Elastic Wave Propagation into Soft Elastic Materials with Thin … · 2016. 6. 23. · Aliénor...

Aliénor BUREL - Junior Seminar June, 26th 2012

Elastic Wave Propagation into Soft Elastic Materials

with Thin Layers

Joint work with Patrick Joly (Project POems INRIA/ENSTA/CNRS)

Sébastien Imperiale (Columbia University, NY)

Marc Duruflé (Project Bacchus INRIA Bordeaux)

Aliénor Burel

(Project POems INRIA + Paris-Sud XI)

Tuesday, 26 June 12

Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012

What is Numerical Analysis? 2

⌦

Tuesday, 26 June 12



⌦

Equation (complicated one) involving an unknown

Physicists

u(t, x)

Tuesday, 26 June 12



TheoricalMathematicians⌦


Physicists

u(t, x)

Tuesday, 26 June 12



TheoricalMathematicians

Solve it

Fields Medal

⌦


Physicists

u(t, x)

Tuesday, 26 June 12



Don’t know how to solve it


Solve it

Fields Medal

⌦


Physicists

u(t, x)

Tuesday, 26 June 12





Solve it

Fields Medal

NumericalMathematicians

⌦


Physicists

u(t, x)

Tuesday, 26 June 12





Solve it

Fields Medal


I know a (good?) approximation of the solution on small parts. What if I cut the domain into small parts and approx the equation on each part?

⌦


Physicists

u(t, x)

Tuesday, 26 June 12





Solve it

Fields Medal



⌦

Mesh


Physicists

u(t, x)

Tuesday, 26 June 12





Solve it

Fields Medal



⌦

Mesh


Physicists

u(t, x)

If my equation is dependent on time, I also choose to cut the

time into small time steps t

Tuesday, 26 June 12


3

⌦

From now, what are the problems ?

Mesh

What is Numerical Analysis?

Tuesday, 26 June 12


3

⌦


Mesh

Size of the mesh

The more squares, the better the approximation


Tuesday, 26 June 12


3

⌦


Mesh

Choice of a scheme

Quality of the approximation of the equation

Size of the mesh



Tuesday, 26 June 12


3

⌦


Mesh

what is it ?

Choice of a scheme


Size of the mesh



Tuesday, 26 June 12


3

⌦


Mesh

what is it ?

It is a way to approximate the equation on each part of the mesh

and on each time step

Choice of a scheme


Size of the mesh



Tuesday, 26 June 12


3

⌦


Mesh

what is it ?



Choice of a scheme


Size of the mesh


Cost of the scheme


Tuesday, 26 June 12


3

⌦


Mesh

what is it ?



Choice of a scheme


Size of the mesh


The equation given by the scheme is supposed to be consistent with the initial problem and the solution to

be stable (doesn’t explode) if the problem is well-posed

Cost of the scheme


Tuesday, 26 June 12


Frame4

We study ultrasonic wave propagation in order to work on Non-Destructive Testing.

Examples: ultrasonic testing of industrial pieces such as rubber, detection of stones in the kidney, ultrasound of organs...

Aim: test the shape or the structure of a piece or an organ without destroying it...

Tuesday, 26 June 12


Two wave types (P and S) in homogeneous isotropic materials with different speeds:

5

Motivations

Pressurewave

Shear wave

Tuesday, 26 June 12


Two wave types (P and S) in homogeneous isotropic materials with different speeds:

5

Motivations

Pressurewave

Shear wave

P-wave speed > S-wave speed

VP > VS

Tuesday, 26 June 12


In soft materials (rubber, human organs...) the P-wavespeed can be very high compared to the S-wavespeed: a hundred times larger...

6

Thesis Motivations

Part 1

In the classical numerical studies, this knowledge is not taken into account.

VP

VS

Tuesday, 26 June 12



6

Thesis Motivations

Part 1 We need to adapt our simulations to the wave characteristics because a certain amount of calculus leads

to:


VP

VS

Tuesday, 26 June 12



6

Thesis Motivations


to:


VP

VS

The cost of the scheme is proportional to:

VP

VS

Tuesday, 26 June 12



6

Thesis Motivations


to:

Compute separately the simulations of the P-waves

and S-waves?In the classical numerical studies, this knowledge is not taken into account.

