Finite Element Method

1
Finite Element Method
for readers of all backgrounds
G. R. Liu and S. S. Quek
CHAPTER 12

• FEM FOR HEAT TRANSFER PROBLEMS

2
CONTENTS

• FIELD PROBLEMS
• WEIGHTED RESIDUAL APPROACH FOR FEM
• 1D HEAT TRANSFER PROBLEMS
• 2D HEAT TRANSFER PROBLEMS
• SUMMARY
• CASE STUDY

3
FIELD PROBLEMS

• General form of system equations of 2D linear
  steady state field problems

(Helmholtz equation)

• For 1D problems

4
FIELD PROBLEMS

• Heat transfer in 2D fin

Note
5
FIELD PROBLEMS

• Heat transfer in long 2D body

Note
Dx kx, Dy tky, g 0 and Q q
6
FIELD PROBLEMS

• Heat transfer in 1D fin

Note
7
FIELD PROBLEMS

• Heat transfer across composite wall

Note
8
FIELD PROBLEMS

• Torsional deformation of bar

Note
Dx1/G, Dy1/G, g0, Q2q
(? - stress function)

• Ideal irrotational fluid flow

Note Dx Dy 1, g Q 0
(? - streamline function and ? - potential
function)
9
FIELD PROBLEMS

• Accoustic problems

P - the pressure above the ambient pressure w
- wave frequency c - wave velocity in the
medium
Note
, Dx Dy 1, Q 0
10
WEIGHTED RESIDUAL APPROACH FOR FEM

• Establishing FE equations based on governing
  equations without knowing the functional.

(Strong form)
Approximate solution
(Weak form)
Weight function
11
WEIGHTED RESIDUAL APPROACH FOR FEM

• Discretize into smaller elements to ensure better
  approximation
• In each element,
• Using N as the weight functions

where
Galerkin method
Residuals are then assembled for all elements and
enforced to zero.
12
1D HEAT TRANSFER PROBLEM
1D fin

• k thermal conductivity
• h convection coefficient
• A cross-sectional area of the fin
• P perimeter of the fin
• temperature, and
• ?f ambient temperature in the fluid

(Specified boundary condition)
(Convective heat loss at free end)
13
1D fin
Using Galerkin approach,
where D kA, g hP, and Q hP?
14
1D fin
Integration by parts of first term on right-hand
side,
Using
15
1D fin
(Strain matrix)
where
(Thermal conduction)
(Thermal convection)
(External heat supplied)
(Temperature gradient at two ends of element)
16
1D fin
For linear elements,
(Recall 1D truss element)
Therefore,
for truss element
(Recall stiffness matrix of truss element)
17
1D fin
for truss element
(Recall mass matrix of truss element)
18
1D fin
or
(Left end)
(Right end)
At the internal nodes of the fin, bL(e) and
bL(e) vanish upon assembly. At boundaries, where
temperature is prescribed, no need to calculate
bL(e) or bL(e) first.
19
1D fin
When there is heat convection at boundary,
E.g.
Since ?b is the temperature of the fin at the
boundary point, ?b ?j
Therefore,
20
1D fin
where
,
For convection on left side,
where
,
21
1D fin
Therefore,
Residuals are assembled for all elements and
enforced to zero KD F
Same form for static mechanics problem
22
1D fin

• Direct assembly procedure

or
Element 1
23
1D fin

• Direct assembly procedure (Contd)

Element 2
Considering all contributions to a node, and
enforcing to zero
(Node 1)
(Node 2)
(Node 3)
24
1D fin

• Direct assembly procedure (Contd)

In matrix form
(Note same as assembly introduced before)
25
1D fin

• Worked example Heat transfer in 1D fin

Calculate temperature distribution using FEM.
4 linear elements, 5 nodes
26
1D fin
Element 1, 2, 3
not required
,
Element 4
,
required
27
1D fin
For element 1, 2, 3
,
For element 4
,
28
1D fin
Heat source
(Still unknown)
?1 80, four unknowns eliminate Q
Solving
29
Composite wall
Convective boundary
at x 0
at x H
All equations for 1D fin still applies except
Recall Only for heat convection
and
vanish.
Therefore,
,
30
Composite wall

• Worked example Heat transfer through composite
  wall

Calculate the temperature distribution across the
wall using the FEM.
2 linear elements, 3 nodes
31
Composite wall
For element 1,
32
Composite wall
For element 2,
Upon assembly,
(Unknown but required to balance equations)
33
Composite wall
Solving
34
Composite wall

• Worked example Heat transfer through thin film
  layers

35
Composite wall
For element 1,
For element 2,
36
Composite wall
For element 3,
37
Composite wall
Since, ?1 300C,
Solving
38
2D HEAT TRANSFER PROBLEM
Element equations
For one element,
Note W N Galerkin approach
39
Element equations
(Need to use Gausss divergence theorem to
evaluate integral in residual.)
(Product rule of differentiation)
Therefore,
Gausss divergence theorem
40
Element equations
2nd integral
Therefore,
41
Element equations
42
Element equations
where
43
Element equations
Define
,
(Strain matrix)
?
44
Triangular elements
Note constant strain matrix
(Or Ni Li)
45
Triangular elements
Note
(Area coordinates)
E.g.
Therefore,
46
Triangular elements
Similarly,
Note b(e) will be discussed later
47
Rectangular elements
48
Rectangular elements
49
Rectangular elements
Note In practice, the integrals are usually
evaluated using the Gauss integration scheme
50
Boundary conditions and vector b(e)
Internal
Boundary
bB(e) needs to be evaluated at boundary
Vanishing of bI(e)
51
Boundary conditions and vector b(e)
Need not evaluate
Need to be concern with bB(e)
52
Boundary conditions and vector b(e)
on natural boundary ?2
Heat flux across boundary
53
Boundary conditions and vector b(e)
Insulated boundary
M S 0 ?
Convective boundary condition
54
Boundary conditions and vector b(e)
Specified heat flux on boundary
55
Boundary conditions and vector b(e)
For other cases whereby M, S ? 0
56
Boundary conditions and vector b(e)
where
,
For a rectangular element,
(Equal sharing between nodes 1 and 2)
57
Boundary conditions and vector b(e)
Equal sharing valid for all elements with linear
shape functions
Applies to triangular elements too
58
Boundary conditions and vector b(e)
for rectangular element
59
Boundary conditions and vector b(e)
Shared in ratio 2/6, 1/6, 1/6, 2/6
60
Boundary conditions and vector b(e)
Similar for triangular elements
61
Point heat source or sink
Preferably place node at source or sink
62
Point heat source or sink within the element
Point source/sink
(Delta function)
?
63
SUMMARY
64
CASE STUDY
Road surface heated by heating cables under road
surface
65
CASE STUDY
Heat convection Mh0.0034 Sff h-0.017
fQ
Repetitive boundary no heat flow across M0, S0
Repetitive boundary no heat flow across M0, S0
Insulated M0, S0
66
CASE STUDY
Surface temperatures