Numerical methods for wave propagation in nonlinear acoustics

1/97

Examples 1D 2D/3D Future Conclusions Ref

Numerical methods for wave propagation innonlinear acoustics

Regis Marchiano

Universite Pierre et Marie CurieInstitut Jean le Rond d’Alembert (UMR UPMC/CNRS 7190)

[email protected] Juin 2010

Regis Marchiano Oleron 2010

2/97


Guidelines...

”[...] In general it is extremely valuable to understand where theequation one is attempting to solve comes from, since a goodunderstanding of the physics (or biology, etc ...) is generallyessential in understanding the development and behavior ofnumerical methods for solving the equation [...]”

R. J. LeVeque, Finite Difference Methods for Ordinaryand Partial Differential Equations[1]


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Propagation of nonlinear waves

Propagation in fluids

Dauxois, Oléron 2007

Propagation in solids

Barrière, Oléron 2007


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Propagation in complex media

Propagation in rocks

Johnson, Oléron 2007

Propagation in granular media


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NDE and medical acoustics

NDE

Van Abeele, Oléron 2007

Medical therapy and imaging

http://www.polyclinique-cotebasquesud.com/images/Lithotritie.jpg


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Shock waves propagation in air (BSN, sonic boom, ...)


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Flow instabilities

Aeroacoustics Musical acoustics


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Various examples of nonlinear acoustics

Which numerical methods ?

Dauxois, Oléron 2007


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Answer:

... It depends ...


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Table of content I

1 Various examples of nonlinear acoustics

2 Numerical Resolution of the Burgers equationNumerical resolution of Burgers equation - case withoutshocksNumerical resolution of Burgers equation - case withshocks

3 Higher dimensionsThe geometrical acousticsThe one-way methodsDirect computation

4 Future ...?

5 Conclusions

6 References


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Examples 1D 2D/3D Future Conclusions Ref no shocks with shocks

The Burgers equation

For plane waves or in 1D in fluids, the simplest equation is theBurgers equation:

∂p

∂x=

β

2ρ0c30

∂p2

∂τ+

δ

2c30

∂2p

∂τ2,

with β the nonlinear term and δ the absorption one. Ifabsorption is neglected, it yields the inviscid Burgers equation:

∂p

∂x=

β

2ρ0c30

∂p2

∂τ,

How to solve that equation numerically? ....There exist severalways.


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Numerical resolution of Burgers equation - casewithout shocks

1 The inviscid Burgers equationFinite Difference MethodPoisson SolutionFrequency Domain Solution

2 The Burgers equationFrequency domainTime DomainHopf-Cole transformation


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Numerical resolution of the inviscid Burgers equation -case without shocks - Finite Difference Method

∂u

∂t+∂f(u)∂x

= 0 (1)

with f(u) = u2/2 for fluids or f(u) = u3/3 for solids.


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Basic Idea [2, 3, 1]: g(x, t)→ gni = g(xi, tn),∂g

∂t≈gn+1i − gni

∆t+O(∆t) or

∂g

∂t≈gn+1i − gn−1

i

2∆t+O(∆t2).

Various schemes exist:Explicit (Natural, Lax-Wendroff, Lax-Friederichs,

Leap-Frog,...)gn+1i − gni

∆t= L(fn)

Implicitgn+1i − gni

∆t= L(fn+1)

Crank-Nicolsongn+1i − gni

∆t=

12(L(gn+1) + L(gn)

)Regis Marchiano Oleron 2010

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Natural scheme for the Burgers equation:

un+1i − uni

∆t= −

Funi+1 − Funi−1

2∆x,

with Funi = f(uni ) and un1 = u(xn)Unstable ![3]Lax-Friederichs scheme:

un+1i =

uni+1 + uni2

− ∆t2∆x

(Funi+1 − Funi−1

),

with Funi = f(uni ) and un1 = u(xn)Conditionally stable [3]: ∆t/∆x < 1/|u|


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Numerical resolution of the inviscid Burgers equation-case without shocks - Finite Difference Method

Nx = Nt = 5000f(u) = u2/2 f(u) = u3/3


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Numerical resolution of the inviscid Burgers equation-case without shocks - Finite Difference Method

Same as previously but ∆x and ∆t are divided by 5(Nx = Nt = 5000).

f(u) = u2/2 f(u) = u3/3

Finite Difference methods for inviscid Burgers equationpresent... strong dissipation!Regis Marchiano Oleron 2010

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Numerical resolution of the inviscid Burgers equation-case without shocks - Poisson Solution

There exists an implicit solution for the inviscid Burgersequation:

u(x, t) = u0(ξ)

x = ξ − t.u0(ξ)

That solution can be used up to the shock formation distance:∂x

∂ξ6= 0.


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Numerical resolution of the inviscid Burgers equation -case without shocks - Poisson Solution

f(u) = u2/2 f(u) = u3/3


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Numerical resolution of the inviscid Burgers equation -case without shocks - Spectral method

The basic idea is closed to the Fubini solution. Solution ofinviscid Burgers equation:

∂p

∂x=

12∂p2

∂t,

is sought as:

p(x, t) =+∞∑

m=−∞Am(x) exp(−imt).

