom and little computational time, and so they can be widely applied to practice.
基金financial support of the National Natural Science Foundation of China (Grant 11572226)
文摘In this article, a direct stress approach based on finite element analysis to determine the stress intensity factor is improved. Firstly, by comparing the rigorous solution against the asymptotic solution for a problem of an infinite plate embedded a central crack, we found that the stresses in a restrictive interval near the crack tip given by the rigorous solution can be used to determine the stress intensity factor, which is nearly equal to the stress intensity factor given by the asymptotic solution. Secondly, the crack problem is solved numerically by the finite element method. Depending on the modeling capability of the software, we designed an adaptive mesh model to simulate the stress singularity. Thus, the stress result in an appropriate interval near the crack tip is fairly approximated to the rigorous solution of the corresponding crack problem. Therefore, the stress intensity factor may be calculated from the stress distribution in the appropriate interval, with a high accuracy.
文摘Although boundary displacement and traction are independent field variables in boundary conditions of an elasticity problem at a non-singular boundary point, there exist definite relations of singularity intensities between boundary displacement derivatives and tractions at a singular boundary point. The analytical forms of the relations at a singular smooth point for 2D isotropic elastic problems have been established in this work. By using the relations, positions of the singular boundary points and the corresponding singularity intensities of the unknown boundary field variables can be determined a priori. Therefore, more appropriate shape functions of the unknown boundary field variables in singular elements can be constructed. A numerical example shows that the accuracy of the BEM analysis using the developed theory is greatly increased.
基金The project supported by National Natural Science Foundation of China.
文摘The paper discusses how to reduce higher singularity order of a boundary integral equation. The approach will be discussed in some detail for plane elasticity.Numerical results for the meshes of unequal length boundary elements are reported.Higher precision for both deflection and force is obtained than that obtained with a general boundary element method.
