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Analyzing 3D Helmholtz Equations by Using the Hybrid Complex Variable Element-Free Galerkin Method
Abstract
In this study, we present the hybrid complex variable element-free Galerkin (HCVEFG) method for solving 3D Helmholtz equations. The dimension splitting method (DSM) will be introduced into the corresponding governing equation, thus a series of 2D forms can be obtained by splitting the problem domain of 3D Helmholtz equation. For every 2D problem, the shape function can be obtained by using the improved complex variable moving least-squares (ICVMLS) approximation, and the essential boundary condition can be imposed by using the penalty method, thus the discretized equations of 2D problems can be derived by using the corresponding Galerkin weak form. These equations can be coupled by using the finite difference method (FDM) in the dimension splitting direction, thus final formulae of the numerical solution for 3D Helmholtz equation can be obtained. In Sec. 4, the relative errors are given, and the convergence is analyzed numerically. The numerical result of these examples illustrates that the calculation speed can be improved greatly when the HCVEFG method is used rather than the improved element-free Galerkin (IEFG) method.
... Complex engineering problems currently can be solved using robust numerical methods such as the finite element method, as well as the spectral element method, the boundary element method, the finite volume method, and meshless methods. Various coupling procedures (hybrid and coupled numerical methods) have been proposed incorporating different numerical methods in order to take advantage of their respective benefits [1][2][3]. ...
... The functionally graded loam is simulated as an M-layered medium described by the relations (28). To check the accuracy, the continuous properties M(z) are substituted into the governing Equations (1) and (2). Therefore, the relative error in the functionally graded loam is decaying with M, while it is stable in the homogeneous media (sand and bedrock). ...
The structure of soils is often heterogeneous with layered strata having distinct permeabilities. An advanced mathematical and numerical coupled model of elastic wave propagation in poroelastic multi-layered soils subjected to subsoil water infiltration is proposed in this study. The coupled model was based on the introduction of an inhomogeneous functionally graded fluid-saturation of the considered soil depending on the infiltration time, which was evaluated employing Richards’ equation. The time-harmonic solution was formulated in terms of the Fourier transform of Green’s matrix and the surface load that excites the vibration. The convergence and efficiency of the proposed approach are demonstrated. An example of dispersion curves for partially saturated porous strata made of loam, sand, and rock at different infiltration times is provided, and it is shown that the characteristics of the surface acoustic waves change with time, which can be further used for inverse problems’ solution.
... Meshless methods developed rapidly in recent decades such as the smoothed particle hydrodynamics [7], the reproducing kernel particle method [8,9], the point interpolation methods [10,11], the elementfree Galerkin method [12,13], etc. We will not go in detail of the literature reviewing about the meshless methods and only discuss some of methods based on the radial basis functions. ...
In this paper, we make the first attempt to apply the Gaussian-cubic basis function in the novel backward substitution method for solving linear and nonlinear problems in irregular domains. To improve the solution accuracy, the weighted parameters of the provided hybrid kernel are varying instead of constant ones in traditional methods. The shape parameters are determined by a symmetric variable shape parameter scheme. Furthermore, a counterintuitive ghost-points method is employed as the strategy for placing the center nodes in the basis functions. To evaluate the accuracy and efficiency of the proposed method, several 2D and 3D numerical examples are considered and the numerical results are compared with the traditional backward substitution method and the results in references. And the method can be used to solve nonlinear problems with some linearization techniques. The superior accuracy and efficiency have been validated.
In this study, by introducing the finite element method (FEM) into the improved element-free Galerkin (IEFG) method, the dimension coupling method (DCM) is proposed for solving three-dimensional (3D) Helmholtz and Poisson’s equations efficiently. The dimensional splitting method is introduced into the corresponding governing equations, thus 3D equations can be split into a series of 2D ones. The IEFG method is selected to discretize those 2D forms, thus the discretized equations are derived easily by using the weak forms. In the third direction, the FEM is selected to couple those discretized equations, thus the final linear equation of 3D equation is derived. In numerical examples, the good convergence of the DCM for Helmholtz and Poisson’s equations is shown. The solutions of numerical examples show that the computational efficiency of the IEFG method is improved significantly without losing the computational accuracy when the DCM is used. In addition, the DCM can enhance the computational efficiency of the hybrid element-free Galerkin (HEFG) method significantly without too many layers when the natural boundary conditions exist in the splitting direction.
