ConfTokyo2014Mini PDE

8/17/2019 ConfTokyo2014Mini PDE

1/26

© Copyright 2014 COMSOL. Any of the images, text, and equations here may be

copied and modified for your own internal use. All trademarks are the property of

their respective owners. See www.comsol.com/trademarks.

Equation-Based Modeling:Building your Equations from scratch

f auuuuc

t

ud

t

ue aa =+∇⋅+−+∇⋅∇−

∂

∂+

∂

∂ β γ α )(

2

2


2/26

Outline

• Demo of using built-in physics interface

• Demo of the same, using PDE interface

• Adding other equation-based modeling features


3/26

A quick demo using the built-in HeatTransfer 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


4/26

Built-in functionality used:

• 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


5/26

Now we will implement the same modelusing the equation-based PDE interface

• 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


6/26

Let’s look at the Heat Transfer equation

in a more general way

0)( =∇⋅∇−

ucud t a

0)( =∇⋅∇−∂

∂T k

t

T C p ρ

Equation of Transient

Heat Transfer in a Solid

Generic Parabolic Equation


7/26

COMSOL provides a general coefficient form

0====== f aea β

f auuuuc

t

ud

t

ue aa =+∇⋅+−+∇⋅∇−

∂

∂+

∂

∂ β γ α )(

2

2

f auuuuct

ud

t

ue

aa

=+∇⋅+−+∇⋅∇−

∂

∂+

∂

∂ β γ α )(

2

2

0)( =∇⋅∇−

ucud t a


8/26

Where to get started with Equation-

based modeling


9/26

Implementing Conductive Heat Transfer

in the Coefficient Form Interface


10/26

Now lets add in the advective term

T C T k t

T C

p p

∇⋅−=∇⋅∇−

∂

∂u ρ ρ )(

∂

∂+

∂

∂=∇⋅

y

T u

x

T uT y xu

u: velocity vector

∂

∂+

∂

∂−=∇⋅∇−

∂

∂

y

T u

x

T uC T k

t

T C y x p p ρ ρ )(

0)( =∇⋅∇−∇⋅+∂

∂

T k T C t

T

C p p u ρ ρ


11/26

Implementing Advective Term, 1st way


12/26

How to evaluate derivatives?

Solution field:u

Spatial 1st derivatives: ux, uy, uz

Spatial 2nd derivatives: uxx, uxy, …, uyz, uzz

Time derivatives: ut, utt

Mixed derivatives: uxt, uytt

Derivatives tangent to surfaces: uTx, uTy, uTz


13/26

Implementing Advective Term, 2nd way


14/26

Implementing Conductive Heat Transfer

in the General Form Interface


15/26

Implementing boundary conditions• All boundary conditions conditions are either:

0

0)(

=−

=−+−+∇⋅

r u

g huuucnMixed (or Robin) condition:

Dirichelet condition:

For Heat Transfer: 0==

hu g uc −=∇⋅ )(n


16/26

All Heat Transfer conditions can be

represented with the same interface

'')( quc =∇⋅−nHeat Flux into domain:

Convective condition: )()( T T huc air −=∇⋅−n

Radiative condition: )()( 44

T T uc amb −=∇⋅− εσ n

0)( =∇⋅− ucnInsulation: Default (natural)boundary condition

A fixed temperature (or Dirichelet) condition could also be used,

but it is often more realistic to use a very high convection coefficient


17/26

Implementing a Heat Flux Condition


18/26

Implementing a Radiative Flux Condition


19/26

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


20/26

Considering a volume of gas

Spinning wafer

• Known volume (mass) of air

• Assume that air is well-mixed

• Assume a heat transfer coefficient

between wafer and air

′′

· 20

Temperature variation

of a well-mixed volume

of air of known mass is:

Integral of heat flux

out of the wafer


21/26

A Global Equation is used to add an

additional degree of freedom to the model

0),,,( =t uuu f tt t


22/26

Implementing the air volume


23/26

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


24/26

The minimum integration operator can

control the heat flux

•

If theminimum

temperature

goes above100°C then

turn off theheat flux


25/26

Features covered…• Setting up an equation from the coefficient form

• Adding additional terms

– Multiple ways of addressing the same problem

• Evaluation of derivatives

• Implementing boundary conditions

• Adding addition Global Equations

• Using coupling variables for feedback control


26/26

Questions?

Documents

ConfTokyo2014Mini PDE