So, the problem to solve is now a coupled ODE set:

dAmdx

= − i2m

+∞∑m=−∞

Am−r(x)Ar(x)

For details see [4, 5, 6]Regis Marchiano Oleron 2010

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For the resolution, the system is truncated :

dAm∂x

= − i2m

+M∑m=−M

Am−r(x)Ar(x),

once the Am are computed, the pressure is reconstructed:

p(x, t) =+∞∑



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M = 10, 20, 30, 40, 50 and 100.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1x 10−6

−6

−4

−2

0

2

4

6x 105

T [s]

P [P

a]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1x 10−6

−6

−4

−2

0

2

4

6x 105

T [s]

P [P

a]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1x 10−6

−6

−4

−2

0

2

4

6x 105

T [s]

P [P

a]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1x 10−6

−6

−4

−2

0

2

4

6x 105

T [s]

P [P

a]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1x 10−6

−6

−4

−2

0

2

4

6x 105

T [s]

P [P

a]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1x 10−6

−6

−4

−2

0

2

4

6x 105

T [s]

P [P

a]


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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1x 10−6

−6

−4

−2

0

2

4

6x 105

T [s]

P [P

a]

M=10M=20M=30M=40M=50M=100

Figure:


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0 10 20 30 40 50 60 70 80 90 100−10

−9

−8

−7

−6

−5

−4

m

log(

Pw)

M=100M=50M=40M=30M=20M=10


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Conclusion

+ −Finite Difference Method simplicity absorption

Poisson Solution nonlinearities continuous wavestransient waves

Spectral method freq. interactions oscillationscontinuous waves transient waves


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Numerical resolution of the Burgers equation - casewithout shocks

The Burgers equation is:

∂p

∂x=

β

2ρ0c30

∂p2

∂τ2+

δ

2c30

∂2p

∂τ2,

Frequency Domain SolutionTime Domain SolutionHopf-Cole transformation


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Numerical resolution of the Burgers equation - casewithout shocks - Frequency domain approximation

As for the inviscid Burgers equation, the solution is sought as:

p(x, t) =+∞∑


There is an additional term in the set of ODE:

dAmdx

= αAm(x)m2 − i

2m

+M∑m=−M

Am−r(x)Ar(x),

The numerical procedure is the same as previously.


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Numerical resolution of the Burgers equation - casewithout shocks - Time domain approximation

Basic idea: resolution in time by using the power of Poissonsolution without drawbacks of numerical absorptionSolution: technique of the operator splitting:

1∂p

∂x=

β

2ρ0c30

∂p2

∂τ2, (implicit Poisson Solution)

2∂p

∂x=

δ

2c30

∂2p

∂τ2. (finite difference method)

!! !

1- Nonlinearitiesimplicit Poisson Solution

2 - Absorptionfinite difference method


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Numerical resolution of the Burgers equation - casewithout shocks - Hopf-Cole approximation

The Burgers equation:

∂u

∂t= u

∂u

∂x+ α

∂2u

∂x2,

The Hopf-Cole transformation:

u(x, t) = −2αθxθ

One obtains the heat equation (linear PDE!):

∂θ

∂t= α

∂2θ

∂x2

It is possible to solve it numerically thanks to classical methods.


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Conclusion

+ −Frequency domain absorption oscillations

nonlinearitiescontinuous waves

Time domain nonlinearities absorptiontransient waves

Pseudo-spectral nonlinearities FFT, IFFTabsorption

Hopf-cole transformation 1D nD


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Numerical resolution of the inviscid Burgers equation -case with shocks - Poisson Solution

f(u) = u2/2 f(u) = u3/3


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Numerical resolution of Burgers equation - case withshocks

The theory (cf 1st lecture by F. Coulouvrat) shows that twoingredients are needed beyond the shock formation distance:

The ’mathematical’ solution of the Burgers equation(whatever the method) ;The weak shock theory ;

From the 1st part, the first point is addressed. For the secondpoints, there are two strategies:

Shock capturing (introduction of numerical dissipation);Shock fitting (explicit treatment of the shock);


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Numerical resolution of Burgers equation - case withshocks - shock capturing

Basic idea: to introduce numerical viscosity in theneighborhood of shocksAn example: McDonald and Ambrosiano scheme (1984):

1 standard scheme for the regular part2 correction of the flux near the shocks

For more details, see Leveque (1992).


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Numerical resolution of Burgers equation - case withshocks - shock fitting

The equal area rule (Landau, 1945)[7] is a good example of themethods of shock fitting.

Formulation géométrique : la loides aires égales

Landau (1944)

Loi des aires égales : le choc est placé de telle façon que les airesdes 2 lobes de part et d’autre du choc sont égales

!

A+ = A"

P

!!C

P -

P +

A -

A +

Not efficientThe numerical cost is high (... but the idea is good)


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The Burgers-Hayes method [8, 9, 10]: Poisson solution for thepotential + max of the potential

!3 !2 !1 0 1 2 30

0.05

0.1

0.15

0.2

0.25

0.3

0.35

!3 !2 !1 0 1 2 3!0.4

!0.3

!0.2

!0.1

0

0.1

0.2

0.3

0.4

!3 !2 !1 0 1 2 3!0.4

!0.3

!0.2

!0.1

0

0.1

0.2

0.3

0.4

Φ(X,T )Φ(X + ∆X,T )Φphys(X + ∆X,T )


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The Burgers-Hayes method needs:1 Compute the potential ;2 Compute the Poisson solution ;3 Compute the max of the Poisson solution (for quadratic

nonlinearities) ;4 Compute the original variable ;

Efficient and elegantfor time domain simulation


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Examples 1D 2D/3D Future Conclusions Ref Rays One-way Direct

The original sin

Now, 1D is a piece of cake, what about 2D/3D ?

Figure: Michel-Ange, Chapelle Sixtine, 1512

The troubles begin !


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A first example

Simulation of sonic boom:

domain 18km×35km×60km.f0 ≈ 3.4Hz; c0 ≈ 340m.s−1 soλ0 ≈ 100msimulation with 100 harmonics impliesλ100 ≈ 1mgood discretization implies ≈ 6 pts/λmeshes are:18.103∗6×35.103∗6×60.103∗6 ≈ 8.1015

time integration dt ≈ 1, 5.10−3s,∆T ≈1min, NT ≈ 40.103

The direct simulationis prohibitive (not necessarily impossible) and alternative wayshave to be found.