In this paper, the government behavior of leasing different land use rights in Beijing, China, is analyzed using data analysis based on the multinomial logit model. The factors that lead the government to lease different land use rights are considered from the aspects of the land features, geographical location of the land, district economic development, government finance and political tenure of the district head, etc. Considering the factors as the variables, the multinomial logit model is presented to analyze the factors that affect the district government behavior on leasing different land use rights. The data of the variables are obtained in Beijing at the district level from 2004 to 2015. From the results, we can see that the area and price of the land, gross domestic product, foreign direct investment, distance of the land from airport, distance of the land from city center, distance of the land from the nearest industrial park, government fiscal deficit and tenure of the district head all influence the district government behavior on leasing land. Finally, the policy implications are proposed. The results and implications can be referenced by other metropolises in China and other developing countries with public ownership of land.
The improved element-free Galerkin (IEFG) method is proposed in this paper for solving 3D Helmholtz equations. The improved moving least-squares (IMLS) approximation is used to establish the trial function, and the penalty technique is used to enforce the essential boundary conditions. Thus, the final discretized equations of the IEFG method for 3D Helmholtz equations can be derived by using the corresponding Galerkin weak form. The influences of the node distribution, the weight functions, the scale parameters of the influence domain, and the penalty factors on the computational accuracy of the solutions are analyzed, and the numerical results of three examples show that the proposed method in this paper can not only enhance the computational speed of the element-free Galerkin (EFG) method but also eliminate the phenomenon of the singular matrix.
To analyze the leasing behavior of residential land in Beijing, the mathematical models of the price and the total area of the leased residential land are presented. The variables of the mathematical models are proposed by analyzing the factors influencing the district government’s leasing behavior for residential land based on the leasing right for residential land in Beijing, China. The regression formulae of the mathematical models are obtained with the ordinary least squares method. By introducing the data of the districts in Beijing from 2004 to 2015 into the mathematical models, the numerical results of the coefficients in the mathematical models are obtained by solving the equations of the regression formulae. After discussing the numerical results of the influencing factors, the district government behavior for leasing residential land in Beijing, China, is investigated. The numerical results show the factors concerning the government and how these factors influence the leased price and the total leased area of residential land for this large city in China. Finally, policy implications for the district government regarding residential land leasing in Beijing are proposed.
In this paper, based on the improved interpolating moving least-squares (IMLS) method and the dimension splitting method, the interpolating dimension splitting element-free Galerkin (IDSEFG) method for three-dimensional (3D) potential problems is proposed. The key of the IDSEFG method is to split a 3D problem domain into many related two-dimensional (2D) subdomains. The shape function is constructed by the improved IMLS method on the 2D subdomains, and the Galerkin weak form based on the dimension splitting method is used to obtain the discretized equations. The discrete equations on these 2D subdomains are coupled by the finite difference method. Take the improved element-free Galerkin (IEFG) method as a comparison, the advantage of the IDSEFG method is that the essential boundary conditions can be enforced directly. The effects of the number of nodes, the direction of dimension splitting, and the parameters of the influence domain on the calculation accuracy are studied through four numerical examples, the numerical solutions of the IDSEFG method are compared with the numerical solutions of the IEFG method and the analytical solutions. It is verified that the numerical solutions of the IDSEFG method are highly consistent with the analytical solution, and the calculation efficiency of this method is significantly higher than that of the IEFG method.
By introducing the dimension splitting method into the reproducing kernel particle method (RKPM), a hybrid reproducing kernel particle method (HRKPM) for solving three-dimensional (3D) wave propagation problems is presented in this paper. Compared with the RKPM of 3D problems, the HRKPM needs only solving a set of two-dimensional (2D) problems in some subdomains, rather than solving a 3D problem in the 3D problem domain. The shape functions of 2D problems are much simpler than those of 3D problems, which results in that the HRKPM can save the CPU time greatly. Four numerical examples are selected to verify the validity and advantages of the proposed method. In addition, the error analysis and convergence of the proposed method are investigated. From the numerical results we can know that the HRKPM has higher computational efficiency than the RKPM and the element-free Galerkin method.