VP

VS

The cost of the scheme is proportional to:

VP

VS

Tuesday, 26 June 12


7

Ultrasonic waves amplitude decreases very fast in the air (doesn’t exist in the vacuum). We need a gel which

allows to keep the solid/solid contact.

Thesis Motivations

Part 2

Tuesday, 26 June 12


7



Thesis Motivations

Part 2

These thin layers are little studied in elastodynamics and a very fine mesh is needed to see something.

Tuesday, 26 June 12


7



Thesis Motivations

Part 2

Collages between materials are very often forgotten in the simulations.


Tuesday, 26 June 12


7



Thesis Motivations

Part 2



“We do as if there were no layer...”

?

Tuesday, 26 June 12


7



Thesis Motivations

Part 2



Necessity of a precise study of these thin layers and the

conditions of its cancellation

“We do as if there were no layer...”

?

Tuesday, 26 June 12


8

Goals

Is this wing safe?

Tuesday, 26 June 12


8

Goals

Is this wing safe?

⌘

Tuesday, 26 June 12


8

Goals

Is this wing safe? Glue: soft materialdifferent wavespeeds

⌘

Tuesday, 26 June 12


8

Goals

Is this wing safe?

?

⌘

Tuesday, 26 June 12


8

Goals

Is this wing safe?

?

⌘

Can we study more precisely the propagation of each wave type into the soft medium?

Part 1

Tuesday, 26 June 12


8

Goals

Is this wing safe?

?

⌘


Part 1

If the width is small enough, can we compute “as if” there were no glue?

Part 2

Tuesday, 26 June 12


PART 1

Using potentials in elastodynamics: a challenge for finite elements

methods

Tuesday, 26 June 12


Motivations10


Tuesday, 26 June 12


Motivations10


Separate mathematically the P-waves and S-waves

Formulas, properties, tricks

Yes, if we know how to...

Tuesday, 26 June 12


Motivations10




Separate numerically the P-waves and S-waves

Verification that our numerical methods are stable

Numerical scheme

Yes, if we know how to... And if we know how to...

Tuesday, 26 June 12


Motivations10




Conservation of energies method

Separate numerically the P-waves and S-waves

Verification that our numerical methods are stable

Numerical scheme

Yes, if we know how to... And if we know how to...

Tuesday, 26 June 12


Homogeneous elastodynamic equation:

11

⇢

@

2u

@t

2� div �(u) = 0, x 2 ⌦, t > 0.

In the isotropic 2D case, Hooke’s law gives:

�(u) = � div u+ 2µ "(u)

Wave propagation in elastodynamics

⌦

Tuesday, 26 June 12


Homogeneous elastodynamic equation:

11

⇢

@

2u

@t

2� div �(u) = 0, x 2 ⌦, t > 0.

In the isotropic 2D case, Hooke’s law gives:

�(u) = � div u+ 2µ "(u)

Wave propagation in elastodynamics

⇢@2u

@t2� (�+ 2µ)r( div u) + µ

��!curl ( curl u) = 0

constantsµ�

��!curl' =

⇣@'

@x2,� @'