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Several ways to follow

Geometrical acoustics ;One way methods ;Nonlinear Wave Equation ;The FDTD.


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Nonlinear geometrical acoustics

High frequency;plane wave assumption;λ << L;λ << LH .


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Numerical implementationFor one ray, one has to solve the following ODE set (18equations):

Trajectory:

dxray

dψ= f(x,n)

dndψ

= g(x,n)

Amplitude:

dxαdψ

=∂f∂x

.xα +∂f∂n

.nαdxβdψ

=∂f∂x

.xβ +∂f∂n

.nβdnαdψ

=∂g∂x

.xα +∂g∂n

.nα.

dnβdψ

=∂g∂x

.xβ +∂g∂n

.nβ.

+ Burgers along each ray


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+ −easy caustics

efficient (fast) shadow zoneinterferences

diffraction

For more details, see Olaf Gainville’s lecture on Friday.


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One-way approximations


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Basic idea of the one-way approximations

Basic idea: factorization of the wave equation.Similar to the derivation of the 1D Burgers equation (see 1stlecture):

∂2p

∂x2− 1c20

∂2p

∂t2= 0, (2)

That equation can be factorized easily:

(∂p

∂x− 1c0

∂p

∂t

) (∂p

∂x+

1c0

∂p

∂t

)= 0 (3)

Propagation towards −x Propagation towards+x


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One way approximation in 1D

The ”one-way” equation is:

∂p

∂x+

1c

∂p

∂t= 0, (4)

Considering that the sound velocity depends on the

instantaneous pressure: c ≈ c0 +βp

ρ0c0, it yields the Burgers

equation:

∂P

∂X=

12∂P 2

∂T(5)

with P = p/P0, X = x/LC , LC = 1/(kβM), M acoustical Machnumber, β ≈ 3.5 (in water), k the wavenumber andT = ω0(t− x/c0)


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One-way approximation in 2D/3D

Let us start from the Lighthill-Westerwelt equation:

∂2pa∂t2

− c20∆pa =β

ρ0c20

∂2p2a

∂t2

That equation can be expressed in delayed time τ = t− x/c0:

2c0∂2pa∂τ∂x

− c20(∂2pa∂x2

+ ∆⊥pa

)=

β

ρ0c20

∂2p2a

∂t2

How to factorize that equation ? formal factorization is possiblebut introduces non integer operators !


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One-way approximation in 2D/3D

Several strategies exist to factorize the wave equation:1 Parabolic approach (KZ, wide angle, NPE, ...)2 Other approches


48/97


One-way approximation in 2D/3D: the parabolicapproximation

Derivation of the KZ equation

2c0∂2pa∂x∂τ

− c20(∂2pa∂x2

+ ∆⊥pa

)=

β

ρ0c20

∂2p2a

∂t2

Parabolic approximation:1c0

∂pa∂τ

>>∂pa∂x

2c0∂2pa∂x∂τ

− c20

∂2pa∂x2

+∂2pa∂y2

=β

ρ0c20

∂2p2a

∂t2.

Khokhlov-Zabolotskaya equation [11] (or NPE [12]) :

2c0∂2pa∂x∂τ

− c20∂2pa∂y2

=β

ρ0c20

∂2p2a

∂τ2.


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One-way approximation in 2D/3D: the wide anglenonlinear equation

Derivation of the wide-angle nonlinear equation:

2c0∂2pa∂x∂τ

− c20(∂2pa∂x2

+ ∆⊥pa

)=

β

ρ0c20

∂2p2a

∂t2

The linear parabolic approximation:∂2pa∂x∂τ

− c02∂2pa∂y2

= 0 so,∂3pa∂x2∂τ

− c02∂3pa∂x∂y2

= 0

It yields the wide-angle nonlinear wave equation [13, 14]:

2c0∂3pa∂x∂τ2

− c20[(

∂

∂τ+c02∂

∂x

)∂2pa∂y2

]=

β

ρ0c20

∂3p2a

∂τ3


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One-way approximation in 2D/3D: go further ?

−2 −1.5 −1 −0.5 0 0.5 1 1.5−1.5

−1

−0.5

0

0.5

1

1.5

KZ

KX

The process can be used again and again to improve theapproximation.

ButThe order of the equation increases (so additional BC arerequired)The improvement is not so important (see results of thenumerical simulations)Regis Marchiano Oleron 2010

51/97


How to solve the KZ equation ?

The KZ equation [11, 15] is:

∂P

∂X∂τ=∂2P

∂Y 2+ µ

∂2P 2

∂τ2,

with µ the nonlinear parameter, α the absorption parameter.3 families of numerical solvers exist:

1 Spectral resolution ;2 Time domain resolution.3 Pseudo-spectral resolution ;


52/97


Spectral resolution of the KZ equation

The solution is sought as:

P (X, τ) =+∞∑

m=−∞Am(X) exp(−imτ).

So, the problem to solve is now a coupled ODE set [5]. For thenumerical resolution, the system has to be truncated.The most famous implementation is known as Bergen code(http://www.uib.no/People/nmajb/aku.html).

+ −Spectral solution absorption oscillations

strong nonlinearities


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Pseudo-Spectral and temporal resolutions the KZequation

The pseudo-spectral and the temporal methods of resolution ofthe KZ equation rely on the split-step procedure [16, 17].

∂2P

∂X∂τ=

∂2P

∂Y 2+ µ

∂2P 2

∂τ2

!! !

Diffraction

FFT + FD + IFFTTemporal solver

Nonlinearities

Temporal solverTemporal solver


54/97


Time domain resolution the KZ equation

This family of solvers has been introduced by Lee and Hamiltonet al. [18].