In this paper, the improved element-free Galerkin (IEFG) method is used for solving 3D advection-diffusion problems. The improved moving least-squares (IMLS) approximation is used to form the trial function, the penalty method is applied to introduce the essential boundary conditions, the Galerkin weak form and the difference method are used to obtain the final discretized equations, and then the formulae of the IEFG method for 3D advection-diffusion problems are presented. The error and the convergence are analyzed by numerical examples, and the numerical results show that the IEFG method not only has a higher computational speed but also can avoid singular matrix of the element-free Galerkin (EFG) method.
We propose a robust optimal 27-point finite difference scheme for the Helmholtz equation in three-dimensional domain. In each direction, a special central difference scheme with 27 grid points is developed to approximate the second derivative operator. The 27 grid points are divided into four groups, and each group is involved in the difference scheme by the manner of weighted combination. As for the approximation of the zeroth-order term, we use the weighted average of all the 27 points, which are also divided into four groups. Finally, we obtain the optimal weights by minimizing the numerical dispersion with the least-square method. In comparison with the rotated difference scheme based on a staggered-grid method, the new scheme is simpler, more practical, and much more robust. It works efficiently even if the step sizes along different directions are not equal. However, rotated scheme fails in this situation. We also present the convergence analysis and dispersion analysis. Numerical examples demonstrate the effectiveness of the proposed scheme.
By employing the improved moving least-square (IMLS) approximation, the improved element-free Galerkin (IEFG) method is presented for the unsteady Schrödinger equation. In the IEFG method, the two-dimensional (2D) trial function is approximated by the IMLS approximation, the variation method is used to obtain the discrete equations, and the essential boundary conditions are imposed by the penalty method. Because the number of coefficients in the IMLS approximation is less than in the moving least-square (MLS) approximation, fewer nodes are needed in the entire domain when the IMLS approximation is used than when the MLS approximation is adopted. Then the IEFG method has high computational efficiency and accuracy. Several numerical examples are given to verify the accuracy and efficiency of the IEFG method in this paper.
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 EFG 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.
Based on the complex network of industry space, this paper investigates the industry structure optimization in Jiangxi Province in China. For the prefecture-level cities in Jiangxi Province, the location quotient is applied to obtain the specialized sub-industries. The conditional probability is used to analyze the interdependencies between industries. The complex network of industry space is presented to illustrate the relatedness among industries and analyze the space and structure of the regional overall industry. The index of transitional potential is proposed to measure the potential of a non-specialized industry which will become a specialized one in the future. The transformation difficulty index of industry is obtained to analyze the difficulty level of the transformation of the original specialized industry set to a new specialized one of a city. The results show that more industries have positive than negative interdependencies with one another in Jiangxi Province; rare industries specialize in cities at higher-level GDP, and common industries dominate the cities at lower-level GDP; it is more difficult for rarer non-specialized industries to become specialized ones, more common non-specialized industries can become specialized ones more easily, and the cities with higher GDP can transform industries more easily. By discussing the industries that have positive interdependencies with existing specialized industries, the detailed direction of industry structure optimization is analyzed. Finally, the policy implications are proposed, and these can be applied to other developing countries.
In this paper, a hybrid reproducing kernel particle method (HRKPM) for three-dimensional (3D) advection-diffusion problems is presented. The governing equation of the advection-diffusion problem includes the second derivative of the field function to space coordinates, the first derivative of the field function to space coordinates and time, so it is necessary to discretize the time domain after discretizing the space domain. By introducing the idea of dimension splitting, a 3D advection-diffusion problem can be transformed into a series of related two-dimensional (2D) ones in the dimension splitting direction. Then, the discrete equations of these 2D problems are established by using the RKPM, and these discrete equations are coupled by using the difference method. Finally, by using the difference method to discretize the time domain, the formula of the HRKPM for solving 3D advection-diffusion problem is obtained. Numerical results show that the HRKPM has higher computational efficiency than the RKPM when solving 3D advection-diffusion problems.
