Research Article | Open Access

Na Zhao, Hongping Ren, "The Interpolating Element-Free Galerkin Method for 2D Transient Heat Conduction Problems", *Mathematical Problems in Engineering*, vol. 2014, Article ID 712834, 9 pages, 2014. https://doi.org/10.1155/2014/712834

# The Interpolating Element-Free Galerkin Method for 2D Transient Heat Conduction Problems

**Academic Editor:**Yumin Cheng

#### Abstract

An interpolating element-free Galerkin (IEFG) method is presented for transient heat conduction problems. The shape function in the moving least-squares (MLS) approximation does not satisfy the property of Kronecker delta function, so an interpolating moving least-squares (IMLS) method is discussed; then combining the shape function constructed by the IMLS method and Galerkin weak form of the 2D transient heat conduction problems, the interpolating element-free Galerkin (IEFG) method for transient heat conduction problems is presented, and the corresponding formulae are obtained. The main advantage of this approach over the conventional meshless method is that essential boundary conditions can be applied directly. Numerical results show that the IEFG method has high computational accuracy.

#### 1. Introduction

In recent years, meshless methods have been successfully developed and applied to solve a variety of science and engineering problems [1–8]. The meshless method, which is based on nodes with a minimum of meshing or no meshing at all, can solve many engineering problems that are not suited to conventional computational methods and has shown some advantages.

Many kinds of meshless methods have been developed, such as smoothed particle hydrodynamics (SPH) [9], diffuse element method (DEM) [10], element-free Galerkin (EFG) method [11], reproducing kernel particle method (RKPM) [12], finite point method (FPM) [13], meshless local Petrov-Galerkin (MLPG) method [14], point collocation method (PCM) [15], radial basis functions (RBF) method [16], meshless finite element method (MFEM) [17], complex variable meshless method (CVMM) [18, 19], boundary node method (BNM) [20], local boundary integral equation (LBIE) method [21], boundary radial point interpolation method (BRPIM) [22], and boundary element-free method (BEFM) [23–25].

The element-free Galerkin (EFG) method is the most important meshless method, the shape function in the EFG method is formed with moving least-squares (MLS) approximation, a disadvantage of the MLS approximation is that the final algebraic equations system is sometimes ill-conditioned, and we cannot obtain a good solution, or even correctly obtain a numerical solution; then Cheng and Peng proposed an improved moving least-squares approximation by orthogonalizing the basis functions in the MLS approximation, and based on it Cheng and Peng put forward a boundary element-free method [23]. Since the shape function of the MLS approximation does not have the properties of Kronecker delta function, the meshless method based on it must use other methods, such as the penalty function method, Lagrange multiplier, to impose essential boundary conditions, which makes the weak form of the problem more complicated and the computational efficiency lower as a result, and Lancaster proposed interpolating moving least-squares (IMLS) method [26], which can obtain the shape function satisfying the property of Kronecker delta function, but, compared with the IMLS method, the shape function of the MLS approximation is much simpler, and thus a few papers on meshless methods based on the IMLS method were published. Ren et al. proposed an improved IMLS method, and based on it the interpolating element-free Galerkin (IEFG) method and the improved boundary element-free method are presented [27–30].

The analysis of transient heat conduction problems is very important to engineering and science. However, analytical solution for this kind of problems is difficult to obtain except for a few simple cases; thus an alternative way is proposed, that is, the numerical solution. Singh et al. analyzed transient nonlinear heat transfer problems in solids by EFG method [31]. Chen and Cheng used complex variable reproducing kernel particle method to solve transient heat conduction problems [32]; its advantage is that 2D problem is solved with 1D basis function. Yang and Gao used radial integration BEM to solve transient heat conduction problems [33]; the features are that thermal material parameters can be functions of spatial coordinates. R. J. Cheng and Y. M. Cheng solved the inverse heat conduction problem [34]; the finite point method is used to obtain the solution of 1D inverse heat conduction problem with a source parameter. Chen and Liew, Baodong and Zheng developed meshless local Petrov-Galerkin approach based on the moving Kriging interpolation for transient heat conduction problems [35, 36]; the main feature is in use of the moving Kriging interpolation as the trail function and the Heaviside step function as the test function. Li et al. combined the complex variable reproducing kernel particle method and the finite element method to solve transient heat conduction problems [37]. Li et al. proposed the MLPG method in conjunction with the modified precise time step integration method for the analysis of transient heat conduction problems [38]. Zhang et al. developed improved element-free Galerkin method for three-dimensional transient heat conduction problems [39]; the effects of scaling parameter, number of nodes, and the time step length were considered in their numerical solutions.

