Virtual Source Function Research Articles

A simple, effective and high-precision boundary meshfree method for solving 2D anisotropic heat conduction problems with complex boundaries

A simple, effective and high-precision boundary meshfree method called virtual boundary meshfree Galerkin method (VBMGM) is used to tackle 2D anisotropic heat conduction problems with complex boundaries. Temperature and heat flux are expressed by virtual boundary element method. The virtual source function is constructed through the utilization of radial basis function interpolation. Calculation model diagram and discrete model diagram of real boundaries, and schematic diagram of VBMGM are demonstrated. Using Galekin method and considering boundary conditions, the integral equation and the discrete formula of VBMGM are given in detail. The benefits of the Galerkin, meshfree, and boundary element methods are all presented in VBMGM. Seven numerical examples of general anisotropic heat conduction problems (including three numerical examples with complex boundaries and four numerical examples with mixed boundary conditions) are computed and contrasted with precise solutions and different numerical methods. The computation time of each example is given. The number of degrees of freedom used in many examples is half or less than that of the numerical method being compared. The suggested method has been demonstrated to be effective and high-precision for solving the 2D anisotropic heat conduction problems with complex boundaries.

A novel boundary-type element-free method for 3D thermal analysis in inhomogeneous media with variable thermal source

Solutions of 3D thermal analysis in inhomogeneous media with variable thermal source are divided into homogeneous and particular solutions by a novel boundary-type element-free method, namely virtual boundary element-free Galerkin method (VBEFGM). The homogeneous solution can be obtained by the virtual boundary element-free method (VBEFM). The virtual source function of the homogeneous solution and the unknown coefficient of the particular solution are constructed by the radial basis function interpolation (RBFI). And VBEFGM for 3D thermal analysis in inhomogeneous media with variable thermal source is real meshfree and does not require the background integral element. Its equation is obtained the Galerkin method of the weighted residual methods. The demonstration of ANSYS Parametric Design Language (APDL) can greatly facilitate the programming for the extraction of the coordinate information and the outer normal cosines on nodes. The calculation scheme and the specific weighted coefficients are deducted in detail for 3D thermal analysis in inhomogeneous media with variable thermal source by VBEFGM. The numerical calculational steps and flow chart are given. Three numerical examples are computed. The results are compared with other numerical methods and the exact solutions. The stability and the accuracy of the proposed method is proved by VBEFGM for 3D thermal analysis in inhomogeneous media with variable thermal source.

A novel boundary-type meshfree method for solving three-dimensional single-domain steady-state heat conduction problem

The virtual boundary meshfree Galerkin method (VBMGM) is further developed to compute three-dimensional single-domain steady-state heat conduction problem. Using VBMGM, three-dimensional problem is reduced into two-dimensional boundary, whose element information can be obtained by the meshing software, namely ANSYS software. It greatly facilitates the programing implementation of the proposed method that ANSYS Parametric Design Language (APDL) is demonstrated for the element information acquisition of the curved surface. The virtual source function on the virtual boundary element method (VBEM) is interpolated by the radial basis function interpolation (RBFI) of the meshfree method. The calculation equation of three-dimensional single-domain steady-state heat conduction problem by VBMGM is constructed by the Galerkin method of the weighted residual method. Therefore, it has the advantages of the VBEM, the meshfree method and the Galerkin method that the proposed method is employed to computed three-dimensional single-domain steady-state heat conduction problem. It is beneficial to further popularize the proposed method to solve other three-dimensional heat conduction problems that the detailed numerical discrete formula of the proposed method for three-dimensional single-domain steady-state heat conduction problem is obtained. The numerical results of two numerical examples are computed and compared with the other numerical method and the extract solutions. The accuracy and stability of VBMGM for three-dimensional single-domain steady-state heat conduction problem are verified.

Calculating the multi-domain transient heat conduction with heat source problem by virtual boundary meshfree Galerkin method

The virtual boundary meshfree Galerkin method (VBMGM) is given to calculate the multi-domain transient heat conduction with heat source problem. The radial basis function interpolation is used to obtain the virtual source function on the virtual boundary, the heat source, the particular solutions of the temperature, and the heat flux in the computational domain. The formula of the proposed method is gotten by the Galerkin method and the control equations of the multi-domain transient heat conduction with heat source. Therefore, there are the advantages of the Galerkin method, the boundary element method, and the meshfree method for the multi-domain transient heat conduction with heat source problem by VBMGM. The numerical meaning of the weighted values are definite for the multi-domain transient heat conduction with heat source problem through VBMGM, such as the partial derivatives of the temperature, the heat flux, the heat source, and the continuity conditions on the public boundary at time. The detailed calculation scheme of VBMGM is deducted for the multi-domain transient heat conduction with heat source problem. Two numerical examples, four subdomains with the time-dependent heat source problem and two subdomains with the constant heat source and mixed boundary condition problem, are computed. The accuracy and the stability of VBMGM for the multi-domain transient heat conduction with heat source problem are verified.

