Download Equation-Based Modeling: Building your Equations from scratch

Equation-Based Modeling:
Building your Equations from scratch
∂ 2u
∂u
ea 2 + d a
− ∇ ⋅ (c∇u + αu − γ ) + β ⋅ ∇u + au = f
∂t
∂t
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Outline
• Demo of using built-in physics interface
• Demo of the same, using PDE interface
• Adding other equation-based modeling features
A quick demo using the built-in Heat
Transfer in Solids physics interface
A spinning wafer gets heated up by a laser moving
back and forth over the surface
Radiation to ambient cools the wafer
Built-in functionality used:
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Heat Transfer in Solids interface
Translational motion feature
Heat Flux boundary condition
Diffuse radiating surface boundary condition
Coupling operators are used to monitor the
average, minimum & maximum temperature
Now we will implement the same model
using the equation-based PDE interface
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Coefficient Form/General Form/Weak Form
Manual boundary conditions
Couplings to domains we do not want to model
Couplings to implement a feedback control system
Let’s look at the Heat Transfer equation
in a more general way
∂T
ρC p
− ∇ ⋅ (k∇T ) = 0
∂t
d a ut − ∇ ⋅ (c∇u ) = 0
Equation of Transient
Heat Transfer in a Solid
Generic Parabolic Equation
COMSOL provides a general coefficient form
∂ 2u
∂u
ea 2 + d a
− ∇ ⋅ (c∇u + αu − γ ) + β ⋅ ∇u + au = f
∂t
∂t
ea = α = γ = β = a = f = 0
∂ 2u
∂u
ea 2 + d a
− ∇ ⋅ (c∇u + αu − γ ) + β ⋅ ∇u + au = f
∂t
∂t
d a ut − ∇ ⋅ (c∇u ) = 0
Where to get started with Equationbased modeling
Implementing Conductive Heat Transfer
in the Coefficient Form Interface
Now lets add in the advective term
∂T
ρC p
+ ρC p u ⋅ ∇T − ∇ ⋅ (k∇T ) = 0
∂t
u: velocity vector
∂T
ρC p
− ∇ ⋅ (k∇T ) = − ρC p u ⋅ ∇T
∂t
 ∂T
∂T 

u ⋅ ∇T =  u x
+ uy
∂y 
 ∂x
 ∂T
∂T 
∂T

ρC p
+ uy
− ∇ ⋅ (k∇T ) = − ρC p  u x
∂y 
∂t
 ∂x
Implementing Advective Term, 1st way
How to evaluate derivatives?
Solution field:
u
Spatial 1st derivatives:
ux, uy, uz
Spatial 2nd derivatives:
uxx, uxy, …, uyz, uzz
Time derivatives:
Mixed derivatives:
Derivatives tangent to surfaces:
ut, utt
uxt, uytt
uTx, uTy, uTz
Implementing Advective Term, 2nd way
Implementing Conductive Heat Transfer
in the General Form Interface
Implementing boundary conditions
• All boundary conditions conditions are either:
Mixed (or Robin) condition:
n ⋅ (c∇u + α u − γ ) + hu − g = 0
Dirichelet condition:
u−r =0
For Heat Transfer:
α= γ = 0
n ⋅ (c∇u ) = g − hu
All Heat Transfer conditions can be
represented with the same interface
Insulation:
Heat Flux into domain:
Convective condition:
Radiative condition:
− n ⋅ (c∇u ) = 0
Default (natural)
boundary condition
− n ⋅ (c∇u ) = q ' '
− n ⋅ (c∇u ) = h(Tair − T )
4
− n ⋅ (c∇u ) = εσ (Tamb
−T 4)
A fixed temperature (or Dirichelet) condition could also be used,
but it is often more realistic to use a very high convection coefficient
Implementing a Heat Flux Condition
Implementing a Radiative Flux Condition
Now lets add:
• Convective heat flux to ambient gas
– A volume of air, that we do not want to model,
gets heated up by the wafer
Considering a volume of gas
• Known volume (mass) of air
• Assume that air is well-mixed
• Assume a heat transfer coefficient
between wafer and air
Spinning wafer
Temperature variation
of a well-mixed volume
of air of known mass is:
·
′′
20
Integral of heat flux
out of the wafer
A Global Equation is used to add an
additional degree of freedom to the model
f (u , ut , utt , t ) = 0
Implementing the air volume
Next lets add:
• Convective heat flux to ambient gas
– A volume of air, that we do not want to model,
gets heated up by the wafer
• A simpler temperature controller
– Heat the wafer until the minimum temperature
goes above 100°C
The minimum integration operator can
control the heat flux
• If the
minimum
temperature
goes above
100°C then
turn off the
heat flux
Features covered…
• Setting up an equation from the coefficient form
• Adding additional terms
– Multiple ways of addressing the same problem
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Evaluation of derivatives
Implementing boundary conditions
Adding addition Global Equations
Using coupling variables for feedback control
Questions?