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Explicit Formulation for FSI Problems

M.SouliUniversité de Lille

Laboratoire de Mécanique de Lille

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• Questions:

• 1) How to develop fluid capability in a Lagrangian structural dynamic code ?

• 2) How to develop the FSI Capability ?

Explicit Formulation for FSI Problems

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1) Lagrangian formulation for momentum and energy equations

fdivtv )(

ste .

How to solve these equations using explicit time integration ?

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Lagrangian formulation for momentum

fdiva )(.

How to solve this equation using explicit time integration ?

fdivtv )(

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Explicit Time integration

extnn fdiva )(.

pnn.

Is known based on the current configurationn

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Explicit Time integration

extnn fdiva )(.

dxNfdxNdivdxNa iextin

in .).(..

pnn.

dsNpdxNfdxNdxNa iiextin

in ....

Using Green’s formula

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Explicit Time integration

dxNNM jiij ..

)().(),( xNtatxa jj

nj

n

nextnn FFaM int.

dsNpdxNfF iinext ..

dxNF inn .int

Mass Matrix

Internal forces

External forces

dxNNM jiij .. Is a consistent mass matrix 8

Explicit Time integration

dxNNM jiij ..

i

nni M

Fa

Non Diag Mass Matrix

dxNNM jij

ii ..Lumped Mass Matrix

In order to have a diagonal linear system we use a diagonal MassMatrix:

nextnn FFF int

For each node i we have:

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The acceleration vector is easily computed once we get the nodal vector force

MFan

n

nnn atvv .2/12/1

tvdd nnn 2/11

The main problem is to compute the nodal force nF

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TIME INTEGRATION FOR EXPLICIT METHOD

2/1, nn vd

2/11, nn vd

The Displacements and Velocities are staggered in time. This Algorithm will give a Second Order Accurate Scheme in Time

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TIME INTEGRATION ALGORITHM

A Central Difference Method is used for time integration

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2/1,2/1,2/1 )(

2)(

2)( torderttfttfff nttnt

nn

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2/1,2/1,2/11 )(

2)(

2)( torderttfttfff nttnt

nn

Subtracting Equation (1) from Equation (2) givesa second order accurate integration rule.

322/1,

1 )(.)( tordertottfff ntnn

The neglected terms are the terms proportional to:

The method is second order accurate in time.

3t

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The acceleration vector is easily computed once we get the nodal vector force

MFan

n

nnn atvv .2/12/1

tvdd nnn 2/11

The main problem is to compute the nodal force nF

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ndd

nn IP

TIME INTEGRATION ALGORITHM

dxNF inn .int

nextnn FFF int

In an explicit time step :External Force is easy to computeInternal Force are not:

The internal Nodal Force is a Function of the Stress including both the Deviatoric parts from the Strength of the Material and the Pressure from the Equation of State

dxBF ntnint

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1

4

2

4

1

3

2

3

1

2

2

2

1

1

2

1

2

4

2

3

2

2

2

1

1

4

1

3

1

2

1

1

0000

0000

xN

xN

xN

xN

xN

xN

xN

xN

xN

xN

xN

xN

xN

xN

xN

xN

B

N are the shape functions

For 2D problem: B=(3,8 ) strain displacement matrixF= (8,3).(3,1)=(8,1)

For 3 D problem B=(6,24) strain displacement matrix

F= (24,6).(6,1)=(24,1)

1 2

4 3

dxBF nV

tn .int

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Stress Calculation

ndd

nn IP

dxBF n

V

tn .int

next

nn FFF int

nextF : Body Force

Boundary Force

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MATERIAL MODELS

LINEAR ELASTICMATERIAL

,..2Gd

)(31

332211

'

VVKP

)1(2EG )21(3

EK

G Shear modulus

K Bulk modulus

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FLUID MATERIAL

'2.

d

MATERIAL MODELS

ddIP.

