1. HOT PLATE CONDUCTION NUMERICAL SOLVER AND VISUALIZER Kurt Hinkle and Ivan Yorgason

2. INTRODUCTION There are analytical methods that, in certain cases, can produce exact mathematical solutions to 2D steady state conduction problems. There are even solutions that are available for simple geometries with specific boundary conditions that can be used simply by plugging in numbers. Sometimes, however, there are geometries and/or boundary conditions that are not covered by the aforementioned solutions. When this occurs, numerical techniques, such as finite-difference, finite- element, and boundary-element methods are used to provide approximate solutions. This project uses the finite-difference form of the heat equation to solve for the temperatures across a square plate.

3. LIMITATIONS AND ASSUMPTIONS 2D steady state conduction Constant wall temperatures No convection Square plate Square elements Temperatures ranging 0ºC - 1000ºC Mesh size ranging 3 - 80

4. METHOD

5. Mesh

6. METHOD 1000ºC 500ºC 0ºC 100ºC 0ºC Initial Values 500ºC 1000ºC 0ºC 100ºC

7. METHOD 1000ºC 500ºC 0ºC 100ºC 0ºC 375ºC 0ºC Calculate First Element Temperature 500ºC 1000ºC 0ºC (1000ºC + 500ºC + 0ºC + 0ºC)/4 = 375ºC ? 100ºC

8. METHOD 1000ºC 500ºC 0ºC 100ºC 80.1ºC 179.7ºC 82.6ºC 140.6ºC 218.8ºC 150.4ºC 343.8ºC 375ºC 360.9ºC 1 st Iteration Complete 500ºC 1000ºC 0ºC 100ºC

9. METHOD 1000ºC 500ºC 0ºC 100ºC 144.6ºC 228.5ºC 116.7ºC 267.2ºC 333.9ºC 222.1ºC 504.3ºC 515.6ºC 438.7ºC 2 nd Iteration Complete 500ºC 1000ºC 0ºC 100ºC

10. METHOD 1000ºC 500ºC 0ºC 100ºC 177.5ºC 259.9ºC 133.4ºC 333.6ºC 395.1ºC 255.9ºC 572.6ºC 584.6ºC 473.7ºC 3 rd Iteration Complete 500ºC 1000ºC 0ºC 100ºC

11. METHOD Differences with finite-difference method Instead of setting up a matrix and inverting it to solve for all temperatures at once, the temperatures are solved for through an iterative process. This iterative process (N^2 algorithm) is limited by a time which is calculated based on the mesh size. Larger mesh sizes are allowed more time to iteratively solve for the element temperatures.

12. FUNCTIONALITY Mesh Size: The number of elements between opposite walls. Temperature: The temperature of the wall. Calculate: Calculates the element temperatures and displays them colorfully. Close: Closes the program. Print: Calculates the element temperatures and once the algorithm is complete, it prints the resulting element temperatures to results.dat in a matrix format along with the wall temperatures.

13. FUNCTIONALITY Live Demo: 14.exe

14. POST PROCESSING

15. FUTURE WORK Allow for other shapes and holes in the geometry Allow for different mesh element types (tetrahedral, etc.) Stop the iterative solver based on a tolerance instead of a time limit Export.jpg of visualized results with results.dat file Have the color scheme be relative to the maximum and minimum temperatures instead of the scale being absolute (1000ºC = red and 0ºC = blue).

16. CONCLUSION Provides quick and accurate results for the given assumptions Graphically displays the results in an understandable and pleasing manner With the option to print the results to a file, further analysis is easily accomplished The finite-difference form of the heat equation is easy to implement programmatically

17. QUESTIONS?