Analysis of 2D heat conduction in nonlinear functionally graded materials using a local semi-analytical meshless method

This paper proposes a local semi-analytical meshless method for simulating heat conduction in nonlinear functionally graded materials. The governing equation of heat conduction problem in nonlinear functionally graded material is first transformed to an anisotropic modified Helmholtz equation by using the Kirchhoff transformation. Then, the local knot method (LKM) is employed to approximate the solution of the transformed equation. After that, the solution of the original nonlinear equation can be obtained by the inverse Kirchhoff transformation. The LKM is a recently proposed meshless approach. As a local semi-analytical meshless approach, it uses the non-singular general solution as the basis function and has the merits of simplicity, high accuracy, and easy-to-program. Compared with the traditional boundary knot method, the present scheme avoids an ill-conditioned system of equations, and is more suitable for large-scale simulations associated with complicated structures. Three benchmark numerical examples are provided to confirm the accuracy and validity of the proposed approach.


Introduction
The functionally graded material is a composite with continuously varying microstructure and mechanical properties and is often applied to reduce the thermal stress or residual stress in dissimilar material joints. With this regard, it is very important to analyze the heat transfer characteristics of functionally graded materials. Generally, it is not easy work to derive an exact expression of temperature distribution in nonlinear functionally graded materials. Therefore, the numerical simulation has become one of the main methods for addressing such problems.
Over the past few decades, various numerical methods have been developed for the analysis of heat transfer in functionally graded materials. The main methods include the finite element method (FEM) [1][2][3][4], the boundary element method (BEM) [5][6][7], and the finite difference method (FDM) [8][9][10]. Among them, the FEM occupies dominant position due to its mature theory system and good stability. These traditional mesh-type methods have their advantages, but at the same time they have many deficiencies, such as the complexity of preprocessing, especially for complex structures.
The LBKM is a domain-type meshfree approach using the non-singular general solution as the basis function. In fact, it can be regarded as a localized version of traditional boundary-type BKM. When this method was first proposed, the artificial boundaries and boundary nodes on it are required for every local subdomain. Later, the LBKM is modified into the local knot method (LKM) without designing the artificial boundaries. Like the local radial basis function collocation method and the element-free Galerkin method [44][45][46], the LKM is a typical localized domain-type meshless method. Unlike them, the LKM is a semi-analytical and strong-form approach, in which the nonsingular general solution is taken as the basis function. Compared with the local radial basis function collocation method, the LKM avoids the issue of selecting shape parameters, and has a higher numerical accuracy. Recently, the LKM has been successfully applied to solving convection-diffusion-reaction equations [47] and acoustic problems [48], and has demonstrated the advantages of truly meshless, high-accuracy, and large-scale calculation.
In view of its above merits, this study makes a first attempt to employ the LKM in conjunction with the Kirchhoff transformation to solve the heat conduction problems of two-dimensional nonlinear functionally graded materials. In the calculation, the Kirchhoff transformation is applied to transform the nonlinear heat conduction equation into a linear equation. The transformed equation is a modified Helmholtz equation whose non-singular general solution is available. The LKM can be directly used to approximate the solution of the modified Helmholtz equation. The solution of the original problem can be acquired by using the inverse Kirchhoff transformation.
The general outline of this paper is as follows. The heat conduction problem of functionally graded nonlinear material is introduced in Section 2. Section 3 describes the procedures of solving the anisotropic modified Helmholtz equation via the LKM. In Section 4, three benchmark numerical examples are presented to demonstrate the feasibility and accuracy of the proposed methodology. Finally, some conclusions are drawn in Section 5.