∂2P

∂X∂τ=

∂2P

∂Y 2+ µ

∂2P 2

∂τ2

∂P

∂X=∫ τ

−∞

∂2P

∂Y 2dτ ′

Implicit scheme: P ki = L(P ki )+Trapezoidal rule

∂2P

∂X∂τ= µ

∂2P 2

∂τ2

Integration:∂P

∂X= µ

∂P 2

∂τThis is the Burgers equation. Itcan be solved by methodspresented in section I.


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Time domain resolution the KZ equation: Exemple

N wave focusing on a cusp caustic [19]

ConvergentWavefront

P

T

N wave


56/97


Pseudo-Spectral resolution the KZ equation

This family of solvers has been introduced by Bakhvalov et al.[20].

∂2P

∂X∂τ=

∂2P

∂Y 2+ µ

∂2P 2

∂τ2

∂2P

∂X∂τ=∂2P

∂Y 2

FFT: iω∂P

∂X=∂2P

∂Y 2

Implicit scheme: P ki = L(P ki )Resolution of a linear system.IFFT: P (Z = ∆Z, τ)

∂2P

∂X∂τ= µ

∂2P 2

∂τ2

Integration:∂P

∂X= µ

∂P 2

∂τThis is the Burgers equation. Itcan be solved by methodspresented in section I.


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Pseudo-Spectral resolution the KZ equation: Example

The reflexion of shock waves on a rigid surface

z

y

θ

θ<15°

Incident wave

Reflected wave

Periodic sawtooth wave


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The reflexion of shock waves on a rigid surfaceθ = 5o θ = 3o θ = 1o

a = 1.81 a = 0.91 a = 0.36

a) KZ code, b) Experiment and c) linear KZ equation


59/97



Piston: (a) analytical solution, (b) KZ simulation

x/ !

y/ !

(a)0 5 10 15

−10

0

10−40

−20

0

x/ !

y/ !

(b)0 5 10 15

−10

0

10−40

−20

0

x/ !

y/ !

(c)0 5 10 15

−10

0

10−40

−20

0

x/ !y/

!

(d)0 5 10 15

−10

0

10−40

−20

0


60/97


Numerical resolution of the wide-angle nonlinearequation

The procedure is the same as for the KZ equation: split-stepprocedure + resolution of the simpler equations

x/ !

y/ !

(a)0 5 10 15

−10

0

10−40

−20

0

x/ !

y/ !

(b)0 5 10 15

−10

0

10−40

−20

0

x/ !

y/ !

(c)0 5 10 15

−10

0

10−40

−20

0

x/ !

y/ !

(d)0 5 10 15

−10

0

10−40

−20

0

x/ !

y/ !

(a)0 5 10 15

−10

0

10−40

−20

0

x/ !y/

!

(b)0 5 10 15

−10

0

10−40

−20

0

x/ !

y/ !

(c)0 5 10 15

−10

0

10−40

−20

0

x/ !

y/ !

(d)0 5 10 15

−10

0

10−40

−20

0

(a) analytical(c) KZ(d) WA


61/97


Summary

+ −Frequency domain absorption oscillations

nonlinearitiescontinuous waves

Time domain nonlinearities absorptiontransient waves

Pseudo-spectral nonlinearities FFT, IFFTabsorption

continuous wavesadditional effects


62/97


Parabolic approximation in heterogeneous media

The parabolic equations (KZ or wide-angle) can be extended totake into account propagation in weakly heterogeneous media[21, 22]Sound speed fluctuations: c(x, y) = c0 + c0(x, y)Generalized KZ equation:

∂2pa∂x∂τ

− c02∂2pa∂y2

=β

ρ0c02

∂2p2a

∂τ2+c20(x, y)

2c03

∂2pa∂τ2

.

!! !

Diffraction Heterogenities Nonlinearities


63/97


Shock wave propagation through heterogeneous medium

c0 ch c0


64/97


Parabolic approximation in heterogeneous media

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,4./0-&!()0/'/&A!+*/'!*2''/! '2.! 3*!,4./0-&!,.4(4-&18&! =(J?!>/7?!$LB?!%&'!-&26! 3/71&'!&1!

(a) Experiment, (b) Wide angle heterogeneous equation and (c)heterogeneous KZ equation


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Numerical Resolution of the KZ equation in 3D

∂2Φ∂X∂τ

= ∆⊥P + µ∂

∂τ

(∂Φ∂τ

)2

+Hc∂2Φ∂τ2

!! !

Diffraction effects

∂2Φ∂X∂τ

= ∆⊥Φ

→ spectral method

Nonlinear effects

∂2Φ∂X∂τ

= µ∂

∂τ

(∂Φ∂τ

)2

→ Burgers-Hayesmethod

Heterogeneouseffects

∂2Φ∂X∂τ

= Hc∂2Φ∂τ2

→ analytical method


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source: P (X = 0, Y, Z, T ) = P0 sin(2πf0T ) exp(−(Y 2 + Z2)σ2

))

Temporal evolution

RMS pressure Phase


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RMS pressureRegis Marchiano Oleron 2010

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Sound speed profil: c(x, y, z) = c0 + 100y

!0.1

0

0.1

0.1

0.2

0.3

0.4

0.5

!0.1

!0.05

0

0.05

0.1

XZ

Y

1490

1492

1494

1496

1498

1500

1502

1504

1506

1508

1510

!0.10

0.10.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55

!0.1

!0.05

0

0.05

0.1

X

Z

Y

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

white curve = acoustical ray


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Summary on the parabolic approaches

Advantagesefficient ;versatile ;very popular ;

