10. FE formulation
- two- and three-dimensional heat flow
Finite Element Method
Differential Equation
Weak Formulation Approximating
Functions
Weighted Residuals
FEM - Formulation
6.1 Constitutive relation
2-dim:
3-dim:
6.2 Heat equation for two and three dimensions - strong form
• Balance equation 2-dim
– Steady state (Stationary) => inflow = outflow
• and
• Eq. (1) may be written
• The region A is arbitrary => Balance equation
A Q eq. (1)
6.2 Heat equation for two and three dimensions - strong form
• Boundary conditions
• Strong form
Fundamental Equations
- two and three dimensional heat flow
Flux vector qn
Gradient
T
Material point Body
Constitutive law
Balance Heat source
Q
Temperature T
Differential eq.
6.3 Weak form of heat flow in two and three dimensions
• Start with balance equation (not diff. eq.)
• 1. multiply with arbitrary weight function v=v(x,y)
• 2. integrate over region
• 3. Integrate first term by parts (Green-Gauss theorem)
6.3 Weak form of heat flow in two and three dimensions
• Insert the rewritten first term
• Use the natural boundary condition:
• Insert Fourier’s law:
FE formulation
-two dimensional heat flow
• Approximate the temperature
• Temperature gradient
• which gives
Na ) (x
T N [N1 N2...Nn]
;
y T x T T
Tn
T T
2 1
a
y N x N
y N x N
y N x N
n n
2 2
1 1
B
Ba
T where B N
FE formulation
-two dimensional heat flow
• Insert approximation in weak form
• Choose the weight function according to Galerkin
• since v is a scalar
• Insert in weak form
Nc v
T
v cTN v cTBT BT ( N )T
FE formulation
-two dimensional heat flow
• As cT is arbitrary constants
• or
• where
Properties of matrices
• K is symmetric
• K is singular
• K is positive semi-definite
• Force balance is fulfilled
Finite element formulations
1. Global form
2. Element form
3. Expanded
element form
Example
Line source (Point source)
• Line source ( a point in 2D ), use
d
–function in 2-dim• 2D integral of
d
–function• Insert source term in fl of the FE formulation
Convection in two-dimensions
• Divide into three boundaries
• Convection may be written
• Insert approximation T=Na
Convection in two-dimensions
• Insert in convective boundary
• The FE equations is re-written
Note that
Fictitious convective layer
Transition elements:
- fictitious elements taking convection into account
𝑞𝑥 = 0
𝑞𝑦 = 𝛼(𝑇 − 𝑇∞)
y k T q
x k T q
y T x T k
k q
q
y y
x x
y x y
x
0 0
𝑇∞
𝑞𝑛 = 𝛼(𝑇 − 𝑇∞)
x y
𝑇 t
𝑇∞ 𝑇
t T k T
y k T q
q
y y
y x
) (
0
t k
k
y x
0
D t
0
0 0
Three dimensional heat flow
• Weak form
• Insert approximation and use Galerkin
Three dimensional heat flow
• Can be written as
• where