Fracture is a highly nonlinear problem, especially under a large deformation in cross-linked elastomers. Existing constitutive theories and fracture models fail to predict the fracture behavior of cross-linked elastomers because of the absence of the random polymer network structure. Meanwhile, few numerical methods can present the realistic fracture process of a polymer network. In this study, we propose a mesoscopic network mechanics method to present the entire large deformation and fracture process of the polymer network in cross-linked elastomers. The microstructure of a cross-linked elastomer is abstracted as a polymer network model where nodes are referred to as crosslinkers and connections between nodes are referred to as polymer chains. A hybrid total energy form of the network model is proposed including the free energy of polymer chains and volumetric deformation energy. In addition, using a proposed stretch criterion of polymer chains to realize chain scission, this mesoscopic network mechanics method is numerically implemented by coding. Two-dimensional network models with uniform and randomly distributed chain lengths, as well as models with different types of pre-cracks, are constructed and simulated under uniaxial tension tests using our method. Our method reproduces the entire hyper-elastic large deformation and fracture process of polymer network models. The random network models show a lower Young's modulus and a less significant strain-hardening stage, indicating that the structural randomness is the primary cause of accuracy degradation for existing hyperelastic constitutive models under large deformations. The results also reveal that the structural randomness of the polymer network is a primary reason for the ductile fracture and the low notch sensitivity of cross-linked elastomers.
In this paper, based on dimension splitting method and the improved interpolating moving least-squares (IMLS) method, a dimension splitting interpolating element-free Galerkin (DSIEFG) method for three-dimensional (3D) transient heat conduction problems is proposed. The main idea of the DSIEFG method is to split a 3D problem domain into a series of related two-dimensional (2D) subdomains. The improved IMLS method is used to construct shape function on 2D subdomains. Finite difference method is used to couple these discretized equations on 2D subdomains and employed in the time domain. Compared with the improved element-free Galerkin (IEFG) method, the advantage of the DSIEFG method is that the essential boundary conditions can be applied directly, which can improve computational accuracy and efficiency. Six examples are chosen to verify the effectiveness and efficiency of the DSIEFG method. The results of DSIEFG are compared with the numerical solutions of the IEFG method, and it is shown that the efficiency and precision of the DSIEFG method are greater than ones of the IEFG method for 3D transient heat conduction problems.
In this paper, to improve the computational speed of the reproducing kernel particle method (RKPM), the dimension splitting reproducing kernel particle method (DSRKPM) is presented for solving three-dimensional (3D) transient heat conduction problems. The idea of the proposed method is transforming the 3D problem into a series of two-dimensional (2D) ones by using the finite difference method in the splitting direction. Since the shape function of the RKPM for 2D problem is much simpler than the one of 3D problem, the DSRKPM has higher computational speed than the RKPM. To demonstrate the applicability of the proposed method, four example problems are presented, and each of them is solved by the DSRKPM and the RKPM, respectively. And the numerical solutions show that the DSRKPM not only greatly improves the computational efficiency, but also has a higher computational accuracy.
In this paper, combining the dimension splitting method with the improved complex variable element-free Galerkin (ICVEFG) method, we present a fast ICVEFG method for three-dimensional wave propagation problems. Using the dimension splitting method, the equations of three-dimensional wave propagation problems are translated into a series of two-dimensional ones in another one-dimensional direction. The new Galerkin weak form of the dimension splitting method for three-dimensional wave propagation problems is obtained. The improved complex variable moving least-square (ICVMLS) approximation is used to obtain the shape functions, and the penalty method is used to apply the essential boundary conditions, finite difference method is used in the one-dimensional direction, then the formulae of the ICVEFG method for three-dimensional wave propagation problems are obtained. The convergence and the corresponding parameters in the ICVEFG method are discussed. Some numerical examples are given to show that the new method has higher computational precision, and can improve the computational efficiency of the conventional meshless methods for three-dimensional problems greatly.
By using the improved moving least-square (IMLS) approximation to present the shape function, the improved element-free Galerkin (IEFG) method is investigated to solve diffusional drug release problems in this paper. In order to get the discretized equation system, Galerkin weak form of a diffusional drug release problem is used with applying essential boundary conditions using the penalty method. The difference method is applied for discretization of time domain. Then the formulae of IEFG method for solving diffusional drug release problems are presented. Three numerical example problems are given to study the convergence of solutions of IEFG method in this paper. The influences of scale parameters of influence domain, penalty factor and node distribution on the accuracy of the solutions of IEFG method are discussed. Compared with finite element method, the correctness of IEFG method in this paper is shown.
