Introduction:
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Mathematics of Nonlinear Acoustics
Research on nonlinear acoustics has recently been driven by the increasing number of industrial and medical applications of high intensity ultrasound ranging from ultrasound cleaning or welding via sonochemistry to lithotripsy and thermotherapy.
Although the classical models of nonlinear acoustics such as the the Westervelt and the Kuznesov equation have already been devised in the 1960's and 70's, the physically correct and mathematically sound modelling of nonlinear wave propagation in this context is a currently a highly active field of research. An important prerequisite for reliable and well-founded numerical simulation and optimization of high intensity ultrasound devices is a mathematical analysis of the underlying partial differential equation (PDE) models in a general spatially three dimensional geometrical setting with appropriate boundary and initial conditions. While recently some results have been achieved concerning the classical models of nonlinear acoustics, an analyis of the qualitative and quantitative behavior also of recently developed models is crucial for assessing the required level of modelling for practically relevant applications. Another important issue is the coupling of nonlinear acoustics to other physical fields (excitation mechanisms, focusing devices, heat generation, interaction with kidney stones in lithotripsy).
Also numerical simulation poses a major challenge due to nonlinearity, coupling to other physical fields, different spatial and temporal scales resulting from different wavelengths within the subdomains, and the fact that one often deals with open domain problems. Finally, the design of high intensity ultrasound devices leads to shape optimization and optimal control problems in the context of the above mentioned PDE models, with state and control constraints arising from physical and technical restrictions.
The focus of this session will be on
- modelling aspects
- qualitative and quantitative analysis of PDE models
- numerical simulation methods
- control and optimization problems
- applications in the context of nonlinear sound propagation. |
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