@x1

⌘tcurlu =

@u1

@x2� @u2

@x1

Classical equation

⌦

"ij(u) =1

2

⇣@ui

@xj+

@uj

@xi

⌘

Tuesday, 26 June 12


Decomposition of the displacement field into P wave and S wave

12

⇢@2u

@t2� (�+ 2µ)r( div u) + µ


Tuesday, 26 June 12



12

u = r'P +��!curl 'S

⇢@2u

@t2� (�+ 2µ)r( div u) + µ


Tuesday, 26 June 12



12


⇢@2u

@t2� (�+ 2µ)r( div u) + µ


⇢@2

@t2(r'P +

��!curl 'S)� (�+ 2µ)r(�'P )� µ

��!curl (�'S) = 0

Tuesday, 26 June 12



12


⇢@2u

@t2� (�+ 2µ)r( div u) + µ


div(��!curl) = 0

div(r) = �

⇢@2

@t2(r'P +

��!curl 'S)� (�+ 2µ)r(�'P )� µ

��!curl (�'S) = 0

Tuesday, 26 June 12



12


⇢@2u

@t2� (�+ 2µ)r( div u) + µ


⇢@2

@t2(r'P +

��!curl 'S)� (�+ 2µ)r(�'P )� µ

��!curl (�'S) = 0

Tuesday, 26 June 12



12


⇢@2u

@t2� (�+ 2µ)r( div u) + µ


curl(r) = 0

curl(��!curl) = ��

⇢@2

@t2(r'P +

��!curl 'S)� (�+ 2µ)r(�'P )� µ

��!curl (�'S) = 0

Tuesday, 26 June 12



12


⇢@2u

@t2� (�+ 2µ)r( div u) + µ


⇢@2

@t2(r'P +

��!curl 'S)� (�+ 2µ)r(�'P )� µ

��!curl (�'S) = 0

Tuesday, 26 June 12



12


⇢@2'P

@t2� (�+ 2µ)�'P = 0

⇢@2u

@t2� (�+ 2µ)r( div u) + µ


⇢@2

@t2(r'P +

��!curl 'S)� (�+ 2µ)r(�'P )� µ

��!curl (�'S) = 0

Tuesday, 26 June 12



12


⇢@2'P

@t2� (�+ 2µ)�'P = 0 ⇢

@2'S

@t2� µ�'S = 0

⇢@2u

@t2� (�+ 2µ)r( div u) + µ


⇢@2

@t2(r'P +

��!curl 'S)� (�+ 2µ)r(�'P )� µ

��!curl (�'S) = 0

Tuesday, 26 June 12



12

Pressure wave Shear wave


⇢@2'P

@t2� (�+ 2µ)�'P = 0 ⇢

@2'S

@t2� µ�'S = 0

⇢@2u

@t2� (�+ 2µ)r( div u) + µ


⇢@2

@t2(r'P +

��!curl 'S)� (�+ 2µ)r(�'P )� µ

��!curl (�'S) = 0

Tuesday, 26 June 12


13

V 2P =

�+ 2µ

⇢V 2S =

µ

⇢

: pressure waves velocityVP : shear waves velocityVS

⇢@2'P

@t2� (�+ 2µ)�'P = 0 ⇢

@2'S

@t2� µ�'S = 0

Decomposition of the displacement field into potentials

1

V 2P

@2'P

@t2��'P = 0

1

V 2S

@2'S

@t2��'S = 0

Tuesday, 26 June 12


Coupling of the P-waves and S-waves14

In a piecewise homogeneous medium, the coupling appear on the boundaries and on the interfaces.

⌦

Tuesday, 26 June 12




P

S

⌦

[u] = 0

[�(u)n ] = 0

8><

>:

Boundary Conditions

u = 0or

�(u)n = 0

Perfect Bounding Interface Conditions

Tuesday, 26 June 12




Main issue: treat these couplings by potentials in a stable way.

P

S

⌦

[u] = 0

[�(u)n ] = 0

8><

>:

Boundary Conditions

u = 0or

�(u)n = 0

Perfect Bounding Interface Conditions

Tuesday, 26 June 12


15

Dirichlet boundary conditions

Example: case of a rigid boundary on � = @⌦

r'P +��!curl 'S = 0 � = @⌦on

u = 0

Tuesday, 26 June 12


15



r'P +��!curl 'S = 0 � = @⌦on

n⌧ ��!curl ' · n = �@'

@⌧

��!curl '⇥ n =

@'

@n�⌦

u = 0

Tuesday, 26 June 12


15



r'P +��!curl 'S = 0 � = @⌦on

n⌧ ��!curl ' · n = �@'

@⌧

��!curl '⇥ n =

@'

