The interpolating moving least-squares （IMLS） method is discussed first in this paper. And the formulae of the IMLS method obtained by Lancaster are revised. Then on the basis of the boundary element-free method （B...The interpolating moving least-squares （IMLS） method is discussed first in this paper. And the formulae of the IMLS method obtained by Lancaster are revised. Then on the basis of the boundary element-free method （BEFM）, combining the boundary integral equation （BIE） method with the IMLS method, the improved boundary element-free method （IBEFM） for two-dimensional potential problems is presented, and the corresponding formulae of the IBEFM are obtained. In the BEFM, boundary conditions are applied directly, but the shape function in the MLS does not satisfy the property of the Kronecker ~ function. This is a problem of the BEFM, and must be solved theoretically. In the IMLS method, when the shape function satisfies the property of the Kronecker 5 function, then the boundary conditions, in the meshless method based on the IMLS method, can be applied directly. Then the IBEFM, based on the IMLS method, is a direct meshless boundary integral equation method in which the basic unknown quantity is the real solution of the nodal variables, and the boundary conditions can be applied directly and easily, thus it gives a greater computational precision. Some numerical examples are presented to demonstrate the method.展开更多
The element-free Galerkin(EFG)method,which constructs shape functions via moving least squares(MLS)approximation,represents a fundamental and widely studied meshless method in numerical computation.Although it achieve...The element-free Galerkin(EFG)method,which constructs shape functions via moving least squares(MLS)approximation,represents a fundamental and widely studied meshless method in numerical computation.Although it achieves high computational accuracy,the shape functions are more complex than those in the conventional finite element method(FEM),resulting in great computational requirements.Therefore,improving the computational efficiency of the EFG method represents an important research direction.This paper systematically reviews significant contributions fromdomestic and international scholars in advancing the EFGmethod.Including the improved element-free Galerkin(IEFG)method,various interpolating EFG methods,four distinct complex variable EFG methods,and a series of dimension splitting meshless methods.In the numerical examples,the effectiveness and efficiency of the three methods are validated by analyzing the solutions of the IEFG method for 3D steadystate anisotropic heat conduction,3D elastoplasticity,and large deformation problems,as well as the performance of two-dimensional splitting meshless methods in solving the 3D Helmholtz equation.展开更多
In this paper,we develop an advanced computational framework for the topology optimization of orthotropic materials using meshless methods.The approximation function is established based on the improved moving least s...In this paper,we develop an advanced computational framework for the topology optimization of orthotropic materials using meshless methods.The approximation function is established based on the improved moving least squares(IMLS)method,which enhances the efficiency and stability of the numerical solution.The numerical solution formulas are derived using the improved element-free Galerkin(IEFG)method.We introduce the solid isotropic microstructures with penalization(SIMP)model to formulate a mathematical model for topology opti-mization,which effectively penalizes intermediate densities.The optimization problem is defined with the numerical solution formula and volume fraction as constraints.The objective function,which is the minimum value of flexibility,is optimized iteratively using the optimization criterion method to update the design variables efficiently and converge to an optimal solution.Sensitivity analysis is performed using the adjoint method,which provides accurate and efficient gradient information for the optimization algorithm.We validate the proposed framework through a series of numerical examples,including clamped beam,cantilever beam,and simply supported beam made of orthotropic materials.The convergence of the objective function is demonstrated by increasing the number of iterations.Additionally,the stability of the iterative process is analyzed by examining the fluctuation law of the volume fraction.By adjusting the parameters to an appropriate range,we achieve the final optimization results of the IEFG method without the checkerboard phenomenon.Comparative studies between the Element-Free Galerkin(EFG)and IEFG methods reveal that both methods yield consistent optimization results under identical parameter settings.However,the IEFG method significantly reduces computational time,highlighting its efficiency and suitability for orthotropic materials.展开更多
The paper begins by discussing the interpolating moving least-squares(IMLS)method.Then the formulae of the IMLS method obtained by Lancaster are revised.On the basis of the boundary element-free method(BEFM),combining...The paper begins by discussing the interpolating moving least-squares(IMLS)method.Then the formulae of the IMLS method obtained by Lancaster are revised.On the basis of the boundary element-free method(BEFM),combining the boundary integral equation method with the IMLS method improved in this paper,the interpolating boundary element-free method(IBEFM)for two-dimensional elasticity problems is presented,and the corresponding formulae of the IBEFM for two-dimensional elasticity problems are obtained.In the IMLS method in this paper,the shape function satisfies the property of Kroneckerδfunction,and then in the IBEFM the boundary conditions can be applied directly and easily.The IBEFM is a direct meshless boundary integral equation method in which the basic unknown quantity is the real solution to the nodal variables.Thus it gives a greater computational precision.Numerical examples are presented to demonstrate the method.展开更多
