MECH300H Introduction to Finite Element Methods

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MECH300H Introduction to Finite Element Methods
Lecture 7 Finite Element Analysis of 2-D Problems
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2-D Discretization
Common 2-D elements
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2-D Model Problem with Scalar Function- Heat
Conduction

• Governing Equation

in W

• Boundary Conditions

Dirichlet BC
Natural BC
Mixed BC
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Weak Formulation of 2-D Model Problem

• Weighted - Integral of 2-D Problem -----

• Weak Form from Integration-by-Parts -----

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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.
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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.
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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
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FEM Implementation of 2-D Heat Conduction Shape
Functions
linear triangular element area coordinates
T1
A2
A3
A1
T3
T2
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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 doesnt 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

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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
f4
f1
f3
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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
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FEM Implementation of 2-D Heat Conduction
Element Equation
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Assembly of Stiffness Matrices
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Imposing Boundary Conditions
The meaning of qi
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3
1
1
1
2
2
3
3
1
1
1
2
2
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Imposing Boundary Conditions
Consider
Equilibrium of flux
FEM implementation
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Calculating the q Vector
Example
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2-D Steady-State Heat Conduction - Example
A
D
AB and BC
CD convection
DA
0.6 m
C
B
0.4 m
y
x