This paper presents an interpolating element-free Galerkin (IEFG) method for solving the two-dimensional (2D) elastic large deformation problems. By using the improved interpolating moving least-squares method to form shape function, and using the Galerkin weak form of 2D elastic large deformation problems to obtain the discrete equations, we obtain the formulae of the IEFG method for 2D elastic large deformation problems. As the displacement boundary conditions can be applied directly, the IEFG method can acquire higher computational efficiency and accuracy than the traditional element-free Galerkin (EFG) method, which is based on the moving least-squares approximation and can not apply the displacement boundary conditions directly. To analyze the influences of node distribution, scale parameter of influence domain and the loading step on the numerical solutions of the IEFG method, three numerical examples are proposed. The IEFG method has almost the same high accuracy as the EFG method, and for some 2D elastic large deformation problems the IEFG method even has higher computational accuracy.
Based on the dimension splitting method (DSM) and the improved complex variable element-free Galerkin (ICVEFG) method, the hybrid complex variable element-free Galerkin (HCVEFG) method for three-dimensional (3D) elasticity is proposed in this paper. The equilibrium equation of a 3D elasticity is written as three sets of equations, which contains the equilibrium equations of any two directions. By introducing the DSM, a set of the equilibrium equation of a 3D elasticity problem is transformed into a series of two-dimensional (2D) problems which are solved by using the ICVEFG method. In splitting direction, the finite difference method is used for coupling those 2D discretized equations. Similarly, the discretized equation of another set of the equilibrium equation can be derived. Then by coupling those two discretized system equations, the numerical solutions of the HCVEFG method for 3D elasticity problems can be obtained. The error and the convergence are analyzed by numerical examples, and the results show that the HCVEFG method has greater computational speed.
In this paper, the interpolating element-free Galerkin (IEFG) method for solving the three-dimensional (3D) elastoplasticity problems is presented. By using the improved interpolating moving least-squares method to form the approximation function, and using the Galerkin weak form of 3D elastoplasticity problems to obtain the discretilized equations, we present the formulae of the IEFG method for the 3D elastoplasticity problems. The method can apply the displacement boundary conditions directly, which results in higher computational efficiency and accuracy. Numerical examples are given to discuss the influences of node distributions, scale parameters of influence domains and the loading steps on the computational accuracy of numerical solutions of the IEFG method. The numerical results show that, comparing with the element-free Galerkin method, the IEFG method for 3D elastoplasticity problems in this paper has higher computational efficiency and accuracy.
By combining the dimension splitting method with the improved element-free Galerkin (IEFG) method, the dimension splitting element-free Galerkin (DSEFG) method is proposed to solve 3D advection-diffusion problems. By using the dimension splitting method, a 3D advection-diffusion problem is transformed a series of 2D ones. The IEFG method is used for solving these related 2D advection-diffusion problems, and the finite difference method (FDM) is applied in the time space and dimension splitting direction. The discretized system equation is obtained depended on the Galerkin weak form of 2D problems, and the essential boundary conditions are imposed with penalty method, then the formulae of the DSEFG method for 3D advection-diffusion problem are obtained. Numerical examples are given to verify the efficiency of the DSEFG method. And the numerical results show that the DSEFG method can improve the computational efficiency and precision of the IEFG method greatly under the similar computational accuracy.
In this paper, the dimension splitting reproducing kernel particle method (DSRKPM) for three‐dimensional (3D) potential problems is presented. In the DSRKPM, a 3D potential problem can be transformed into a series of two‐dimensional (2D) ones in the dimension splitting direction. The reproducing kernel particle method (RKPM) is used to solve each 2D problem, the essential boundary conditions are imposed by penalty method, and the discretized equation is obtained from Galerkin weak form of potential problems. Finite difference method (FDM) is used in the dimension splitting direction. Then by combining a series of the equations of the RKPM for solving 2D problems, the final equation of the DSRKPM for 3D potential problems is obtained. Five example problems on regular or irregular domains are selected to show that the DSRKPM has higher computational efficiency than the RKPM and the improved element‐free Galerkin (IEFG) method for 3D potential problems.