Rubber-like materials that are commonly used in structural applications are modelled using hyperelastic material models.Most of the hyperelastic materials are nearly incompressible,which poses challenges,i.e.,volumetr...Rubber-like materials that are commonly used in structural applications are modelled using hyperelastic material models.Most of the hyperelastic materials are nearly incompressible,which poses challenges,i.e.,volumetric locking during numerical modelling.There exist many formulations in the context of the finite element method,among which the mixed displacementpressure formulation is robust.However,such a displacement-pressure formulation is less explored in meshfree methods,which mitigates the problem associated with mesh distortion during large deformation.This work addresses this issue of alleviating volumetric locking in the element-free Galerkin method(EFGM),which is one of the popular meshfree methods.A two-field mixed variational formulation using the perturbed Lagrangian approach within the EFGM framework is proposed for modelling nearly incompressible hyperelastic material models,such as Neo-Hookean and Mooney-Rivlin.Taking advantage of the meshless nature of the EFGM,this work introduces a unique approach by randomly distributing pressure nodes across the geometry,following specific guidelines.A wide spectrum of problems involving bending,tension,compression,and contact is solved using two approaches of the proposed displacement-pressure node formulation involving regular and irregular pressure node distribution.It is observed that both approaches give accurate results compared to the reference results,though the latter offers flexibility in the pressure nodal distribution.展开更多
The moving least-square approximation is discussed first.Sometimes the method can form an ill-conditioned equation system,and thus the solution cannot be obtained correctly.A Hilbert space is presented on which an ort...The moving least-square approximation is discussed first.Sometimes the method can form an ill-conditioned equation system,and thus the solution cannot be obtained correctly.A Hilbert space is presented on which an orthogonal function system mixed a weight function is defined.Next the improved moving least-square approximation is discussed in detail.The improved method has higher computational efficiency and precision than the old method,and cannot form an ill-conditioned equation system.A boundary element-free method(BEFM)for elastodynamics problems is presented by combining the boundary integral equation method for elastodynamics and the improved moving least-square approximation.The boundary element-free method is a meshless method of boundary integral equation and is a direct numerical method compared with others,in which the basic unknowns are the real solutions of the nodal variables and the boundary conditions can be applied easily.The boundary element-free method has a higher computational efficiency and precision.In addition,the numerical procedure of the boundary element-free method for elastodynamics problems is presented in this paper.Finally,some numerical examples are given.展开更多
A novel extended traction boundary element-free method is proposed to analyze the crack problems of two-dimensional infinite magnetoelectroelastic solid.An extended traction boundary integral equation only involving C...A novel extended traction boundary element-free method is proposed to analyze the crack problems of two-dimensional infinite magnetoelectroelastic solid.An extended traction boundary integral equation only involving Cauchy singularity is firstly derived.Then,the extended dislocation densities on the crack surface are expressed as the combination of a characteristic term and unknown weight functions,and the radial point interpolation method is adopted to approximate the unknown weight functions.The numerical scheme of the extended traction boundary element-free method is further established,and an effective numerical procedure is used to evaluate the Cauchy singular integrals.Finally,the stress intensity factor,electric displacement intensity factor and magnetic induction intensity factor are computed for some selected crack problems that contain straight,curved and branched cracks,and good numerical results are obtained.At the same time,the fracture properties of these crack problems are discussed.展开更多
This paper presents a 3D mathematical model for transfer and transform of water contaminants in reservoir,describes the discrete methods and the compute process of using an element-free collocation method.By these mod...This paper presents a 3D mathematical model for transfer and transform of water contaminants in reservoir,describes the discrete methods and the compute process of using an element-free collocation method.By these models and methods,the water quality conditions of Miyun reservoir area,the quantified description of three dimensions concerning the distribution and change of diversified contaminants are obtained within the assigned time.The distributions of total nitrogen and total phosphorus of the reservoir area in 2003,2004 and 2005 are simulated and forecasted by the boundary conditions in 2003,so the main development trend is shown.It is found through calculation that:The contribution of the bed mud with regard to total phosphorus is comparatively marked.On the whole,the influence factor of the bed mud in the reservoir area and other comprehensive factors to total phosphorus are quite great.The influence of the living things or other function in the surface water of the Chaohe River valley in the east of the reservoir on the total phosphorus is comparatively conspicuous.Within the reservoir area in spring,summer and autumn,the concentration distribution of total nitrogen is basically in the trend of a slow progressive decrease,but the concentration of total phosphorus increases progressively.It is just contrary in autumn,winter and spring,the concentration of total nitrogen increases and the concentration of total phosphorus decreases.On the whole of the year's development,the total nitrogen is basically in a steady state,and the total phosphorus is in the increasing trend with years,but the kind of the trend is not conspicuous.展开更多
In this paper, an improved interpolating moving least-square （IIMLS） method is presented. The shape function of the IIMLS method satisfies the property of the Kronecker 5 function. The weight function used in the II...In this paper, an improved interpolating moving least-square （IIMLS） method is presented. The shape function of the IIMLS method satisfies the property of the Kronecker 5 function. The weight function used in the IIMLS method is nonsingular. Then the IIMLS method can overcome the difficulties caused by the singularity of the weight function in the IMLS method. The number of unknown coefficients in the trial function of the IIMLS method is less than that of the moving least-square （MLS） approximation. Then by combining the IIMLS method with the Galerkin weak form of the potential problem, the improved interpolating element-free Galerkin （IIEFG） method for two-dimensional potential problems is presented. Compared with the conventional element-free Galerkin （EFG） method, the IIEFG method can directly use the essential boundary conditions. Then the IIEFG method has higher accuracy. For demonstration, three numerical examples are solved using the IIEFG method.展开更多