Drawbacksangular limitation ;no backscattering ;


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Beyond the parabolic

Is it possible to keep the advantages of parabolic approach,while avoiding the drawbacks ?Existing answers:

1 Christopher and Parker phenomenological approach [23] ;2 Wojcik et al.,’s TAWE method [24];3 HOWARD method.

These methods are still one-way methods


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Christopher and Parker phenomenological approach[23]

problem, in order to gain advantages in computational effi- ciency, to develop a formalism that does not require the parabolic approximation, and to allow for multiple layer propagations. Basically, our methodology utilizes an incremental propagation of an axially symmetric field from some plane at axial distance z from the source to a parallel plane at distance z + Az. The concept is illustrated in Fig. 1. The normal velocity radial profiles ui (z,r) of a fundamental (solid line) and higher harmonics u, (z,r) (dashed lines), n = 2 to N, are propagated together in Az steps. Note in Fig. 1 that to highlight the nonlinear effect over Az, the initial field consists of the fundamental harmonic only.

First, the diffraction effect is accounted for by utilizing the (Fourier) convolution theorem and a discrete Hankel transform (DHT). The recently developed DHT •8 offers great computational savings 19 as compared to the two-dimension- al discrete fast Fourier transform. Computing diffraction in- volves multiplying the DHT of each harmonic radial profile by its appropriate linear transfer function H, (z,R) [the DHT of the point spread function h• (z,r) ], which is equivalent to convolution in the original spatial domain, as implied by Huygen's principle. An inverse DHT gives the desired diffracted harmonic radial profile u'• (z + Az, r) (the prime notation indicates intermediate results) for each harmonic present in the propagation. This diffraction operator is de- picted in Fig. 2. Note also the presense of attenuation in the form of an attenuated transfer function. The nonlinear effect is accounted for by advancing the diffracted and discretely sampled field forward on a point-by-point basis over an

equivalent incremental Az distance without diffraction, but with accretion and depletion of harmonics according to a temporal frequency domain solution to Burgers' equation (FDSBE).20-22 That is, for some radial position ri, the normal velocities u', (z + Az, ri) are modified by nonlinear mechanisms to u, (z + Az, r• ) as if the acoustic velocity rep- resented a plane wave traveling in the direction of the phase front at ri. This concept is illustrated in Fig. 3.

The model also accounts for the effects of refraction and reflection (but not multiple reflections) in the case of propagation through multiple, parallel layers of fluid medium. •9 The model's use of a novel harmonic-limiting scheme for the FDSBE 23 makes possible some previously intractable high- intensity (shocked) propagations. The model's approach appears to be a departure from other nonlinear models in its extensive use of transform (spatial and temporal) tech- niques. There are well-known computational advantages in- herent in transform operations, but also pitfalls in imple- mentations as discussed extensively in our companion paper on linear propagation. 19 These pitfalls can be avoided by appropriately windowing the point spread function h or its transform H.

Conceptually, our approach is somewhat akin to the work of Pestorius and Blackstock, 24'25 who treated the nondif- fraeting propagation of plane waves in two domains using incremental advances. In the time domain, waveform distor-

DIFFRACTION & ATTENUATION SUBSTEP

INCREMENTAL PROPAGATION I Partk•e Velocity

U • U • HANKEL TRANSFORM J r R

MULTIPLY BY ATTENUATED

INVERSE DIFFRACTION NONUNEAR HANKEL TRANSFORM 8tJBSl'EP 8UBSTEP

FIG. 1. Illustration depicting the concept of incremental field propagation using substeps for the diffraction and nonlinear operators. Shown are radial (transaxial) plots of normal velocity magnitude of the fundamental (solid lines) and the higher harmonics (dotted lines) at different stages in the propagation substeps. To highlight the nonlinear effect over Az, the figure depicts an initial single harmonic field.

FIG. 2. The diffraction substep with its discrete Hankel transform useage (here applied to one harmonic using the RFSC approach).

489 J. Acoust. Soc. Am., Vol. 90, No. 1, July 1991 P.T. Christopher and K. d. Parker: Nonlinear diffractive field 489

Christopher and Parker, JASA, 1991

1 Resolution of thewave equation

2 Resolution of theBurgers equation


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Woicjik et al.’s TAWE method [24]

TAWE = Time Averaged Wave Envelopes method

the given point in space as the superposition of sinusoidalwaves unbounded in time. The vertical lines in Fig. 2(A)

correspond to the coefficients Cl of the decomposition.They are calculated after the substitution of Eq. (5) into

d0

),( zPm x

QmP

),( zzPm ∆+x

x

x

y

dmin (x, z) P(x, z)

z

nonlinear step

diffraction & absorption step

S

dmax(x, z)

z∆

Fig. 1. Explanation of the incremental wave propagation using incremental steps for the diffraction, absorption and non-linear distortion operators. Hered0 denotes the diameter of the source; dmax and dmin are the maximum and minimum distances of the field point (x,z) from the source. The pressure at (x,z)is propagated to the field point (x,z + Dz) in two steps.

n

V(x, z, n) = F [ P(x, z, τ) ]

V1

V2Vm

0 2πτ

P(x, z,τ)

Cn(x, z)

… + … +

A

B

C

n max

dn

mnim ezP

−),τ,(x

niezP 2

2 ), τ,(−

xni

ezP 1), τ,(1−

x

Nn~

11= Nn~

22=Nmnm~=

Fig. 2. Decomposition of the disturbance, P(x,z,s). (C) shows the representation of P(x,z,s) as Re[R(x,z,s)]. (B) shows the decomposition of P(x,z,s) intothe sum of M wavelet-like sinusoidal pulses, Pmðx; z; sÞ expð#im~NsÞ. (A) shows the Fourier spectrum, V(x,z,n). The N spectral components, Cn(x,z)(shown as the solid lines – in (A)) are calculated in the CM. The Fourier spectra V mðx; z; n# m~NÞ of the individual wavelets (shown as thin solid lines – in(A)), are calculated in the TAWE method and sum into the total spectrum identical to V(x,z,n).

