2 T. x + 2 T. , T( x, y = 0) = T 1

Transcription

1 LAB 2: Conduction with Finite Difference Method Objective: The objective of this laboratory is to introduce the basic steps needed to numerically solve a steady state two-dimensional conduction problem using the finite difference method. The impact of mesh refinement on accuracy will also be investigated by comparing to the analytical solution obtained in Lab 1. Background: We will write a simple code for solving two-dimensional conduction problems using the finite difference method to outline the basic steps required. In the following labs we will modify this code to handle more complicated boundary conditions, improve its performance, and include variable thermal conductivity. We will solve the same problem as the one for which we obtained an analytical solution last week in lab. Recall that we solved for the steady-state two-dimensional temperature distribution, T(x, y), in a solid where the governing equation is the 2-D heat diffusion equation (or Laplace s equation) and the boundary conditions are specified as follows 2 T x + 2 T 2 y = 0 (1) 2 T( x = 0,y) = T 1, T( x = L x, y) = T 1, T( x, y = 0) = T 1, and T( x, y = L y ) = T 2. (2) For our finite difference code there are three main steps to solve problems: 1. Define the mesh 2. Formulate the finite difference form of the governing equation 3. Solve the system of linear equations simultaneously y y = L y T (y = L y ) = T 2 T (x = 0) = T 1 T (x = L x ) = T 1 x T (y = 0) = T 1 x = L x Figure 1. Schematic of two-dimensional domain for conduction heat transfer.

2 For step 1, use a mesh with regular node spacing x = y and with nodes located on the boundaries. For the x-direction, let N x +1 be the number of nodes and i = 1, 2,..., N x + 1 be the index for nodes. For the y-direction, let N y +1 be the number of nodes and j = 1, 2,..., N y + 1 be the index for nodes. Thus, the x and y-coordinates can be evaluated using the following x i = ( i 1) Δx and y j = ( j 1) Δy (3) For step 2 for this problem, we only need to formulate a finite difference equation for interior nodes because temperature is constant on the boundaries. From our derivation in class, Equation (1) can be approximated using finite differences for x = y as T i, j = 1 ( 4 T + T + T + T i 1, j i+1, j i, j 1 i, j +1) (4) which states that the temperature at any interior node is the average of the surrounding nodes. For the boundary nodes, simply set temperature to apply each boundary condition: T 1, j = T 1, T Nx +1, j = T 1, T i,1 = T 1, and T i,ny +1 = T 2. (5) Note that the corners are called singularities where the temperature jumps from T 1 to T 2 and that its value can be set to T 1, T 2 or (T 1 + T 2 )/2. This will not influence your solution, but will affect the appearance of your contour plot. For our fixed temperature boundary conditions, we now have a set of (N x - 1)(N y - 1) linear equations to solve. If the boundary conditions involve specified heat transfer or convection then we would have a set of (N x + 1)(N y + 1) linear equations to solve. For step 3 we will use Gauss- Seidel iteration that consists of the following steps (where p is the iteration number): a. Assume an initial guess for each T p i, j (where p = 0 for initial iteration) For our example, a good initial guess would be either T 1, T 2, or 0 C. Picking a good initial guess will speed up convergence b. Set tolerance, tol, for convergence and maximum number of iterations, p max c. Solve for an updated T p+1 i, j at each node Always use most recently calculated values for each update Perform calculations in a systematic order to improve efficiency For increasing i and j in for loops, Equation (4) for interior nodes becomes +T p+1 +T p ( i, j 1 ) i, j+1 T p+1 i, j = 1 4 T p+1 p i 1, j +T i+1, j 2

3 d. Check for convergence Calculate normalized residual, Res, for the p + 1 iteration For increasing i and j in for loops, for interior nodes use Res p+1 = N y N x p+1 r i, j j=2 i=2 N y N x p+1 T i, j j=2 i=2, r p+1 i, j = 1 4 T p+1 i 1, j +T p+1 i+1, j +T p+1 p+1 p+1 ( i, j 1 +T i, j+1 ) T i, j Check if normalized residual drops below set tolerance, Res p+1 tol If this is false, return to step c If this is true or the maximum number of iterations has been reached, stop Once the temperature has converged, post-process the data by plotting out the solution and checking for agreement with the analytical solution. Laboratory: For this laboratory, you will write a MATLAB m-file to calculate the temperature using the finite-difference method, make a contour plot of the results, and compare your answer to an analytical solution for the same problem. Below are some general instructions on how to write your program. Please refer to the online MATLAB help files as needed for more complete information. 1. Open MATLAB and create an m-file named ME554_Lab_02.m (or anything else equally boring). Write a header for your program using % for comment lines. Your header should include the program name, description, your name, date created, date last modified, and a variable list with definitions. Continue to use many comment lines throughout your program to describe each subsequent set of commands. At the beginning of your code, include a clear statement to clear all previous variables and a clc statement to clear the display. 2. Set values for the geometry parameters: L x, L y, N x, N y, x, and y. Make sure x = y and that L x = N x x and L y = N y y, thus all of these parameters cannot be set indepently. Set them in such a way that it is impossible to input a value that is not correct. 3. Set values for the boundary condition parameters: T 1 and T Set a tolerance for convergence of the Gauss-Seidel iteration. Typically, a value of tol = 10-9 is sufficient for good accuracy. NOTE: recall that 10-9 = 1 x 10-9 = 1e-9 5. Set the maximum number of iterations for the Gauss-Seidel iteration to p max = 10 9 so that the program will quit eventually if there is a problem with convergence. You may need to increase this later for problems that take a large number of iterations to converge. 3