In this paper, the improved complex variable moving least-squares （ICVMLS） approximation is presented. The ICVMLS approximation has an explicit physics meaning. Compared with the complex variable moving least-squar...In this paper, the improved complex variable moving least-squares （ICVMLS） approximation is presented. The ICVMLS approximation has an explicit physics meaning. Compared with the complex variable moving least-squares （CVMLS） approximations presented by Cheng and Ren, the ICVMLS approximation has a great computational precision and efficiency. Based on the element-free Galerkin （EFG） method and the ICVMLS approximation, the improved complex variable element-free Galerkin （ICVEFG） method is presented for two-dimensional elasticity problems, and the corresponding formulae are obtained. Compared with the conventional EFC method, the ICVEFG method has a great computational accuracy and efficiency. For the purpose of demonstration, three selected numerical examples are solved using the ICVEFG method.展开更多
This paper presents the dimension split element-free Galerkin （DSEFG） method for three-dimensional potential problems, and the corresponding formulae are obtained. The main idea of the DSEFG method is that a three-d...This paper presents the dimension split element-free Galerkin （DSEFG） method for three-dimensional potential problems, and the corresponding formulae are obtained. The main idea of the DSEFG method is that a three-dimensional potential problem can be transformed into a series of two-dimensional problems. For these two-dimensional problems, the improved moving least-squares （IMLS） approximation is applied to construct the shape function, which uses an orthogonal function system with a weight function as the basis functions. The Galerkin weak form is applied to obtain a discretized system equation, and the penalty method is employed to impose the essential boundary condition. The finite difference method is selected in the splitting direction. For the purposes of demonstration, some selected numerical examples are solved using the DSEFG method. The convergence study and error analysis of the DSEFG method are presented. The numerical examples show that the DSEFG method has greater computational precision and computational efficiency than the IEFG method.展开更多
In this paper, based on the element-free Galerkin （EFG） method and the improved complex variable moving least- square （ICVMLS） approximation, a new meshless method, which is the improved complex variable element-f...In this paper, based on the element-free Galerkin （EFG） method and the improved complex variable moving least- square （ICVMLS） approximation, a new meshless method, which is the improved complex variable element-free Galerkin （ICVEFG） method for two-dimensional potential problems, is presented. In the method, the integral weak form of control equations is employed, and the Lagrange multiplier is used to apply the essential boundary conditions. Then the corresponding formulas of the ICVEFG method for two-dimensional potential problems are obtained. Compared with the complex variable moving least-square （CVMLS） approximation proposed by Cheng, the functional in the ICVMLS approximation has an explicit physical meaning. Furthermore, the ICVEFG method has greater computational precision and efficiency. Three numerical examples are given to show the validity of the proposed method.展开更多
In this paper, based on the conjugate of the complex basis function, a new complex variable moving least-squares approximation is discussed. Then using the new approximation to obtain the shape function, an improved c...In this paper, based on the conjugate of the complex basis function, a new complex variable moving least-squares approximation is discussed. Then using the new approximation to obtain the shape function, an improved complex variable element-free Galerkin（ICVEFG） method is presented for two-dimensional（2D） elastoplasticity problems. Compared with the previous complex variable moving least-squares approximation, the new approximation has greater computational precision and efficiency. Using the penalty method to apply the essential boundary conditions, and using the constrained Galerkin weak form of 2D elastoplasticity to obtain the system equations, we obtain the corresponding formulae of the ICVEFG method for 2D elastoplasticity. Three selected numerical examples are presented using the ICVEFG method to show that the ICVEFG method has the advantages such as greater precision and computational efficiency over the conventional meshless methods.展开更多
We first give a stabilized improved moving least squares （IMLS） approximation, which has better computational stability and precision than the IMLS approximation. Then, analysis of the improved element-free Galerkin...We first give a stabilized improved moving least squares （IMLS） approximation, which has better computational stability and precision than the IMLS approximation. Then, analysis of the improved element-free Galerkin method is provided theoretically for both linear and nonlinear elliptic boundary value problems. Finally, numerical examples are given to verify the theoretical analysis.展开更多
The paper presents the improved element-free Galerkin （IEFG） method for three-dimensional wave propa- gation. The improved moving least-squares （IMLS） approx- imation is employed to construct the shape function, w...The paper presents the improved element-free Galerkin （IEFG） method for three-dimensional wave propa- gation. The improved moving least-squares （IMLS） approx- imation is employed to construct the shape function, which uses an orthogonal function system with a weight function as the basis function. Compared with the conventional moving least-squares （MLS） approximation, the algebraic equation system in the IMLS approximation is not ill-conditioned, and can be solved directly without deriving the inverse matrix. Because there are fewer coefficients in the IMLS than in the MLS approximation, fewer nodes are selected in the IEFG method than in the element-free Galerkin method. Thus, the IEFG method has a higher computing speed. In the IEFG method, the Galerkin weak form is employed to obtain a dis- cretized system equation, and the penalty method is applied to impose the essential boundary condition. The traditional difference method for two-point boundary value problems is selected for the time discretization. As the wave equations and the boundary-initial conditions depend on time, the scal- ing parameter, number of nodes and the time step length are considered for the convergence study.展开更多