),(EEOSP

)(. vdivKPCompressible Flow

Quasi compressible Flow

Incompressible Flow )(2 vdivP

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The Deformation for Solids is due to DisplacementGradient or Strain .

The Deformation for Fluid is due to Velocity Gradient

or Strain Rate

In Solid Mechanics Displacements are the DependantVariables.

In Fluid Mechanics Velocities are the DependantVariables.

A Fluid unlike the Solid cannot sustain FiniteDeformation under the action of constant Shear Stress.

When a Shear Stress is applied to a Fluid, The Fluidwill deform continuously so long as the Shear Stress isapplied.

The Viscosity is a Measure of the Resistance of theFluid to flow.

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Solid element

Fluid element

'2d

Shear Force

'2 Gd

Shear Force

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ENERGY EQUATION

ste :'

P

For Ideal gas and s=0 :

)(. vdivPte

dxeEk

nn 11 .dxeEk

nn .

We integrate over an element k:

The internal energy at time steps

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ENERGY EQUATION

nn VVV 1

tVVPdxvdivP

k..)(.

)(1. 1 nn

k

EEtdxte

The energy difference between the 2 time steps is due to Pdv work

VPEE nn .1

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For problems involving Shock Waves an artificial Pressure Q must be added to ensure Mesh-Resolvable-Shock.

Without Dissipation spurious Velocity Oscillationsdevelop behind the Shock. .

Shock Viscosity

without bulk viscosity

with bulk viscosity

VQPEE nn ).(1

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xudivcQ ).(..

If div(u) < 0 Compression:

0Q

If div(u) > 0 Tension:

speed of sound of the material

Density of the material

c

x Element size

R.D. Ritchmyer and K.W. Morton, Difference Methods for Initial Value Problems, Interscience, Wiley, 1969.

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Pressure Computation

Compressible fluid : Equation of state for gas

eP0

)1(

Quasi compressible fluid : Pressure penalization

Incompressible fluid : Implicit Pressure

)(2 vdivP

)(. vdivKtP

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NAVIER STOKES EQUATIONS

v

).()(. uvvdivt

fvuvdivtv ).()(

).(. uvte

Conservation of Mass Momentum and Energy.

Fluid velocity

u Mesh velocity

General Navier Stokes Equations:

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vvdivt )(.

fvvdivtv )(

evte :

NAVIER STOKES EQUATIONS

Navier Stokes Equations:

Navier-Stokes Equations are the Eulerian Formulation of FluidDynamic Equations.The Conservative Equations are solved on a fixed Mesh.

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fdivtv )(

:te

Navier Stokes Equations: Lagrangian Formulation

Mass conservation satisfied

NAVIER STOKES EQUATIONS

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PROERTIES OF FLUID and solid MATERIAL

Fluid Material

Solid Material

Shear Stress

Viscosity

Shear Stress

Shear Modulus G

Strain

Strain rate

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Split Operator Technique to solve the Conservative Equations.

First a Lagrangian Phase: Mesh velocity = Fluid velocity

fdivtv )(

:te

In The Lagrangian Phase the Mass Equation is automatically satisfied.

If the Mesh Deformation are “reasonable”, we should run a Lagrangian Formulation.

We only used the Eulerian or Advection Phase if the Mesh is severely distorted.

The Lagrangian Equations are cheaper and simpler to solve than Eulerian Formulation or Navies-Stokes Equations

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CycleLagrangien

Cycled’advection

Time step

Split Operator For Navier-Stokes

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Split: time step into 2 phasesLagrangian cycle

Advection cycle+

Time step

:dtde Fdivdtdv

evg ,,:

0g)0,(g

0)vg(divg

Split Operator For Navier-Stokes

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Structure and Fluid Computation

Loop over solid elements And fluid elements

-Compute

dxBFk

tint

-Compute Kinematics: velocity, displacement

Loop over fluid elements only-New nodes position using ALE remesh

-advection of Mass, Momentum, Energy

End Loop

End Loop

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EULER FORMULATION

1 2 3

1 2 3

1 2 3

flux

Initial mesh

Lagrangianmesh

Eulerianmesh

1D Advection

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1

1

2

2

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ELEM2Material is moving from element 1 to element 2.