Problem statement
Consider the steady-state heat conduction problem in a two-dimensional nonlinear functionally graded material with the domain () ( ( , ) ) 0, , (1) with the following boundary conditions: , , (4) where () In this study, we focus on the exponentially functionally graded material whose heat conductivity matrix could be represented as follows  (5) where ( ) 0 aT  , 1  and 2  are the material parameters, is a symmetric positive definite matrix. By using the Kirchhoff transformation Equations (1)-(4) can be converted to , Eq (7) can be simplified to the following formula with 11 22 12 21 det( ) 0  , an isotropic modified Helmholtz equation will be obtained, namely, The non-singular general solution of Eq (13) is available [49], and thus the non-singular general solution of Eq (11) can be derived by using the inverse transformation of Eq (12), where 22 12 ( , ) xx

Local knot method
This section will introduce the implementation of the LKM in solving Eq (7), with the help of the non-singular general solution (15). According to the basic theory of the LKM, ( ) 1 are discretized inside the domain Ω and along its boundary Γ , here  Figure 1 shows the schematic diagram of the LKM, which indicates the distribution of the central node (0) x and its m supporting points ( () , 1, 2,...,  Based on the essential ideas of moving least squares (MLS) approximation, in each local subdomain, the residual function can be defined as follows where () p  is the weight function, many kinds of weight functions can be adopted. We use the spline weight function [50] in the present study.
where p d is the distance between the central node (0) x and the pth supporting node () On the basis of the MLS theory, the undetermined coefficients From Eq (20), the following linear system can be consisted The unknown vector b in Eq (22) could be recast as In accordance with Eqs (21) Substituting Eq (24) into Eq (17) as 0, p = the temperature at central node (0) x is expressed as (25) in which ) . In addition, the normal heat flux at the boundary node can be calculated by (27) in which (28) or for brevity x is a boundary node with Robin boundary condition, we have Substituting Eq (29) into Eq (30) yields For the interior nodes, the temperature distribution should satisfy Eq (26), namely, In Eqs (34) and (35), the subscript "p" (indicates the node number) is used to distinguish the different boundary nodes. By using given boundary data and combining Eqs (32)-(35), the following sparse system of linear equations is obtained represents the unascertained vector of variables at all nodes, and To summarize and to make clearer the procedure of the developed method for solving the heat conduction problem in nonlinear functionally graded material, a computational flow chart is given in Figure 2. Volume 6, Issue 11, 12599-12618.

Numerical examples
In this section, three typical numerical examples are provided to verify the applicability and accuracy of the proposed method for solving the heat conduction problem of two-dimensional nonlinear functionally graded materials. In order to estimate the accuracy, the measured errors are defined as For the sake of investigating the influence of the total number of nodes on the calculation results, 2300 regular and irregular nodes are used in the calculation (as shown in Figure 3). Irregular nodes are derived by jiggling the regular nodes, i.e., assigning a perturbation on regular nodes along the x and y directions. The temperature distribution in the domain can be calculated with different numbers of supporting node (10 40 m  ). Figure 4 shows the error curves as the number of supporting node increases, under regular/irregular nodal distributions. As can be seen from Figure 4, although the errors show fluctuation for some specific numbers of supporting node, it can still be seen that the global errors and the maximum errors generally have a decreasing trend with the increase of supporting nodes. In this example, 35 m = is a relatively optimal value to achieve the highest accuracy. This indicates that the number of supporting nodes has a certain influence on the calculation results, but a relatively ideal numerical result can be obtained in a larger range.   Figure 5 displays the 3D error surface and the 2D error plane. The red dots in Figure 5 According to Eq (41), we can easily estimate the number of supported nodes according to the node spacing. This example uses 2300 nodes, and thus h  is about 0.038. It can be observed from Figure 5 Figure 6 shows the variations of errors with the increase of the total number of nodes under the regular and irregular nodes, where 30 m = . We can observe that both the maximum errors and global errors are less than 2 5.621 10 −  , and both decrease with the augment of the total number of nodes, indicating that the LKM has the good accuracy and convergence in dealing with the heat transfer problem of nonlinear functionally graded materials. It can also be obtained from Figure 6 that the calculation precision with regular node distribution is higher than that with irregular node distribution, which verifies that the proposed method, as a local meshless method, can simulate the heat conduction problem with arbitrary node distribution, and the calculation accuracy with regular node distribution is slightly better.  To validate the accuracy and condition number of the developed method, we compared the LKM with the BKM. Table 1 lists the errors and the condition numbers of the above two methods with increasing number of boundary nodes. It is worth noting that the BKM is a boundary meshless method, while the LKM is a local meshless method. In the BKM, the number of boundary nodes (NB) is set to the same value as the LKM to ensure the fairness of comparison. From Table 1, it is observed that the numerical veracity of the LKM is marginally better than that of the BKM for various values of NB. Moreover, the condition number of the LKM is obviously less than that of the BKM. It could be carefully concluded that the presented LKM is accurate and stable for solving two-dimensional nonlinear heat conduction problems.

