Comparison of different solvers for two-dimensional steady heat conduction equation ME 412 Project 2

Transcription

1 Comparison of different solvers for two-dimensional steady heat conduction equation ME 412 Project 2 Jingwei Zhu March 19, 2014 Instructor: Surya Pratap Vanka 1 Project Description The purpose of this project is to develop a computer program to solve the steady, two-dimensional heat conduction equation listed below: where T is the temperature, x and y are spatial coordinates. 2 T x T y 2 = 0 (1) The equation is discretized using second order central differencing scheme. Different point iterative solvers (Jacobi, Gauss-Seidel, Successive Over-relaxation) and line inversion will be used to solve the discrete equations and their performance will be compared. We apply the computer program to achieve the steady temperature distribution in a square plate with top wall at T = 1[ C], and the others at T = 0[ C]. Grids with 11 11, and nodes are used. Convergence rate and error in discretization as a function of the grid size will be studied. Analytical solution is used as the exact solution. The analytical solution is given by a series summation found in text books [1] on first level heat transfer. 2 Discretization Scheme We used an explicit finite difference equation with second-order central differencing of the spatial derivatives to discretize the derivatives. The scheme can be described with the following equation: 1 x 2 (T i+1,j 2T i,j + T i 1,j ) + 1 y 2 (T i,j+1 2T i,j + T i,j 1 ) = 0 (2) 1

2 where x and y are the spatial intervals of the finite difference (uniform) grid. The index i and j denote the number of nodes in the x direction and y direction respectively. Boundary conditions are shown in the following equations: T i=1,j = 0 C, j [1, N y ]; (3) T i=nx,j = 0 C, j [1, N y ]; (4) T i,j=1 = 0 C, i [2, N x 1]; (5) T i,j=ny = 1 C, i [2, N x 1] (6) where N x and N y are the total number of grid points in the x direction and y direction. The relationship between x, y and N x, N y is shown as follows: x = L x N x 1 ; y = L y N y 1 (7) where L x is the length of the rectangular domain and L y is the width of the domain. In this project, we set L x and L y to 1m and 1m respectively. We applied four iterative scheme (Jacobi scheme, Gauss-Seidel scheme, Successive Over-relaxation scheme and line inversion) and ran simulations with 11 11, 21 21, grids. We also assumed that the simulation converges when maximum norm of changes in the temperature of nodes between iterations is less than C. The temperature distribution can be regarded as steady after convergence is reached. Discretization errors arise from truncation of the expansions in the differencing scheme. They can be estimated as the difference between the exact solution and the numerical solution. The exact solution can be achieved through Equation 8: T i,j = 2T s [1 ( 1) n ] n=1 nπsinh( nπly L x ) 1) nπ y(j 1) sin(nπ x(i )sinh( ) (8) L x L x where T s = 1 C is the temperature at the top wall. We approximated the exact solution by summing finite terms in Equation 8. Figure 1 shows temperature distribution achieved by Equation 8 on a grid when number of series terms summed is set to 15 and 30 respectively. It can be observed that when n = 15, there exists some oscillation on the contour of temperature, which is not reasonable for an exact solution. Therefore in this project, the exact solution is approximated 2

3 by summing the first 30 terms in Equation 8. 3 Iterative Schemes Equation 2 can be written in a compass notation as: a P T i,j = a W T i 1,j + a E T i+1,j + a S T i,j 1 + a N T i,j+1 (9) where a W = a E = 1/ x 2, a S = a N = 1/ y 2 and a P = 2/ x 2 + 2/ y Jacobi Scheme Jacobi scheme seeks to obtain a solution to the set of Equations 9 by iteratively updating the values of the unknowns: T r+1 i,j = (a W T r i 1,j + a E T r i+1,j + a S T r i,j 1 + a N T r i,j+1)/a P (10) where r denotes the iteration number. Since the value at all neighbors are known, all T i,j can be simultaneously updated. Therefore, Jacobi scheme is highly parallelizable and can be efficiently executed on vector and parallel computers. 3.2 Gauss-Seidel Scheme Gauss-Seidel scheme is also a point iterative solver. It updates the values of the unknowns with: T r+1 i,j = (a W T r+1 i 1,j + a ET r i+1,j + a S T r+1 i,j 1 + a NT r i,j+1)/a P (11) where r denotes the iteration number. The Gauss-Seidel scheme has to be applied row by row or column by column. The computation of Ti,j r+1 uses only the elements of T r+1 that have already been computed and only the elements of T r that have not yet to be advanced to iteration r + 1. It is not parallelizable and can not be efficiently executed on vector and parallel computers. 3.3 Successive Over-relaxation Scheme Successive Over-relaxation scheme is an acceleration scheme over traditional point and line iterative solvers. It updates the values of the unknowns with: R r+1 i,j = (a W T r+1 i 1,j + a ET r i+1,j + a S T r+1 i,j 1 + a NT r i,j+1)/a P ; (12) Ti,j r+1 = ωri,j r+1 + (1 ω)ti,j r (13) where r denotes the iteration number and the optimal value of parameter ω depends on the specific problem. For the heat conduction problem with Dirichlet boundary conditions and N N grid, ω 3

