A New Truly Meshless Method for Heat Conduction in Solid Structures

Abstract

Abstract-In this study a new meshless method is presented for the analysis of heat transfer in heterogeneous solid structures. The presented meshless method is based on the integral form of energy equation for the sub-particles in the domain of the material. A micromechanical model based on the presented meshless method is presented for analysis of heat transfer, temperature distribution and steady-state effective thermal conductivities of fiber-matrix type of composite materials. Because the domain integration is eliminated in the presented meshless formulation, the computational efforts in presented method are decreased substantially. A small area of the composite system called the representative volume element (RVE) is considered as the solution domain. The fully bonded fiber-matrix interface is considered and contact thermal resistant is neglected in the fiber-matrix interface and so the continuity of temperature and reciprocity of heat flux is satisfied in the fiber-matrix interface. A direct interpolation method is employed for enforcement the appropriate boundary conditions to the RVE. Numerical results are presented for temperature distribution, heat flux and thermal conductivity. Numerical results show that presented meshless method is simple, effective, accurate and less costly method in micromechanical modeling of heat conduction in heterogeneous materials. Keywords-Truly meshless method; Heat transfer, Micromechanical model; Thermal conductivity, Fibrous composites; 1-INTRODUCTION During the past decade, the idea of using meshless methods for solution of boundary value problems has received much attention and significant progress were achieved on meshless methods. In the meshless methods, no predefined mesh of elements needed between the nodes for the construction of trial or test function. Therefore, the one of the main objectives of the meshless methods is to eliminate or alleviate various difficulties related to elements such as meshing and remeshing of domain. Various meshless methods In this paper, a new truly meshless method based on the integral form of energy balance equation is developed to study the heat conduction problem in an anisotropic and heterogeneous medium. A direct method is introduced for treatment of material discontinuity in the present meshless method. The continuity of temperature and reciprocity of heat flux in the interface of material discontinuity can be enforced by this method. As a practical problem a micromechanical model is developed to study the heat conduction in the UD composite material. This micromechanical model is introduced to investigate the temperature distribution, heat flux and effective transverse thermal conductivity of UD fiber-matrix type of composites. It is assumed the fibers are circular in cross section and is packed in square array in the matrix. The governing equations of the problem over the representative volume element (RVE) are solved employing the presented meshless method. Numerical results are presented for temperature distribution, temperature gradient, heat flux and transverse thermal conductivity of UD composite materials. The results show that the present meshless method is efficient and less costly method in thermal analysis of heterogeneous material. 2-SOLUION PROCEDURE In the actual unidirectional fibrous composites the fibers are distributed randomly in cross-section of the composite. Usually micromechanical modeling of composite with the real fiber arrangement is more expensive. So in most of micromechanical models a periodic arrangement of fibers in the composite is assumed, and for which the composite can be represented by Representative Volume Element (RVE). Usually the smallest repeating area of the cross-section of periodic composite is chosen as the RVE. In this study as shown in 2-1 Meshless Formulation Consider an arbitrary local particle (sub-domain) named. in which ρ is the mass density, c is the specific heat capacity of the medium, n is the outward normal vector of the boundary ∂Ω s I and q i is the heat flux in the x i direction. g shows the rate of energy generation per unit volume in the domain. For a heat conduction problem the Dirichlet and Neumann boundary conditions can be written as; in which T is the temperature, T is the prescribed temperature on the Γ Τ and q is the prescribed heat flux on the Γ q . The boundary of the local sub-domain in general consists of three parts, According to the Fourier's law of heat conduction the heat flux q i can be obtained by the flowing empirical low; where k ik is the coefficient of thermal conduction for a general anisotropic material and index follow a comma i.e. , k indicated partial derivative respect to x k . Substitution of In this study a meshless method is presented for discretization of (5). In the meshless methods a kind of interpolation schemes is needed for approximation of the field variable over the randomly located nodes within the domain. There are a number of local interpolation schemes, such as moving least-square (MLS) approximation, partition of unity method (PUM), Shepard function, reproducing kernel particle Copyright © 2010 by ASME Downloaded From: https://proceedings.asmedigitalcollection.asme.org on 06/28/2019 Terms of Use: http://www.asme.org/about-asme/terms-of-use 3 method (RKPM), etc.One of the well known methods for this purpose is the moving least squares (MLS) technique (Atluri and Zhu, 1998) which is briefly described in this section. 2-2 Moving leas Square Approximation The first step in the solution procedure is to approximate the field variable u(x) over a number of randomly located nodes within the domain. One of the well-known methods for this purpose is the moving least squares (MLS) technique, which is described in [30]. In this method, the nodal interpolation form of u(x) may be expressed as; 2-3 Numerical Discreizaion To obtain the discrete form of (5), the MLS approximation in (9) is used to approximate the test function T from the fictitious nodal values T . Substitution of where φ J ,k is the partial derivative of φ J (x) respect to x k and can be found in detail in . Equation (13) may be written in the form of discretized system of linear equations as; ˆˆ, 1,..., For the nodes that located on the Dirichlet boundary, the temperature boundary condition is enforced as; where I T is the prescribed value of temperature on node I that located on essential boundary conditions. Therefore, for the nodes that located on Dirichlet boundary, the stiffness anf force matrix will be; ( ) For the nodes that are located on Neumann boundary the flux conditions can be enforced as So the stiffness matrix for these nodes will be , ( ) In this method K IJ is bonded and asymmetric. It should be mentioned that by ignoring the heat generation term, for steady state problems the domain integration is totally eliminated from formulation. 3-MATERIAL DISCONINUIY In meshless methods there is no mesh of elements and the material interface cannot be defined based of elements. In this paper the inhomogeneous material in the RVE is considered as two homogeneous bodies. The influence domain of each node is confined within the domain of material of that node. In order to treat material discontinuity at the fiber and matrix interface, two sets of nodes are assigned on the interface at the same location with different material properties. One set is dedicated to the fiber known as I f while the other set is related to the matrix denoted by I m . In this study, the fully bounded fiber-matrix interface is assumed. For the nodes that lie on the interface, the continuity of temperature and reciprocity of heat flux must be satisfied as follows