314 J. Wojcik et al. / Ultrasonics 44 (2006) 310–329

the given point in space as the superposition of sinusoidalwaves unbounded in time. The vertical lines in Fig. 2(A)

correspond to the coefficients Cl of the decomposition.They are calculated after the substitution of Eq. (5) into

d0

),( zPm x

QmP

),( zzPm ∆+x

x

x

y

dmin (x, z) P(x, z)

z

nonlinear step

diffraction & absorption step

S

dmax(x, z)

z∆

Fig. 1. Explanation of the incremental wave propagation using incremental steps for the diffraction, absorption and non-linear distortion operators. Hered0 denotes the diameter of the source; dmax and dmin are the maximum and minimum distances of the field point (x,z) from the source. The pressure at (x,z)is propagated to the field point (x,z + Dz) in two steps.

n

V(x, z, n) = F [ P(x, z, τ) ]

V1

V2Vm

0 2πτ

P(x, z,τ)

Cn(x, z)

… + … +

A

B

C

n max

dn

mnim ezP

−),τ,(x

niezP 2

2 ), τ,(−

xni

ezP 1), τ,(1−

x

Nn~

11= Nn~

22=Nmnm~=

Fig. 2. Decomposition of the disturbance, P(x,z,s). (C) shows the representation of P(x,z,s) as Re[R(x,z,s)]. (B) shows the decomposition of P(x,z,s) intothe sum of M wavelet-like sinusoidal pulses, Pmðx; z; sÞ expð#im~NsÞ. (A) shows the Fourier spectrum, V(x,z,n). The N spectral components, Cn(x,z)(shown as the solid lines – in (A)) are calculated in the CM. The Fourier spectra V mðx; z; n# m~NÞ of the individual wavelets (shown as thin solid lines – in(A)), are calculated in the TAWE method and sum into the total spectrum identical to V(x,z,n).

314 J. Wojcik et al. / Ultrasonics 44 (2006) 310–329

[Wojcik et al., Ultrasonics, 2006]


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HOWARD

HOWARD = Heterogeneous One-Way Approximation forResolution of Diffraction

2c0∂2pa∂x∂τ

−c20(∂2pa∂x2

+∂2pa∂y2

)− β

ρ0(x)c20(x)∂2p2

a

∂τ2

−(

1− c20c20(x)

)∂2pa∂τ2

+c20(x)ρ0(x)

([∂pa∂x− 1c0

∂pa∂τ

]∂ρ0

∂x+∂pa∂y

∂ρ0

∂y

)=0

No approximation on the diffraction term ;High frequency approximation on the nonlinear term ;Wide angle approximation on the heterogeneous term(Heterogeneous terms are assumed to be weak).


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HOWARD

Numerical resolution: the splitting operator technique

2c0∂2pa∂x∂τ

−c20(∂2pa∂x2

+∂2pa∂y2

)− β

ρ0(x)c20(x)∂2p2

a

∂τ2

−(

1− c20c20(x)

)∂2pa∂τ2

+c20(x)ρ0(x)

([∂pa∂x− 1c0

∂pa∂τ

]∂ρ0

∂x+∂pa∂y

∂ρ0

∂y

)=0

!! !

Diffraction

Spectral method

Heterogeneities

Finite difference

Nonlinearities

Burgers-Hayes


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Step 1: diffraction

Diffraction: Spectral resolution

2c0∂2Φ∂x∂τ

−c20(∂2pa∂x2

+∂2pa∂y2

)= 0

Angular spectrum method [?, 23, 25]:ˆpa(X, kY , ω) = FY {FT {Φ(X,Y, T )}}

D∂2 ˆpa∂X2

+ iω∂ ˆpa∂X−K2

Yˆpa = 0.

2nd order ODE => 2 solutions (propagation towards +x et −x).

ˆpa(X+∆X,KY , ω) =

ˆpa(X,KY , ω) exp

−iω +√

4DK2Y − ω2

2D

∆X if 4DK2Y ≥ ω2

ˆpa(X,KY , ω) exp

−iω + i√ω2 − 4DK2

Y

2D

∆X if 4DK2Y < ω2.

[1] Goodman et al., 1976 ; [2] Christopher et al., 1991, [3] Marchiano et al., 2008


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Dispersion

−2 −1.5 −1 −0.5 0 0.5 1 1.5−1.5

−1

−0.5

0

0.5

1

1.5

KZ

KX


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Step 2 : heterogeneities

heterogeneities: Finite differences method (pseudo-spectral method)

2c0∂2Φ∂x∂τ

−(

1− c20c20(x)

)∂2Φ∂τ2

+c20(x)ρ0(x)

([∂Φ∂x− 1c0

∂Φ∂τ

]∂ρ0

∂x+∂Φ∂y

∂ρ0

∂y

)=0

Resolution in Fourier space with an implicit scheme (possibilityto add absorption).


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HOWARD

x/

y/

(a: analytique)

0 5 10 15

15

10

5

0

5

10

15 40

30

20

10

0

x/

y/

(b : HOWARD)

0 5 10 15

15

10

5

0

5

10

15 40

30

20

10

0

x/

y/

(c : KZ)

0 5 10 15

15

10

5

0

5

10

15 40

30

20

10

0

x/

y/

(d : Grand angle)

0 5 10 15

15

10

5

0

5

10

15 40

30

20

10

0


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Propagation of sonic boom in turbulent atmosphere


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Propagation of sonic boom in turbulent atmosphere


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Conclusions

Today, one-way methods are the standard for manyapplications. But remember that there is a severe drawback(due to their advantages!): they are one-way.