In this study, the improved element-free Galerkin (IEFG) method is presented to the three-dimensional elastoplasticity problems. The improved least-squares (IMLS) approximation is used to obtain the shape function, the Galerkin weak form of three-dimensional elastoplasticity problems considering the nonlinear stress–strain relationship is used to form the discrete equation system, and penalty method is applied to the displacement boundary conditions, then the formula of the IEFG method for three-dimensional elastoplasticity problems are obtained. Some numerical examples are given to discuss the convergence of the IEFG method in this paper and the influences of the weight function, the scaling parameter, the penalty factor, the node distribution and the step number on the computational accuracy of the numerical solutions of the IEFG method. Comparing with the element-free Galerkin (EFG) method, the method in this study has a greater computational efficiency.
Salt concentration-sensitive hydrogel is a material that not only has potential in novel applications of mechanical engineering, but also is very common in nature. Several theoretical studies of this type of hydrogel have been conducted in the past several years. However, existing research on the theory of salt concentration-sensitive hydrogel usually assumes the concentration of the solution to be independent of the deformation, ignoring the fact that correlations exist between them. In this study, we present a theory of salt concentration-sensitive hydrogel aiming at predicting its deformation behavior by linking the concentration of the solution with the deformation based on Donnan equilibrium and large-deformation theory. To address the difficulties raised by the process of free-energy differentiation, a mathematical method to obtain stress without the aforementioned assumption is developed. Then, two typical cases, namely a free-swelling case and a constrained-swelling case, are studied to demonstrate the usefulness of this theory. We also compare the present theory with existing ones to emphasize the suitability of the present theory for the salt concentration-sensitive hydrogel. Furthermore, because of the novel mathematical process, the present theory can be realized with the finite-element method (FEM), whose results are then compared with the analytical results. Finally, to verify the robustness of the FEM formulation, we conduct an interesting bilayer swelling experiment, which shows a reasonable agreement with the FEM result. We also compare the theoretical prediction with the experimental data that are available in the literature.
In this study, based on a nonsingular weight function, the improved element-free Galerkin (IEFG) method is presented for solving elastoplastic large deformation problems. By using the improved interpolating moving least-squares (IMLS) method to form the approximation function, and using Galerkin weak form based on total Lagrange formulation of elastoplastic large deformation problems to form the discretilized equations, which is solved with the Newton-Raphson iteration method, we obtain the formulae of the IEFG method for elastoplastic large deformation problems. In numerical examples, the influences of the penalty factor, scale parameter of influence domain and weight functions on the computational accuracy are analyzed, and the numerical solutions show that the IEFG method for elastoplastic large deformation problems has higher computational efficiency and accuracy.
In this paper, combining the dimension splitting method with the improved complex variable element-free Galerkin method, a hybrid improved complex variable element-free Galerkin (H-ICVEFG) method is presented for three-dimensional advection-diffusion problems. Using the dimension splitting method, a three-dimensional advection-diffusion problem is transformed into a series of two-dimensional ones which can be solved with the improved complex variable element-free Galerkin (ICVEFG) method. In the ICVEFG method, the improved complex variable moving least-squares (ICVMLS) approximation is used to obtain the shape functions, and the penalty method is used to apply the essential boundary conditions. Finite difference method is used in the one-dimensional direction. And Galerkin weak form of three-dimensional advection-diffusion problems is used to obtain the final discretized equations. Then the H-ICVEFG method for three-dimensional advection-diffusion problems is presented. Numerical examples are provided to discuss the influences of the weight functions, the effects of the scale parameter, the penalty factor, the number of nodes, the step number and the time step on the numerical solutions. And the advantages of the H-ICVEFG method with higher computational accuracy and efficiency are shown.