@n�⌦

�

� = @⌦

u · n = 0

u · ⌧ = 0

@'P

@n� @'S

@⌧= 0

@'S

@n+

@'P

@⌧= 0

After projection along the normal and the tangent to , we get(·n) (⇥n)

� = @⌦

(() )

(() )

u = 0

Tuesday, 26 June 12


The energy identity16

@'P

@n� @'S

@⌧= 0

@'S

@n+

@'P

@⌧= 0 @⌦on

8>>><

>>>:

⇢

@

2'P

@t

2� (�+ 2µ)�'P = 0, x 2 ⌦, t > 0

⇢

@

2'S

@t

2� µ�'S = 0, x 2 ⌦, t > 0

1

V 2S

@2'S

@t2��'S = 0

1

V 2P

@2'P

@t2��'P = 0

Tuesday, 26 June 12



@'P

@n� @'S

@⌧= 0

@'S

@n+

@'P

@⌧= 0 @⌦on

Discretization of these boundary conditions?

8>>><

>>>:

⇢

@

2'P

@t

2� (�+ 2µ)�'P = 0, x 2 ⌦, t > 0

⇢

@

2'S

@t

2� µ�'S = 0, x 2 ⌦, t > 0

1

V 2S

@2'S

@t2��'S = 0

1

V 2P

@2'P

@t2��'P = 0

Tuesday, 26 June 12


We demonstrated the following energy equality:

Proposition:

Let

E is positive and conservative:

E(t) :=1

2

Z

⌦

⇣ 1

V 2P

��@'P

@t

��2 + |r'P

��2⌘

+

Z

⌦

⇣ 1

V 2S

��@'S

@t

��2 + |r'S

��2⌘

@

@tE(t) = 0.


@'P

@n� @'S

@⌧= 0

@'S

@n+

@'P

@⌧= 0 @⌦on

8>>><

>>>:

⇢

@

2'P

@t

2� (�+ 2µ)�'P = 0, x 2 ⌦, t > 0

⇢

@

2'S

@t

2� µ�'S = 0, x 2 ⌦, t > 0

1

V 2S

@2'S

@t2��'S = 0

1

V 2P

@2'P

@t2��'P = 0

Tuesday, 26 June 12


We demonstrated the following energy equality:

Proposition:

Let

E is positive and conservative:

E(t) :=1

2

Z

⌦

⇣ 1

V 2P

��@'P

@t

��2 + |r'P

��2⌘

+

Z

⌦

⇣ 1

V 2S

��@'S

@t

��2 + |r'S

��2⌘

@

@tE(t) = 0.

The energy identity

�Z

�

⇣@'S

@⌧'P � @'P

@⌧'S

⌘�.

16

@'P

@n� @'S

@⌧= 0

@'S

@n+

@'P

@⌧= 0 @⌦on

8>>><

>>>:

⇢

@

2'P

@t

2� (�+ 2µ)�'P = 0, x 2 ⌦, t > 0

⇢

@

2'S

@t

2� µ�'S = 0, x 2 ⌦, t > 0

1

V 2S

@2'S

@t2��'S = 0

1

V 2P

@2'P

@t2��'P = 0

Tuesday, 26 June 12


17

The key to demonstrate the previous proposition is:

8 'P ,'S 2 H1(⌦)2

Z

⌦

⇣|r'P |2 + |r'S |2

⌘�

Z

�

⇣@'S

@⌧'P � @'P

@⌧'S

⌘Lemma:

The energy identity

=

Z

⌦

��r'P +��!curl 'S

��2

Tuesday, 26 June 12


17


8 'P ,'S 2 H1(⌦)2

Z

⌦

⇣|r'P |2 + |r'S |2

⌘�

Z

�

⇣@'S

@⌧'P � @'P

@⌧'S

⌘Lemma:

2

Z

⌦r'P ·

��!curl 'S

The energy identity

=

Z

⌦


��2

Tuesday, 26 June 12


17


8 'P ,'S 2 H1(⌦)2

Z

⌦

⇣|r'P |2 + |r'S |2

⌘�

Z

�

⇣@'S

@⌧'P � @'P

@⌧'S

⌘Lemma:

2

Z

⌦r'P ·

��!curl 'S

The energy identity

We found an energy identity so we are confident in the fact that we can find the same kind of equality for the discrete scheme.