The present paper is motivated by the MLS method and the IMLS method; this is the fundamental principle of the EFG method and the IEFG method; then combining the shape function constructed by the IMLS method and Galerkin weak form of the 2D transient heat conduction problem, the interpolating element-free Galerkin (IEFG) method for 2D transient heat conduction problems is presented, and the corresponding formulae are obtained. For the purposes of demonstration, some selected numerical examples are solved using the IEFG method.

#### 2. The Interpolating Moving Least-Squares (IMLS) Method

In the MLS approximation, it is assumed that a function is to be approximated and that its values are given.

An approximating function of is where are monomial basis functions and are the coefficients of the basis functions .

In general, the basis functions are as follows in 2D space.

Linear basis:

Quadratic basis:

Let be a weight function with compact support, and define a functional and are the nodes in the influence domain of point and is the sum of the squares of the residuals of all data points in the influence domain.

When , we can obtain the coefficients : where matrices and are

The expression of the approximation function is then where is called the shape function and

The normalized weight function is Let Assume which is a weighted average of the function values at nodes in the influence domain of .

In the moving least-squares method, approximating function need not interpolate the data points; then by orthogonalizing the last basis functions to the first one, and taking a singular weight function in the points, we establish the interpolating moving least-squares (IMLS) method.

We define the following inner product of function and :

Normalize at point ; we have for ; generate the basis functions orthogonal to as follows:

In the MLS approximation, while we use the new basis functions , the corresponding approximating function is Then where

In the IMLS method, the singular weight function is selected as where and is the radius of the influence domain.

In [4], we have proved that the approximating function in (15) can interpolate at all points , ; that is, .

#### 3. 2D Transient Heat Conduction Analysis by Interpolating Element-Free Galerkin (IEFG) Method

##### 3.1. Governing Equation and Its Galerkin Weak Form

In general, the governing equation for two-dimensional transient heat conduction in isotropic solid body with spatially varying conductivity occupying a region can be written as where represents temperature, is time, is the density of material, is the specific heat capacity, is heat generation rate, and and are thermal conductivities in - and -directions, respectively.

The initial condition is the Dirichlet boundary condition is the Neumann boundary condition is and the Robin boundary condition is where is the unit outward normal to the boundary , and are the prescribed temperature and the given heat fluxes on the corresponding boundaries, is the convection heat transfer coefficient, and is the environmental temperature.

Consider

The weak form of (19), (22), and (23) is The functional can be written as Let ; then where is the variational operator, and is a differential operator,

##### 3.2. Discretization of the Weak Form

We employed nodes in the domain , and the union of its compact support domain , , must cover .

From the approximating function (14), the temperature at an arbitrary field point in the domain at a given time can be expressed as where is the number of nodes in which the compact support domains cover the field point .

Moreover where Substituting (29), (32), and (34) into (27) yields The first term of (36) can be written as is a matrix of , where

The second term of (36) can be written as where is a matrix of , where

The third term of (36) can be written as

The fourth term of (36) can be written as

The fifth term of (36) can be written as

The sixth term of (36) can be written as is a matrix of , where

Substituting (37), (40), (43), (46), (49), and (52) into (36) yields Because of the arbitrariness of , we have where

Now, (27) has been discretized to be the ordinary differential equation (55), in which the time is the only variable.

The traditional two-point difference method is selected for the time discretization, Selecting , then (55) can be written as that is, where

Substituting the boundary condition (21) into (59) directly, we can obtain the unknowns at nodes by solving (59).

Above all, the IEFG method is presented for two-dimensional transient heat conduction problems.

#### 4. Numerical Examples

We select four numerical examples to demonstrate the applicability of the IEFG method in transient heat conduction problem.

For the purpose of convergence studies, the root-mean-square (RMS) error is defined as where is the number of sample points.