By transforming a 3D problem into some related 2D problems, the dimension splitting element-free Galerkin (DSEFG) method is proposed to solve 3D transient heat conduction problems. The improved element-free Galerkin (IEFG) method is used for 2D transient heat conduction problems, and the finite difference method is applied in the splitting direction. The discretized system equation is obtained based on the Galerkin weak form of 2D problem; the essential boundary conditions are imposed with the penalty method; and the finite difference method is employed in the time domain. Four exemplary problems are chosen to verify the efficiency of the DSEFG method. The numerical solutions show that the efficiency and precision of the DSEFG method are greater than ones of the IEFG method for 3D problems.
The paper presents a hybrid element‐free Galerkin (HEFG) method for solving wave propagation problems. By introducing the dimension split method, the three‐dimensional wave propagation problems are transformed into a series of two‐dimensional ones in other one‐dimensional direction. The two‐dimensional problems are solved using the improved element‐free Galerkin (IEFG) method, and the finite difference method is used in one‐dimensional splitting direction and the time space. Then the formulae of HEFG method for three‐dimensional wave propagation problems are obtained. Numerical examples are selected to show the effectiveness and the advantage of HEFG method. The convergence and error analysis of HEFG method are discussed according to the numerical results under different splitting directions, weight functions, node distributions, scale parameters of the influence domain, penalty factor and time steps. The numerical results are given to show that the convergence and advantages of HEFG method over the improved element‐free Galerkin (IEFG) method. Comparing with IEFG method, HEFG method has greater computational precision and speed for three‐dimensional wave propagation problems.
In this paper, the interpolating moving least-squares (IMLS) method based on a nonsingular weight function is applied to obtain the approximation function. The penalty method is applied to impose the displacement boundary condition, and Galerkin weak form of elastic large deformation problems based on total Lagrange formulation is used to form the final equations which is solved with the Newton–Raphson iteration method, then the improved element-free Galerkin (IEFG) method based on a nonsingular weight function for elastic large deformation problems is presented. The IMLS method can overcome the disadvantage of singular weight functions in the traditional MLS method, then the IEFG method in this paper has high computational accuracy and efficiency, which are shown by numerical examples of elastic large deformation problems. And the influences of the weight functions, scale parameter of influence domain, step number and penalty factor on the numerical results are discussed.
In this paper, the interpolating moving least-squares (IMLS) method based on a nonsingular weight function is used to construct the approximation function, the weak form of the problem of inhomogeneous swelling of polymer gels is used to obtain the final discretized equations, and penalty method is applied to impose the displacement boundary condition, then an improved element-free Galerkin (IEFG) method for the problem of the inhomogeneous swelling of polymer gels is presented. Three selected examples of inhomogeneous swelling of polymer gels solved with the IEFG method are given in this paper. The accuracy of the numerical solutions of the IEFG method are discussed by using different weight functions, penalty factor, scale parameter of influence domain, node distribution and step number. Numerical results of the IEFG method for inhomogeneous swelling of polymer gels show that this method has great precision, and it can solve large deformation problems of polymer gels effectively.
Two-dimensional wave propagation problems in this paper are analyzed with improved complex variable element-free Galerkin (ICVEFG) method. Improved complex variable moving least-squares (ICVMLS) approximation is applied to construct the shape function, and modified Galerkin weak form of wave propagation problems is used for obtaining the final system equations, then all equations of ICVEFG method for wave propagation problems are obtained. The convergences of ICVEFG method about node distribution and time step are analyzed, and the influences of weight functions, scale parameter of the influence domain and penalty factor on the numerical solutions are discussed. Example problems show that ICVEFG method has higher computational speed with higher precision.
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.
In this paper, by combining the dimension splitting method and the improved complex variable element-free Galerkin method, the dimension splitting - improved complex variable element-free Galerkin (DS-ICVEFG) method is presented for three-dimensional transient heat conduction problems. Using the dimension splitting method, a three-dimensional transient heat conduction problem is translated into a series of two-dimensional ones which can be solved with the improved complex variable element-free Galerkin (ICVEFG) method. In the ICVEFG method for each two-dimensional problem, the improved complex variable moving least-square (ICVMLS) approximation is used to obtain the shape functions, and the penalty method is used to apply the essential boundary conditions. Finite difference method is used in the one-dimensional direction. And the Galerkin weak form of three-dimensional transient heat conduction problem is used to obtain the final discretized equations. Then the DS-ICVEFG method for three-dimensional transient heat conduction problems is presented. Three numerical examples are given to show that the new method has higher computational precision and efficiency.