=

Z

⌦


��2

Tuesday, 26 June 12


17


8 'P ,'S 2 H1(⌦)2

Z

⌦

⇣|r'P |2 + |r'S |2

⌘�

Z

�

⇣@'S

@⌧'P � @'P

@⌧'S

⌘Lemma:

2

Z

⌦r'P ·

��!curl 'S

The energy identity

And why are you happy with that?


=

Z

⌦


��2

Tuesday, 26 June 12


17


8 'P ,'S 2 H1(⌦)2

Z

⌦

⇣|r'P |2 + |r'S |2

⌘�

Z

�

⇣@'S

@⌧'P � @'P

@⌧'S

⌘Lemma:

2

Z

⌦r'P ·

��!curl 'S

The energy identity

And why are you happy with that?

Because if a quantity involving my discrete potentials is conserved in time, it means I can find a scheme that won’t explode.


=

Z

⌦


��2

Tuesday, 26 June 12


18

Finite Elements discretization

We choose two space steps and

We take Vh = VhP ⇥ VhS discretization spaces

two meshes

or two different ordershP hS

)

An energy-preserving scheme and a code

Remark : Since we don’t want any of the two waves to be more penalizing than the other, it will be natural to choose our approximation spaces so that we can play with either the space step or the order of the scheme.

Tuesday, 26 June 12


A numerical illustration19

P-wave S-wave

The modulus of the displacement field is represented(in color levels) as a function of time


VP ' 3 VS

Tuesday, 26 June 12


A numerical illustration20

Our method

The modulus of the displacement field is represented(in color levels) as a function of time

Classical elastodynamics

method

A numerical illustration


20

Tuesday, 26 June 12


PART 2

Thin layers approximation for time-domain elastodynamics

Tuesday, 26 June 12


22

GoalIf the width of the layer is small enough, can we compute “as if” there were no glue?

What does that mean?

⌘

Tuesday, 26 June 12


22



⌘

We need more computation points inside the layer if we want to “see” something

Tuesday, 26 June 12


22



⌘

We need more computation points inside the layer if we want to “see” something

We increase the calculation cost

Aim: find relationships between the unknowns on the upper boundary and on

the bottom boundary

Tuesday, 26 June 12


23

How do we do that?

⌘

u�

u+

u⌘

Tuesday, 26 June 12


23

How do we do that?

⌘

Interface conditions

[u] = 0

[�(u)n] = 0

We use the

u�

u+

u⌘

Tuesday, 26 June 12


23

How do we do that?


[u] = 0

[�(u)n] = 0

We use the

1

SCALED

U

u�

u+

Tuesday, 26 June 12


23

How do we do that?


[u] = 0

[�(u)n] = 0

We use the

We develop in the unknowns and⌘ U

U = U0 + ⌘U1 + ⌘2U2 + ⌘3U3...

u±

u± = u0± + ⌘u1

± + ⌘2u2± + ⌘3u3

±...

+

1

SCALED

U

u�

u+

Tuesday, 26 June 12


23

How do we do that?


[u] = 0

[�(u)n] = 0

We use the

We develop in the unknowns and⌘ U

U = U0 + ⌘U1 + ⌘2U2 + ⌘3U3...

u±

u± = u0± + ⌘u1

± + ⌘2u2± + ⌘3u3

±...

+

⇢@2U

@t2� (�+ 2µ)r⌘( div⌘U) + µ

��!curl⌘( curl⌘U) = 0

+

1

We use the scaled equation

SCALED

U

u�

u+

Tuesday, 26 June 12


24

How do we do that?[u] = 0

[�(u)n] = 0

U = U0 + ⌘U1 + ⌘2U2 + ⌘3U3...

u± = u0± + ⌘u1

± + ⌘2u2± + ⌘3u3

±...

