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4. Strong and weak formulations - one dimensional heat equation
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Direct Approach
Repitition
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Finite Element Method
Differential Equation
Weak Formulation
Approximating Functions
Weighted Residuals
FEM - Formulation
TodayOne-dim. Heat equation
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Vector
Classification of Problems
Scalar
1-D
2-D
3-D
T(x)
u(x)
T(x,y)
T(x,y,z)
u(x)
u(x) = v(x)
(x)
u(x) =
u(x,y,z)
v(x,y,z)
w(x,y,z)
u(x,y)
v(x,y)u(x) =
Heat flow
Spring elements
2D Flow
3D Flow
Beam
2D - solid
3D - solid
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4.1 One-dimensional heat equation- strong form (stationary flow)
• T(x) – temperature distribution [K]
• A(x) – cross section area [m]
• Q – internal heat source [J/s m] or [W/m]
Strong Form : • Differential equation• Boundary conditions• Region
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Balance equation
• Study an infinitely small part of the 1-D body
H - Heat flow [J/s] or [W]
Heat balance:(Input positive)
or
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Balance equation, contdbut
The balance equation becomes:
H - Heat flow [J/s] or [W]q – Heat flux [W/m2]
Compare with stresses:A
N
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Constitutive relation(Material law)
• Fourier’s law (1822):
(Heat flows from hotter to cooler)
T
x
20
10 Direction of heat flow
k = thermal conductivity [W/mK]
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Differential Equation
• Inserting constitutive relation into balance eq.:
Natural, (Neumann)
Essential, (Dirichlet)
Known heat flux:
Known temperature:
Boundary conditions
Boundary conditions must always be known at all boundaries
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Fundamental Equations- One dimensional heat flow
Flux vector qn
Gradient T
Material point Body
Constitutive law
dx
dTkq
Balance
QAqdx
d)(
Heat source Q
Temperature T
Differential eq.
0
Q
dx
dTAk
dx
d
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Strong Form
1. differential equation
2. region
3. boundary conditions
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Integration by Parts(mathematical reminder)
• By definition we have: (Fundamental Theorem of Calculus)
• Assume that
• Eq. (2) in (1) implies
• or
(1)
(2), differentiate
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4.4 Weak form of one-dimensional heat flow
• Start with differential eq.
• Multiply with arbitrary weight function, v(x)
• Integrate over region
• Integrate by parts
=
=choose:
(3)
(4)
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• Eq. (4) in (3) results in
• Inserting x=0 and x=L in the boundary term and using that
• the boundary term may be written as
4.4 Weak form of one-dimensional heat flow
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4.4 Weak form of one-dimensional heat flow
• the weak form of one dimensional heat flow is obtained
1. integral equation
2. essential boundary condition (Temperature boundary cond.)
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4.4 Advantages of the weak formulation
• only first derivatives of the temperature
• simpler approximating functions can be used
• discontinuities are allowed
+