Post on 03-Aug-2021
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
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
What is Numerical Analysis? 2
⌦
Equation (complicated one) involving an unknown
Physicists
u(t, x)
Tuesday, 26 June 12
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
What is Numerical Analysis? 2
TheoricalMathematicians⌦
Equation (complicated one) involving an unknown
Physicists
u(t, x)
Tuesday, 26 June 12
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
What is Numerical Analysis? 2
TheoricalMathematicians
Solve it
Fields Medal
⌦
Equation (complicated one) involving an unknown
Physicists
u(t, x)
Tuesday, 26 June 12
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
What is Numerical Analysis? 2
Don’t know how to solve it
TheoricalMathematicians
Solve it
Fields Medal
⌦
Equation (complicated one) involving an unknown
Physicists
u(t, x)
Tuesday, 26 June 12
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
What is Numerical Analysis? 2
Don’t know how to solve it
TheoricalMathematicians
Solve it
Fields Medal
NumericalMathematicians
⌦
Equation (complicated one) involving an unknown
Physicists
u(t, x)
Tuesday, 26 June 12
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
What is Numerical Analysis? 2
Don’t know how to solve it
TheoricalMathematicians
Solve it
Fields Medal
NumericalMathematicians
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?
⌦
Equation (complicated one) involving an unknown
Physicists
u(t, x)
Tuesday, 26 June 12
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
What is Numerical Analysis? 2
Don’t know how to solve it
TheoricalMathematicians
Solve it
Fields Medal
NumericalMathematicians
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?
⌦
Mesh
Equation (complicated one) involving an unknown
Physicists
u(t, x)
Tuesday, 26 June 12
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
What is Numerical Analysis? 2
Don’t know how to solve it
TheoricalMathematicians
Solve it
Fields Medal
NumericalMathematicians
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?
⌦
Mesh
Equation (complicated one) involving an unknown
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
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
3
⌦
From now, what are the problems ?
Mesh
What is Numerical Analysis?
Tuesday, 26 June 12
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
3
⌦
From now, what are the problems ?
Mesh
Size of the mesh
The more squares, the better the approximation
What is Numerical Analysis?
Tuesday, 26 June 12
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
3
⌦
From now, what are the problems ?
Mesh
Choice of a scheme
Quality of the approximation of the equation
Size of the mesh
The more squares, the better the approximation
What is Numerical Analysis?
Tuesday, 26 June 12
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
3
⌦
From now, what are the problems ?
Mesh
what is it ?
Choice of a scheme
Quality of the approximation of the equation
Size of the mesh
The more squares, the better the approximation
What is Numerical Analysis?
Tuesday, 26 June 12
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
3
⌦
From now, what are the problems ?
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
Quality of the approximation of the equation
Size of the mesh
The more squares, the better the approximation
What is Numerical Analysis?
Tuesday, 26 June 12
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
3
⌦
From now, what are the problems ?
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
Quality of the approximation of the equation
Size of the mesh
The more squares, the better the approximation
Cost of the scheme
What is Numerical Analysis?
Tuesday, 26 June 12
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
3
⌦
From now, what are the problems ?
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
Quality of the approximation of the equation
Size of the mesh
The more squares, the better the approximation
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
What is Numerical Analysis?
Tuesday, 26 June 12
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
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
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
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
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
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
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
Two wave types (P and S) in homogeneous isotropic materials with different speeds:
5
Motivations
Pressurewave
Shear wave
Tuesday, 26 June 12
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
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
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
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
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
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 We need to adapt our simulations to the wave characteristics because a certain amount of calculus leads
to:
In the classical numerical studies, this knowledge is not taken into account.
VP
VS
Tuesday, 26 June 12
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
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 We need to adapt our simulations to the wave characteristics because a certain amount of calculus leads
to:
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
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
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 We need to adapt our simulations to the wave characteristics because a certain amount of calculus leads
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
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
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
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
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
These thin layers are little studied in elastodynamics and a very fine mesh is needed to see something.
Tuesday, 26 June 12
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
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
Collages between materials are very often forgotten in the simulations.
These thin layers are little studied in elastodynamics and a very fine mesh is needed to see something.