4 (a) n=15 (b) n=30 Figure 1: Approximation of exact solution to the steady two-dimensional heat conduction equation 4

5 is usually set to 2 1+π/N. In this project, for N x N y grid, ω = 2 1+π/ N x N y. Successive Over-relaxation scheme is usually superior to Gauss-Seidel and Jacobi schemes in terms of convergence speed. 3.4 Line Inversion Scheme Line inversion scheme updates the unknowns with: F r i,j = a S T r i,j 1 + a N T r i,j+1; (14) a P T r+1 i,j a W T r+1 i 1,j a ET r+1 i+1,j = F r i,j (15) where r denotes the iteration number. During iteration r +1, Fi,j r is computed with available values of last iteration r in storage. Then each row of the grid is updated by TDMA (Tri-Diagonal Matrix Algorithm) method. 4 Results and Discussion Figure 2 displays the history of maximum norm of solution errors from iteration to iteration of the four iterative schemes (Jacobi, Gauss-Seidel, Successive Over-relaxation and line inversion) for different grid sizes. Table 1 shows the number of iterations needed to convergence for the four iterative schemes with different grid sizes Jacobi G-S SOR Line Inversion Table 1: Number of iterations needed to convergence for the four iterative schemes with different grid sizes It can be observed from Figure 2 and Table 1 that Successive Over-relaxation scheme has the highest convergence speed among the four iterative schemes while convergence speed of Jacobi scheme is the lowest. However, Jacobi scheme is highly parallelizable and can run efficiently on parallel computers. Table 2 displays the discretization errors for different schemes with different grid sizes. It is achieved by calculating root mean square of the solution errors, which is the difference between the exact 5

6 (a) (b) (c) Figure 2: History of maximum norm of solution errors between iterations of four iterative schemes for the three grids 6

7 Figure 3: Discretization errors against spatial interval for the four iterative schemes solution and the numerical solution given by iterative solvers. Plots of discretization errors against spatial interval of the grid for the four iterative schemes are shown in Figure 3. ( x) 2 against spatial interval is also ploted in the same diagram for reference. It is a little bit surprising to see that with Jacobi scheme and Gauss-Seidel scheme, when the grid becomes finer, the discretization errors can even increase. This phenomenon might be caused by the inaccuracy of exact solution computation since we only sum up the first 30 terms of the series expansion Jacobi G-S SOR Line Inversion Table 2: Discretization errors for the four iterative schemes with different grid sizes Solutions provided by different iterative schemes are very close such that the difference can not be figured out by looking at the temperature contours. For reference, steady temperature distribution solved by Jacobi scheme for different grid sizes is shown in Figure 4. The converged temperature on the vertical centerline for the three grids is shown in Figure 5. It 7

8 (a) (b) (c) Figure 4: Temperature distribution solved by Jacobi scheme for the three grids 8

9 Figure 5: Converged temperature on the vertical centerline for the three grids can be observed that difference in grid size does not bring about much difference in the converged temperature profile. 5 Conclusions In this project we successfully developed a program to solve two-dimensional steady temperature distribution in a square domain by using four iterative schemes with different grid sizes. Convergence was assumed to be reached when maximum norm of changes in the temperature of nodes between iterations is less than C. Among the four schemes, Jacobi scheme has the lowest convergence speed. But it is highly parallelizable and can run efficiently on parallel computers. Successive Over-relaxation scheme with appropriate ω has the highest convergence speed. Temperature contours and plots of temperature profile along the vertical centerline for different grid size were also provided. The accuracy of solution given by the four iterative schemes is close to each other. References [1] Joel H. Ferziger and Milovan Peric, Computational Methods for Fluid Dynamics, 3rd edition. 9