1. MECH593 Introduction to Finite Element Methods
Finite Element Analysis of 2-D Problems
Dr. Wenjing Ye

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 -----
Ω 𝜕 𝜕𝑥 𝜅𝑤 𝜕𝑇 𝜕𝑥 𝑑𝑥𝑑𝑦= Γ 𝜅𝑤 𝜕𝑇 𝜕𝑥 𝑛 𝑥 𝑑𝑠
Ω 𝜕 𝜕𝑦 𝜅𝑤 𝜕𝑇 𝜕𝑦 𝑑𝑥𝑑𝑦= Γ 𝜅𝑤 𝜕𝑇 𝜕𝑦 𝑛 𝑦 𝑑𝑠
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 -----
Ω 𝜕𝑤 𝜕𝑥 𝜅 𝜕𝑇 𝜕𝑥 + 𝜕𝑤 𝜕𝑦 𝜅 𝜕𝑇 𝜕𝑦 −𝑤𝑄 𝑥,𝑦 𝑑𝑥𝑑𝑦 − Γ 𝑤 𝜅 𝜕𝑇 𝜕𝑥 𝑛 𝑥 + 𝜅 𝜕𝑇 𝜕𝑦 𝑛 𝑦 𝑑𝑠 =0
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 (T3) T1
Derivation of linear triangular shape functions: T2
Let T3
Interpolation properties
Same

8. FEM Implementation of 2-D Heat Conduction – Shape Functions
linear triangular element – local (area) coordinates T1
𝜙 1 = 1 𝑥 𝑦 2 𝐴 𝑒 𝑥 2 𝑦 3 − 𝑥 3 𝑦 2 𝑦 2 − 𝑦 3 𝑥 3 − 𝑥 2 = 𝐴 1 𝐴 𝑒 =𝜉 A2 A3 A1 T2
𝜙 2 = 1 𝑥 𝑦 2 𝐴 𝑒 𝑥 3 𝑦 1 − 𝑥 1 𝑦 3 𝑦 3 − 𝑦 1 𝑥 1 − 𝑥 3 = 𝐴 2 𝐴 𝑒 =𝜂 T3 T1
𝜉=1
𝜙 3 = 1 𝑥 𝑦 2 𝐴 𝑒 𝑥 1 𝑦 2 − 𝑥 2 𝑦 1 𝑦 1 − 𝑦 2 𝑥 2 − 𝑥 1 = 𝐴 3 𝐴 𝑒 =1−𝜉−𝜂=𝜁
𝜂=1 T2 T3
𝜉=0 f1 f2 f3
𝜂=0

9. FEM Implementation of 2-D Heat Conduction – Shape Functions
quadratic triangular element (T6) – local (area) coordinates T1 T4 T6
𝜙 1 =𝜉 2𝜉−1
𝜙 2 =𝜂 2𝜂−1
𝜙 3 =𝜁 2𝜁−1
𝜙 4 =4𝜉𝜂
𝜙 5 =4𝜂𝜁
𝜙 6 =4𝜉𝜁 T2 T5 T3 T1
𝜉=1
𝜂=1 T4 T6
𝜉=0.5 T2 T3 T5
𝜉=0
𝜂=0
Serendipity Family – nodes are placed on the boundary
for triangular elements, incomplete beyond quadratic

10. 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

11. Formulation of 2-D 4-Node Rectangular Element – Bi-linear Element (Q4)
Let h
𝜙 1 = −𝜉 1−𝜂 4 3
𝜙 2 = 𝜉 1−𝜂 x
𝜙 3 = 𝜉 1+𝜂 1 2
𝜙 4 = −𝜉 1+𝜂
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

12. Formulation of 2-D 4-Node Rectangular Element – Bi-linear Element (Q4)
Physical domain (physical element)
Reference domain (master element) h h 3 4 4 3 x x y 1 1 2 2 x
𝑥= 𝜙 1 𝑥 1 + 𝜙 2 𝑥 2 + 𝜙 3 𝑥 3 + 𝜙 4 𝑥 4
𝑦= 𝜙 1 𝑦 1 + 𝜙 2 𝑦 2 + 𝜙 3 𝑦 3 + 𝜙 4 𝑦 4

13. FEM Implementation of 2-D Heat Conduction – Element Equation
Weak Form of 2-D Model Problem -----
Ω 𝑒 𝜕𝑤 𝜕𝑥 𝜅 𝜕𝑇 𝜕𝑥 + 𝜕𝑤 𝜕𝑦 𝜅 𝜕𝑇 𝜕𝑦 −𝑤𝑄 𝑥,𝑦 𝑑𝑥𝑑𝑦 + Γ 𝑒 𝑤 𝑞 𝑛 𝑑𝑠 =0
𝑇= 𝑗=1 𝑛 𝑇 𝑗 𝜙 𝑗
Assume approximation:
and let w(x,y)=fi(x,y) as before, then
Ω 𝑒 𝜕 𝜙 𝑖 𝜕𝑥 𝜅 𝜕 𝜕𝑥 𝑗=1 𝑛 𝑇 𝑗 𝜙 𝑗 + 𝜕 𝜙 𝑖 𝜕𝑦 𝜅 𝜕 𝜕𝑦 𝑗=1 𝑛 𝑇 𝑗 𝜙 𝑗 − 𝜙 𝑖 𝑄 𝑥,𝑦 𝑑𝑥𝑑𝑦 + Γ 𝑒 𝜙 𝑖 𝑞 𝑛 𝑑𝑠 =0
𝑗=1 𝑛 𝐾 𝑖𝑗 𝑇 𝑗 = Ω 𝑒 𝜙 𝑖 𝑄 𝑥,𝑦 𝑑𝑥𝑑𝑦 − Γ 𝑒 𝜙 𝑖 𝑞 𝑛 𝑑𝑠
where

14. Linear Triangular Element Equation
𝑗=1 𝑛 𝐾 𝑖𝑗 𝑇 𝑗 = Ω 𝑒 𝜙 𝑖 𝑄 𝑥,𝑦 𝑑𝑥𝑑𝑦 − Γ 𝑒 𝜙 𝑖 𝑞 𝑛 𝑑𝑠
where 𝒍 𝒊𝒋 is the length vector from the ith node to the jth node.

15. Assembly of Stiffness Matrices
𝐹 𝑖 𝑒 = Ω 𝑒 𝜙 𝑖 𝑄 𝑥,𝑦 𝑑𝑥𝑑𝑦 − Γ 𝑒 𝜙 𝑖 𝑞 𝑛 𝑑𝑠 = 𝑗=1 𝑛 𝑒 𝐾 𝑖𝑗 𝑒 𝑇 𝑗 𝑒
𝑈 1 = 𝑇 , 𝑈 2 = 𝑇 𝑇 , 𝑈 3 = 𝑇 𝑇 , 𝑈 4 = 𝑇 , 𝑈 5 = 𝑇 ,

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

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

18. Calculating the q Vector
Example:

19. 2-D Steady-State Heat Conduction - Example
AB:
CD: convection
DA and BC:
0.6 m C B
0.4 m y x