A Meshless Method for the Numerical Solution of a Two-Dimension IHCP

This paper uses the collocation method and radial basis functions (RBFs) to analyze the solution of a two-dimension inverse heat conduction problem (IHCP). The accuracy of the method is tested in terms of Error and RMS errors. Also, the stability of the technique is investigated by perturbing the additional specification data by increasing the amounts of random noise. The results of numerical experiments are compared with the analytical solution in illustrative examples to confirm the accuracy and efficiency of the presented scheme.


Introduction
In many industrial applications one wishes to determine the temperature on the surface of a body, where the surface itself is inaccessible for measurements.In this case, it is necessary to determine the surface temperature from a measured temperature history at a fixed location inside the body.This is called an IHCP and has been an interesting subject recently [1].Inverse problems have practical implications in thermal transport systems which involve conduction, convection, and radiation.In thermal radiation [2], for example, identifying the distribution of the radiation source has been stimulated by a wide range of applications, including thermal control in space technology, combustion, high-temperature forming and coating technology, solar energy utilization, high-temperature engines, and furnace technology [3].The importance of inverse heat conduction problems and appropriate solution algorithms are established in numerous works and the books (see, e.g., [1,[4][5][6][7] and the references therein).Several techniques have been proposed for solving a one-dimensional IHCP [8][9][10][11][12][13][14].Among the methods proposed for higher dimensional IHCP, boundary element [15], finite difference [16], and finite element [17] have been widely adopted for problems in two-dimension.Besides, the sequential function specification method [1,15] and differential method [18] have also been used in solving the IHCP.There is, however, still a need on numerical scheme for two-dimensional IHCP.The traditional meshdependent finite difference and finite element methods are so far the principal numerical tool of choice for the modeling and simulation of the IHCP.A major disadvantage of these methods, however, is their mesh-dependent characteristics which normally requires enormous computational effort and induces numerical instability when large number of grids or elements are required.
In this paper, a two-dimensional IHCP is solved by RBFs as a truly meshless/meshfree method.A meshfree method does not require a mesh to discretize the domain or boundary of the problem under consideration and the approximate solution is constructed entirely based on a set of scattered nodes.It is considered as the main advantage of these methods over the mesh-dependent techniques.

Problem Formulation
In this section, we consider the following two-dimension IHCP, in the dimensionless form: and the overspecified condition: where  1 ∈ (0, 1) is known,  fin represent the final time of interest for the time evolution of the problem, and ,  1 , ℎ 0 , ℎ 1 , and  are known functions in their domain satisfying the compatibility conditions while (, , ) and  0 (, ) remain to be determined from some interior temperature measurements.
The existence and uniqueness of the solution of this problem are discussed in [28].The problem ( 1)-( 2) may be divided into two separate problems.

Radial Basis Functions
Radial basis functions are very efficient instruments for interpolating a scattered set of points.The use of the radial basis function for solving partial differential equations has some advantages over mesh-dependent methods, such as finite difference methods, finite element methods, spectral methods, finite volume methods, and boundary element methods.The use of radial basis functions as a meshless method for numerical solution of partial differential equations is based on the collocation method.Because of the collocation technique, this method does not need to evaluate any integral.

Definition of the Space Radial Basis Functions.
Let R + = { ∈ R,  ≥ 0}, ‖ ⋅ ‖ 2 denote the Euclidean norm and let  : R + → R be a continuous function with (0) ≥ 0. A radial basis function on R  ;  = 1, 2, 3 is a function of the form which depended only on the distance between x ∈ R  and a fixed point x  ∈ R  [30].So that the radial basis function   is radially symmetric about the center x  .Some best-known RBFs are listed in Table 1, where  = ‖x − x  ‖ 2 and  is a free positive parameter, often referred to as the shape parameter, to be specified by the user.Despite many research works, which have been done to find algorithms for selecting the optimum values of  [31][32][33], the optimal choice of shape parameter is an open problem, which is still under intensive investigation.
The standard radial basis functions are categorized into two major classes [34,35].
These basis functions are infinitely differentiable and involve a parameter  (such as multiquadric (MQ), inverse multiquadric (IMQ), and Gaussian) which needs to be selected so that the required accuracy of the solution is attained.
Advances in Numerical Analysis 3 These basis functions are not infinitely differentiable.These basis functions are shape parameter free and have comparatively less accuracy than the basis functions discussed in Class 1. Examples are thin plate splines.

Convergence Analysis and Error Bound
This section covers the error analysis of the proposed method.Also the sufficient conditions are presented to guarantee the convergence of RBFs, when applied to solve the differential equations.

Approximation Error.
Here, we are concerned with the error of the approximation of a given three-variate function by its expansion in terms of radial basis functions.

Convergence Analysis.
We have the following theorem about the convergence of RBFs interpolation.
Proof.A complete proof is given by authors [37,38].
It can be seen that not only RBFs themselves but also any of their order derivatives has a good convergence.

Test Examples
In order to illustrate the performance of the RBFs method in solving IHCPs and justify the accuracy and efficiency of the method presented in this paper, we consider the following examples.For two examples, the true solutions are available.We tested the accuracy and stability of the method presented in this paper by performing the mentioned method for different values of , , and .To study the convergence behavior of the RBFs method, we applied the following laws.
(1) The error Error is described using (2) The root mean square (RMS) is described using where (  ,   ,   ) are interpolate nodes,  is the exact value, and Ũ is the RBFs approximation.
In order to investigate the stability of the numerical method, the additional specification data has been perturbed as where  is the relative (percentage) noise level and rand() is a random number between (0, 1).
Table 6 shows the comparison between the exact solution, RBFs solution and approximate solution result from method in [28] by Tikhonov regularization 0th, 1st, and 2nd, and SVD with noiseless data.Table 7 shows this comparison with noisy data.Furthermore, Table 8 shows the  and RMS error values for (, , ) and  0 (, ) on the intervals  ∈ [0, 0.1],  ∈ [0, 1], and  ∈ [0, 1] for various values of , , and .The corresponding results obtained for (, , ) on the intervals  ∈ [0.1, 1],  ∈ [0, 1], and  ∈ [0, 1] are presented in Table 9.It can be obtained from Tables 8 and 9 that the accuracy increases with the increase of the number of collocation points.Also absolute error with different radial basis functions is depicted in Figure 2.

Conclusion
In the paper, an application of the RBFs for the solution of a two-dimension IHCP is presented.Problem consists in the calculation of temperature distribution in the domain, as well as in the reconstruction of functions describing the temperature on the boundary, when the temperature measurements in the domain are known.The present study successfully applies the numerical method to IHCPs and has been found stable with respect to small perturbation in the input data.The results of computing by radial basis function Advances in Numerical Analysis 9 method are compared to that by the method in [28].The results of numerical examples demonstrate that this method is more accurate than the method in [28].

Figure 1 :
Figure 1: Graph of absolute error for  and  0 by using GA-RBF with  = 0.1 for Example 1.

Table 1 :
Some well-known functions that generate RBFs.