Design Method of High-Order Kalman Filter for Strong Nonlinear System Based on Kronecker Product Transform

In this paper, a novel design idea of high-order Kalman filter based on Kronecker product transform is proposed for a class of strong nonlinear stochastic dynamic systems. Firstly, those augmenting systems are modeled with help of the Kronecker product without system noise. Secondly, the augmented system errors are illustratively charactered by Gaussian white noise. Thirdly, at the expanded space a creative high-order Kalman filter is delicately designed, which consists of high-order Taylor expansion, introducing magical intermediate variables, representing linear systems converted from strongly nonlinear systems, designing Kalman filter, etc. The performance of the proposed filter will be much better than one of EKF, because it uses more information than EKF. Finally, its promise is verified through commonly used digital simulation examples.

Residue Regulating Homotopy Method for Strongly Nonlinear Oscillator

It is highly desired yet challenging to obtain analytical approximate solutions to strongly nonlinear oscillators accurately and efficiently. Here we propose a new approach, which combines the homtopy concept with a “residue-regulating” technique to construct a continuous homotopy from an initial guess solution to a high-accuracy analytical approximation of the nonlinear problems, namely the residue regulating homotopy method (RRHM). In our method, the analytical expression of each order homotopy-series solution is associated with a set of base functions which are pre-selected or generated during the previous order of approximations, while the corresponding coefficients are solved from deformation equations specified by the nonlinear equation itself and auxiliary residue functions. The convergence region, rate and final accuracy of the homotopy are controlled by a residue-regulating vector and an expansion threshold. General procedures of implementing RRHM are demonstrated using the Duffing and Van der Pol-Duffing oscillators, where approximate solutions containing abundant frequency components are successfully obtained, yielding significantly better convergence rate and performance stability compared to the other conventional homotopy-based methods.

Ion dynamics driven by a strongly nonlinear plasma wake

In plasma wakefield accelerators, the wave excited in the plasma eventually breaks and leaves behind slowly changing fields and currents that perturb the ion density background. We study this process numerically using the example of a FACET experiment where the wave is excited by an electron bunch in the bubble regime in a radially bounded plasma. Four physical effects underlie the dynamics of ions: (1) attraction of ions toward the axis by the fields of the driver and the wave, resulting in formation of a density peak, (2) generation of ion-acoustic solitons following the decay of the density peak, (3) positive plasma charging after wave breaking, leading to acceleration of some ions in the radial direction, and (4) plasma pinching by the current generated during the wavebreaking. Interplay of these effects result in formation of various radial density profiles, which are difficult to produce in any other way.

Nonlinear Young’s Modulus of New Red Sandstone: Experimental Studies

Young’s modulus of New Red Sandstone was investigated experimentally to gain insight into its nonlinear nature. A large experimental programme was carried out by applying a controllable quasi-static and dynamic uniaxial loading to 286 dry sandstone samples of four different sizes. The static and dynamic tests, similar to those aiming at determining the uniaxial compressive strength, were conducted using the state-of-the-art experimental facilities at the University of Aberdeen including a custom-built small experimental rig for inducing a dynamic uniaxial compressive load via a piezoelectric transducer. The obtained results have confirmed a complex nature of Young’s modulus of sandstone. Specifically, under a harmonic dynamic loading, it shows strongly nonlinear behaviour, which is hardening and softening with respect to frequency and amplitude of the dynamic loading, respectively.

A Simplified Lindstedt-Poincaré Method for Saving Computational Cost to Determine Higher Order Nonlinear Free Vibrations

In order to improve the Lindstedt-Poincaré method to raise the accuracy and the performance for the application to strongly nonlinear oscillators, a new analytic method by engaging in advance a linearization technique in the nonlinear differential equation is developed, which is realized in terms of a weight factor to decompose the nonlinear term into two sides. We expand the constant preceding the displacement in powers of the introduced parameter so that the coefficients can be determined to avoid the appearance of secular solutions. The present linearized Lindstedt-Poincaré method is easily implemented to provide accurate higher order analytic solutions of nonlinear oscillators, such as Duffing and van Der Pol nonlinear oscillators. The accuracy of analytic solutions is evaluated by comparing to the numerical results obtained from the fourth-order Runge-Kotta method. The major novelty is that we can simplify the Lindstedt-Poincaré method to solve strongly a nonlinear oscillator with a large vibration amplitude.

Avalanche transport of energetic-ions in magnetic confinement plasmas: Nonlinear multiple wave-number simulation

Large burst activity, identified as toroidal Alfv\'{e}n eigenmode (TAE) avalanche, occurs frequently in neutral-beam heated plasmas in National Spherical Torus Experiment (NSTX). Based on the typical experimental observation of TAE avalanche on NSTX, a self-consistent nonlinear multiple wave-number ($k_{\parallel}\simeq n/R$, where $n$ toroidal mode-number and $R$ major radius) simulation associated with TAE avalanches is performed using the experimental parameters and profiles before the occurrence of TAE avalanche as the M3D-K input. The wave-wave nonlinear coupling among different modes and the resonant interaction between different modes and energetic-ions during TAE avalanches are identified in the nonlinear multiple wave-number simulations. The resonance overlap during the TAE avalanche is clearly observed in the simulation. It is found that the effective wave-wave coupling and a sufficiently strong drive are two important ingredients for the onset of TAE avalanches. TAE avalanche is considered to be a strongly nonlinear process and it is always accompanied by the simultaneous rapid frequency-chirping and large amplitude bursting of multiple modes and significant energetic-ion losses. The experimental phenomenon is observed on NSTX and is qualitatively reproduced by the simulation results in this work. These findings indicate that the onset of avalanche is triggered by nonlinearity of the system, and are also conducive to understanding the underlying mechanism of avalanche transport of energetic particles in the future burning plasmas, such as ITER.