Modeling, Analysis and Simulation of Nonlinear Ultrasound

Nonlinear sound propagation plays a role in many highly relevant medical ultrasound applications. The project deals with the analysis (and some numerics) of partial differential equations PDEs modeling nonlinear acoustics. Its purpose is to contribute to substantial progress in our understanding of novel ultrasound imaging techniques involving nonlinear wave propagation, such as harmonic imaging and nonlinearity parameter tomography. Their optimized as well as safe use requires a thorough understanding of these nonlinear phenomena by means of appropriate mathematical models that are capable of capturing all relevant physical effects. Compared to the linear regime, where model simplifications allow to reduce the imaging problem to a signal processing task, nonlinearity requires a fundamentally different modeling approach based on physical balance and constitutive laws leading to partial differential equations (PDEs). The present FWF project is concerned with a number of crucial aspects involving PDE modeling, analysis and numerics, that are targeted at the requirements in the mentioned applications. These are existence of low regularity solutions for classical and advanced models of nonlinear acoustics with nonsmooth coefficients (as relevant in imaging) modeling and analysis of fractionally damped wave equations (as relevant in medical ultrasonics) adaptive discretization methods for fractionally damped nonlinear wave equations (as required for efficient simulation) The planned work relies on previous achievements of the applicant and co-workers in the FWF project P24970 Mathematics of Nonlinear Acoustics: Analysis, Numerics, and Optimization 2012- 2015 and subsequently.