Advanced Numerical Method usedin Composite Materials Modellings

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Advanced Numerical Method usedin Composite Materials Modellings



Advanced Numerical Modelling.

Scope and modelling methods.

Scope of the present presentation consist in quick listening of theimportance of numerical study on establishing the properties,loadings system and perspectives of the numerical simulation

The present modelling situation is based on the levels:Macro modellingMezoscale modellingNanoscale modelling



Transport phenomena modelling Phase change and global properties modelling Phase change and local properties modelling Solid state transformations and residual stresses and

Advanced Numerical Modelling.

Scope and modelling methods.

s ra ns es a s ng Structure stresses, strains and deflections calculation Dynamics of structure, cracks generation an growth,fatigue verifications Schematic aspects of a structure modelling.



Macro modelling

Transport Equations Mass Conservation Momentum Conservation

Incompressible fluids Compressible fluids

Ener Conservation

Phase change

a) (solid liquid, liquid solid, liquid vapors)b) (solid states transformation and properties)

Residual stresses, strains and deflections

Stress and strain analysis using loading system and limitconditions



TRANSPORT PHENOMENON MODEL

General aspects of equations involved in mass, momentum and energytransport have the general expression:

+

−

=

genaration

of rate

outputs

of rate

inputs

of rate

onsaccumulati

of rate

−

=

outputs

massof rate

inputs

massof rate

nacumulatio

massof rate

For an infinitesimal volume element dΩΩΩΩ is ρdΩΩΩΩ that must be integrateover the element to obtain the rate of mass accumulation

∫∫∫ ΩΩ⋅

∂

∂d

t ρ

Rate of mass changed from the infinitesimal surface d Γ can beexpressed by

influx/efflux = (-/+) (ρρρρ u) (d Γ ΓΓ Γ cos θ θθ θ ),



where θ is the angle between the velocity vector u and the normal outwardunit vector n to the surface dΓ .

Or, using the vector algebra:

Γ ⋅⋅⋅=⋅⋅⋅Γ ⋅=⋅Γ ⋅ d nunud d u )()cos()cos)((rrrr

ρ θ ρ θ ρ

By integrals over the whole surface of the element d Γ give


∫∫Γ Γ ⋅⋅⋅− d nu )(

rr

ρ

The integral form of the equation of mass transfer become:

0)( =Γ ⋅⋅⋅+Ω⋅

∂

∂∫∫∫∫∫ Γ Ω

d nud

t

rr ρ ρ



Differential form of mass balance:

a) The aid of Gauss divergence theoremb) The direct concept of mass conservation applied to 3D differentialcontrol volume as applied in figure 2


Mass Balance

∫∫ ∫∫∫Γ ΩΩ⋅⋅⋅∇=Γ ⋅⋅⋅ d ud nu )()(

rrr ρ ρ

0)( =Ω⋅

⋅⋅∇+

∂

∂∫∫∫ Ω

d ut

r ρ

ρ

As the integral must vanish on the arbitrary control volume and theintegrant is a continuum function, it follow that the integral must be equalwith zero. So:

0)( =⋅⋅∇+∂

∂

ut

r

ρ

ρ



And for an incompressible fluid, ρ = constant

0=⋅∇ ur

b) Use the 3D differential control volume,the left hand side of general equation, the rate of mass accumulationwithin the differential control volume (∆x∆y∆z) can be expressed as:


Mass Balance

)( z y x

t

∆⋅∆⋅∆⋅

∂

ρ

The influx in the control volume on the three directions is:

( ) z yuu x x x x x ∆⋅∆⋅⋅−⋅

∆+ ρ ρ

x zuu y y y y y

∆⋅∆⋅⋅−⋅∆+

ρ ρ

( ) y xuu z z z z z

∆⋅∆⋅⋅−⋅∆+

ρ ρ

The total net rate of mass influx is the sum of directional mass input and

the differential eq. become:

Figure 2



( ) ( )