The present paper deals with the numerical solution of a two-dimensional linear hyperbolic equation by using the element-free Galerkin （EFG） method which is based on the moving least-square approximation for the tes...The present paper deals with the numerical solution of a two-dimensional linear hyperbolic equation by using the element-free Galerkin （EFG） method which is based on the moving least-square approximation for the test and trial functions. A variational method is used to obtain the discrete equations, and the essential boundary conditions are enforced by the penalty method. Compared with numerical methods based on mesh, the EFG method for hyperbolic problems needs only the scattered nodes instead of meshing the domain of the problem. It neither requires any element connectivity nor suffers much degradation in accuracy when nodal arrangements are very irregular. The effectiveness of the EFG method for two-dimensional hyperbolic problems is investigated by two numerical examples in this paper.展开更多
Through the construction of a new ramp function, the element-flee Galerkin method and finite element coupling method were applied to the whole field, and was made fit for the structure of element nodes within the inte...Through the construction of a new ramp function, the element-flee Galerkin method and finite element coupling method were applied to the whole field, and was made fit for the structure of element nodes within the interface regions, both satisfying the essential boundary conditions and deploying meshless nodes and finite elements in a convenient and flexible way, which can meet the requirements of computation for complicated field. The comparison between the results of the present study and the corresponding analytical solutions shows this method is feasible and effective.展开更多
The improved element-free Galerkin （IEFG） method of elasticity is used to solve the topology optimization problems. In this method, the improved moving least-squares approximation is used to form the shape function....The improved element-free Galerkin （IEFG） method of elasticity is used to solve the topology optimization problems. In this method, the improved moving least-squares approximation is used to form the shape function. In a topology opti- mization process, the entire structure volume is considered as the constraint. From the solid isotropic microstructures with penalization, we select relative node density as a design variable. Then we choose the minimization of compliance to be an objective function, and compute its sensitivity with the adjoint method. The IEFG method in this paper can overcome the disadvantages of the singular matrices that sometimes appear in conventional element-free Galerkin （EFG） method. The central processing unit （CPU） time of each example is given to show that the IEFG method is more efficient than the EFG method under the same precision, and the advantage that the IEFG method does not form singular matrices is also shown.展开更多
Based on the improved interpolating moving least-squares （ⅡMLS） method and the Galerkin weak form, an improved interpolating element-free Galerkin （ⅡEFG） method is presented for two-dimensional elasticity proble...Based on the improved interpolating moving least-squares （ⅡMLS） method and the Galerkin weak form, an improved interpolating element-free Galerkin （ⅡEFG） method is presented for two-dimensional elasticity problems in this paper. Compared with the interpolating moving least-squares （IMLS） method presented by Lancaster, the ⅡMLS method uses the nonsingular weight function. The number of unknown coefficients in the trial function of the ⅡMLS method is less than that of the MLS approximation and the shape function of the ⅡMLS method satisfies the property of Kronecker δ function. Thus in the ⅡEFG method, the essential boundary conditions can be applied directly and easily, then the numerical solutions can be obtained with higher precision than those obtained by the interpolating element-free Galerkin （IEFG） method. For the purposes of demonstration, four numerical examples are solved using the ⅡEFG method.展开更多
By introducing the dimensional splitting(DS)method into the multiscale interpolating element-free Galerkin(VMIEFG)method,a dimension-splitting multiscale interpolating element-free Galerkin(DS-VMIEFG)method is propose...By introducing the dimensional splitting(DS)method into the multiscale interpolating element-free Galerkin(VMIEFG)method,a dimension-splitting multiscale interpolating element-free Galerkin(DS-VMIEFG)method is proposed for three-dimensional(3D)singular perturbed convection-diffusion(SPCD)problems.In the DSVMIEFG method,the 3D problem is decomposed into a series of 2D problems by the DS method,and the discrete equations on the 2D splitting surface are obtained by the VMIEFG method.The improved interpolation-type moving least squares(IIMLS)method is used to construct shape functions in the weak form and to combine 2D discrete equations into a global system of discrete equations for the three-dimensional SPCD problems.The solved numerical example verifies the effectiveness of the method in this paper for the 3D SPCD problems.The numerical solution will gradually converge to the analytical solution with the increase in the number of nodes.For extremely small singular diffusion coefficients,the numerical solution will avoid numerical oscillation and has high computational stability.展开更多
基金Project supported by the National Natural Science Foundation of China (Grant No 10871124)Innovation Program of Shanghai Municipal Education Commission (Grant No 09ZZ99)Shanghai Leading Academic Discipline Project (Grant No J50103)
文摘The interpolating moving least-squares （IMLS） method is discussed first in this paper. And the formulae of the IMLS method obtained by Lancaster are revised. Then on the basis of the boundary element-free method （BEFM）, combining the boundary integral equation （BIE） method with the IMLS method, the improved boundary element-free method （IBEFM） for two-dimensional potential problems is presented, and the corresponding formulae of the IBEFM are obtained. In the BEFM, boundary conditions are applied directly, but the shape function in the MLS does not satisfy the property of the Kronecker ~ function. This is a problem of the BEFM, and must be solved theoretically. In the IMLS method, when the shape function satisfies the property of the Kronecker 5 function, then the boundary conditions, in the meshless method based on the IMLS method, can be applied directly. Then the IBEFM, based on the IMLS method, is a direct meshless boundary integral equation method in which the basic unknown quantity is the real solution of the nodal variables, and the boundary conditions can be applied directly and easily, thus it gives a greater computational precision. Some numerical examples are presented to demonstrate the method.
