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Apply integration by parts to the second order derivative in space, using
the non–reflecting boundary condition (5). This yields a set of equations
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(f) Performance
Beowulf Cluster: 24 dual Pentium-III 500MHz nodes, with 512MB memory each.
) to muscle (
(e) X–Wave in inhomogeneous
medium
Memory req.
1140MB
Origin 2000 (1 CPU)
Bessel Beam 1.66 hours
X–Wave
4.35 hours
Beowulf Cluster (18 CPUs)
Bessel Beam 3.04 minutes
X–Wave
3.33 minutes
For the linear case, the problem formulates into a linear problem of the type
at each timestep, which is usually solved with a Krylov subspace
method such as Conjugate Gradients.
Computers used: SGI Origin 2000, used as scalar computer for its large amount of shared memory.
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Inhomogeneous medium: abrupt boundary change, from human fat (
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(d) Pulsed X–Wave
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(c) Pulsed Bessel Beam
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Finite difference approximation of the time derivative in the linear model (1)
gives
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(b) Bessel Beam in inhomogeneous medium
(a) Parameters
6
Domain: 10mm 10mm 7mm
Frequency: 2.5MHz
Speed of sound:
,
Space: ca. 10 times oversampling
relative to
(150 150 100 nodes)
Time: 20 times oversampling
(
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Circular symmetry make these well suited for annular arrays.
Here is a constant, is the frequency, is the distance from the centerline of the transducer, is the propagation distance from the transducer surface, and , are design parameters with
. See [1, 2, 3]
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We assume that there is no flux through the body–wall.
On the transducer surface
, we use the real parts of (2), (3) to generate
the pressure excitation.
On the remaining part of the boundary, we adopt a first order non–reflecting
boundary [4]
a
is the speed of sound.
is a domain in
where the problem is solved
One variant of Bessel beams and X-Waves are those of order zero
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Bessel Beams and X–Waves are limited diffraction solutions to the lossless
linear wave equation
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Assume that
is the solution at time
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Boundary Conditions
Assume
We use a Galerkin Finite Element method in the space domain.
Numerical Method
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Model
The second order wave equation needs two initial conditions. We start with
rest at atmospheric pressure:
Initial Conditions
Linear Propagation
Abstract
We are building a finite element solver for acoustics waves using the numerical library Diffpack. The long term objective is to compare limited diffraction beams, such as Bessel Beams and X–Waves travelling in linear and nonlinear media.
Åsmund Ødegård, Paul D. Fox, Sverre Holm, Aslak Tveito
Department of Informatics, University of Oslo, P.O. Box 1080, Blindern, N-0316 Oslo, Norway
Finite Element Modelling of Pulsed Bessel Beams and X–Waves using Diffpack
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Using the conditions (13), we can set
and .
Assume
is computed.
Set
. We can now compute a hopefully better approximation to
by Newton–Raphson iterations
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is the Jacobi matrix with components
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and
are the rejective expressions evaluated at
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On convergence, we set
and restart the iteration.
where
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[1] J.-Y. Lu. Designing limited diffraction beams. IEEE Transaction on Ultrasonics,
Ferroelectrics and Frequency control, 44(1), 1997.
[2] S. Holm. Bessel and conical beams and approximation in annular arrays.
IEEE Transaction on Ultrasonics, Ferroelectrics and Frequency control, 45(3),
1998.
[3] P.D. Fox and S. Holm. Effects of parameter mismatch in ultrasonic bessel
transducers. Proc. IEEE Norwegian Signal Processing Symposium NORSIG’99, 1999.
[4] Bjorn Engquist and Andrew Majda. Absorbing boundary conditions for the
numerical simulation of waves. Mathematics of computation, 31(139), 1977.
[5] S. Makarov and M. Ochmann. Nonlinear and thermoviscous phenomena in
acoustics, part i. ACUSTICA - acta acustica, 82, 1996.
[6] S. Makarov and M. Ochmann. Nonlinear and thermoviscous phenomena in
acoustics, part ii. ACUSTICA - acta acustica, 83, 1997.
[7] Hans Petter Langtangen. Computational Partial Differential Equations, Numerical methods and Diffpack programming, volume 2 of Lecture notes in
computational science and engineering. Springer–Verlag, 1999.
References
shows the reflection from the boundary. To the right, we view the
maximum field intensity at each point.
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This is a 2D simulation of a Bessel beam on a 10mm 8mm domain. The picture to the left is a snapshot of the solution, which
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Derive a better absorbing boundary condition for the nonlinear
model.
Implement the full nonlinear model, i.e. with the pressure—
velocity potential relation (9) instead of (12). This require a separate grid of the transducer.
Implement a parallel version of the nonlinear solver.
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Further Work
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Nonlinear simulations
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Diffpack is an object oriented numerical Finite Element library[7].
Simulators are built as C++ classes. The simulator class is derived from the finite element manager class in Diffpack.
Support for a wide range of linear solvers, finite elements, flexible grid representations. Also support for adaptivity and parallel
computations.
The linear solver is now also implemented as a parallel solver,
with nice performance boost.
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Explore the possibilities of adaptivity for both the linear and nonlinear model, to make full–scale simulations possible.
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Diffpack Simulator
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Let
be a finite element basis such that
Further, multiply (15) with
, and apply integration by parts,
using the condition (14). We obtain the nonlinear problem
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Using the relation (12), we can derive initial conditions for the nonlinear model, from (4):
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Initial Conditions
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the nonlinearity parameter and
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is the velocity potential,
is the absorption coefficient.
Alternative relation to (9):
has components given by
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We use the same approach as for the linear model, Galerkin Finite Element method in space and Finite Differences in time.
Assume that
is the velocity potential at
and
approximate the time derivatives with centred finite differences.
This gives the semi–discrete problem
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Numerical Method
, where
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A formulation of the lossy nonlinear propagation conditions is defined in [5, 6] by the system
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On the transducer surface
, the boundary condition is set as
in the linear case, using (12).
On the rest of the boundary, we adopt the absorbing condition (5)
as
Boundary Conditions
Model
Nonlinear Propagation
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