( ) ( ) y xuu x zuu

z yuu z y x

t

z z z z z y y y y y

x x x x x

∆⋅∆⋅⋅+⋅+∆⋅∆⋅⋅−⋅+

∆⋅∆⋅⋅−⋅=∆⋅∆⋅∆⋅

∂

∂

∆+∆+

∆+

ρ ρ ρ ρ

ρ ρ ρ

Dividing by ∆x ∆y ∆z and taking the limit as ∆x ∆y ∆z


Mass Balance

( ) ( ) ( ) 0=⋅∂

∂+⋅∂

∂+⋅∂

∂+∂

∂ z y x u

zu

yu

xt ρ ρ ρ ρ

In vector form:

( )0=⋅⋅∇+

∂

∂

ut

r

ρ

ρ Figure 2



Momentum Transfer Governing Equation

Integral Form of Balance Equation

+

−

=

systemon

acting forces

of sum

out

momentum

of rate

in

momentum

of rate

onaccumulati

momentum

of rate


The first and second term of right-hand site of equation consist intwo component, convective and viscous flux transfer

+

−

=

systemon

acting forces

of sum

momentum

viscousnet

of rate

momentum

convectivenet

of rate

onaccumulati

momentum

of rate



( ) Ω⋅⋅∂

∂∫∫∫Ω

d ut

r ρ

∫∫Γ Γ ⋅⋅⋅⋅− d nuu )(

rrr ρ

Γ ⋅⋅− d nr

τ

- Rate of momentum accumulation

- Rate of net convectiv momentum

- Rate of net viscous momentum



Similar with the mass balance consideration:

∫∫ ∫∫∫Γ ΩΩ⋅+Γ ⋅⋅− d f d nP b

r

∫∫∫Ω Ω⋅∇− d P f b )(

( ) ( ) ( )∫∫∫ ∫∫ ∫∫ ∫∫∫Ω Γ Γ ΩΩ⋅∇−+Γ ⋅⋅−Γ ⋅⋅⋅⋅−=Ω⋅⋅

∂

∂d P f d nd nuud u

t b

rrrrrτ ρ ρ

- Sum of forces acting on system



Differential Form of Momentum Balance Equation

( )

( ) b f Pnuut

u

+⋅∇−⋅∇−⋅⋅⋅⋅−∇=∂

⋅∂

τ ρ

ρ rrrr

)(2

b f Puuut

u+∇−∇=∇⋅⋅+

∂

∂ rrrr

µ ρ ρ

1 2 3 (4)



zu

yu

xu

t Dt

D z y x

∂

∂+

∂

∂+

∂

∂+

∂

∂=

b f Pu Dt

u D+∇−∇=

rr

2 µ ρ

Observation:

mass ( ρ ρρ ρ ) x accelera ţ ion(Du/Dt ) = viscous forces – externalforces



Boundary Conditions.

1) Prescribed inlet or outlet conditions2) Free sleep conditions3) No-sleep conditions4) At liquid/liquid interfaces the momentum flux and speeds





Governing Equations of Energy Transfer

+

−

=

systemon

acting forces

of sum

momentum

viscousnet

of rate

momentum

convectivenet

of rate

onaccumulati

momentum

of rate

General expression of energy balance


+

−

−

=

)5()4(

)3()2()1(

generation

heat of rate

donework

of rate

out energyinenergyonaccumulati

The therms (1),(3) includes the thermal, kinetic and potential energy perunit volume of the fluid, with equation

++⋅= energy potential

u

T C E V 2

2

ρ



Term 4 include the work done by the fluid on surroundings, so heconsider the pressure, viscous heating and shaft work

Term 5 include the heat generation caused by Joule effect, chemicalreactions and phase transformation.

In the application, the kinetics and potential energy, therms (1), (2), (3)are neglected as compared with thermal energy. The therm (4) can be



neglected too in some applications and the aspect of energy balanceequation is reduced to thermal equation.

+

−

=

generation

heat of rate

out energy

thermalof rate

inenergy

thermalof rate

onaccumulatienergy

thermalof rate

The first and second right-hand site of the equation can be writetogether using the convection and conduction expression of heat flow.