Example 2
The second example considers a heat transfer problem on a square functionally graded material plate with four circular holes of the same size, as shown in Figure 8 The exact solution is indicated as  Under 1226 N = (as shown in Figure 8(b)) and 60 m = , Figure 9 illustrates the comparison of numerical results on the computational domain. It can be found from the figure that the numerical results obtained by the LKM are extremely in agreement with the analytical solutions. Furthermore,

AIMS Mathematics
Volume 6, Issue 11, 12599-12618. the absolute errors are less than 5 8.306 10 −  , and the maximum error appears at the boundary d. It can be seen from Table 2, both global errors and maximum errors decrease with increasing number of supporting node, showing a convergence trend. As can be expected, the CPU times gradually increase with increasing number of supporting nodes, but not too much even for the relatively larger value 60 m = .   Finally, we set 60 m = , Figure 10 depicts the absolute errors of the LKM under different numbers ( 960, 1890, 2620 N = ) of total nodes. Noted that the error becomes smaller and smaller as the number of total nodes increases, indicating a convergence trend. The above numerical experiment with complicated geometries and mixed boundary conditions confirms the capacity, accuracy, and convergence of the developed methodology in solving the 2D heat conduction in nonlinear functionally graded material.

Example 3
In the last example, the LKM is applied to a nonlinear heat conduction problem in an irregular domain. Figure 11 shows the geometry model of the problem. In this case, we use the same analytical solution and parameters as for example 2. To investigate the influence of the number of supporting nodes on the computational accuracy, we set N to be 1228 (irregular nodes) and plot the relative errors of the temperatures at all points in Figure 12, when m is equal to 30, 40, and 50. It is noticed that the error gradually decreases with increasing number of supporting nodes. To investigate the convergence of the LKM for nonlinear heat conduction problem in an irregular structure, Figure 13 shows the variations of global errors and maximum errors as the increase of node number. Numerical results in the figure demonstrate the good convergence properties of the proposed scheme. Furthermore, even if the number of nodes exceeds 100000, the numerical simulation work is still performed on a regular laptop. A great number of numerical tests indicate that the proposed methodology in this paper is accurate, stable and convergent, and is suitable for large-scale simulations in an arbitrary domain.

Conclusions
This paper firstly presented a novel local semi-analytical meshless method, the local knot method (LKM) in conjunction with the Kirchhoff transformation, to numerically simulate heat conduction problems of nonlinear functionally graded materials. As a local semi-analytical meshless algorithm, the LKM uses the non-singular general solution as the basis function. Compared with the traditional BKM, this method not only retains the simplicity and high accuracy, but also avoids an ill-conditioned system of equations, and is more appropriate for large-scale simulations associated with complex structures.
Numerical experiments including simply/multi-connected and regular/irregular domains shown that the proposed method is accurate, stable, and convergent. Furthermore, the resultant matrix is wellconditioned and has a smaller condition number than the traditional BKM. Compared with the existing methods for solving the heat conduction problem of nonlinear functionally graded materials, the present scheme could be regarded as a highly competitive one.