1. Dr. Jennifer Parham-Mocello

2. What is an IDE? IDE – Integrated Development Environment Software application providing conveniences to computer programmers for software development. Consists of editor, compiler/interpreter, building tools, and a graphical debugger. Heat Diffusion / Finite Difference Methods2

3. Eclipse Java, C/C++, and PHP IDE Uses Java Runtime Environment (JRE) Install JRE/JDK - http://www.oracle.com/technetwork/java/index.html Need C/C++ compiler Install Wascana (Windows version) http://www.eclipselabs.org/p/wascana Heat Diffusion / Finite Difference Methods3

4. Using Eclipse Example – Open HelloWorld C++ project File -> New -> C++ Project Enter Project Name Building/Compiling Projects Project -> Build All Run -> Run Console Heat Diffusion / Finite Difference Methods4

5. Heat Diffusion Heat Diffusion / Finite Difference Methods5

6. 6 Heat Diffusion Equation Describes the distribution of heat (or variation in temperature) in a given region over time. For a function u(x, t) of one spatial variables(x) and the time variable t, the heat diffusion equation is: 1D or Material Parameters – thermal conductivity (k), specific heat (c), density (  )

7. Conceptual and theoretical basis Conservation of mass, energy, momentum, etc. Rate of flow in - Rate of flow out = Rate of heat storage Heat Diffusion / Finite Difference Methods7 1D 2D 3D

8. x=0.0 x=4.0 Physical Parameters Boundary Conditions Initial Conditions Wire with perfect insulation, except at ends Example 1D Heat Diffusion Problem Heat Diffusion / Finite Difference Methods8

9. Outline of Solution Discretization (spatial and temporal) Transformation of theoretical equations to approximate algebraic form Solution of algebraic equations Heat Diffusion / Finite Difference Methods9

10. Discretization Spatial - Partition into equally-spaced nodes Temporal - Decide on time stepping parameters x=0.0 x=1.0x=0.33x=0.67 u0u0 u1u1 u2u2 u3u3 Let t o = 0.0, t n = 10.0, and  t = 0.1 Heat Diffusion / Finite Difference Methods10

11. Approximate Theoretical with Algebra Finite difference approximations for first and second derivatives u0u0 u i-1 u i+1 unun u1u1 u n-1 uiui Heat Diffusion / Finite Difference Methods11

12. Approximation Heat Diffusion / Finite Difference Methods12

13. Algorithm for t=0,t n for each node, i predict u t+  t endfor Predicting u t+  t at each node Explicit solution Implicit solution (system of equations) Heat Diffusion / Finite Difference Methods13

14. Explicit Solution Heat Diffusion / Finite Difference Methods14

15. Simulation (Explicit Solution) Physical Parameters Boundary Conditions Initial Conditions u0u0 u1u1 u2u2 u3u3 Heat Diffusion / Finite Difference Methods15

16. Extension Two Dimensions u i,j u i,j-1 u i,j+1 u i+1,j u i-1,j Heat Diffusion / Finite Difference Methods16

17. Implement 1-D Heat Diffusion Open New C++ Project Name the Project Open New C++ source code file File -> New -> Source File Name C++ File (remember extension,.C,.c++,.cpp) Heat Diffusion / Finite Difference Methods17

18. EXTRAS Heat Diffusion / Finite Difference Methods18

19. Shorthand Notations Gradient (“Del”) Operator Heat Diffusion / Finite Difference Methods19

20. Divergence (Gradient of a vector field) Heat Diffusion / Finite Difference Methods20

21. Laplacian Operator Heat Diffusion / Finite Difference Methods21

22. Heat Diffusion Equation - rewritten LHS represents spatial variations RHS represents temporal variation Heat Diffusion / Finite Difference Methods22