Analytical and Numerical Analysis of Functionally Graded Heat Conduction Based on Dirichlet Boundary Conditions

An analytical and numerical solution for the one dimensional of heat conduction in a slab exposed to different temperature at both ends is presented. The distribution of heat throughout the transient direction obeys to functionally graded (FG) temperature based on Dirichlet boundary conditions. The variation of functionally graded temperature can be described by any form of continuous function. In this case, where the external heat fluxes are not directly definite based on the Dirichlet or mixed boundary conditions, the fluxes that concluded over the slab faces are free to vary until the equilibrium condition is reached. By numerically solving the resulting heat-conduction equation, the distribution of temperature which vary with time through the slab is obtained. The obtained analytical results are presented graphically and the influence of the gradient variation of the temperature on shape formed with changed time is investigated. *Corresponding author: Essa S, Department of Civil Engineering, Erbil Technical Engineering College, Erbil, Iraq, Tel: 9647504823149; E-mail: saad_khalis@epu.edu.krd Received October 19, 2016; Accepted October 31, 2016; Published November 11, 2016 Citation: Essa S (2016) Analytical and Numerical Analysis of Functionally Graded Heat Conduction Based on Dirichlet Boundary Conditions. J Material Sci Eng 5: 293. doi:10.4172/2169-0022.1000293 Copyright: © 2016 Essa S. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.


Introduction
Functionally graded materials (FGMs) are defined as the perfect materials in mechanical, thermal and corrosive resistant properties. Different from fiber-matrix laminated composites, FGMs do not have the problems of de-bonding resulting from large inter-laminar and thermal stresses. The impression of FGMs was firstly introduced by Japanese researchers in the mid-1980s, as ultra-high temperature resistant materials for various engineering fields for instance aircraft, space vehicles, and nuclear reactors. FGMs consist of two or more materials which microscopically inhomogeneous and spatial composite materials such as a pair of ceramic-metal. The mechanical properties of the material structure changes gradually throughout the thickness with varying continuously and smoothly from top to the bottom surface. Noda [1] showed many topics range from thermoelastic to thermoinelastic problems. He suggested that temperature dependent properties of the material should be taken into account in order to achieve more accurate analysis.
For a historical era, FGMs studying focused on the analyses of thermal stress in the ceramic coatings, static deformation, and forced vibration. Noda and Jin [2,3] presented a steady thermal stress for a crack elastic solid based on nonhomogeneous infinite, and concluded that effects of the thermal stress intensity for cracks in FGMs. Cho and Oden [4] used a Crank-Nicolson and Galerkin scheme to investigate a parametric study of thermal stress characteristics. Reddy [5] proposed a theoretical formulation and finite element models for the analysis of functionally graded plates (FGPs). Praveen and Reddy [6] studied responses of FGPs based on static and dynamic thermoelastic responses and concluded that, the differences of pure ceramic or metal plates depend on responses of FGPs. Yang and Shen [7] presented a free and forced vibration analyses of functionally graded plates subjected to impulsive lateral loads under thermal environments. Dirichlet boundary conditions (DBCs) generally hold two forms: homogeneous Dirichlet boundary conditions (HDBCs) and inhomogeneous Dirichlet boundary conditions (IDBCs). The former can be considered as a special case of the latter with zero imposed value. Zhang and Zhao [8] developed a weighted finite cell method (FCM) with high computing accuracy which extended to define boundary value function so that the inhomogeneous Dirichlet boundary conditions (IDBCs) are imposed exactly.

Theoretical Formulation
There are many models for expressing the variation of material properties in FGMs. The most commonly used of these models is the power law distribution. In this study, the new expression of heat transfer through in slab is assumed based on two parameters as: Where, m and k are the parameters which are used to define the variation.
Here, φ 0 =ɸ(x) is initial value of the variation at x=a Eq. (1) is a nonlinear function and its variation (shape) is controlled by using two parameters. One may observe that, the adjustment of the parameters m and k is not easy for describing the desired variations. Figure 1 shows the variation of temperature of metal in the transverse direction of slab.