⇢@2U

@t2� (�+ 2µ)r⌘( div⌘U) + µ


Tuesday, 26 June 12


U 0+⌘U 1

+⌘ 2U 2

+⌘ 3U 3

=u± 0

+⌘u

± 1+⌘ 2u± 2

+⌘ 3u± 3,

µ �@1 (U 0

2 +⌘U 12 +

⌘ 2U 22 +

⌘ 3U 32 ) + 1

⌘ @2 (U 0

1 +⌘U 1

1 +⌘ 2U 21 +

⌘ 3U 31 ) �

=

µ± �@1 (u 0

2± +⌘u 12± +

⌘ 2u 22± +

⌘ 3u 32± ) +

@2 (u 0

1± +⌘u 11± +

⌘ 2u 21± +

⌘ 3u 31± ) �

,

(�+2µ) 1⌘ @

2 (U 02 +⌘U 1

2 +⌘ 2U 22 +

⌘ 3U 32 ) +

�@1 (U 0

1 +⌘U 1

1 +⌘ 2U 21 +

⌘ 3U 31 ) =

(�± +

2µ± )@

2 (u 02± +

⌘u 12± +

⌘ 2u 22± +

⌘ 3u 32± ) +

�± @

1 (u 01± +

⌘u 11± +

⌘ 2u 21± +

⌘ 3u 31± ).

24


[�(u)n] = 0

U = U0 + ⌘U1 + ⌘2U2 + ⌘3U3...

u± = u0± + ⌘u1

± + ⌘2u2± + ⌘3u3

±...

⇢@2U

@t2� (�+ 2µ)r⌘( div⌘U) + µ


�(u) =

✓(�+ 2µ)@1u1 + �@2u2 µ(@1u2 + @2u1)

µ(@1u2 + @2u1) (�+ 2µ)@2u2 + �@1u1

◆r⌘ div⌘ u =

0

B@@21u1 +

1

⌘@1@2u2

1

⌘@1@2u1 +

1

⌘2@22u2

1

CA

��!curl⌘curl⌘ u =

0

B@

1

⌘@1@2u2 �

1

⌘2@22u1

�@21u2 +

1

⌘@1@2u1

1

CA⇢@2(U

0 + ⌘U1 + ⌘2U

2 + ⌘3U3)

@t2

� (�+ 2µ)

0

B@@21(U

01 + ⌘U

11 + ⌘2U

21 + ⌘3U

31 ) +

1

⌘@1@2(U

02 + ⌘U

12 + ⌘2U

22 + ⌘3U

32 )

1

⌘@1@2(U

01 + ⌘U

11 + ⌘2U

21 + ⌘3U

32 ) +

1

⌘2@22(U

02 + ⌘U

12 + ⌘2U

22 + ⌘3U

31 )

1

CA

+ µ

0

B@

1

⌘@1@2(U

02 + ⌘U

12 + ⌘2U

22 + ⌘3U

32 )�

1

⌘2@22(U

01 + ⌘U

11 + ⌘2U

21 + ⌘3U

31 )

�@21(U02 + ⌘U

12 + ⌘2U

22 + ⌘3U

32 ) +

1

⌘@1@2(U

01 + ⌘U

11 + ⌘2U

21 + ⌘3U

31 )

1

CA = 0

A[u0] =0,

A[u1] = < �(u0) > �BJ < @1u0 >

A[u2] = < �(u1) > �BJ < @1u1 >

A[u3] = < �(u2) > �BJ < @1u2 >

� 1

12

⇢[@2

t u1]� J

⇣A� (�+ µ)2A�1

⌘J [@2

1u1]� (�+ µ)JA�1@1

⇣⇢ < @2

t u0 > �JAJ < @2

1u0 >

⌘�

A[u4] = < �(u3) > �BJ < @1u3 >

� 1

12

⇢[@2

t u2]� J(A� (�+ µ)2A�1)J [@2

1u2]� (�+ µ)JA�1@1

⇣⇢ < @2

t u1 > �JAJ < @2

1u1 >

⌘�

[�(u0)] =0,

[�(u1)] =⇢ < @2t u

0 > �JAJ < @21u

0 > �JB[@1u1],

[�(u2)] =⇢ < @2t u

1 > �JAJ < @21u

1 > �JB[@1u2],

[�(u3)] =⇢ < @2t u

2 > �JAJ < @21u

2 > �JB[@1u3]