+

=

)3()2()1(

heat of rate

conduction

byinenergy

thermalof ratenet

convection

byinenergy

thermalof ratenet

onaccumulati

energy

thermalof rate



+

)4(

generation

Similarly with the mass balance integral equation the integral form of energy

balance equation get the form( ) ( )( ) ( ) ∫∫∫∫∫ ∫∫ ∫∫ ΩΩ Γ Γ

Ω⋅+Γ ⋅⋅−Γ ⋅⋅⋅⋅−=Ω⋅⋅⋅∂

∂d gd nqd nuT C d T C

t V V

&rrrr

ρ ρ

(1) (2) (3) (4)



Terms (2) and (3) called “surface phenomenon” get the phenomenon acrossthe frontiers of the volume control (named control surfaces). Indicate the netinfluxes of thermal energy due to convection – bulk fluid flow and conduction.If g is the heat generate per unit volume that is considerate constant over the



. .



Differential Form of Energy Balance Equation

( ) ( ) Ψ++⋅∇−⋅⋅⋅−∇=⋅⋅∂∂ gqT C T C t

V V &r

ρ ρ

gT T uC T

C V v&

r+∇⋅+∇⋅⋅⋅−=

∂⋅ 2λ ρ ρ

Or, for sub-sonic fluid speed



Or:gT T u

t

T C V &

r+∇⋅=

∇⋅+

∂

∂⋅ 2λ ρ

And, using the notation convention for derivation the aspect of

balance of energy becomegT

Dt

DT C V &+∇⋅=⋅ 2λ ρ

Boundary and limits conditions0) Knotweed initial temperature

0),,(00=== t for z y xT T or T T



1) Prescribed temperature

bbbb z z y y x xboundarytheonT T ==== ;;

2) Prescribed heat flux

b

b

x

x x

T

q x

T

=∂

−

=∂

∂−

=

λ

λ



b

b

b

b

z

z z

y

y y

q z

T

y

=∂

∂−

∂

=

=

λ

3) Prescribed convective fluxes on boundary

)(

)(

)(

S

b

S

b

S

b

z z

z z

y y

y y

x x

x x

T T h z

T

T T h y

T

T T h x

T

−=∂

∂−

−=∂

∂−

−=∂

∂−

∞

=

∞

=

∞

=

∞

∞

∞

λ

λ

λ

4) Radiation heat flux)( S incr

x x

T T h xT

b

−=∂∂−

=

λ ( )( )22

S incS incr T T T T h ++⋅= σ ε



Similarities among the transport phenomenon


If we analyze carefully the three transport equations we shall see a lot ofsimilitude between them. Good understands and solve of one of them

make easy the understand and solve of all of them. For that, let to seethe similarity and differences between the three phenomenons.

Aspects of Fluxes Diffusion

Aspects of fluxes diffusion from Oy direction

Mass transfer(Fick low )

dy

d D j A

AB y A

ρ −=

,

Momentum transfer( Newton viscous low)

dy

du x yx µ τ −=

Heat transfer(Fourier conduction)

dy

dT q y y λ −=

Where, λ,DAB,µ are transport coefficients of the T,ρA and ux.



If the Measure Units of this coefficients become similar, then theequations become full analogues. For mass and momentum, the MU is

similar, m2 /s. For heat conduction, [J/m2 /s/ oK]

ρ

λ α

⋅=

pC

⋅⋅

The heat flux expression become:



dy

qp

y −= α

General expression of diffusive fluxes become

gradient eqtransfer tydifuzibili flux .×−=

( )

dy

d D j A

AB y A

ω ρ ⋅−=

,

( )dy

ud x

yx

⋅−=

ρ υ τ



Convective transfer

( )S A AC

S A A

y

A

AB

y y A

y D

n ,,

,,

0

0,ρ ρ

ρ ρ

ρ

−⋅Κ=

−

∂

∂−

≡ ∞

∞

=

=

Fluxes transfer for mass, momentum, energy



( )S f

S

x

y yxuuC

uu

x

u

−⋅=

−∂

∂

−≡ ∞

∞

=

'

