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Transcript
Equation-Based Modeling: Building your Equations from scratch ∂ 2u ∂u ea 2 + d a − ∇ ⋅ (c∇u + αu − γ ) + β ⋅ ∇u + au = f ∂t ∂t © 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. 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: • • • • • 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 • • • • 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 • • • • Evaluation of derivatives Implementing boundary conditions Adding addition Global Equations Using coupling variables for feedback control Questions?