Mathematics of nonlinear acoustics: Analysis, numerics, and optimization

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. Our work in this field is motivated, e.g., by applications in lithotripsy, where a better understanding and control of the physical effects via mathematical analysis, numerical simulation, and optimization should lead to a considerable reduction of lesion and complication risks. 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. This has so far only been done for the classical models of nonlinear acoustics. In this project we plan to analyze the qualitative and quantitative behavior of recently developed models, which is crucial for assessing the required level of modelling for practically relevant applications. Another important issue which we plan to investigate is the coupling of nonlinear acoustics to other physical fields (excitation mechanisms, focusing devices, heat generation, interaction with kidney stones). 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 we deal with open domain problems. Here we are going to use domain decomposition methods (nonmatching grids, mortar elements) for coupling and work on operator splitting methods for efficient and robust time integration as well as on absorbing boundary conditions for simulating unbounded wave propagation using a truncated computational domain. 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. Here we plan to derive theory based first and second order sensitivities for use in efficient mathematical optimization methods, and investigate multilevel methods based on natural model hierarchies.

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. Our work in this field has particularily been motivated by applications in lithotripsy, where a better understanding and control of the physical effects via mathematical analysis, numerical simulation, and optimization can lead to a considerable reduction of lesion and complication risks.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. This had so far only been carried out for the classical models of nonlinear acoustics that miss to fully take into account certain higher order physical effects, though. In this project we have analyzed the qualitative and quantitative behavior of recently developed models, which is crucial for assessing the required level of modelling for practically relevant applications. Another important issue which we have investigated is the coupling of nonlinear acoustics to other physical fields (excitation mechanisms, focusing devices, heat generation, interaction with kidney stones).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 we deal with open domain problems. Here we have been investigating applicability of domain decomposition methods for coupling and developed operator splitting methods for efficient and robust time integration as well as absorbing boundary conditions for simulating unbounded wave propagation using a truncated computational do- main.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. Here we have derived and mathematically justified first and second order sensitivities, that lay the basis for efficient and reliable numerical optimization methods.