基金supported by the National Natural Science Foundation of China(Grant No.12271341).
文摘The element-free Galerkin(EFG)method,which constructs shape functions via moving least squares(MLS)approximation,represents a fundamental and widely studied meshless method in numerical computation.Although it achieves high computational accuracy,the shape functions are more complex than those in the conventional finite element method(FEM),resulting in great computational requirements.Therefore,improving the computational efficiency of the EFG method represents an important research direction.This paper systematically reviews significant contributions fromdomestic and international scholars in advancing the EFGmethod.Including the improved element-free Galerkin(IEFG)method,various interpolating EFG methods,four distinct complex variable EFG methods,and a series of dimension splitting meshless methods.In the numerical examples,the effectiveness and efficiency of the three methods are validated by analyzing the solutions of the IEFG method for 3D steadystate anisotropic heat conduction,3D elastoplasticity,and large deformation problems,as well as the performance of two-dimensional splitting meshless methods in solving the 3D Helmholtz equation.
基金supported by the Graduate Student Scientific Research Innovation Project through Research Innovation Fund for Graduate Students in Shanxi Province(Project No.2024KY648).
文摘In this paper,we develop an advanced computational framework for the topology optimization of orthotropic materials using meshless methods.The approximation function is established based on the improved moving least squares(IMLS)method,which enhances the efficiency and stability of the numerical solution.The numerical solution formulas are derived using the improved element-free Galerkin(IEFG)method.We introduce the solid isotropic microstructures with penalization(SIMP)model to formulate a mathematical model for topology opti-mization,which effectively penalizes intermediate densities.The optimization problem is defined with the numerical solution formula and volume fraction as constraints.The objective function,which is the minimum value of flexibility,is optimized iteratively using the optimization criterion method to update the design variables efficiently and converge to an optimal solution.Sensitivity analysis is performed using the adjoint method,which provides accurate and efficient gradient information for the optimization algorithm.We validate the proposed framework through a series of numerical examples,including clamped beam,cantilever beam,and simply supported beam made of orthotropic materials.The convergence of the objective function is demonstrated by increasing the number of iterations.Additionally,the stability of the iterative process is analyzed by examining the fluctuation law of the volume fraction.By adjusting the parameters to an appropriate range,we achieve the final optimization results of the IEFG method without the checkerboard phenomenon.Comparative studies between the Element-Free Galerkin(EFG)and IEFG methods reveal that both methods yield consistent optimization results under identical parameter settings.However,the IEFG method significantly reduces computational time,highlighting its efficiency and suitability for orthotropic materials.
基金supported by the National Natural Science Foundation of China(Grant No.10871124)the Innovation Program of Shanghai Municipal Education Commission(Grant No.09ZZ99)the ShanghaiLeading Academic Discipline Project(Grant No.J50103)
文摘The paper begins by discussing the interpolating moving least-squares(IMLS)method.Then the formulae of the IMLS method obtained by Lancaster are revised.On the basis of the boundary element-free method(BEFM),combining the boundary integral equation method with the IMLS method improved in this paper,the interpolating boundary element-free method(IBEFM)for two-dimensional elasticity problems is presented,and the corresponding formulae of the IBEFM for two-dimensional elasticity problems are obtained.In the IMLS method in this paper,the shape function satisfies the property of Kroneckerδfunction,and then in the IBEFM the boundary conditions can be applied directly and easily.The IBEFM is a direct meshless boundary integral equation method in which the basic unknown quantity is the real solution to the nodal variables.Thus it gives a greater computational precision.Numerical examples are presented to demonstrate the method.
