1. The Finite Element Approach to Thermal Analysis Appendix A

2. Training Manual Inventory #001445 March 15, 2001 A-2 Finite Element Approach Assume a simple polynomial variation of temperature within each element. Typically, linear, quadratic and mixed cubic terms may be included depending on the element type. The assumed polynomial is such that temperature continuity exists within the element and on inter-element boundaries. Write the polynomial in terms of the unknown values of the element nodal temperatures:

3. Training Manual Inventory #001445 March 15, 2001 A-3 Finite Element Approach ( continued ) Calculate the thermal gradients and thermal flux in each element in terms of the element nodal temperatures.

4. Training Manual Inventory #001445 March 15, 2001 A-4 Finite Element Approach ( continued ) Substituting the assumed temperature variation into the integral equation and noting that each term is multiplied by the virtual temperature and hence that term cancels on both sides, yields

5. Training Manual Inventory #001445 March 15, 2001 A-5 Finite Element Approach ( continued ) This equation can be rewritten in simplified form as:

6. Training Manual Inventory #001445 March 15, 2001 A-6 Finite Element Approach ( continued ) Where,

7. Training Manual Inventory #001445 March 15, 2001 A-7 Finite Element Approach ( continued ) The system equations are formed by assembling the element contributions

8. Training Manual Inventory #001445 March 15, 2001 A-8 Finite Element Approach ( continued ) Dimensional Analysis The prior equations allow one to quickly determine if we are using dimensionally-consistent units:

9. Training Manual Inventory #001445 March 15, 2001 A-9 Example: 3-Noded Triangle Element The finite element approach to heat transfer will be demonstrated in an example using a simplistic 3-noded triangular solid element. Use of a 4-noded solid is usually preferred, but in this case the linear triangular element is being used because of its more simplistic shape functions. Physical System: 1”x 1” isotropic planar solid Convection Boundary Condition; h f, T B Constant Temperature Boundary, T s = 0 Symmetry

10. Training Manual Inventory #001445 March 15, 2001 A-10 Example: 3-Noded Triangle Element (Continued) Finite Element Model: 2 triangular elements 4 nodes Derive Element 1 Matrices: 1 2 4 3 x y 1 2 1 2 3 1 Element shape functions

11. Training Manual Inventory #001445 March 15, 2001 A-11 Example: 3-Noded Triangle Element (Continued) Derive gradient- temperature matrix Define isotropic material property matrix Element conductance matrix

12. Training Manual Inventory #001445 March 15, 2001 A-12 Example: 3-Noded Triangle Element (Continued) Convection contribution to conductance matrix Convective nodal heat flow vector

13. Training Manual Inventory #001445 March 15, 2001 A-13 Example: 3-Noded Triangle Element (Continued) Use similar technique to derive element 2 matrices and combine to form global matrices Matrices may be partitioned as shown since T 3 = T 4 = zero Solve simultaneously to obtain unknown temperatures Q 3 and Q 4 are reaction heat flow rates

14. Training Manual Inventory #001445 March 15, 2001 A-14 Example: 3-Noded Triangle Element (Continued) Solve for reaction heat flow rates at node 3 of Element 1 Calculate thermal gradient vector for Element 1 Calculate thermal flux vector in Element 1 Gradient and flux are constant within element.