Tuesday, 26 June 12
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
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
Collages between materials are very often forgotten in the simulations.
These thin layers are little studied in elastodynamics and a very fine mesh is needed to see something.
Tuesday, 26 June 12
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
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
Collages between materials are very often forgotten in the simulations.
These thin layers are little studied in elastodynamics and a very fine mesh is needed to see something.
“We do as if there were no layer...”
?
Tuesday, 26 June 12
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
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
Collages between materials are very often forgotten in the simulations.
These thin layers are little studied in elastodynamics and a very fine mesh is needed to see something.
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
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
8
Goals
Is this wing safe?
Tuesday, 26 June 12
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
8
Goals
Is this wing safe?
Tuesday, 26 June 12
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
8
Goals
Is this wing safe?
⌘
Tuesday, 26 June 12
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
8
Goals
Is this wing safe? Glue: soft materialdifferent wavespeeds
⌘
Tuesday, 26 June 12
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
8
Goals
Is this wing safe? Glue: soft materialdifferent wavespeeds
⌘
Tuesday, 26 June 12
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
8
Goals
Is this wing safe?
?
⌘
Tuesday, 26 June 12
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
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
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
8
Goals
Is this wing safe?
?
⌘
Can we study more precisely the propagation of each wave type into the soft medium?
Part 1
If the width is small enough, can we compute “as if” there were no glue?
Part 2
Tuesday, 26 June 12
Aliénor BUREL - Junior Seminar June, 26th 2012
PART 1
Using potentials in elastodynamics: a challenge for finite elements
methods
Tuesday, 26 June 12
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
Motivations10
Can we study more precisely the propagation of each wave type into the soft medium?
Tuesday, 26 June 12
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
Motivations10
Can we study more precisely the propagation of each wave type into the soft medium?
Separate mathematically the P-waves and S-waves
Formulas, properties, tricks
Yes, if we know how to...
Tuesday, 26 June 12
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
Motivations10
Can we study more precisely the propagation of each wave type into the soft medium?
Separate mathematically the P-waves and S-waves
Formulas, properties, tricks
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
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
Motivations10
Can we study more precisely the propagation of each wave type into the soft medium?
Separate mathematically the P-waves and S-waves
Formulas, properties, tricks
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
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
Motivations10
Can we study more precisely the propagation of each wave type into the soft medium?
Separate mathematically the P-waves and S-waves
Formulas, properties, tricks
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
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
Motivations10
Can we study more precisely the propagation of each wave type into the soft medium?
Separate mathematically the P-waves and S-waves
Formulas, properties, tricks
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
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
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
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
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
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
Decomposition of the displacement field into P wave and S wave
12
⇢@2u
@t2� (�+ 2µ)r( div u) + µ
���!curl ( curl u) = 0
Tuesday, 26 June 12
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
Decomposition of the displacement field into P wave and S wave
12
u = r'P +���!curl 'S
⇢@2u
@t2� (�+ 2µ)r( div u) + µ
���!curl ( curl u) = 0
Tuesday, 26 June 12
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
Decomposition of the displacement field into P wave and S wave
12
u = r'P +���!curl 'S
⇢@2u
@t2� (�+ 2µ)r( div u) + µ
���!curl ( curl u) = 0
⇢@2
@t2(r'P +
���!curl 'S)� (�+ 2µ)r(�'P )� µ
���!curl (�'S) = 0
Tuesday, 26 June 12
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
Decomposition of the displacement field into P wave and S wave
12
u = r'P +���!curl 'S
⇢@2u
@t2� (�+ 2µ)r( div u) + µ
���!curl ( curl u) = 0
div(��!curl) = 0
div(r) = �
⇢@2
@t2(r'P +
���!curl 'S)� (�+ 2µ)r(�'P )� µ
���!curl (�'S) = 0
Tuesday, 26 June 12
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
