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COMPUTER MODELLING

2. Governing Physics

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Computer Modelling

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Governing PhysicsMaterials Processing

Fluid Flow + Free surfaceChemical ReactionsElectromagnetic FieldsHeat Transfer

SolidificationStress

Many processes governed by interactions of

the above

MultiphysicsMultiscale

0.1 - 1m

Casting

Refining

Soldering

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Computer Modelling

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Partial Differential Equations Enabling Techn ology for Modelling

PDEs are used extensively to represent real world phenomena and

processes.

Heat transfer in nuclear reactors.Airflow around an aircraft.Structural dynamics of a bridge.Movement of money in financial markets.Etc.

Modelling and simulation of such processes requires solution of these

PDEs.

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Computer Modelling Reynolds transport theorem

Let Fbe a quantity of our interest, and Vis any region, which may be fixed

or moving. We are interested in

The case in 1D (Leibnizs theorem):

In general,

whereA is the boundary ofV, and uA

is the velocity of the boundary, and nthe unit outward normal toA

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d

dt

V(t)

F dV =?

ddt

b(t)x=a(t)

F(x, t) dx =ba

Ft

dx+ dbdt

F(b, t)

dadt

F(a, t)

d

dtV(t)

F dV= V(t)

F

tdV+

A(t)

FuA

n dA

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Computer Modelling Reynolds transport theorem: special cases

1. When Vis fixed in space:

2. When Vmoves with the particles of the continuum, then

where u is the velocity of the continuum

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d

dt

V

F dV=

V

F

tdV

d

dt

V

F dV=

V

F

tdV+

V

Fu n dA

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Computer Modelling Mass balance (continuity equation)

For a fixed volume V, the change of total mass within it is equal to the

influx of mass; hence,

Using Gauss divergence theorem,

as a result,

But Vis arbitrary, as a result,

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V

tdV=

A

u n dA

A

u n dA =

V

(u) dV

V

t + (

u)

dV = 0

t+ (u) = 0

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Computer Modelling

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Momentum balance

Momentum balance (fluids & solids):

dv

dt=

v

t+ v v

where

dv

dt= + b

: stress tensor

b: body force per unit volume

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Computer Modelling

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Governing Equations - Elasticity

Momentum balance in Linear isotropic solid:

Constitutive law and geometrical equations:

Substituting, we solve the momentum balance eqn indisplacements:

: strain

,: Lamparameters

v

dt= + b

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Computer Modelling Stress Equations

Equilibrium equations, quasistatic :

Thermal and Creep/plastic strains:

Thermal strains (linear):

Plasticity (nonlinear behaviour)

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Computer Modelling

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Governing Equations - Fluid FlowMomentum balance in Newtonian fluids

: Constitutive law (linear relation between stress-strain

rate)

Substituting constitutive law in momentum balance:

v

dt= + b

d: strainrate

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Computer Modelling

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Heat Transfer + SolidificationThermal Energy Conservation

Terms:-- latent heat,

f -- Liquid fraction"SOLID

MUSHY ZONE

LIQUID

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Computer Modelling Transient and steady heat conduction

1. Transient heat conduction, with only the solid phase. In this case,f=0, v

0

If and c are constant, k is isotropic (diagonal, in this case) and

constant, then we can non-dimensionalize the above equation as

2.If we have a heat source terms, we then have

3. In the stead state, /t = 0, and we have Poissons equation:

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Computer Modelling

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Electromagnetics

Low frequency --- high frequency waves.

Interaction of Magnetic and Electric Fields. Phenomena governed by

Maxwells equations

E : Electric field (volt/meter)

B : Magnetic induction or magnetic

flux density (tesla)D : Electric displacement or electricflux density (coulomb/meter2)

H : Magnetic field or magnetic fluxvector (ampere/meter)

J : Current density (ampere/meter2) : Charge density (coulomb/meter3)

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Computer Modelling

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Electromagnetics

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Computer Modelling

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Electromagnetics

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Computer Modelling Governing Equations generic form

The governing equations can be expressed in a standard form:

Or as we see from MATLABs PDE Toolbox:

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Computer Modelling Governing Equations generic form

Given a point P-Domain of dependence:region from which solution at point P depends.-Range of influence: part of the domain in which the solution depends on the

solution at point P.

Figures from:Numerical methods for engineers and ScientistsJ.D. Hoffman. Mc Graw-Hill. 1993.

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Computer Modelling Summary

Elliptic eqs. Parabolic eqs. Hyperbolic eqs.

Physical problem Equilibrium(Laplace eq.)

Evolution (diffusion) Evolution (waves,convection)

Characteristics 2 complex 2 real (repeated) 2 real (different)

Domain of

dependenceEntire domain Present and past

Past between

characteristics

Range ofinfluence

Entire domain Present and future Future betweencharacteristics

Numerical

problemEfficiency Stability Stability

Boundary

conditionsDirichlet, Neumann, Mixed

Initial conditions No Yes Yes

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Computer Modelling boundary conditions

Example of types of boundary conditions:

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u(temperature) prescribed (Dirichlet)

ku/ nprescribedku/ nprescribed

ku/ n(heat flux) prescribed (Neumann)

note: Some people like to write

Robin: