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The particular and homogeneous solutions of two-dimension (2D) thermal-elastic problems are expressed by using the virtual boundary element method (VBEM) and the superposition principle. Accordingly, two matrix equations are needed and formed for particular and homogeneous solutions by the novel nonsingular boundary element method with the radial basis function interpolation (RBFI), namely virtual boundary meshfree Galerkin method (VBMGM). Considering the boundary conditions of heat conduction and combining the Galerkin method, virtual source functions of particular solutions can be solved by first using VBMGM. Applying boundary conditions of transformation, virtual source functions of homogeneous solutions can be obtained by second employing VBMGM. The coefficients of two matrix equations are symmetrical. Formed equations are nonsingular and have merits of virtual boundary element method, meshfree method, and Galerkin method. The detailed expressions, calculation steps, and simplified flow chart are given in detail, that is convenient for other scholars to study other more complex thermo-elastic problems through the method of this paper. Three examples are calculated. Comparing their calculation results with other methods, the stability and the precision of VBMGM for 2D thermal-elastic problems are validated. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

A new boundary-type meshfree collection method, namely the virtual boundary element-free Galerkin method (VBEFGM), is demonstrated for three-dimension (3-D) multi-domain steady-state heat conduction problems. The virtual source functions of the virtual nodes are constructed by the radial basis function interpolation. And the virtual and real boundaries are discreted into the virtual and real nodes. 3-D multi-domain steady-state heat conduction problems are solved without the background integral element by VBEFGM. Using the Galerkin method to obtain the formula of VBEFGM, the proposed method has the advantages of the virtual boundary element-free method, the meshfree collection method and the Galerkin method. It is easily programed and extended to solve other more complicated 3-D heat conduction problems, since the detailed calculation scheme is deduced and shown. Two numerical examples of the boundary conditions varying with the space are computed. The computational results are compared with other numerical methods. The accuracy and stability of the proposed method are validated for 3-D multi-domain steady-state heat conduction problems. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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 + Δ t 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. [ABSTRACT FROM AUTHOR]

A 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. [ABSTRACT FROM PUBLISHER]

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. [ABSTRACT FROM AUTHOR]

In 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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

• A novel nonlinear energy sink with internal oscillator (NES-IO) is applied to the low frequency vibration control. • The vibration reduction mechanism of the NES-IO is analyzed. • The correctness of the analytical results is verified by numerical results. • The NES-IO has obvious advantages under large energy vibration. • Parameter selection of oscillators is very important for optimizing the absorber. This paper concerns an investigation into the excessive vibration control of laminated composite plates using a nonlinear energy sink with internal oscillator (NES-IO). The proposed NES-IO can suppress vibration more effectively than traditional NES measured by both energy dissipation and frequency responses. Based on the first-order shear deformation theory and the Galerkin method, the dynamic model of the nonlinear laminated composite plate with the NES-IO is established. Then, the mechanism of the NES-IO is analyzed, and the internal oscillator improves the vibration reduction performance of the absorber, especially with large amplitude excitations. Through comparing the NES-IO with the traditional NES, it is found that the effective vibration reduction range of the NES-IO under shock excitation is wider. Moreover, the slow flow equations of the system are derived by the complexification-averaging method, and the result is verified by the Runge-Kutta method. It is also found that the NES-IO has a great advantage in suppressing the occurrence of higher branch responses through the frequency-responses analysis. Furthermore, the influences of the parameters of the grounded and internal oscillators on the vibration suppression are also discussed to optimize the NES-IO structure. [ABSTRACT FROM AUTHOR]