0

µ τ

( )S S

y

y y T T hT T

y

T

−⋅=

−

∂

∂−

≡ ∞

∞

=

=

0

0

λ

ρ

difference potentialt coefficien

transfer

zoneboundary

theon flux

×

−=

Expression of convective fluxes

Aspects of fluxes convection



+

+

=

)4()3()2()1(

sgeneration

of rate

fluxesviscouse

or diffusive

of rate

fluxes

convective

of rate

onsaccumulati

of rate



( ) ∫∫∫∫∫∫ ∫∫ ∫∫ ΩΦ

Ω Γ Γ Φ Ω⋅+Γ ⋅⋅−Γ ⋅⋅Φ⋅⋅−=Ω⋅Φ⋅

∂

∂d gd n f d nud

t )()(

rrrr ρ ρ

( ) ( ) ΦΦ +⋅∇−Φ⋅⋅⋅−∇=∂

Φ⋅∂ g f ut

rr ρ ρ

Results, integral and differential unified equations





The specifically volume of the liquid and solid is not equal and the

shape of solid domain can differ from the shape of liquid domain,the flow of liquid phase around the solid interface can generatedifferent concentrations and give segregationsThe latent heat of transformation modifies the heat gradients in



the neighborhood of the interface and the solidification speed andthe microstructure will be modified.When the alloys are in process of solidification, the species arerejected or absorbed inside the solid phase and the defects at microscale can appear



S

S n f

L

Ln

T R H

n

T

∂

∂=⋅⋅+

∂

∂λ ρ λ

( )S

S S n L

L

L L

n

C D Rk C

n

C D

∂

∂=−+

∂

∂0

*1

Whereλ , λ thermal conductibilit in li uid


Phase Change Inside Control Volume

CL,CS the species concentration inliquid phaseDL,DS diffusion constant in liquidrespective solid phaseC*L the liquid equilibrium species

concentrationKo the CS /CL concentration relationRm the raze of the solid phase

Boundary Heat flow distribution

Boundary Species massdistribution



Solvers variants

•Fixed grids and use the solid fraction proportion•Variables rids



•Transformed grids





( )t

f QT

t

T C S

LV ∂

∂⋅+∇⋅⋅∇=

∂

∂⋅ ρ λ ρ

Composites

Fixed grid method

t T

T f Q

t f Q S

L

S

L∂∂

∂∂⋅=

∂∂⋅ ρ ρ

( )T t

T

T

f QC S

LV

∇⋅⋅∇=∂

∂

∂

∂− λ ρ ( ) ( )

≤=

∈∈

≥=

S S

S LS

LS

T T f

T T T f

T T f

1

,...,1,...,0

0





L Lm L C mT T ⋅+=

))(1()1(00

0

T T k

T T

k C

C C f

m

L

L

L

S −−

−=

−

−=

2

m L LS

L

T T Q f Q

−

−⋅

−=

∂−

Level model

Lm

Scheil model

2)()1(

Lm

m L LS

LT T

T T

k

Q

T

f Q

−

−⋅

−=

∂

∂−

Brody-Fleming model

−

−−⋅−=

−k

m

Lm

S T T

T T k f

1

1

1)1( α

1

1

1

1

−

−

−⋅

−⋅

−=

∂

∂−

k

m L

m

m

LS

LT T

T T

T T k

Q

T

f Q

a

f S t Dλ

α ⋅⋅≅4 ( ) ( )( ) k

k

m

k Lm LS

L

T T T T

k k Q

T f Q

−

−

−

−−⋅

−⋅+⋅=

∂∂−

1

2

1

1

11 α





Linear model

S L

LS

T T T T f

−−=

S L

LS

LT T

Q

T

f Q

−=

∂

∂−




Applications in Civil Engineering

Large-capacity ground-supported tanks are used to store a variety ofliquids, e.g. water for drinking and cooling energetically and industrial

systems, fire fighting, petroleum, chemicals, and liquefied natural gas.Satisfactory performance of tanks during strong ground shaking is crucialfor modern facilities. Tanks that were inadequately designed or detailedhave suffered extensive damage during past earthquakes [2-7] or external

Dynamics of fluid inside a reservoir.

.

tanks walls during the earthquake, explosions, tsunami and other naturalor military exceptional loads plays essential role in reliable and durabledesign of structure resistance tanks, which are made from steel orconcrete and working at the soil level, inside the soil or over the soil level.From the last big earthquake the knowledge of the fluid movement inside

the waste reservoirs and tank become more important until now





Applications in Civil EngineeringDynamics of fluid inside a reservoir.

The geometry and type ofboundary conditions inposed to

fluid.The earthquake accelerationspectras on OX an OYdirections

l) The flow is 2-D, incompressible, and initial laminar.2) Each thermal property of incompressible fluid is constant.3) The walls deformations are small and the structure move between the

earthquake with the instant acceleration of the quake.4) The time computed effects of the quake on the tank is double that totalearthquake time5) The heat dissipation and the turbulent indices are calculated only in the fluidcontrol volumes of the bulk.

