Modeling, Analysis and Simulation of No...

Nonlinear sound propagation plays a role in many highly relevantmedical ultrasound applications.

The project deals with the analysis (and some numerics) of partialdifferential equations PDEs modeling nonlinear acoustics. Its purposeis to contribute to substantial progress in our understanding ofnovel ultrasound imaging techniques involving nonlinear wavepropagation, such as harmonic imaging and nonlinearity parametertomography. 

Their optimized as well as safe use requires a thorough understandingof these nonlinear phenomena by means of appropriate mathematicalmodels that are capable of capturing all relevant physical effects.

Compared to the linear regime, where model simplifications allow toreduce the imaging problem to

a signal processing task, nonlinearity requires a fundamentallydifferent modeling approach based on physical balance andconstitutive laws leading to partial differential equations (PDEs).

The present FWF project is concerned with a number of crucial aspectsinvolving PDE modeling,

analysis and numerics, that are targeted at the requirements in thementioned applications. These are

• existence of low regularity solutions for classical and advancedmodels of nonlinear acoustics with

nonsmooth coefficients (as relevant in imaging)

• modeling and analysis of fractionally damped wave equations (asrelevant in medical ultrasonics)

• adaptive discretization methods for fractionally damped nonlinearwave equations (as required

for efficient simulation)