Dynamic mode decomposition for multiscale nonlinear physics.

We present a data-driven method for separating complex, multiscale systems into their constituent timescale components using a recursive implementation of dynamic mode decomposition (DMD). Local linear models are built from windowed subsets of the data, and dominant timescales are discovered using spectral clustering on their eigenvalues. This approach produces time series data for each identified component, which sum to a faithful reconstruction of the input signal. It differs from most other methods in the field of multiresolution analysis (MRA) in that it (1) accounts for spatial and temporal coherencies simultaneously, making it more robust to scale overlap between components, and (2) yields a closed-form expression for local dynamics at each scale, which can be used for short-term prediction of any or all components. Our technique is an extension of multi-resolution dynamic mode decomposition (mrDMD), generalized to treat a broader variety of multiscale systems and more faithfully reconstruct their isolated components. In this paper we present an overview of our algorithm and its results on two example physical systems, and briefly discuss some advantages and potential forecasting applications for the technique.

Analysis and Design of Capacitive Parametric Ultrasonic Transducers for Efficient Ultrasonic Power Transfer Based on a 1-D Lumped Model.

There is an increasing interest in wireless power transfer for medical implants, sensor networks, and consumer electronics. A passive capacitive parametric ultrasonic transducer (CPUT) can be suitable for these applications as it does not require a dc bias or a permanent charge. In this paper, we present a 1-D lumped parameter model of the CPUT to study its operation and investigate relevant design parameters for power transfer applications. The CPUT is modeled as an ultrasound-driven piston coupled to an RLC resonator resulting in a system of two coupled nonlinear ordinary differential equations. Simulink is used along with an analytical approximation of the system to obtain the voltage across the capacitor and displacement of the piston. Parametric resonance threshold and ultrasound-to-electrical conversion efficiency are evaluated, and the dependence of these performance metrics on the load resistance, input ultrasound intensity, forcing frequency, electrode coverage area, gap height, and the mechanical Q-factor are studied. Based on this analysis, design guidelines are proposed for highly efficient power transfer. Guided by these results, practical device designs are obtained through COMSOL simulations. Finally, the feasibility of using the CPUT in air is predicted to set the foundation for further research in ultrasonic wireless power transfer, energy harvesting, and sensing.

Explicit symplectic approximation of nonseparable Hamiltonians: Algorithm and long time performance.

Explicit symplectic integrators have been important tools for accurate and efficient approximations of mechanical systems with separable Hamiltonians. This article proposes for arbitrary Hamiltonians similar integrators, which are explicit, of any even order, symplectic in an extended phase space, and with pleasant long time properties. They are based on a mechanical restraint that binds two copies of phase space together. Using backward error analysis, Kolmogorov-Arnold-Moser theory, and additional multiscale analysis, an error bound of O(Tδ^{l}ω) is established for integrable systems, where T,δ,l, and ω are, respectively, the (long) simulation time, step size, integrator order, and some binding constant. For nonintegrable systems with positive Lyapunov exponents, such an error bound is generally impossible, but satisfactory statistical behaviors were observed in a numerical experiment with a nonlinear Schrödinger equation.

An improved wave-vector frequency-domain method for nonlinear wave modeling.

In this paper, a recently developed wave-vector frequency-domain method for nonlinear wave modeling is improved and verified by numerical simulations and underwater experiments. Higher order numeric schemes are proposed that significantly increase the modeling accuracy, thereby allowing for a larger step size and shorter computation time. The improved algorithms replace the left-point Riemann sum in the original algorithm by the trapezoidal or Simpson's integration. Plane waves and a phased array were first studied to numerically validate the model. It is shown that the left-point Riemann sum, trapezoidal, and Simpson's integration have first-, second-, and third-order global accuracy, respectively. A highly focused therapeutic transducer was then used for experimental verifications. Short high-intensity pulses were generated. 2-D scans were conducted at a prefocal plane, which were later used as the input to the numerical model to predict the acoustic field at other planes. Good agreement is observed between simulations and experiments.

Control of a model of DNA division via parametric resonance.

We study the internal resonance, energy transfer, activation mechanism, and control of a model of DNA division via parametric resonance. While the system is robust to noise, this study shows that it is sensitive to specific fine scale modes and frequencies that could be targeted by low intensity electro-magnetic fields for triggering and controlling the division. The DNA model is a chain of pendula in a Morse potential. While the (possibly parametrically excited) system has a large number of degrees of freedom and a large number of intrinsic time scales, global and slow variables can be identified by (1) first reducing its dynamic to two modes exchanging energy between each other and (2) averaging the dynamic of the reduced system with respect to the phase of the fastest mode. Surprisingly, the global and slow dynamic of the system remains Hamiltonian (despite the parametric excitation) and the study of its associated effective potential shows how parametric excitation can turn the unstable open state into a stable one. Numerical experiments support the accuracy of the time-averaged reduced Hamiltonian in capturing the global and slow dynamic of the full system.

Statistics of dislocation slip avalanches in nanosized single crystals show tuned critical behavior predicted by a simple mean field model.

We show that slowly sheared metallic nanocrystals deform via discrete strain bursts (slips), whose size distributions follow power laws with stress-dependent cutoffs. We show for the first time that plasticity reflects tuned criticality, by collapsing the stress-dependent slip-size distributions onto a predicted scaling function. Both power-law exponents and scaling function agree with mean-field theory predictions. Our study of 7 materials and 2 crystal structures, at various deformation rates, stresses, and crystal sizes down to 75 nm, attests to the universal characteristics of plasticity.

Evaluation of a wave-vector-frequency-domain method for nonlinear wave propagation.

A wave-vector-frequency-domain method is presented to describe one-directional forward or backward acoustic wave propagation in a nonlinear homogeneous medium. Starting from a frequency-domain representation of the second-order nonlinear acoustic wave equation, an implicit solution for the nonlinear term is proposed by employing the Green's function. Its approximation, which is more suitable for numerical implementation, is used. An error study is carried out to test the efficiency of the model by comparing the results with the Fubini solution. It is shown that the error grows as the propagation distance and step-size increase. However, for the specific case tested, even at a step size as large as one wavelength, sufficient accuracy for plane-wave propagation is observed. A two-dimensional steered transducer problem is explored to verify the nonlinear acoustic field directional independence of the model. A three-dimensional single-element transducer problem is solved to verify the forward model by comparing it with an existing nonlinear wave propagation code. Finally, backward-projection behavior is examined. The sound field over a plane in an absorptive medium is backward projected to the source and compared with the initial field, where good agreement is observed.