They are not adapted for nonlinear propagation in complexmedia.


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Direct computation

Very intensive but ...Various approaches exist:Sparrow and Raspet, JASA, 1991 [?]

FDTD: Finite Difference Time DomainResolution of a system of 1st order equations (nonlinear +dissipative effects)2D simulation of the propagation of spark pulse inhomogeneous media(NX ×NY )=322×322, NT = 4000.Computer: Cray 2.


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Direct computation

Sparrow and Raspet, JASA, 1991(a) {c)

• ..... :=-:'•.. •,•;•.,--•. "'".":::'•-:.¾!:.':...! '.: .-::' ::' :-.'.-:'• : ": -' : .... ' :' ' ':: :.'.':'%'•½A•. ' :'.-*:•,;-'! :.:.::-:;:-,:.-::.:7.:::.-:-: ================================= '- :--'•':.::-- ";:-i"i:"-i:::':::" :-':::Y ::: i- !: •:-..'- :i-...':..:'.-i:?:'-..:::+: i-:..:-'

.i:::-:: ß ' '.': :':':':i:i:.:•; ' ::.i..%%t.--. ........ ' ........... '.: ,:-: .:::

.... . ..... . :•.-..:•.•.. .;..-:.:-:-:-:-.-:-: :.:.:.:.:-:.:.: :...::.'.!::.::

:.:!::..::... :.. " '"'::.'...•.: ..i '- ' '. .... -' ":':'.':..!..ii,:•.•: • ::.:'%"v ' ß 'i : ß ' ß: ': ':"': ":" :' "i'."::'":".:: :"';'""::•'"::•':'"""" ' -•:•' ':'& •i :'"" •:i':::':' :' ' ' '

lb) (d)

ß .. -•.."?•.• .- :,'.•-:,: •.',,-.,':i..i' :.-.'.i: ,.ii :ii E: i::. •:i :':i:.:::.:r':' :', ': "::':':"':'*'½.:,, i:• ::"• '"' •:'" ' ::::::::::::::::::::::::::::::::::::: •':" :'"':"'"':•:'":'"'""•':

'::.4. ,:•, • ; ' .:" :i.• .-'- ' -,-':' .il, :.:.i :L .' :': ': .ii':: "i:: , i .:: ,: :i: i ::: ':::' ' ' : .... :'• • """: :'::"::' :::':::':':'::":: ': ':':'"'":: ,'--•::, "::.i'"::."!: J!;:•! :::.•:":.::::.i:i:L•:.?• i:: !::'::::::.:i::.

FIG. 4. Mach stem development for one of the simulation runs. Each snapshot is 4 cm long in d and extends from I 0175 to 3'0 cm high in Zl (a) The initial condition at time zero. (b)-(d) Snapshots at times 40, 80, and ! 20Fs in a coordinate system moving forward at speed 343 m/s. Positive acoustic pressures are the darker shades, while the negative acoustic pressures are light.

2.4 ÷

2.2

2

1.8

1.6

1.4

1.2

amplificatiønb• factor

a

d

incident angle in degrees

2'0 4'0 6'0 8'0

FIG. 5. Oblique reflection amplification factor versus angle in degrees for fixed free-field incident peak pressures. When the angle is 0 deg, the wave front is locally normally incident on the hard surface, and when the angle is 90 deg, the wave front is locally grazing. (a) Free-field incident peak pres- sureis 15.0 kPa, 177.5 dB, (b) I0.0 kPa, 174 dB, (c) 5.0 kPa, 168 dB, and (d) 2.5 kPa, 162 dB.

add the effects of molecular relaxation dissipation. In the future the numerical technique will model the propagation of acoustic waves from explosions over natural ground sur- faces and the results will be compared to published experi- mental data.

ACKNOWLEDGMENTS

This research was supported by the United States Army Construction Engineering Research Laboratory (USA- CERL), Champaign, IL. The National Center for Super- computing Applications (NCSA), Champaign, IL provided the supercomputing resources.

• D. H. Trivett and A. L Van Buren, "Propagation of plane, cylindrical, and spherical finite amplitude waves," J. Acoust. Soc. Am. 69, 943-949 (1981).

2S. I. Aanonsen, T. Barkve, J. Naze Tj6tta and S. TjCtta, "Distortion and

2690 J. Acoust. Soc. Am., Vol. 90, No. 5, November 1991 V. W, Sparrow and R. Raspet: Finite amplitude wave propagation 2690

Downloaded 18 Jun 2010 to 134.157.34.104. Redistribution subject to ASA license or copyright; see http://asadl.org/journals/doc/ASALIB-home/info/terms.jsp


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Direct computation

Pinton et al., IEEE Trans. Ultrason. Ferroelectr. Freq. Control,2010 [26]

FDTD: Finite Difference Time DomainResolution of the Westervelt equation (2nd orderderivatives, nonlinear + dissipative effects)3D simulation of nonlinear propagation of ultrasonic wavesgenerated by a transducer(LX × LY × LZ)=1.5cm×1.5cm×1.5cm, f0 = 2MHz,λ = 0.075cm.(NX ×NY ×NZ)=833×833×833, NT = 10000.Cluster: 56 proc, 128GB, Linux, MPI

For more details, see Gianmarco’s talk on thursday.


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Direct computation

cm

cm

10 5 0 5 10

0

5

10

1535

30

25

20

15

10

5

0


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New trends ?