##### 4.1. The Transient Heat Conduction without Heat Generation

The first example considered is the 2D transient heat conduction in a square domain , the thermal conductivities W/m·°C, the specific heat capacity J/kg·°C, and the density . Its governing equation is

The boundary condition is

The initial temperature is

The analytical solution of this problem is

As shown in Figure 1, regular distribution of nodes is arranged in the domain and time step is in the computing process. The numerical results and the analytical solution of temperature at when , , , and are shown in Figure 2. In addition, the values of root-mean-square errors at different times when the time step is are given in Figure 3, which shows that numerical solution converges as the time increases.

##### 4.2. The Transient Heat Conduction without Heat Generation

Consider

The boundary conditions are

The initial condition is

The analytical solution of this problem is

The distribution of nodes in this example is the same as that in Figure 1. The time step is chosen as . The numerical results and the analytical solution of temperature at , , and when are plotted in Figure 4; we can conclude that the numerical results are in good agreement with the analytical solutions.

##### 4.3. The Transient Heat Conduction with Heat Generation

The third example is the transient heat conduction equation with heat generation; consider

The boundary conditions are

The initial temperature is

The analytical solution of this problem is

As Figure 5 shows, nodes are distributed in . The time step is . The numerical results and the analytical solution at when , , and are shown in Figure 6; it is obvious that the numerical results are in excellent agreement with the analytical solutions.

##### 4.4. The Transient Heat Conduction with Heat Generation and Lateral Heat Loss

Consider where

The boundary conditions are:

The initial temperature is

The analytical solution of this problem is

As Figure 7 shows, nodes are distributed in the rectangular domain . The time step is . The numerical results and the analytical solution at and when , , and are shown in Figures 8 and 9, respectively; it can be found that IEFG method works well for transient heat conduction problems with heat generation and even with lateral heat loss.

#### 5. Conclusions

The present study is concentrated on the interpolating element-free Galerkin (IEFG) method for 2D transient heat conduction problems; compared with the conventional EFG method, the essential boundary conditions are applied naturally and directly in the IEFG method, and thus the IEFG method gives a greater computational efficiency. Numerical results show that the IEFG method has high computational accuracy.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgment

The work is supported by the Natural Science Foundation of Shanxi Province of China (2013011022-6).