基金supported by the DST-SERB and VSSC,ISRO of the project titled“Functionality Enhancement through Design and Development of Advanced Finite Element Algorithms for STR tools”under IMPRINT.IIC(IMP/2019/000276)scheme.
文摘Rubber-like materials that are commonly used in structural applications are modelled using hyperelastic material models.Most of the hyperelastic materials are nearly incompressible,which poses challenges,i.e.,volumetric locking during numerical modelling.There exist many formulations in the context of the finite element method,among which the mixed displacementpressure formulation is robust.However,such a displacement-pressure formulation is less explored in meshfree methods,which mitigates the problem associated with mesh distortion during large deformation.This work addresses this issue of alleviating volumetric locking in the element-free Galerkin method(EFGM),which is one of the popular meshfree methods.A two-field mixed variational formulation using the perturbed Lagrangian approach within the EFGM framework is proposed for modelling nearly incompressible hyperelastic material models,such as Neo-Hookean and Mooney-Rivlin.Taking advantage of the meshless nature of the EFGM,this work introduces a unique approach by randomly distributing pressure nodes across the geometry,following specific guidelines.A wide spectrum of problems involving bending,tension,compression,and contact is solved using two approaches of the proposed displacement-pressure node formulation involving regular and irregular pressure node distribution.It is observed that both approaches give accurate results compared to the reference results,though the latter offers flexibility in the pressure nodal distribution.
基金supported by the National Natural Science Foundation of China(Grant No.10571118)the Shanghai Leading Academic Discipline Project(Grant No.Y0103).
文摘The moving least-square approximation is discussed first.Sometimes the method can form an ill-conditioned equation system,and thus the solution cannot be obtained correctly.A Hilbert space is presented on which an orthogonal function system mixed a weight function is defined.Next the improved moving least-square approximation is discussed in detail.The improved method has higher computational efficiency and precision than the old method,and cannot form an ill-conditioned equation system.A boundary element-free method(BEFM)for elastodynamics problems is presented by combining the boundary integral equation method for elastodynamics and the improved moving least-square approximation.The boundary element-free method is a meshless method of boundary integral equation and is a direct numerical method compared with others,in which the basic unknowns are the real solutions of the nodal variables and the boundary conditions can be applied easily.The boundary element-free method has a higher computational efficiency and precision.In addition,the numerical procedure of the boundary element-free method for elastodynamics problems is presented in this paper.Finally,some numerical examples are given.
基金supported by the National Natural Science Foundation of China (10772123,11072160)Natural Science Foundation for Outstanding Young People of Hebei Province (A2009001624),China
文摘A novel extended traction boundary element-free method is proposed to analyze the crack problems of two-dimensional infinite magnetoelectroelastic solid.An extended traction boundary integral equation only involving Cauchy singularity is firstly derived.Then,the extended dislocation densities on the crack surface are expressed as the combination of a characteristic term and unknown weight functions,and the radial point interpolation method is adopted to approximate the unknown weight functions.The numerical scheme of the extended traction boundary element-free method is further established,and an effective numerical procedure is used to evaluate the Cauchy singular integrals.Finally,the stress intensity factor,electric displacement intensity factor and magnetic induction intensity factor are computed for some selected crack problems that contain straight,curved and branched cracks,and good numerical results are obtained.At the same time,the fracture properties of these crack problems are discussed.
基金supported by the National Major Basic Research Development Project(Grant No.G1999045705).
文摘This paper presents a 3D mathematical model for transfer and transform of water contaminants in reservoir,describes the discrete methods and the compute process of using an element-free collocation method.By these models and methods,the water quality conditions of Miyun reservoir area,the quantified description of three dimensions concerning the distribution and change of diversified contaminants are obtained within the assigned time.The distributions of total nitrogen and total phosphorus of the reservoir area in 2003,2004 and 2005 are simulated and forecasted by the boundary conditions in 2003,so the main development trend is shown.It is found through calculation that:The contribution of the bed mud with regard to total phosphorus is comparatively marked.On the whole,the influence factor of the bed mud in the reservoir area and other comprehensive factors to total phosphorus are quite great.The influence of the living things or other function in the surface water of the Chaohe River valley in the east of the reservoir on the total phosphorus is comparatively conspicuous.Within the reservoir area in spring,summer and autumn,the concentration distribution of total nitrogen is basically in the trend of a slow progressive decrease,but the concentration of total phosphorus increases progressively.It is just contrary in autumn,winter and spring,the concentration of total nitrogen increases and the concentration of total phosphorus decreases.On the whole of the year's development,the total nitrogen is basically in a steady state,and the total phosphorus is in the increasing trend with years,but the kind of the trend is not conspicuous.
