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1. Boundary stabilization of the linear MGT equation with partially absorbing boundary data and degenerate viscoelasticity

   https://doi.org/10.3934/dcdss.2022020  

   Bongarti, Marcelo; Lasiecka, Irena; Rodrigues, José H. (January 2022, Discrete and Continuous Dynamical Systems - S)

   The Jordan–Moore–Gibson–Thompson (JMGT) equation is a well-established and recently widely studied model for nonlinear acoustics (NLA). It is a third–order (in time) semilinear Partial Differential Equation (PDE) with a distinctive feature of predicting the propagation of ultrasound waves at finite speed. This is due to the heat phenomenon known as second sound which leads to hyperbolic heat-wave propagation. In this paper, we consider the problem in the so called "critical" case, where free dynamics is unstable. In order to stabilize, we shall use boundary feedback controls supported on a portion of the boundary only. Since the remaining part of the boundary is not "controlled", and the imposed boundary conditions of Neumann type fail to saitsfy Lopatinski condition, several mathematical issues typical for mixed problems within the context o boundary stabilizability arise. To resolve these, special geometric constructs along with sharp trace estimates will be developed. The imposed geometric conditions are motivated by the geometry that is suitable for modeling the problem of controlling (from the boundary) the acoustic pressure involved in medical treatments such as lithotripsy, thermotherapy, sonochemistry, or any other procedure involving High Intensity Focused Ultrasound (HIFU). 

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2. Anomalous Nonlinear Dynamics Behavior of Fractional Viscoelastic Beams

   https://doi.org/10.1115/1.4052286  

   Suzuki, Jorge L.; Kharazmi, Ehsan; Varghaei, Pegah; Naghibolhosseini, Maryam; Zayernouri, Mohsen (November 2021, Journal of Computational and Nonlinear Dynamics)

   Abstract Fractional models and their parameters are sensitive to intrinsic microstructural changes in anomalous materials. We investigate how such physics-informed models propagate the evolving anomalous rheology to the nonlinear dynamics of mechanical systems. In particular, we study the vibration of a fractional, geometrically nonlinear viscoelastic cantilever beam, under base excitation and free vibration, where the viscoelasticity is described by a distributed-order fractional model. We employ Hamilton's principle to obtain the equation of motion with the choice of specific material distribution functions that recover a fractional Kelvin–Voigt viscoelastic model of order α. Through spectral decomposition in space, the resulting time-fractional partial differential equation reduces to a nonlinear time-fractional ordinary differential equation, where the linear counterpart is numerically integrated through a direct L1-difference scheme. We further develop a semi-analytical scheme to solve the nonlinear system through a method of multiple scales, yielding a cubic algebraic equation in terms of the frequency. Our numerical results suggest a set of α-dependent anomalous dynamic qualities, such as far-from-equilibrium power-law decay rates, amplitude super-sensitivity at free vibration, and bifurcation in steady-state amplitude at primary resonance. 

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3. A Robust Bubble Growth Solution Scheme for Implementation in CFD Analysis of Multiphase Flows

   https://doi.org/10.3390/computation11040072  

   Pang, Hao; Ngaile, Gracious (April 2023, Computation)

   Although the full form of the Rayleigh–Plesset (RP) equation more accurately depicts the bubble behavior in a cavitating flow than its reduced form, it finds much less application than the latter in the computational fluid dynamic (CFD) simulation due to its high stiffness. The traditional variable time-step scheme for the full form RP equation is difficult to be integrated with the CFD program since it requires a tiny time step at the singularity point for convergence and this step size may be incompatible with time marching of conservation equations. This paper presents two stable and efficient numerical solution schemes based on the finite difference method and Euler method so that the full-form RP equation can be better accepted by the CFD program. By employing a truncation bubble radius to approximate the minimum bubble size in the collapse stage, the proposed schemes solve for the bubble radius and wall velocity in an explicit way. The proposed solution schemes are more robust for a wide range of ambient pressure profiles than the traditional schemes and avoid excessive refinement on the time step at the singularity point. Since the proposed solution scheme can calculate the effects of the second-order term, liquid viscosity, and surface tension on the bubble evolution, it provides a more accurate estimation of the wall velocity for the vaporization or condensation rate, which is widely used in the cavitation model in the CFD simulation. The legitimacy of the solution schemes is manifested by the agreement between the results from these schemes and established ones from the literature. The proposed solution schemes are more robust in face of a wide range of ambient pressure profiles. 

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4. Tropical cyclone rapid intensification and the excitation of inertia-gravity waves on the edges of an evolving potential vorticity structure

   https://doi.org/10.3389/feart.2022.1038351  

   Schubert, Wayne H.; Taft, Richard K. (November 2022, Frontiers in Earth Science)

   The problem of tropical cyclone rapid intensification is reduced to a potential vorticity (PV) equation and a second order, inhomogeneous, partial differential equation for the azimuthal wind. The latter equation has the form of a Klein-Gordon equation, the right-hand side of which involves the radial derivative of the evolving PV field. When the PV field evolves rapidly, inertia-gravity waves are excited at the edges of the evolving PV structure. In contrast, when the PV field evolves slowly, the second order time derivative term in the Klein-Gordon equation is negligible, inertia-gravity waves are not excited, and the equation reduces to an invertibility principle for the PV. The above concepts are presented in the context of an axisymmetric shallow water model, in both its linear and nonlinear forms. The nonlinear results show a remarkable sensitivity of vortex intensification to the percentage of mass that is diabatically removed from the region inside a given absolute angular momentum surface. 

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5. Robust Neural Network Approach to System Identification in the High-Noise Regime

   Negrini, Elisa; Citti, Giovanna; Capogna, Luca (June 2023, Springer Verlag)

   We present a new algorithm for learning unknown gov- erning equations from trajectory data, using a family of neural net- works. Given samples of solutions x(t) to an unknown dynamical system x ̇ (t) = f (t, x(t)), we approximate the function f using a family of neural networks. We express the equation in integral form and use Euler method to predict the solution at every successive time step using at each iter- ation a different neural network as a prior for f. This procedure yields M-1 time-independent networks, where M is the number of time steps at which x(t) is observed. Finally, we obtain a single function f(t,x(t)) by neural network interpolation. Unlike our earlier work, where we numer- ically computed the derivatives of data, and used them as target in a Lipschitz regularized neural network to approximate f, our new method avoids numerical differentiations, which are unstable in presence of noise. We test the new algorithm on multiple examples in a high-noise setting. We empirically show that generalization and recovery of the governing equation improve by adding a Lipschitz regularization term in our loss function and that this method improves our previous one especially in the high-noise regime, when numerical differentiation provides low qual- ity target data. Finally, we compare our results with other state of the art methods for system identification. 

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