#### References

- T. Belytschko, Y. Krongauz, D. Organ, M. Fleming, and P. Krysl, “Meshless methods: an overview and recent developments,”
*Computer Methods in Applied Mechanics and Engineering*, vol. 139, no. 1–4, pp. 3–47, 1996. View at: Publisher Site | Google Scholar | Zentralblatt MATH - R. Cheng and Y. Cheng, “Error estimates for the finite point method,”
*Applied Numerical Mathematics*, vol. 58, no. 6, pp. 884–898, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH - R.-J. Cheng and Y.-M. Cheng, “A meshless method for the compound KdV-Burgers equation,”
*Chinese Physics B*, vol. 20, no. 7, Article ID 070206, 2011. View at: Publisher Site | Google Scholar - H. Ren and J. Cheng, “The complex variable interpolating moving least-squares methed,”
*Applied Mathematics and Computation*, vol. 219, no. 4, pp. 1724–1736, 2012. View at: Publisher Site | Google Scholar - L. Chen and Y.-M. Cheng, “The complex variable reproducing kernel particle method for two-dimensional elastodynamics,”
*Chinese Physics B*, vol. 19, no. 9, Article ID 090204, 2010. View at: Publisher Site | Google Scholar - W. Jufeng, S. Fengxin, and C. Yumin, “An improved interpolating element-free Galerkin method with nonsingular weight function for two-dimensional potential problems,”
*Chinese Physics B*, vol. 21, no. 9, Article ID 090204, 2012. View at: Publisher Site | Google Scholar - B. Dai and Y. Cheng, “An improved local boundary integral equation method for two-dimensional potential problems,”
*International Journal of Applied Mechanics*, vol. 2, no. 2, pp. 421–436, 2010. View at: Publisher Site | Google Scholar - F.-N. Bai, D.-M. Li, J.-F. Wang, and Y.-M. Cheng, “An improved complex variable element-free Galerkin method for two-dimensional elasticity problems,”
*Chinese Physics B*, vol. 21, no. 2, Article ID 020204, 2012. View at: Publisher Site | Google Scholar - R. A. Gingold and J. J. Monaghan, “Smoothed particle hydrodynamics: theory and application to nonspherical stars,”
*Monthly Notices of the Royal Astronomical Society*, vol. 18, pp. 375–389, 1977. View at: Google Scholar - B. Nayroles, G. Touzot, and P. Villon, “Generalizing the finite element method: Diffuse approximation and diffuse elements,”
*Computational Mechanics*, vol. 10, no. 5, pp. 307–318, 1992. View at: Publisher Site | Google Scholar | Zentralblatt MATH - T. Belytschko, Y. Y. Lu, and L. Gu, “Element-free Galerkin methods,”
*International Journal for Numerical Methods in Engineering*, vol. 37, no. 2, pp. 229–256, 1994. View at: Publisher Site | Google Scholar | Zentralblatt MATH - W. K. L. Wing Kam Liu, S. J. Sukky Jun, and Y. F. Z. Yi Fei Zhang, “Reproducing kernel particle methods,”
*International Journal for Numerical Methods in Fluids*, vol. 20, no. 8-9, pp. 1081–1106, 1995. View at: Publisher Site | Google Scholar | Zentralblatt MATH - E. Oñate, S. Idelsohn, O. C. Zienkiewicz, and R. L. Taylor, “A finite point method in computational mechanics. Applications to convective transport and fluid flow,”
*International Journal for Numerical Methods in Engineering*, vol. 39, no. 22, pp. 3839–3866, 1996. View at: Google Scholar | Zentralblatt MATH - S. N. Atluri and T. Zhu, “A new Meshless Local Petrov-Galerkin (MLPG) approach in computational mechanics,”
*Computational Mechanics*, vol. 22, no. 2, pp. 117–127, 1998. View at: Publisher Site | Google Scholar | Zentralblatt MATH - N. R. Aluru, “A point collocation method based on reproducing kernel approximations,”
*International Journal for Numerical Methods in Engineering*, vol. 47, no. 6, pp. 1083–1121, 2000. View at: Google Scholar | Zentralblatt MATH - H. Wendland, “Meshless Galerkin methods using radial basis functions,”
*Mathematics of Computation*, vol. 68, no. 228, pp. 1521–1531, 1999. View at: Publisher Site | Google Scholar | Zentralblatt MATH - S. R. Idelsohn, E. Onate, N. Calvo, and F. Del Pin, “The meshless finite element method,”
*International Journal for Numerical Methods in Engineering*, vol. 58, no. 6, pp. 893–912, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH - Y. Cheng and J. Li, “Complex variable meshless method for fracture problems,”
*Science in China G*, vol. 49, no. 1, pp. 46–59, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH - M. Peng, P. Liu, and Y. Cheng, “The complex variable element-free Galerkin (CVEFG) method for two-dimensional elasticity problems,”
*International Journal of Applied Mechanics*, vol. 1, no. 2, pp. 367–385, 2009. View at: Publisher Site | Google Scholar - M. K. Chati, S. Mukherjee, and Y. X. Mukherjee, “The boundary node method for three-dimensional linear elasticity,”