Element 2 is gaining Material Flux > 0

Element 1 is loosing Material Flux < 0

For each Face j of each element:The Flux is the volume of the Material moving from Element 1 toElement 2 through the Face j

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ADVECTION ALGORITHMS

Mass Advection.

For Each Element the New Eulerian density iscomputed according to the following AdvectionScheme.

jfaces

jlllee FluxVV ...

e Density of Eulerian Element

eV Volume of Eulerian Element

l Density of Lagrangian Element

lV Volume of Lagrangian Element

jl Density of Adjacent Lagrangian Element j

jFlux Volume Flux Through Adjacent Element jnt36

General Advection Rules

For Each Variable s, the advection of the QuantityseVe is computed using the General Advection Rule.

jfaces

jlllee FluxsVsVs ...

Internal Energy Advection.

VEs

Mass advection s

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X-Momentum Advection u-velocity.

Step 1: Compute the element u-velocity

8,181j

jl

eleml uu

Step 2: Compute the Variable x-Momentum

eleml

eleml

eleml us .

Step 3: The New Element Centered x-Momentum Flux is:

jfacesj

jlllee FluxssVsV ...

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The Change in x-Momentum resulting from Advection :

jfacesj

jlx FluxsM .

,1

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Y-Momentum Advection v-velocity.

Step 1: Compute the element v-velocity

8,181j

jl

eleml vv

Step 2: Compute the Variable y-Momentum

eleml

eleml

eleml vs .

Step 3: Tne New Element Centered y-Momentum Flux is:

jfacesj

jlllee FluxssVsV ...

,1The Change in y-Momentum resulting from Advection :

jfacesj

jly FluxsM .

,1

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Z-Momentum Advection w-velocity.

Step 1: Compute the element w-velocity

8,181j

jl

eleml ww

Step 2:Compute the Variable z-Momentum

eleml

eleml

eleml ws .

Step 3:Tne New Element Centered z-MomentumFlux is:

jfacesj

jlllee FluxssVsV ...

,1

The Change in z-Momentum resulting fromAdvection :

jfacesj

jlz FluxsM .

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Nodal Mass.Eulerian Nodal Mass at Node iMass Distribution from Elements to Nodes.

elemsj

je

ie mm

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Eulerian Nodal Velocity at Node IThe New Velocity is computed using the new Nodal Mass, and Distributing the Momentum from Element to the 8 Nodes.

8,181j

jx

il

il

ie

ie umum

8,181j

jy

il

il

ie

ie vmvm

jx Momentum Advected from Adjacent Element through Face j.

1 8

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i i i i je e l l z

j ,

m w m w

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ADVECTION TYPE

The Accuracy Order for the advection is determined by the Method used to compute the

Value of the Variable jls at the Face j in the

advection Scheme.

jfacesj

jlllee FluxsVsVs ...

,1

For an “Exact” advection the Value ofjls

Should be taken right at the Face of the Element.

This is not possible since the Variable is defined at the Center of the Element, and is constant on the Element: Density, Internal Energy, Momentum, Stress andPlastic Strain. 42

First Order Advection

Donor Cell Algorithm or Upwind Algorithm

The Variable Value jls is the Average Value in the Element from which the

Material is flowing, the Donor Cell.

The Donor Cell algorithm is Stable, Monotonic, Conservative and Simple

Monotonicity requires that range of the solution variables does not increase during Advection.

Problem: The Donor Cell algorithm is DISSIPATIVE. Numerical Dissipation.

The Dissipation occurs because Material that just entered the element, and should still be near the face, is smeared over the whole element .