Exact Solution of the Heat Equation
Consider a one-dimensional diffusion equation which is a partial differential equation for the temperature T(x,t): Where, c(x) is the specific heat of the material, κ is the constant of proportionality (thermal conductivity) of the material and S(x,t) represents a given source of heat energy per unit volume. The equation simplifies when κ and c are independent of position: Where, χ is the material thermal diffusivity, The initial temperature in the slab heat equation is a first-order PDE in time: The second order equation in space requires two boundary conditions to satisfy the solution. By putting the faces in a suitable thermal contact at a specified temperature T 1 (t) and T 2 (t) so the Dirichlet boundary conditions are achieved as: The equilibrium solution ɸ(x) that satisfies these boundary conditions, as well as the time-independent heat equation: Using the direct integration (analytically), equation goes to general solution: The constants C 1 and C 2 are chosen to match the boundary conditions. For a temperature balance to exist, there can be no net heat energy infused into the section by the source: , else the temperature should increase or decrease as the overall energy content in the slab varies with time. The relation between the equilibrium temperature and the deviation from equilibrium ∆T(x,t) is: By Substitution Eq. (10) into the heat equation Eq. (3) and application of Eq. (8) indicates that ∆T(x,t) satisfies the homogeneous heat equation: With homogeneous boundary conditions, and initial condition: We will only consider the case where κ and C are constants, so that Eq. (11) becomes the diffusion equation with no heat source, We look for a solution of the form Which can be separated into two ODEs with respect to independent x and t. By putting of each separated equations of Eq. (14) are equal to a constant λ, After applying Dirichlet boundary conditions, solution of Eq. (16) for the eigenmodes is, And; The solution of Eq. (15) provides the time-dependent amplitude for each eigenmode as, Therefore, the general solution of the heat equation away from equilibrium is, The constant D is absorbed into the Fourier coefficient A n which can be found by matching the initial condition, Eq. (12), Equation (21) is a Fourier sine series and the coefficient A n is determined as, Dirichlet boundary conditions in a uniform slab are applied to get equations (20) and (22) which is the solution for the deviation from equilibrium. In addition, Eq. (22) is solved numerically by using trapezoidal method which can be seen in the numerical example. substituting the exact solution T in the discrete system. This means the numeric solution T ∆x converges to exact solution in a given norm if

The Crank-Nicolson Method
The consistency between the solutions of the continuous and discrete problems does not guarantee also have a tendency to zero. On the other hand, the difference between the differential and discrete operators on a smooth sufficient function tends to zero.

Numerical Example
In this section, the distribution of temperature throughout the transverse direction of slab is tested based on assumed temperature gradient. The properties of slab are selected as the parameters in Eq (1) Table 1.

Conclusions
In this paper, simple method to solve the transient response problem of a functionally graded temperature variation in the 1-D Space derivative approximation which is used for is just an average of approximations in points ( , ) i j x t and 1 ( , ) i j x t + , Introducing 2 t x α = χ ∆ ∆ one can rewrite Eq. (24) as, The terms which appear in the right-hand side of Eq. (25) are known. Hence, Eq. (25) form a tridiagonal linear system (AT=b) which can be solve simultaneously to find the temperature at every node at any point in time.
The complex stability analysis will be procedure in some simple steps. The equation (24) can be written as, ( )  T  T  T  T  T  T  T T  t x The equation (26) has a consistency,  T  T  T  T  T  T  T T To make a stability analysis, the scheme will be written into two form stages:  T  T T  T  T  t  x   T  T  T  T T t x + + − Where,   slab has been proposed. The assumed temperature is embedded into the diffusion equation with no source of heat then exact solution is obtained. In addition, the integral part of A n coefficient in Eq. (22) is solved numerically using trapezoidal method and the difference is exhibited in Table 1. In this table, the differences of temperature between the two methods are too small. Crank-Nicolson (CN) method is applied to solve the diffusion equation based on the given variation ɸ(x). The assumption of variation is selected for two variables k and m. The temperature at any point goes to take the assumed variation with time. By fixing the numerical method at the shape of variation function, the times that the exact solution is needed to get the numerical solution are computed. Using these methods, the time required to reach the assumed model can be calculated at any position with the given Dirichlet boundary condition.