Solving the single-domain transient heat conduction with heat source problem by virtual boundary meshfree Galerkin method

The virtual boundary meshfree Galerkin method (VBMGM) is demonstrated to solve the single-domain transient heat conduction with heat source problem. Both the heat source in the computational domain and the virtual source function on the virtual boundary are interpolated by using the radial basis function interpolation. The equation of the proposed method is formed by the Galerkin method. Consequently, VBMGM have the advantages of the Galerkin method, the boundary element method and the meshfree method for the single-domain transient heat conduction with heat source problem. And the numerical definitions of the weighted values in VBMGM are definite, such as the partial derivatives of the temperature, the heat flux and the heat source expression at t+Δt time. The detailed equation of VBMGM is deducted for the single-domain transient heat conduction with heat source problem. Two numerical examples, the square domain with constant heat source and the square domain with variable boundary condition and variable heat source, are presented. The accuracy and the stability of VBMGM for the single-domain transient heat conduction with heat source problem are proved.

A new boundary meshfree method for calculating the multi-domain constant coefficient heat conduction with a heat source problem

ABSTRACTA new high-precision boundary meshfree method, namely virtual boundary meshfree Galerkin method (VBMGM), for calculating the multi-domain constant coefficient heat conduction with a heat source problem is given. In the paper, the radial basis function interpolation is used to solve the virtual source function of virtual boundary and the heat source within each subdomain. Simultaneously, the equation of VBMGM for multi-domain constant coefficient heat conduction with a heat source problem is obtained by the Galerkin method. Therefore, the proposed method has common advantages of the boundary element method, meshfree method, and Galerkin method. Coefficient matrix of this specific expression is symmetrical and the specific expression of VBMGM for the multi-domain constant coefficient heat conduction with a heat source problem is given. Two numerical examples are given. The numerical results are also compared with other numerical methods. The accuracy and feasibility of the method for the multi-domain constant coefficient heat conduction with a heat source problem are proved.

Solving the multi-domain variable coefficient heat conduction problem with heat source by virtual boundary meshfree Galerkin method

Virtual boundary meshfree Galerkin method (VBMGM) is developed to solve the multi-domain variable coefficient heat conduction problem with heat source. Both the virtual source function on the virtual boundary and heat source within each subdomain are approximate by the radial basis function interpolation. The equation of the proposed technique is derived by one of the weighted residual methods, namely the Galerkin method, for multi-domain variable coefficient heat conduction problem with heat source. Therefore, VBMGM simultaneously possesses the merits of Galerkin method, boundary element method and meshfree method. And the numerical meanings of the weighted values are definite, such as the partial derivative heat source expressions, the temperature, the heat flux and the continuity conditions, in VBMGM for multi-domain variable coefficient heat conduction problem with heat source. The specific and detailed equation of VBMGM is obtained for multi-domain variable coefficient heat conduction problem with heat source. The accuracy and the stability of the proposed technique are validated by two numerical examples for multi-domain variable coefficient heat conduction problem with heat source.

A new boundary-type mesh-free method for solving the multi-domain heat conduction problem

ABSTRACTIn this article, a new high-precision boundary-type mesh-free method, called the virtual boundary mesh-free Galerkin method (VBMGM), is given for analyzing the heat conduction problem consisting of multiple media. A control equation of the proposed method is established by the the Galerkin method of the weighted residual methods. The virtual source function is approached through the radial basis function interpolation in the mesh-free method. Thus, the main feature of the proposed technique has the advantages of the mesh-free method, the boundary-element method, and the Galerkin method, a symmetrical coefficient matrix. The numerical meanings of the weighted values in the VBMGM equation, such as the partial derivatives the temperature, the heat flux, the temperature connection condition, and the heat flux connection condition, are clear. Numerical examples of a thick-walled cylinder consisting of two media and a two-layered rectangular plate are provided. The feasibility and the accuracy of the proposed technique are proved.

Calculating the single-domain heat conduction with heat source problem by virtual boundary meshfree Galerkin method

The virtual boundary meshfree Galerkin method (VBMGM) is presented to calculate the single-domain heat conduction with heat source problem. The virtual source function of the virtual boundary and the heat source in the computational domain both are interpolated by the radial basis function interpolation. The equation of the proposed method is obtained by the Galerkin method of the weighted residual methods. Consequently, VBMGM possesses the advantages of Galerkin method, boundary element method and meshfree method. And the numerical meanings of the weighted values in VBMGM, such as the partial derivative the temperature, the heat flux and the heat source expression, are definite. The detailed and specific equation of VBMGM is deducted. The numerical examples, namely the temperatures of the rectangular domain and the square domain, are computed. And the accuracy and the stability of the proposed method for the single-domain heat conduction with heat source problem are validated.