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0

X0 X1 X2

1

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Second Order Van Leer Algorithm

X0 X1 X2 X3

S is a second order approximation of the slope.The Value of Variable at the Face j is Given by:

)()(21

1010 xx

xxS

Parabola Fit

0

2

1

Slope at the Element Face

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M onotonicity: In order to prevent new extrem e values in the Numerical Profile of the Element D ensity Value:

maxmin SSS

W here:

),,( 210min MinS

),,( 210max MaxS

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ALE FORMULATION

1 2 3

1 2 3

1 2 3

Initial mesh

Lagrangianmesh

ALEmesh flux

1D Advection

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1 2

1 2

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The Flux is the Volume of Material moving from Element 1 to Element 2

The Flux Material belongs initially to Lagrangian Element 1 . After Rezoning now belongs to Eulerian Element 2.

Element 1 is loosing Material Element 2 is gaining Material.

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EQUIPOTENTIAL ALGORITHM.

0. L

x1 x2 x3 x4 x5 x6 x7 x8 x9

0. L

1 2 3 4 5 6 7 8 9

For 1 Dimensional Problem, the Equipotential transforms a non-uniform Mesh to a Uniform Mesh,by keeping the nodes at Boundary at the Initial location or the Lagrangian location.

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Equipotential Equation

In General Meshes, the Equipotential method keeps the boundary nodes unchanged, and iterates on the Internal Nodes to converge to a UNIFORM INSIDE MESH.

This can be done by solving the Equipotential Equation for the Node Displacement u:

02u

Where the Laplacian operator is defined by:

2

2

2

2

2

22

zyx

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Volume of Fluid Method

For free surface problems, a VOF method needs to be used:

First a Lagrangain phase is performed

Second a advection phase:

Advection of:

Mass Momentum Energy Volume fraction

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For each Element Loop over the Material Group i

Strain rate i Internal energy Ei

Deviatoric or Shear Stress iPressure from Equation of State Pi

Stress i=-Pi + i Internal Nodal force:

dxBFV

iT

i .

i : Volume Fraction of Material in the Element.

.11

Ngroups

ii

Volume of Fluid Method

Lagrangian Phase

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Interface reconstruction

2.08.0

old

2

old1

5.01n1 2.02n

1

7.03n1 5.04n

1

oldV

nodal averagedvolume fractions

element based volume fractions

xxn 1

11ˆ

n̂

oldold1 V

oldold

2V

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Structure

Fluid

Fluid Structure Coupling

Merge Fluid and Structure Nodes

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Contact Algorithm

Structure

ContactInterface

Fluid

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Impact Problem: Euler-Lagrange Lagrange

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Structure

Fluid

Fluid Structure Coupling

2) Euler Lagrange Coupling

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F

F=K.d

Slave node

Master node

n

Contact Algorithm

d

n

Step n-1:

Step n:

-F

d d.n>0 no penetration

d.n<0 penetration

K is a stiffness value, spring stiffness

spring

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F1

F

F2

F=K.d

Slave segmentMaster node

n

Contact Algorithmnode to segment

d

n

Step n-1:

Step n:

The value of K is based on the mass of the fluid and structure nodes

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(Contact Algorithm Zhong,1993)

tcAk

A :Contact area

c :Speed of sound

t :time step

:Fluid density

Stiffness value

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zoom

kdF

nd

k

zoom

Structure

1nd

svfv

Step n:

Step n-1:

sF

fF

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Penalty Method for Airbag

1xp

2xp

gas

AirBag

)(tc

air

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Void Material

Elastic Material

Rigid Plate

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ALE Moving Mesh

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ALE Moving Mesh

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Leaking

Lagrangian structure

high pressure

low pressure

only a few fluid nodes are exposed to coupling forces

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Leakage

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No Leakage

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1 pt coupling quadrature coupling

quadrature coupling

Leaking

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1 quadrature couplingLeakage

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4 quadrature couplingNo Leakage