1. MECH300H Introduction to Finite Element Methods
Lecture 7
Finite Element Analysis of 2-D Problems

2. 2-D Discretization
Common 2-D elements:

3. 2-D Model Problem with Scalar Function - Heat Conduction
Governing Equation
in W
Boundary Conditions
Dirichlet BC:
Natural BC:
Mixed BC:

4. Weak Formulation of 2-D Model Problem
Weighted - Integral of 2-D Problem -----
Weak Form from Integration-by-Parts -----

5. Weak Formulation of 2-D Model Problem
Green-Gauss Theorem -----
where nx and ny are the components of a unit vector, which is normal to the boundary G.

6. Weak Formulation of 2-D Model Problem
Weak Form of 2-D Model Problem -----
EBC: Specify T(x,y) on G
NBC: Specify on G
where is the normal
outward flux on the boundary G at the segment ds.

7. FEM Implementation of 2-D Heat Conduction – Shape Functions
Step 1: Discretization – linear triangular element T1
Derivation of linear triangular shape functions: T3
Let T2
Interpolation properties
Same

8. FEM Implementation of 2-D Heat Conduction – Shape Functions
linear triangular element – area coordinates T1 A2 A3 A1 T3 T2 f1 f2 f3

9. Interpolation Function - Requirements
Interpolation condition
Take a unit value at node i, and is zero at all other nodes
Local support condition
fi is zero at an edge that doesn’t contain node i.
Interelement compatibility condition
Satisfies continuity condition between adjacent elements over any element boundary that includes node i
Completeness condition
The interpolation is able to represent exactly any displacement field which is polynomial in x and y with the order of the interpolation function

10. Formulation of 2-D 4-Node Rectangular Element – Bi-linear Element
Let
Note: The local node numbers should be arranged in a counter-clockwise sense. Otherwise, the area
Of the element would be negative and the stiffness matrix can not be formed. f2 f1 f4 f3

11. FEM Implementation of 2-D Heat Conduction – Element Equation
Weak Form of 2-D Model Problem -----
Assume approximation:
and let w(x,y)=fi(x,y) as before, then
where

12. FEM Implementation of 2-D Heat Conduction – Element Equation

13. Assembly of Stiffness Matrices

14. Imposing Boundary Conditions
The meaning of qi: 3 3 1 1 1 2 2 3 3 1 1 1 2 2

15. Imposing Boundary Conditions
Consider
Equilibrium of flux:
FEM implementation:

16. Calculating the q Vector
Example:

17. 2-D Steady-State Heat Conduction - Example
AB and BC:
CD: convection
DA:
0.6 m C B
0.4 m y x