Various new trends exist. Let us focus on the parallelization.Again, various paradigms exist:

1 SPMD = Single Program Multiple Data (OpenMPIstandard)

2 SIMD = Single Instruction Multiple Data (OpenMPstandard)

3 CPU vs. GPU


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OpenMP vs MPI

Shared memory

P1 P2 P3 P4

SIMD (OpenMP)

P1 P2 P3 P4

SPMD (MPI)

M1 M2 M3 M4

Message PassingInterface


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MPI example: Domain Decomposition


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GPU vs CPU

CPU = Central Processing UnitIn June 2010: a standard configuration has ≈ 10 cores.Numerous clusters exist through the world from ≈ 100 cores inthe labs to ≈ 100.000 cores for the best ones (TOP 500 : 1st

Oak Ridge National Laboratory 224162 cores ! )GPU = Graphics Processing UnitIn June 2010: on 1 GPU = (≈ 500 cores, 6Go memory, doubleprecision). Possibility to have a cluster of GPU.

New technologies...... need new methodologies to take advantage of the availablepower. Think out of the box !


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Conclusions

Lot of methods exist ;No-one is the best one ;Everything begins with a good modeling ;Follow the physics to find the appropriate method ;Numerical methods are fun (numerical experiments) ;Think out of the box !


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References I

R. J. Leveque, Finite Difference Methods for Ordinary andPartial Differential Equations (SIAM) (2007).

G. Cohen, Higher-order numerical methods for transientwave equations (Springer-Verlag) (2002).

D. Euvrard, Resolution numerique des equations auxderivees partielles (Masson) (1994).

A. Korpel, “Frequency approach to non-linear dispersivewaves”, J. Acoust. Soc. Am. 67, 1954–1958 (1980).

S. I. Aanonsen, T. Barkve, J. N. Tjotta, and S. Tjotta,“Distortion and harmonic generation in the nearfield of afinite amplitude sound beam”, J. Acoust. Soc. Am. 75,749–768 (1984).


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References II

R. Fernando, “Modelisation et simulation du buzz sawnoise”, Ph.D. thesis, Universite Pierre et Marie Curie(2010).

L. Landau, “On shock waves at large distances from theplace of their origin”, J. Phys. USSR 9, 496–500 (1945).

J. M. Burgers, “A mathematical model illustrating the theoryof turbulence”, Adv. Appl. Mech. 1, 171–199 (1948).

W. D. Hayes, R. C. Haefeli, and H. E. Kulsrud, “Sonic boompropagation in a stratified atmosphere with computerprogram”, Technical Report CR-1299, NASA (1969).

F. Coulouvrat, “A quasi-analytical shock solution for generalnonlinear progressive waves”, Wave Motion online (2008).


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References III

E. A. Zabolotskaya and R. V. Khokhlov, “Quasi-plane wavesin the nonlinear acoustics of confined beams”, Sov. Phys.Acoust. 15, 35–40 (1969).

B. E. McDonald and W. A. Kuperman, “Time domainformulation for pulse propagation including nonlinearbehavior at a caustic”, J. Acoust. Soc. Am. 81, 1406–1417(1987).

J. F. Claerbout, Fundamentals of geophysical dataprocessing: with applications to petroleum pospecting(McGraw-Hill (New-York)) (1976).

L. Ganjehi, “Ondes de choc acoustiques en milieuheterogene, des ultrasons au bang sonique”, Ph.D. thesis,Univeriste Pierre et Marie Curie (2008).


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References IV

V. P. Kuznetsov, “Equation of nonlinear acoustics”, Sov.Phys. Acoust. 16, 467–470 (1970).

W. F. Ames, Numerical methods for partial differentialequations (Academic Press (San Diego)) (1977).

R. J. Leveque, Numerical Methods for Conservation Laws(Birkhauser) (1992).

Y. S. Lee and M. F. Hamilton, “Time-domain modeling ofpulsed finite-amplitude sound beams”, J. Acoust. Soc. Am.97 (1995).

R. Marchiano, F. Coulouvrat, and J. L. Thomas, “Nonlinearfocusing of acoustic shock waves at a caustic cusp”, J.Acoust. Soc. Am. 117, 566 – 577 (2005).


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References V

N. S. B. et al., Nonlinear Theory of Sound Beams(American Institute of Physics) (1987).

P. Blanc-Benon, P. Lipkens, L. Dallois, M. F. Hamilton, andD. T. Blackstock, “Propagation of finite amplitude soundthrough turbulence: Modeling with geometrical acousticsand the parabolic approximation”, J. Acoust. Soc. Am.487–498 (2002).

L. Ganjehi, R. Marchiano, F. Coulouvrat, and J. L. Thomas,“Evidence of the wave front folding of sonic boom by alaboratory scale deterministic experiment of shock waves ina heterogeneous medium”, J. Acoust. Soc. Am. 124, 57–71(2008).


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References VI

T. Christopher and K. Parker, “New approaches tononlinear diffractive field propagation”, J. Acoust. Soc. Am.90, 488–499 (1991).

J. Wojcik, A. Nowicki, P. A. Lewin, P. E. Bloomfield,T. Kujawska, and L. Filipczynski, “Wave envelopes methodfor description of nonlinear acoustic wave propagation”,Ultrasonics 44, 310–329 (2006).

R. Marchiano, F. Coulouvrat, L. Ganjehi, and J. L. Thomas,“Numerical investigation of the properties of nonlinearacoustical vortices through weakly heterogeneous media”,Physical Review E 77, 016605 (2008).


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References VII

G. Pinton, J. Dahl, S. Rosenzweig, and G. E. Trahey, “Aheterogeneous nonlinear attenuating full- wave model ofultrasound”, IEEE Trans. Ultrason. Ferroelectr. Freq.Control 56, 474–488 (2010).


Numerical methods for wave propagation in nonlinear acoustics

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