� A�1

12

⇣(⇢@2

t � JAJ@21)

�⇢ < @2

t u0 > �JAJ < @2

1u0 > �(JB +BJ)[@1u

1]� ⌘

Tuesday, 26 June 12


U 0+⌘U 1

+⌘ 2U 2

+⌘ 3U 3

=u± 0

+⌘u

± 1+⌘ 2u± 2

+⌘ 3u± 3,

µ �@1 (U 0

2 +⌘U 12 +

⌘ 2U 22 +

⌘ 3U 32 ) + 1

⌘ @2 (U 0

1 +⌘U 1

1 +⌘ 2U 21 +

⌘ 3U 31 ) �

=

µ± �@1 (u 0

2± +⌘u 12± +

⌘ 2u 22± +

⌘ 3u 32± ) +

@2 (u 0

1± +⌘u 11± +

⌘ 2u 21± +

⌘ 3u 31± ) �

,

(�+2µ) 1⌘ @

2 (U 02 +⌘U 1

2 +⌘ 2U 22 +

⌘ 3U 32 ) +

�@1 (U 0

1 +⌘U 1

1 +⌘ 2U 21 +

⌘ 3U 31 ) =

(�± +

2µ± )@

2 (u 02± +

⌘u 12± +

⌘ 2u 22± +

⌘ 3u 32± ) +

�± @

1 (u 01± +

⌘u 11± +

⌘ 2u 21± +

⌘ 3u 31± ).

24


[�(u)n] = 0

U = U0 + ⌘U1 + ⌘2U2 + ⌘3U3...

u± = u0± + ⌘u1

± + ⌘2u2± + ⌘3u3

±...

⇢@2U

@t2� (�+ 2µ)r⌘( div⌘U) + µ


�(u) =

✓(�+ 2µ)@1u1 + �@2u2 µ(@1u2 + @2u1)

µ(@1u2 + @2u1) (�+ 2µ)@2u2 + �@1u1

◆r⌘ div⌘ u =

0

B@@21u1 +

1

⌘@1@2u2

1

⌘@1@2u1 +

1

⌘2@22u2

1

CA

��!curl⌘curl⌘ u =

0

B@

1

⌘@1@2u2 �

1

⌘2@22u1

�@21u2 +

1

⌘@1@2u1

1

CA⇢@2(U

0 + ⌘U1 + ⌘2U

2 + ⌘3U3)

@t2

� (�+ 2µ)

0

B@@21(U

01 + ⌘U

11 + ⌘2U

21 + ⌘3U

31 ) +

1

⌘@1@2(U

02 + ⌘U

12 + ⌘2U

22 + ⌘3U

32 )

1

⌘@1@2(U

01 + ⌘U

11 + ⌘2U

21 + ⌘3U

32 ) +

1

⌘2@22(U

02 + ⌘U

12 + ⌘2U

22 + ⌘3U

31 )

1

CA

+ µ

0

B@

1

⌘@1@2(U

02 + ⌘U

12 + ⌘2U

22 + ⌘3U

32 )�

1

⌘2@22(U

01 + ⌘U

11 + ⌘2U

21 + ⌘3U

31 )

�@21(U02 + ⌘U

12 + ⌘2U

22 + ⌘3U

32 ) +

1

⌘@1@2(U

01 + ⌘U

11 + ⌘2U

21 + ⌘3U

31 )