Virtual Boundary Meshless with Trefftz Method for the Steady-State Heat Conduction Crack Problem

This article presents a virtual boundary meshless with Trefftz method for calculation of the two-dimensional steady-state heat conduction crack problem, which is based on the idea of multidomain combination. The virtual source function is constructed by the radial basis functions approximation in the subdomain not containing the crack. The temperature or heat flux is calculated by crack special solution of Trefftz function in the crack subdomain. Thus, the proposed method has the advantages of both the boundary-type meshless method and the Trefftz method. No nodes or elements are distributed on the crack, because the boundary condition of the crack is automatically satisfied by employing the crack special solution of the Trefftz function. The heat stress intensity factor is simply and directly calculated.

A New High-Precision Boundary-Type Meshless Method for Calculation of Multidomain Steady-State Heat Conduction Problems

This article presents the idea for calculating 2-D steady-state heat conduction problems with multidomain combination by employing the virtual boundary meshless least-square method. Being different from the conventional virtual boundary-element method (VBEM), this method incorporates the point interpolation method (PIM) with the compactly supported radial basis function (CSRBF) to approximately construct the virtual source function of the VBEM. Thus, the proposed method has the advantages of both the boundary-type meshless method and the virtual boundary element method. Since the configuration of the virtual boundary requires a certain preparation, the integration along the virtual boundary can be carried out over the smooth simple curve that can be structured beforehand (for 2-D problems) to reduce the complexity and difficulty of calculus without loss of accuracy, while the “vertex question” existing in the BEM can be avoided. Numerical examples show that the proposed method is more precise than several other numerical methods while selecting fewer degrees of freedom. In addition, its numerical stability is also verified by computing several cases.

Solving multi-domain 2D heat conduction problems by the least squares collocation method with RBF interpolation on virtual boundary

Abstract This paper is concerned with the virtual boundary meshless least squares collocation method of two dimensional steady-state heat conduction problems in multi-domain. In the proposed method, each region with different heat transfer properties is considered as a piecewise homogeneous in a heterogeneous system. Being different from the conventional virtual boundary element method (VBEM), this method incorporates the point interpolation method (PIM) with the compactly supported radial basis function (CSRBF) to approximately construct the virtual source function of the VBEM. Consequently, this method has the advantages of boundary-type meshless methods. In addition, it does not have to deal with singular integral and has the symmetric coefficient matrix, and the pre-processing operation is also very simple. Since the mathematical model for steady-state heat conduction can be represented by a generalized Poisson equation, this problem is solved by finding an approximate particular solution to the Poisson equation and making use of the virtual boundary meshless least squares collocation method to solve the resulting Laplace equation. This method can be used to analyze multi-domain composite structures with each subdomain having different heat transfer properties or materials or geometries. Since the configuration of virtual boundary has a certain preparability, the integration along the virtual boundary can be carried out over the smooth simple curve that can be structured beforehand (for 2D problems) to reduce the complicity and difficulty of calculus without loss of accuracy, while “Vertex Question” existing in BEM can be avoided. In the end, several numerical examples are analyzed using the proposed method and some other commonly used methods for verification and comparison purposes.

Virtual boundary meshless least square collocation method for calculation of 2D multi-domain elastic problems

A virtual boundary meshless least square collocation method is developed for calculation of two-dimensional multi-domain elastic problems in this paper. This method is different from the conventional virtual boundary element method (VBEM) since it incorporates the point interpolation method (PIM) with the compactly supported radial basis function (CSRBF) to approximately construct the virtual source function of the VBEM. Consequently, this method has the advantages of boundary-type meshless methods. In addition, it does not have to deal with singular integral and has the symmetric coefficient matrix, and the pre-processing operation is also very simple. This method can be used to analyze multi-domain composite structures with each subdomain having different materials or geometries. Since the configuration of virtual boundary has a certain preparability, the integration along the virtual boundary can be carried out over the smooth simple curve that can be structured beforehand (for 2D problems) to reduce the complicity and difficulty of calculus without loss of accuracy, while “Vertex Question” existing in BEM can be avoided. In the end, several numerical examples are analyzed using the proposed method and some other commonly used methods for verification and comparison purposes. The results show that this method leads to faster convergence and higher accuracy in comparison with the other methods considered in this study.