Decomposition of the displacement field into P wave and S wave
12
u = r'P +���!curl 'S
⇢@2u
@t2� (�+ 2µ)r( div u) + µ
���!curl ( curl u) = 0
⇢@2
@t2(r'P +
���!curl 'S)� (�+ 2µ)r(�'P )� µ
���!curl (�'S) = 0
Tuesday, 26 June 12
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
Decomposition of the displacement field into P wave and S wave
12
u = r'P +���!curl 'S
⇢@2u
@t2� (�+ 2µ)r( div u) + µ
���!curl ( curl u) = 0
curl(r) = 0
curl(��!curl) = ��
⇢@2
@t2(r'P +
���!curl 'S)� (�+ 2µ)r(�'P )� µ
���!curl (�'S) = 0
Tuesday, 26 June 12
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
Decomposition of the displacement field into P wave and S wave
12
u = r'P +���!curl 'S
⇢@2u
@t2� (�+ 2µ)r( div u) + µ
���!curl ( curl u) = 0
⇢@2
@t2(r'P +
���!curl 'S)� (�+ 2µ)r(�'P )� µ
���!curl (�'S) = 0
Tuesday, 26 June 12
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
Decomposition of the displacement field into P wave and S wave
12
u = r'P +���!curl 'S
⇢@2'P
@t2� (�+ 2µ)�'P = 0
⇢@2u
@t2� (�+ 2µ)r( div u) + µ
���!curl ( curl u) = 0
⇢@2
@t2(r'P +
���!curl 'S)� (�+ 2µ)r(�'P )� µ
���!curl (�'S) = 0
Tuesday, 26 June 12
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
Decomposition of the displacement field into P wave and S wave
12
u = r'P +���!curl 'S
⇢@2'P
@t2� (�+ 2µ)�'P = 0
⇢@2u
@t2� (�+ 2µ)r( div u) + µ
���!curl ( curl u) = 0
⇢@2
@t2(r'P +
���!curl 'S)� (�+ 2µ)r(�'P )� µ
���!curl (�'S) = 0
Tuesday, 26 June 12
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
Decomposition of the displacement field into P wave and S wave
12
u = r'P +���!curl 'S
⇢@2'P
@t2� (�+ 2µ)�'P = 0 ⇢
@2'S
@t2� µ�'S = 0
⇢@2u
@t2� (�+ 2µ)r( div u) + µ
���!curl ( curl u) = 0
⇢@2
@t2(r'P +
���!curl 'S)� (�+ 2µ)r(�'P )� µ
���!curl (�'S) = 0
Tuesday, 26 June 12
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
Decomposition of the displacement field into P wave and S wave
12
Pressure wave Shear wave
u = r'P +���!curl 'S
⇢@2'P
@t2� (�+ 2µ)�'P = 0 ⇢
@2'S
@t2� µ�'S = 0
⇢@2u
@t2� (�+ 2µ)r( div u) + µ
���!curl ( curl u) = 0
⇢@2
@t2(r'P +
���!curl 'S)� (�+ 2µ)r(�'P )� µ
���!curl (�'S) = 0
Tuesday, 26 June 12
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
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
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
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
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
Coupling of the P-waves and S-waves14
In a piecewise homogeneous medium, the coupling appear on the boundaries and on the interfaces.
P
S
⌦
[u] = 0
[�(u)n ] = 0
8><
>:
Boundary Conditions
u = 0or
�(u)n = 0
Perfect Bounding Interface Conditions
Tuesday, 26 June 12
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
Coupling of the P-waves and S-waves14
In a piecewise homogeneous medium, the coupling appear on the boundaries and on the interfaces.
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
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
15
Dirichlet boundary conditions
Example: case of a rigid boundary on � = @⌦
r'P +���!curl 'S = 0 � = @⌦on
u = 0
Tuesday, 26 June 12
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
15
Dirichlet boundary conditions
Example: case of a rigid boundary on � = @⌦
r'P +���!curl 'S = 0 � = @⌦on
n⌧ ��!curl ' · n = �@'
@⌧
��!curl '⇥ n =
@'
@n�⌦
u = 0
Tuesday, 26 June 12
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
15
Dirichlet boundary conditions
Example: case of a rigid boundary on � = @⌦
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
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
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
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
The energy identity16
@'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
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
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 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
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
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 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
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
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 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
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
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
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
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
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
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
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
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:
2
Z
⌦r'P ·
���!curl 'S
The energy identity
=
Z
⌦
���r'P +���!curl 'S
���2
Tuesday, 26 June 12
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
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:
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
⌦
���r'P +���!curl 'S
���2
Tuesday, 26 June 12
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
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:
2
Z
⌦r'P ·
���!curl 'S
The energy identity
And why are you happy with that?