4 6. Initialize the temperature matrix to T 1 using the following code: T = T1*ones(Nx+1, Ny+1); 7. Set the fixed temperatures at the boundary nodes to the correct temperatures. Because all of the temperatures have been set to T1, the bottom and side boundaries are already specified correctly. The top boundary condition given by Equation (5) can be set using: T(2:Nx,Ny+1) = T2; T(1,Ny+1) = (T1 + T2)/2; T(Nx+1,Ny+1) = (T1 + T2)/2; 8. Begin the Gauss-Seidel iteration loop using either a for or while loop such as for p = 1:pmax Enter code outlined in steps 9, 10, and 11 inside this for loop. 9. Update the value of Τ i,j at each interior node. To do this, use two nested for loops: for j = 2:Ny for i = 2:Nx Insert Equation (4) here. Note that for Gauss-Seidel iteration since you always want to use the most recently calculated temperatures for your updated values, you only need a single two-dimensional matrix for Τ i,j. 10. Calculate the residual for each node, r i,j, after updating all the temperatures using two additional for loops similar to those in Step 9. Calculate the normalized residual using: Res = sum(sum(r(2:nx,2:ny)))/sum(sum(t(2:nx,2:ny))); 11. Print out the progress and check for conversion by comparing residual to tolerance using: fprintf( Iter = %8.0d - Res = %4.2e \n, p, Res); if (Res < tol) break % Exit iteration loop because of convergence 4

5 12. Check if the main iteration loop exited by reaching the maximum number of iterations: if (p == pmax) disp( Warning: code did not converge ) 13. Print out temperature data to the screen using fprintf to produce formatted output. The following lines work well: disp( Temperatures in brick in deg. C = ) for j = Ny+1:-1:1 fprintf('%7.2f', T(:,j)) fprintf('\n') These lines print out the temperatures in the order you expect for a xy-coordinate system with typical orientation. The %7.2f sets the format for the temperatures as floating point with seven spaces and two numbers after the decimal place. The \n adds a carriage return between lines. 14. Make a contour plot of the resulting temperatures. If you would like, you can add the plot within the main iteration loop allowing yourself to visualize how the temperatures converge. If you do this, put a pause(0.001) statement after the contourf statement to allow the plot to display properly. Assignment Submit your lab report as a single pdf file using PolyLearn that contains the items listed below. 1. Set the following values in your code: L x = L y = 1 m, T 1 = 0 C, and T 2 = 100 C (which are the same as for Lab 1). Include tables for temperature values at each location for N x = N y = 5 and N x = N y = 10. Label them Table 1 and Table 2 along with a descriptive caption above the table. Include x and y locations for each temperature on your table so you can determine which temperatures are evaluated at the same location. Use 4 decimal places for temperature data evaluated using double precision. 2. Compare the values for each case at the same coordinate locations. Are the temperatures the same or different (be quantitative) at each location where they overlap? Briefly explain why this result makes sense. 3. Include contour plots for the two cases in step 2. For each figure include Figure 1 and Figure 2 along with a descriptive caption below the figure. 4. Briefly explain the two reasons these contour plots are different and which one of these reasons has a more significant impact on the visual appearance of the contour plot. 5

6 5. Test the effect of mesh refinement on the accuracy of your results by comparing your numerical results to the analytical solution from Lab 1. In particular, for the same geometry and boundary conditions above, solve for the temperature at the node located at x = y = 0.4 m (add lines to your code to do this) for N x = N y = 5, 10, 20, 30, 40, and 50. Make a table and plot temperature at this location versus number of nodes. Add the analytical value to this plot as a horizontal line. Make sure you output the temperature data to four decimal places. Call these Table 3 and Figure 3 and include descriptive captions. Does the numerical solution appear to be converging to the analytical solution (be quantitative)? How many nodes are required for your mesh to achieve sufficient accuracy? How can you determine if your mesh has been sufficiently refined? 6. When will round-off error start to impact your solution? To help you answer this question use the MATLAB function eps() to determine the accuracy of both double and single precision floating point calculations and compare this to the magnitude of your residuals. 7. Include a copy of your final m-file. 6