1

CA = 0

A[u0] =0,

A[u1] = < �(u0) > �BJ < @1u0 >

A[u2] = < �(u1) > �BJ < @1u1 >

A[u3] = < �(u2) > �BJ < @1u2 >

� 1

12

⇢[@2

t u1]� J

⇣A� (�+ µ)2A�1

⌘J [@2

1u1]� (�+ µ)JA�1@1

⇣⇢ < @2

t u0 > �JAJ < @2

1u0 >

⌘�

A[u4] = < �(u3) > �BJ < @1u3 >

� 1

12

⇢[@2

t u2]� J(A� (�+ µ)2A�1)J [@2

1u2]� (�+ µ)JA�1@1

⇣⇢ < @2

t u1 > �JAJ < @2

1u1 >

⌘�

[�(u0)] =0,

[�(u1)] =⇢ < @2t u

0 > �JAJ < @21u

0 > �JB[@1u1],

[�(u2)] =⇢ < @2t u

1 > �JAJ < @21u

1 > �JB[@1u2],

[�(u3)] =⇢ < @2t u

2 > �JAJ < @21u

2 > �JB[@1u3]

� A�1

12

⇣(⇢@2

t � JAJ@21)

�⇢ < @2

t u0 > �JAJ < @2

1u0 > �(JB +BJ)[@1u

1]� ⌘

We do quite long calculus...

Tuesday, 26 June 12


25

Effective Interface Conditions

U

u�

u+[ui

±] = F (ui�1± )

[�(ui±)n] = G(ui�1

± )

It leads to conditions for each i = 0,1,2,3...

Tuesday, 26 June 12


25


U

u�

u+[ui

±] = F (ui�1± )

[�(ui±)n] = G(ui�1

± )


We reconstruct each jump:

[u±] = [u0±] + ⌘[u1

±] + ⌘2[u2±] + ⌘3[u3

±]...

with the previous conditions.

[�(u±)n] = [�(u0±)n] + ⌘[�(u1

±)n] + ⌘2[�(u2±)n]...

Tuesday, 26 June 12


25


U

u�

u+[ui

±] = F (ui�1± )

[�(ui±)n] = G(ui�1

± )



[u±] = [u0±] + ⌘[u1

±] + ⌘2[u2±] + ⌘3[u3

±]...


[�(u±)n] = [�(u0±)n] + ⌘[�(u1

±)n] + ⌘2[�(u2±)n]...

So that, given an initial condition on the bottom part for instance, if I know every on the bottom boundary, I can deduce on the upper boundary.ui

� u+

Tuesday, 26 June 12


25


U

u�

u+[ui

±] = F (ui�1± )

[�(ui±)n] = G(ui�1

± )


u0�


[u±] = [u0±] + ⌘[u1

±] + ⌘2[u2±] + ⌘3[u3

±]...


[�(u±)n] = [�(u0±)n] + ⌘[�(u1

±)n] + ⌘2[�(u2±)n]...

So that, given an initial condition on the bottom part for instance, if I know every on the bottom boundary, I can deduce on the upper boundary.ui

� u+

given source

Tuesday, 26 June 12


26


This method allows us to approximate the displacement on the layer interfaces. If I know what’s going on on the upper

interface, then I know the displacement on the bottom interface.

So if the width of the interface is small enough, we can compute “as if” there were nothing.

Q: are we sure this approximation will lead to stable conditions?

Tuesday, 26 June 12


27

Scaled asymptotic expansion

Partially


Tuesday, 26 June 12


27


Partially

Yes, up to the order 2: this approximation preserves an energy


Tuesday, 26 June 12


27


Partially

Yes, up to the order 2: this approximation preserves an energy

No, with orders greater than 2, for now

Work in progress...


Tuesday, 26 June 12


To be continued...28

Part 1

Part 2

Finally: use of these two methods together?

• Dirichlet 3D

• Neumann condition? Unstable in time for the moment

• Transmission? Neumann needed...

• Stability of the “high” orders

• 3D

• Numerical scheme + code

Tuesday, 26 June 12


Thank you for your attention

Tuesday, 26 June 12

Elastic Wave Propagation into Soft Elastic Materials with Thin … · 2016. 6. 23. · Aliénor...

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