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
⌦
���r'P +���!curl 'S
���2
Tuesday, 26 June 12
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
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:
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.
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
⌦
���r'P +���!curl 'S
���2
Tuesday, 26 June 12
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
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
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
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
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
A numerical illustration19
P-wave S-wave
The modulus of the displacement field is represented(in color levels) as a function of time
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
VP ' 3 VS
Tuesday, 26 June 12
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
A numerical illustration19
P-wave S-wave
The modulus of the displacement field is represented(in color levels) as a function of time
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
VP ' 3 VS
Tuesday, 26 June 12
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
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
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
20
Tuesday, 26 June 12
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
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
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
20
Tuesday, 26 June 12
Aliénor BUREL - Junior Seminar June, 26th 2012
PART 2
Thin layers approximation for time-domain elastodynamics
Tuesday, 26 June 12
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
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
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
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
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
22
GoalIf the width of the layer is small enough, can we compute “as if” there were no glue?
What does that mean?
⌘
We need more computation points inside the layer if we want to “see” something
Tuesday, 26 June 12
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
22
GoalIf the width of the layer is small enough, can we compute “as if” there were no glue?
What does that mean?
⌘
We need more computation points inside the layer if we want to “see” something
Tuesday, 26 June 12
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
22
GoalIf the width of the layer is small enough, can we compute “as if” there were no glue?
What does that mean?
⌘
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
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
23
How do we do that?
⌘
u�
u+
u⌘
Tuesday, 26 June 12
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
23
How do we do that?
⌘
Interface conditions
[u] = 0
[�(u)n] = 0
We use the
u�
u+
u⌘
Tuesday, 26 June 12
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
23
How do we do that?
Interface conditions
[u] = 0
[�(u)n] = 0
We use the
1
SCALED
U
u�
u+
Tuesday, 26 June 12
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
23
How do we do that?
Interface conditions
[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
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
23
How do we do that?
Interface conditions
[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
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
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
��!curl⌘( curl⌘U) = 0
Tuesday, 26 June 12
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
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
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
��!curl⌘( curl⌘U) = 0
�(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
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
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
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
��!curl⌘( curl⌘U) = 0
�(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
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
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
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
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...
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
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
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...
We reconstruct each jump:
[u±] = [u0±] + ⌘[u1
±] + ⌘2[u2±] + ⌘3[u3
±]...
with the previous conditions.
[�(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
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
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...
u0�
We reconstruct each jump:
[u±] = [u0±] + ⌘[u1
±] + ⌘2[u2±] + ⌘3[u3
±]...
with the previous conditions.
[�(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
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
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...
u0�
We reconstruct each jump:
[u±] = [u0±] + ⌘[u1
±] + ⌘2[u2±] + ⌘3[u3
±]...
with the previous conditions.
[�(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
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
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...
u0�
We reconstruct each jump:
[u±] = [u0±] + ⌘[u1
±] + ⌘2[u2±] + ⌘3[u3
±]...
with the previous conditions.
[�(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
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
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...
u0�
We reconstruct each jump:
[u±] = [u0±] + ⌘[u1
±] + ⌘2[u2±] + ⌘3[u3
±]...
with the previous conditions.
[�(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
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
26
Effective Interface Conditions
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
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
27
Scaled asymptotic expansion
Partially
Q: are we sure this approximation will lead to stable conditions?
Tuesday, 26 June 12
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
27
Scaled asymptotic expansion
Partially
Yes, up to the order 2: this approximation preserves an energy
Q: are we sure this approximation will lead to stable conditions?
Tuesday, 26 June 12
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
27
Scaled asymptotic expansion
Partially
Yes, up to the order 2: this approximation preserves an energy
No, with orders greater than 2, for now
Work in progress...
Q: are we sure this approximation will lead to stable conditions?
Tuesday, 26 June 12
Aliénor Burel Elastic wave propagation in soft elastic material June, 26th 2012
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
Aliénor BUREL - Junior Seminar June, 26th 2012
Thank you for your attention
Tuesday, 26 June 12