Work reduced Hypothesis


Γ



( ) ( ( )

S x x x x

x x x

v

x

v

x

v

t

k k j j

iik

k

j

j

i

i

+

∂

∂Γ

∂

∂+

∂

∂Γ

∂

∂

+

∂

∂Γ

∂

∂=

∂

∂+

∂

∂+

∂

∂+

∂

∂

ϕ ϕ

ϕ ϕ ϕ ϕ ϕ



ϕ Γ S

Continuity ρ 0 0

Momentum OX

vi

velocity on xi direction

µ

Momentum OY v j

velocity on x j direction

µ

Momentum OZ vk µ

Energy T

temperature

j x

jg x +∂

∂ϕ

j x

j

g x

+∂

∂ϕ

c ρ

λq&

k x

k

g x

+∂

∂ϕ






Boundary conditions

,

be accelerating with the same acceleration that acts on thewall. In our case, with the consideration that the walls are rigidand there deformation is are neglected, the acceleration willbe equal with the acceleration gives by earthquake spectralaccelerations.




Boundary condition on cells in contact with the walls

For fluid flow analysis, either the slip wall condition or the no-slipwall condition is adopted according to the size of a cell and themagnitude of velocity. When the walls move, the convectioncoefficient between fluid and solid boundary is calculated, basedon Re and Pr numbers in the fluid cells. The heat flow e uation is



used only to calculate the fluid temperature distribution variation inthe quake action. For water and other liquids the temperature isnot important but for oils and liquefied gases the knowledge offluid temperature and pressure becomes important. The equationsfor boundary domain in contact with solid walls become




viuv

av y

pv

y

vu

x

v

t

v

au x

pv

y

uu

x

u

t

u

Y

X

rrv⋅+⋅=

⋅+∆+∂

∂−

∂

∂−

∂

∂−=

∂

∂

⋅+∆+∂

∂−

∂

∂−

∂

∂−=

∂

∂

ρ µ ρ

ρ µ ρ



Where a X and a Y are the spectrum of quakeaccelerations transmitted to the walls. In accord withthe 3) hypothesis, that accelerations will be take incalculus equal with the earthquake accelerations




The free boundary surface of the fluid

t

T

x

vmn

x

v

x

vmn

mn x

vmn

T

j

j

x x

i

j

j

i x x

x x

i

i x x

i ji j

ji ji

∂

∂=

∂

∂+

∂

∂+

∂

∂

++∂

∂

γ µ

2)

(2

Tangential stress condition



The free boundary

Normal stress condition

a

j

j

x x

i

j

j

i x x

i

i x x

x

vmn

x

v

x

xnn

x

vmn

j j jiiiφ µ =

∂

∂+

∂

∂+

∂

∂+

∂

∂2

φa

= (p ext

)/ ρ + γ T

/R m γ T is the surface tension function of temperature

p ext is the pressure of gas phase inside the tank

R m is the local mean radius of the free surface




Initial and load conditions

Because the problem is time dependent, the initial conditions consistin imposing null speed value for v x and v y and the gas pressure equal



.

considerate equal with 200

C and the walls reservoir streams null too.After the first time step, the quake event is considerate and thespectra given in the figure 2 was applied on the liquid/solid boundaryaccordingly with figure




Solution and results



,

2s, 3s, 4s for two reservoirgeometry (H/L) and filling rate






Speed on OY directions for 1s – 10s and for sec. 35 and 40





Pressure dynamicsinside the fluid





Time history of heat

generated by viscous frictionInside the fluid bulk




Turbulence factor time


history


http://zeta-re10-5.avi/




Speed on OX axis time variation








Applications in Civil EngineeringWind loads simulation

The wind loading is considered by Devenport(1998) in threecategories:

a) Extraneously-induced loading based on naturally turbulentoncoming wind. The weak of upstream obstructions enhance thiscategories buffeting.b) Unstable flow phenomenon such as separations,reattachments and vortex shedding generate a secondary type of

forces.

c) The movement-induced excitation of the body generate by thedeflection of the structure create fluid flow too. This phenomenonwith a strong unsteady states character gives the complexity ofthe fluid flow around the flexible tall structures. The modern

design of flexible tall structures must request to earth quakesevents and wind loads, cases that represent a state of the art ofthe civil engineering.





