Research Article  Open Access
A Meshless Method for the Numerical Solution of a TwoDimension IHCP
Abstract
This paper uses the collocation method and radial basis functions (RBFs) to analyze the solution of a twodimension 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.
1. 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, hightemperature forming and coating technology, solar energy utilization, hightemperature 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, 4–7] and the references therein). Several techniques have been proposed for solving a onedimensional IHCP [8–14]. 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 twodimension. 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 twodimensional 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 meshdependent characteristics which normally requires enormous computational effort and induces numerical instability when large number of grids or elements are required.
In this paper, a twodimensional 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 meshdependent techniques.
In the recent rapid development of meshless computational schemes, the RBF has been successfully developed as an efficient meshless scheme for solving various kinds of partial differential equations (PDEs) and ordinary differential equations (ODEs). For example, see [19–25]. Also some applications of this approach in solving inverse problems can be found in [20, 21, 26, 27].
2. Problem Formulation
In this section, we consider the following twodimension IHCP, in the dimensionless form: and the overspecified condition: where is known, represent the final time of interest for the time evolution of the problem, and , , , , and are known functions in their domain satisfying the compatibility conditions while and 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.
The first problem is This problem may be analyzed as a direct problem, for the portion of the body from to with known boundary conditions. There is a unique stable solution to the direct problem (3)–(8) and may be found in [29]. The second problem is the following IHCP: Therefore, for solving the inverse problem (1)–(2), we will investigate the direct problem (3)–(8) and IHCP (9)–(14). In the next section, the above problems will be considered; the heat in the body and heat at the boundary will be obtained by solving these problems numerically.
3. 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 meshdependent 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.
3.1. Definition of the Space Radial Basis Functions
Let , denote the Euclidean norm and let be a continuous function with . A radial basis function on is a function of the form which depended only on the distance between and a fixed point [30]. So that the radial basis function is radially symmetric about the center . Some bestknown RBFs are listed in Table 1, where 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 [31–33], 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].
Class 1. Infinitely smooth RBFs.
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.
Class 2. Infinitely smooth (except at centers) RBFs.
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.
4. Numerical Procedures
Now we use the RBFs for discretization of both time and space variables. Let , , be a set of scattered nodes. Then, the function of two spatial dimensions to be interpolated can be represented by RBFs as where for a radial function and ; , , and are unknown constants that must be found.
4.1. The Application of RBFs in IHCP (9)–(14)
For this problem, let , ; , ; and , . Now (9)–(14) are approximated using (16). The collocation technique is used for finding unknown constants .
We collocate (9) in points ; , ; we have Now, collocation (10) in points yields By collocating (12) in points , we obtain By collocation (13) in points , we can write and by collocation (14) in points ; , , we have Equations (17)–(21) give a system of linear algebraic equations with the unknown coefficients . Solving this linear system, the approximate temperature distribution of the problem (9)–(14) is obtained. Knowing the temperature distribution, we can easily determine the function describing the boundary condition:
4.2. The Application of RBFs in Direct Problem (3)–(8)
Now, let , , and . The unknown function in (3)–(8) can be approximated by using (16). The collocation technique is used for finding unknown coefficients .
We collocate (3) in points ; we get Now, collocation (4) in points , yields By collocation (5) in points , we obtain By collocation (6) in points , we can write By collocation (7) in points , we have and by collocation (8) in points , we have Equations (23)–(28) give a system of linear algebraic equations with the unknown coefficients . Solving this linear system, the approximate solution of the problem (3)–(8) is obtained.
5. 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.
5.1. Approximation Error
Here, we are concerned with the error of the approximation of a given threevariate function by its expansion in terms of radial basis functions.
Suppose that , where (also note that the solution ); the inner product in this space is defined by and the norm is as follows: Let be the set of radial basis functions and and let be an arbitrary element in . Since is a finite dimensional vector space, has the unique best approximation out of such as [36], such that Moreover, since , there exist unique coefficients such that
Theorem 1. Let be a Hilbert space and let be a closed subspace of such that and is any basis for . Let be an arbitrary element in and let be the unique best approximation to out of . Then [36] where
5.2. Convergence Analysis
We have the following theorem about the convergence of RBFs interpolation.
Theorem 2. Assume are nodes in which is convex; let when for any satisfies ; we have where is RBFs, the constant depends on the RBFs, is space dimension, and and are nonnegative integer.
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.
6. 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 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 is a random number between .
6.1. Example 1
We solve the problem (1)–(2) with [28]: , , , , , and for which the exact solution is and .
Table 2 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 3 shows this comparison with noisy data. Also, Table 4 shows the Error and RMS error values for and on the intervals , , and for various values of , , and . The corresponding results obtained for on the intervals , , and are presented in Table 5. Figure 1 shows the plot of error for and on the intervals , , and . It can be obtained from Tables 4 and 5 and Figure 2 that the accuracy increases with the increase of the number of collocation points.




(a) for
(b) for , ,
(c) for
(d) for , ,
(a) for with
(b) for with
(c) for with
(d) for with
6.2. Example 2
As the second example, consider (1)–(2) with [28]: , , , , , and for which the exact solution is and .
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 on the intervals , , and for various values of , , and . The corresponding results obtained for on the intervals , , and 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.