*International Journal for Numerical Methods in Engineering*, vol. 46, no. 8, pp. 1163–1184, 1999. View at: Google Scholar | Zentralblatt MATH - T. Zhu, J. Zhang, and S. N. Atluri, “Meshless numerical method based on the local boundary integral equation (LBIE) to solve linear and non-linear boundary value problems,”
*Engineering Analysis with Boundary Elements*, vol. 23, no. 5, pp. 375–389, 1999. View at: Publisher Site | Google Scholar - Y. T. Gu and G. R. Liu, “A boundary radial point interpolation method (BRPIM) for 2-D structural analyses,”
*Structural Engineering and Mechanics*, vol. 15, no. 5, pp. 535–550, 2003. View at: Publisher Site | Google Scholar - Y. Cheng and M. Peng, “Boundary element-free method for elastodynamics,”
*Science in China G*, vol. 48, no. 6, pp. 641–657, 2005. View at: Publisher Site | Google Scholar - M. Peng and Y. Cheng, “A boundary element-free method (BEFM) for two-dimensional potential problems,”
*Engineering Analysis with Boundary Elements*, vol. 33, no. 1, pp. 77–82, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH - Y. Cheng, K. M. Liew, and S. Kitipornchair, “Reply to “Comments on Boundary element-free method (BEFM) and its application to two-dimensional elasticity problems”,”
*International Journal for Numerical Methods in Engineering*, vol. 78, no. 10, pp. 1258–1260, 2009. View at: Publisher Site | Google Scholar - P. Lancaster and K. Salkauskas, “Surfaces generated by moving least square methods,”
*Mathematics of Computation*, vol. 37, pp. 141–158, 1981. View at: Publisher Site | Google Scholar - H. Ren and Y. Cheng, “The interpolating element-free Galerkin (IEFG) method for two-dimensional elasticity problems,”
*International Journal of Applied Mechanics*, vol. 3, no. 4, article 735, 2011. View at: Publisher Site | Google Scholar - H. Ren and Y. Cheng, “The interpolating element-free Galerkin (IEFG) method for two-dimensional potential problems,”
*Engineering Analysis with Boundary Elements*, vol. 36, no. 5, pp. 873–880, 2012. View at: Publisher Site | Google Scholar - H.-P. Ren, Y.-M. Cheng, and W. Zhang, “An improved boundary element-free method (IBEFM) for two-dimensional potential problems,”
*Chinese Physics B*, vol. 18, no. 10, pp. 4065–4073, 2009. View at: Publisher Site | Google Scholar - R. Hongping, C. Yumin, and Z. Wu, “An interpolating boundary element-free method (IBEFM) for elasticity problems,”
*Science China*, vol. 53, no. 4, pp. 758–766, 2010. View at: Publisher Site | Google Scholar - A. Singh, I. V. Singh, and R. Prakash, “Meshless element free Galerkin method for unsteady nonlinear heat transfer problems,”
*International Journal of Heat and Mass Transfer*, vol. 50, no. 5-6, pp. 1212–1219, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH - L. Chen and Y.-M. Cheng, “Complex variable reproducing kernel particle method for transient heat conduction problems,”
*Acta Physica Sinica*, vol. 57, no. 10, pp. 6047–6055, 2008. View at: Google Scholar | Zentralblatt MATH - K. Yang and X.-W. Gao, “Radial integration BEM for transient heat conduction problems,”
*Engineering Analysis with Boundary Elements*, vol. 34, no. 6, pp. 557–563, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH - R.-J. Cheng and Y.-M. Cheng, “The meshless method for solving the inverse heat conduction problem with a source parameter,”
*Acta Physica Sinica*, vol. 56, no. 10, pp. 5569–5574, 2007. View at: Google Scholar | Zentralblatt MATH - L. Chen and K. M. Liew, “A local Petrov-Galerkin approach with moving Kriging interpolation for solving transient heat conduction problems,”
*Computational Mechanics*, vol. 47, no. 4, pp. 455–467, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH - D. Baodong and B. Zheng, “Numerical solution of transient heat conduction problems using improved meshless Local Petrov-Galerkin method,”
*Applied Mathematics and Computation*, vol. 219, pp. 10044–10052, 2013. View at: Publisher Site | Google Scholar - C. Li, M. Heping, and C. Yumin, “Combining the complex variable reproducing kernel particle method and the finite element method for solving transient heat conduction problems,”
*Chinese Physics B*, vol. 22, no. 5, Article ID 050202, 2013. View at: Publisher Site | Google Scholar - Q.-H. Li, S.-S. Chen, and G.-X. Kou, “Transient heat conduction analysis using the MLPG method and modified precise time step integration method,”
*Journal of Computational Physics*, vol. 230, no. 7, pp. 2736–2750, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH - Z. Zhang, J. Wang, Y. Cheng, and K. M. Liew, “The improved element-free Galerkin method for three-dimensional transient heat conduction problems,”
*Science China Physics, Mechanics & Astronomy*, vol. 56, no. 8, pp. 1568–1580, 2013. View at: Publisher Site | Google Scholar

#### Copyright

Copyright © 2014 Na Zhao and Hongping Ren. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.