基金Project supported by the National Natural Science Foundation of China (Grant No. 11171208)the Shanghai Leading Academic Discipline Project, China (Grant No. S30106)
文摘In this paper, an improved interpolating moving least-square （IIMLS） method is presented. The shape function of the IIMLS method satisfies the property of the Kronecker 5 function. The weight function used in the IIMLS method is nonsingular. Then the IIMLS method can overcome the difficulties caused by the singularity of the weight function in the IMLS method. The number of unknown coefficients in the trial function of the IIMLS method is less than that of the moving least-square （MLS） approximation. Then by combining the IIMLS method with the Galerkin weak form of the potential problem, the improved interpolating element-free Galerkin （IIEFG） method for two-dimensional potential problems is presented. Compared with the conventional element-free Galerkin （EFG） method, the IIEFG method can directly use the essential boundary conditions. Then the IIEFG method has higher accuracy. For demonstration, three numerical examples are solved using the IIEFG method.
基金supported by the National Natural Science Foundation of China (Grant No.11026223)the Shanghai Leading Academic Discipline Project,China (Grant No.S30106)the Innovation Fund Project for Graduate Student of Shanghai University,China (Grant No.SHUCX112359)
文摘In this paper, the improved complex variable moving least-squares （ICVMLS） approximation is presented. The ICVMLS approximation has an explicit physics meaning. Compared with the complex variable moving least-squares （CVMLS） approximations presented by Cheng and Ren, the ICVMLS approximation has a great computational precision and efficiency. Based on the element-free Galerkin （EFG） method and the ICVMLS approximation, the improved complex variable element-free Galerkin （ICVEFG） method is presented for two-dimensional elasticity problems, and the corresponding formulae are obtained. Compared with the conventional EFC method, the ICVEFG method has a great computational accuracy and efficiency. For the purpose of demonstration, three selected numerical examples are solved using the ICVEFG method.
基金supported by the National Natural Science Foundation of China (Grants 11571223, 51404160)Shanxi Province Science Foundation for Youths (Grant 2014021025-1)
文摘This paper presents the dimension split element-free Galerkin （DSEFG） method for three-dimensional potential problems, and the corresponding formulae are obtained. The main idea of the DSEFG method is that a three-dimensional potential problem can be transformed into a series of two-dimensional problems. For these two-dimensional problems, the improved moving least-squares （IMLS） approximation is applied to construct the shape function, which uses an orthogonal function system with a weight function as the basis functions. The Galerkin weak form is applied to obtain a discretized system equation, and the penalty method is employed to impose the essential boundary condition. The finite difference method is selected in the splitting direction. For the purposes of demonstration, some selected numerical examples are solved using the DSEFG method. The convergence study and error analysis of the DSEFG method are presented. The numerical examples show that the DSEFG method has greater computational precision and computational efficiency than the IEFG method.
基金Project supported by the National Natural Science Foundation of China (Grant No. 11171208)the Shanghai Leading Academic Discipline Project, China (Grant No. S30106)the Innovation Fund Project for Graduate Student of Shanghai University,China (Grant No. SHUCX112359)
文摘In this paper, based on the element-free Galerkin （EFG） method and the improved complex variable moving least- square （ICVMLS） approximation, a new meshless method, which is the improved complex variable element-free Galerkin （ICVEFG） method for two-dimensional potential problems, is presented. In the method, the integral weak form of control equations is employed, and the Lagrange multiplier is used to apply the essential boundary conditions. Then the corresponding formulas of the ICVEFG method for two-dimensional potential problems are obtained. Compared with the complex variable moving least-square （CVMLS） approximation proposed by Cheng, the functional in the ICVMLS approximation has an explicit physical meaning. Furthermore, the ICVEFG method has greater computational precision and efficiency. Three numerical examples are given to show the validity of the proposed method.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11171208 and U1433104)
文摘In this paper, based on the conjugate of the complex basis function, a new complex variable moving least-squares approximation is discussed. Then using the new approximation to obtain the shape function, an improved complex variable element-free Galerkin（ICVEFG） method is presented for two-dimensional（2D） elastoplasticity problems. Compared with the previous complex variable moving least-squares approximation, the new approximation has greater computational precision and efficiency. Using the penalty method to apply the essential boundary conditions, and using the constrained Galerkin weak form of 2D elastoplasticity to obtain the system equations, we obtain the corresponding formulae of the ICVEFG method for 2D elastoplasticity. Three selected numerical examples are presented using the ICVEFG method to show that the ICVEFG method has the advantages such as greater precision and computational efficiency over the conventional meshless methods.
基金Project supported by the National Natural Science Foundation of China(Grant No.11471063)the Chongqing Research Program of Basic Research and Frontier Technology,China(Grant No.cstc2015jcyj BX0083)the Educational Commission Foundation of Chongqing City,China(Grant No.KJ1600330)
文摘We first give a stabilized improved moving least squares （IMLS） approximation, which has better computational stability and precision than the IMLS approximation. Then, analysis of the improved element-free Galerkin method is provided theoretically for both linear and nonlinear elliptic boundary value problems. Finally, numerical examples are given to verify the theoretical analysis.
