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We consider an undetermined coefficient inverse problem for a nonlinear partial differential equation describing high‐intensity ultrasound propagation as widely used in medical imaging and therapy. The usual nonlinear term in the standard model using the Westervelt equation in pressure formulation is of the form ppt. However, this should be considered as a low‐order approximation to a more complex physical model where higher order terms will be required. Here we assume a more general case where the form taken is f(p) pt and f is unknown and must be recovered from data measurements. Corresponding to the typical measurement setup, the overposed data consist of time trace observations of the acoustic pressure at a single point or on a one‐dimensional set Σ representing the receiving transducer array at a fixed time. Additionally to an analysis of well‐posedness of the resulting pde, we show injectivity of the linearized forward map from f to the overposed data and use this as motivation for several iterative schemes to recover f. Numerical simulations will also be shown to illustrate the efficiency of the methods. [ABSTRACT FROM AUTHOR]

We propose a hysteretic model for electromechanical coupling in piezoelectric materials, with the strain and the electric field as inputs and the stress and the polarization as outputs. This constitutive law satisfies the thermodynamic principles and exhibits good agreement with experimental measurements. Moreover, when it is coupled with the mechanical and electromagnetic balance equations, the resulting PDE system is well-posed under the hypothesis that hysteretic effects take place only in one preferred direction. We prove the existence and uniqueness of its global weak solutions for each initial data with prescribed regularity. One of the tools is a new Lipschitz continuity theorem for the inverse Preisach operator with time dependent coefficients. [ABSTRACT FROM AUTHOR]

Abstract: In automotive industry, reliability analysis of smart power semiconductor devices is mandatory for safety reasons. To reliably forecast the lifetime of devices, a mathematical description of the material degradation due to applied stress is required. In this work, the degradation process of a metal film of interest under thermo‐mechanical stress is studied. Therefore, a bi‐layer test structure (metal film on substrate) is used. Degradation refers to the formation, growth and coalescence of voids, initializing cracks that propagate through the metal film. Since the film's elastic‐plastic deformation is triggered by random diffusional activities with drift, stochastic differential equations (SDEs) are used to model the evolution of the microstructural heterogeneity in the metallization. The choice of an efficient numerical method to estimate the unknown parameters of the inverse problem is complex due to ill‐posedness and rare measurements. To address these challenges, an approach by Dunker and Hohage of 2014 is followed, where parameter identification in SDE models is performed on basis of the Kolmogorov forward (Fokker‐Planck) equation [1]. The mentioned approach has to be adapted to our application. Firstly, a stochastically consistent cost functional is defined, taking into account the different type of measurements and secondly, instead of estimating a function, a finite set of model parameters has to be recovered. For efficient computational optimization an adjoint approach is used to calculate the gradient of the cost functional. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim) [ABSTRACT FROM AUTHOR]