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6. Strong and weak forms - two- and three-dimensional heat flow
Finite Element Method
Differential Equation
Weak Formulation
Approximating Functions
Weighted Residuals
FEM - Formulation
Heat flow in two and three dimensions
Fundamental Equations- two and three dimensional heat flow
Flux vector qn
Gradient T
Material point Body
Constitutive law
BalanceHeat source
Q
Temperature T
Differential eq.
?
Gradient
6.1 Heat flux vector
• Heat flux, q (w/m2)
• Heat flux over boundary, qn (w/m2)
qn positive when heat is leaving the body since nis directed out from boundary
qt
• Heat supply, Q (W/m3)
– Q is a scalar and is positive if heat is supplied
• Temperature, T (°K)
• Temperature gradient, (°K/m)T
6.1 Constitutive relation
• Heat conductivity, D (W/m °K)
• Two possibilities
– scalar, k
– matrix,
1-dim
2-dim ̶ ?
Fourier’s law
Cool
Hot
Heat flows from hot to cool regions
6.1 Constitutive relation
• Isotropic material – scalar, k
• Orthotropic material, (wood, fibre reinforced plastic)
– matrix,
• Anisotropic material
– matrix,
•
• D positive definite =>
Hot
Cool
Cool
Hot
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 Qeq. (1)
6.2 Heat equation for two and three dimensions - strong form
• Differential equation– Insert constitutive equation in balance equation
• D is orthotropic
• D = k, isotropic material
• tk is constant
• No heat supply, Q=0
quasi-harmonic equations
Poisson equation
Laplace equation(harmonic equation)
6.2 Heat equation for two and three dimensions - strong form
• Boundary conditions (randvillkor)
• Strong form
6.2 Heat equation for two and three dimensions - strong form
• Strong form in three-dimensions
Fundamental Equations- two and three dimensional heat flow
Flux vector qn
Gradient T
Material point Body
Constitutive law
BalanceHeat 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
• Insert const. eq. and the natural bc:
6.3 Weak form of heat flow in two and three dimensions
• and in three dimensions