基金supported by the National Natural Science Foundation of China (11171208)Shanghai Leading Academic Discipline Project (S30106)
文摘The paper presents the improved element-free Galerkin （IEFG） method for three-dimensional wave propa- gation. The improved moving least-squares （IMLS） approx- imation is employed to construct the shape function, which uses an orthogonal function system with a weight function as the basis function. Compared with the conventional moving least-squares （MLS） approximation, the algebraic equation system in the IMLS approximation is not ill-conditioned, and can be solved directly without deriving the inverse matrix. Because there are fewer coefficients in the IMLS than in the MLS approximation, fewer nodes are selected in the IEFG method than in the element-free Galerkin method. Thus, the IEFG method has a higher computing speed. In the IEFG method, the Galerkin weak form is employed to obtain a dis- cretized system equation, and the penalty method is applied to impose the essential boundary condition. The traditional difference method for two-point boundary value problems is selected for the time discretization. As the wave equations and the boundary-initial conditions depend on time, the scal- ing parameter, number of nodes and the time step length are considered for the convergence study.
基金Project supported by the Natural Science Foundation of Ningbo, China (Grant Nos 2009A610014, 2009A610154, 2008A610020 and 2007A610050)
文摘The present paper deals with the numerical solution of a two-dimensional linear hyperbolic equation by using the element-free Galerkin （EFG） method which is based on the moving least-square approximation for the test and trial functions. A variational method is used to obtain the discrete equations, and the essential boundary conditions are enforced by the penalty method. Compared with numerical methods based on mesh, the EFG method for hyperbolic problems needs only the scattered nodes instead of meshing the domain of the problem. It neither requires any element connectivity nor suffers much degradation in accuracy when nodal arrangements are very irregular. The effectiveness of the EFG method for two-dimensional hyperbolic problems is investigated by two numerical examples in this paper.
文摘Through the construction of a new ramp function, the element-flee Galerkin method and finite element coupling method were applied to the whole field, and was made fit for the structure of element nodes within the interface regions, both satisfying the essential boundary conditions and deploying meshless nodes and finite elements in a convenient and flexible way, which can meet the requirements of computation for complicated field. The comparison between the results of the present study and the corresponding analytical solutions shows this method is feasible and effective.
基金supported by the National Natural Science Foundation of China(Grant Nos.11571223 and U1433104)
文摘The improved element-free Galerkin （IEFG） method of elasticity is used to solve the topology optimization problems. In this method, the improved moving least-squares approximation is used to form the shape function. In a topology opti- mization process, the entire structure volume is considered as the constraint. From the solid isotropic microstructures with penalization, we select relative node density as a design variable. Then we choose the minimization of compliance to be an objective function, and compute its sensitivity with the adjoint method. The IEFG method in this paper can overcome the disadvantages of the singular matrices that sometimes appear in conventional element-free Galerkin （EFG） method. The central processing unit （CPU） time of each example is given to show that the IEFG method is more efficient than the EFG method under the same precision, and the advantage that the IEFG method does not form singular matrices is also shown.
基金Project supported by the National Natural Science Foundation of China(Grant No.11171208)the Shanghai Leading Academic Discipline Project,China(Grant No.S30106)
文摘Based on the improved interpolating moving least-squares （ⅡMLS） method and the Galerkin weak form, an improved interpolating element-free Galerkin （ⅡEFG） method is presented for two-dimensional elasticity problems in this paper. Compared with the interpolating moving least-squares （IMLS） method presented by Lancaster, the ⅡMLS method uses the nonsingular weight function. The number of unknown coefficients in the trial function of the ⅡMLS method is less than that of the MLS approximation and the shape function of the ⅡMLS method satisfies the property of Kronecker δ function. Thus in the ⅡEFG method, the essential boundary conditions can be applied directly and easily, then the numerical solutions can be obtained with higher precision than those obtained by the interpolating element-free Galerkin （IEFG） method. For the purposes of demonstration, four numerical examples are solved using the ⅡEFG method.
基金supported by the Natural Science Foundation of Zhejiang Province,China(Grant Nos.LY20A010021,LY19A010002,LY20G030025)the Natural Science Founda-tion of Ningbo City,China(Grant Nos.2021J147,2021J235).
文摘By introducing the dimensional splitting(DS)method into the multiscale interpolating element-free Galerkin(VMIEFG)method,a dimension-splitting multiscale interpolating element-free Galerkin(DS-VMIEFG)method is proposed for three-dimensional(3D)singular perturbed convection-diffusion(SPCD)problems.In the DSVMIEFG method,the 3D problem is decomposed into a series of 2D problems by the DS method,and the discrete equations on the 2D splitting surface are obtained by the VMIEFG method.The improved interpolation-type moving least squares(IIMLS)method is used to construct shape functions in the weak form and to combine 2D discrete equations into a global system of discrete equations for the three-dimensional SPCD problems.The solved numerical example verifies the effectiveness of the method in this paper for the 3D SPCD problems.The numerical solution will gradually converge to the analytical solution with the increase in the number of nodes.For extremely small singular diffusion coefficients,the numerical solution will avoid numerical oscillation and has high computational stability.