Combining the dimension splitting method with the improved complex variable element-free Galerkin method, a hybrid improved complex variable element-free Galerkin (H-ICVEFG) method is presented for three-dimensional potential problems. Using the dimension splitting method, a three-dimensional potential problem is transformed into a series of two-dimensional ones which can be solved with the improved complex variable element-free Galerkin (ICVEFG) method. In the ICVEFG method for each two-dimensional problem, the improved complex variable moving least-square (ICVMLS) approximation is used to obtain the shape functions, and the penalty method is used to apply the essential boundary conditions. Finite difference method is used in the one-dimensional direction. And Galerkin weak form of three-dimensional potential problem is used to obtain the final discretized equations. Then the H-ICVEFG method for three-dimensional potential problems is presented. Four numerical examples are given to show that the new method has higher computational efficiency.
Shape memory polymers (SMPs) are a class of polymeric smart materials that have the capacity to return from a deformed state (impermanent shape) to their original state (permanent shape) by temperature stimulus. In this work, we propose a novel phase-transition-based viscoelastic model including the time factor for shape memory polymers (SMPs), which has a clearer physical significance. To describe the phase transition phenomenon of SMPs, our new model defines different constitutive structures for above and below transformation temperature separately. As the proposed viscoelastic model is based on multiplicative thermoviscoelasticity, it can not only be used for different types of SMP materials, but also can be used to treat large strain problems. To validate the model's availability and show the model's capability of reproducing the shape memory effect (SME), two testing examples are predicted with this new constitutive model. The prediction results of the simulation are in good agreement with the available experimental results.
The interpolating reproducing kernel particle method is a meshless method with discrete points interpolation character. Coupling this method with the minimum potential energy principle of space axisymmetrical problems of elastic mechanics, the interpolating smoothed particle method (ISPM) is formed. The ISPM, which is a meshless method with discrete points interpolation character, can refrain from quadric error of fitting calculation in stress post-processing by obtaining global domain continuous stress fields directly. This method not only has the advantage in directly exerting boundary conditions just like the finite element method, but is also a new numerical method which has greater computational efficiency and precision than it in solving space axisymmetrical problems of elastic mechanics. Numerical examples are given to show the validity of the new meshless method in the paper.
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 δ 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.
In this paper, the moving least-squares (MLS) approximation and the interpolating moving least-squares (IMLS) method proposed by Lancaster are discussed first. A new method for deriving the MLS approximation is presented, and the IMLS method is improved. Compared with the IMLS method proposed by Lancaster, the shape function of the improved IMLS method in this paper is simpler so that the new method has higher computing efficiency. Then combining the shape function of the improved IMLS method with Galerkin weak form of the potential problem, the interpolating element-free Galerkin (IEFG) method for the two- dimensional potential problem is presented, and the corresponding formulae are obtained. Compared with the conventional element-free Galerkin (EFG) method, the boundary conditions can be applied directly in the IEFG method, which makes the computing efficiency higher. For the purposes of demonstration, some selected numerical examples are solved using the IEFG method.
An analysis of moving least squares (m.l.s.) methods for smoothing and interpolating scattered data is presented. In particular, theorems are proved concerning the smoothness of interpolants and the description of m.l.s. processes as projection methods. Some properties of compositions of the m.l.s. projector, with projectors associated with finite-element schemes, are also considered. The analysis is accompanied by examples of univariate and bivariate problems.
An element-free Galerkin method which is applicable to arbitrary shapes but requires only nodal data is applied to elasticity and heat conduction problems. In this method, moving least-squares interpolants are used to construct the trial and test functions for the variational principle (weak form); the dependent variable and its gradient are continuous in the entire domain. In contrast to an earlier formulation by Nayroles and coworkers, certain key differences are introduced in the implementation to increase its accuracy. The numerical examples in this paper show that with these modifications, the method does not exhibit any volumetric locking, the rate of convergence can exceed that of finite elements significantly and a high resolution of localized steep gradients can be achieved. The moving least-squares interpolants and the choices of the weight function are also discussed in this paper.