Nonlinear acoustics: modelling, applications and numerics

The workshop will take place on the 25th June online.

Speakers:

Organisers: Lehel Banjai and Vanja Nikolić (Radboud)

Abstracts

Srinath Rajagopal:
Measurement Traceability in Medical Ultrasound

The earliest application of ultrasound in medicine dates back to 1940s when its therapeutic effects were demonstrated by successfully destroying brain tissue in animal models. It was nearly after a decade later the first diagnostic capability of ultrasound in the detection of breast carcinoma was reported. The ultrasound-induced damage to tissue in therapy did not go unnoticed as the diagnostic use of ultrasound continued to rise in the 1950s and 1960s especially in monitoring of foetal development. In 1980s US Food and Drug Administration initiated the regulation of diagnostic ultrasound equipment. The regulation placed restriction on the ultrasound exposure level, which has been adopted by the manufacturers globally.

The ultrasound exposure levels are quantified by the measurement of the two key quantities, pressure, and power. These two quantities represent potential mechanical and thermal damage to tissue under certain excitation conditions. Manufacturers are required to perform measurements under a number of different operational conditions to demonstrate equipment safety. The devices used to make measurements of pressure and power must be traceable to International System of Units (SI) via their calibration at a National Measurement Institute.

Hydrophones are used to make the measurement of the dynamic pressure of ultrasound, whereas Radiation Force Balance (RFB) is used to measure the ultrasound power. The primary standards and dissemination methods implemented at NPL for ultrasound pressure and power will be described.

Pedro Jordan:
Nonlinear Acoustics: Fundamental Concepts and Shock Applications

In this lecture, I will review the fundamental concepts and
equations of nonlinear acoustics theory, derive equations of motion in
terms of the scalar velocity potential, and present a number of example
problems involving shock phenomena.

Barbara Kaltenbacher:
Nonlinear acoustics: some parameter asymptotics and absorbing boundary conditions

In this talk, after a short introduction on models of nonlinear acoustics, we will focus on two topics: The first is an analytic one and concerns the asymptotics with respect to certain crucial parameters in the partial differential equations. The second is from numerics, concerning the question of how absorbing boundary conditions can be designed to aviod spurious reflections in trucated domains.
This is joint work with Vanja Nikolic, Radboud University, and Igor Shevchenko, Imperial College London.

Bernhard Maier:
Error analysis for space and time discretizations of quasilinear
wave-type equations

The Westervelt equation is a prominent model for the propagation of
ultrasonic waves. From a mathematical perspective, it corresponds to a
quasilinear wave-type equation. This problem class includes not only
important models from nonlinear acoustics, but also from optics,
elastodynamics, fluid dynamics, etc. Collecting characteristic
properties of such problems, we first introduce a framework for the
analysis of numerical discretizations in space and time. In this very
general framework, we then prove wellposedness and a rigorous error
estimate for the full discretization of quasilinear wave-type problems
with a generic space discretization and a variant of the implicit
midpoint rule for the integration in time. Returning to the initial
motivation of this talk, we finally use these abstract results to deduce
novel error estimates for the full discretization of the Westervelt
equation without damping.