The Eurocode and most used standards are highlighted by Allsop (2009) asthe most flexible and inclusive code for normal buildings. The quasi-staticmethods offered by these codes are only applicable for buildings withstructural properties such that they are not susceptible to dynamic excitation

(Metha, 1998). Thus, the tall buildings, those with high slenderness ratiosand/or asymmetric planes, exceed limitations and are advised to be testedin the wind tunnel.





Numerical analysis

The domain of computing is established in the figure and the wind

input speed diagram in the figure 2. The system of partialdifferential equations is give on the forms of mass conservation ,momentum conservation and energy conservation

r0=⋅∇+

∂V

t

( ) ( ) g pV V V t

rrrr⋅+−∇=×⋅∇+⋅

∂

∂ ρ ρ ρ

( ) ( ) ( )T V p E E t

∇⋅∇=⋅+⋅∇+⋅∂

∂ λ ρ ρ

r

Index

Ox

Value

[m]

Index

Oy

Value

[m]

Index

Oz

Value

[m]

L 96 Y 96 Z 96

L1 6 Y/2 48 H1 21

L2 21 Y1 36 H2 69

L3 46 Y2 33 H3 12

L4 3 Y3 24

L5 36 Y4 24

L6 18

Domaingeometryandnotations

Values ofdomaingeometry





)()()( qqq hg f q ++=

( )

+

+

=

=

p E u

uw

uv

pu

u

f

E

w

v

u

q q

2

)( ;;

ρ

ρ

ρ

ρ

ρ

ρ

ρ

ρ

( ) ( )

+

+

=

+

+=

p E w

pw

vwuw

w

h

p E v

vw

pvuv

v

g qq

2

)(

2

)( ;

ρ

ρ ρ

ρ

ρ

ρ ρ

ρ

( )222

2

1

1 wvu

p

E +++−= ργ

v

p

c

c=γ

γ -law polytrophic gas considered in the present

Where c p

and c v

are the specific heat at constant pressure, respectively constant

volume





1.Boundary and Initial Conditions and input particularities

Buildings domain geometry Wind input time variation onsurface P1





Results

Pressure on façade tall building





a) t=0.152484 s b) t=0.299869 s c) t=0.446137 s d) t=0.592342 s

Wind speed time variations (OX axis)







Façade pressure time variation history

a) t=0.299869 s b) t=0.446137 s c) t=0.592342 s d) t=0.731194 s

e) t=0.871237 s f) t=0.985921 s g) t=1.107586 s h) t=1.245531 s





i) t=1.390198 s j) t=1.534835 s k) t=1.679116 s h) t=5.39784 s

The map of pressure on the facade of the tall building for

different moment of the aplication.





Wind loads simulation

Time dynamics of the pressure on the tall building façade






a) t=0.871237 s a) t=1.390198 s a) t=2.114628 s a) t=5.39784 s

The pressure map on the back surface of tall building for diverse moments of the loadsapplication.






Pressure Dynamics on the Back Surgace of Tall Building








Pressure Dynamics on the left side tall building surface






a) t=0.871237 s b) t=1.390198 s c) t=2.114628 s d) t=5.39784 s

The pressure maps on the left surface of the tall building for diverse moments of

application.






Pressure dynamics on the right surface of the tall building

C






Speed on OX axis (u) in the middle plane of the modelled area

A li ti i Ci il E i i






Wind speed on OY axis (v) in middle plane







•Conclsions

The gas dynamics modelling based on Euler PDE system of equation cansolve the problems of wind loads on tall structures without using theNavier-Stokes PDE system of equation with diverse turbulence flow



mo e s or accurate ow ynam cs.

A combination between the two modelles can be used because the gasspeeds are low in the case of wind loads and the gas is practicalincompressible. The turbulence area of flow, that in the civil engineeringhave a huge area of the domain (60-80%) in the cases of wind loads ontall buildings can be simulate using the Euler system of equations and the

complicated turbulences models can be avoided





Thank you dear students and



for that short time together

Documents

Advanced Numerical Method usedin Composite Materials Modellings