Pre-Analysis: Example: Steady One-Dimensional Heat Conduction in A Bar
Pre-Analysis: Example: Steady One-Dimensional Heat Conduction in A Bar
Pre-Analysis: Example: Steady One-Dimensional Heat Conduction in A Bar
1. Mathematical model
2. Numerical solution procedure
3. Hand-calculations of expected results/trends
x
z
L
We are interested in finding the temperature
distribution in the bar due to heat conduction
1
Energy Conservation for an
Infinitesimal Control Volume
Infinitesimal
“Control Volume”
Δ
x
Δ
Δ
2
Numerical Solution:
Discretization
• Reduce the problem to determining temperature
values at selected locations (“nodes”)
T
1 2 3 4 x
x
We have assumed a shape for ( ) consisting of piecewise
polynomials
3
How to Derive System of Algebraic
Equations?
Piecewise Weighted Integral Piecewise polynomial
polynomial Form approximation for T
approximation for T
∫ ∫ + =0
(x) is an
+ =0 + =0 arbitrary piecewise
polynomial function
( ) is an
System of algebraic arbitrary function
eqs. in nodal System of algebraic eqs.
temperatures in nodal temperatures
∫ + dx = 0
(x) is an
arbitrary piecewise
polynomial function
4
Integration by Parts
• ∫ + =0
• w k − ∫ k +
∫ =0
dT dT
w k − k + =0
1 2 3 4
w + + 0.5 Δ − +
+ + + Δ +
+ + + Δ +
w + + 0.5 Δ + =0
5
dT dT
w k − k + =0
1 2 3 4
⋯+w + + 0.5 Δ +
dT dT
w k − k + =0
1 2 3 4
6
dT dT
w k − k + =0
1 2 3 4
w + + 0.5 Δ − +
+ + + Δ +
+ + + Δ +
w + + 0.5 Δ + =0
={ }
dT dT
w k − k + =0
1 2 3 4
={ }
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dT dT
w k − k + =0
1 2 3 4
w + + 0.5 Δ − +
+ + + Δ +
+ + + Δ +
w + + 0.5 Δ + =0
dT dT
w k − k + =0
1 2 3 4
+ = 0.5 Δ − ={ }
+ + = Δ
+ + = Δ
+ = 0.5 Δ +
8
Essential Boundary Conditions
=T + = 0.5QΔ −
+ + = Δ
+ + = Δ
+ = 0.5 Δ +
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Comparison of / between Finite-
Element and Exact Solutions
• Error in / > Error in
• Energy is not conserved for
each element
=− =
− 5.5 W/m
• Energy is
conserved for
the bar
10
How to Improve the Polynomial
Approximation?
• Increase no. of elements Original Mesh
• Increase order of polynomial 4
1 2 3
within each element
– Use more nodes per Refined Mesh
element
1 2 3 4 5 6 7
Second-Order Element
11
Finite-Element Analysis: Summary of
the Big Ideas
• Mathematical model to be solved is usually a boundary value
problem
• Reduce the problem to solving selected variable(s) at selected
locations (nodes)
• Assume a shape for selected variable(s) within each element
• Derive system of algebraic equations relating neighboring nodal
values
• Invert this system to determine selected variable(s) at nodes
• Derive everything else from selected variable(s) at nodes
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