Maria Haase

Finite element approximation to two-dimensional sine-Gordon solitons

Abstract The paper presents a finite element algorithm for the numerical solution of the sine-Gordon equation in two spatial dimensions, as it arises, for example, in rectangular large-area Josephson junctions. The dispersive nonlinear partial differential equation of the system allows for soliton-type solutions, an ubiquitous phenomenon in a large variety of physical problems. A semidiscrete Galerkin approach based on simple four-noded bilinear finite elements in combination with a generalized Newmark integration scheme is used throughout the paper and is tested in a variety of cases. Comparisons with finite difference solutions show the superior performance of the proposed algorithm leading to very accurate, numerically stable and physically consistent solitary wave solutions. The results support the confidence in the present numerical model which should be capable to treat also more complex situations involving soliton-type interactions.

An engineer's guide to soliton phenomena: Application of the finite element method

Abstract The paper attempts an elementary survey of the physical and mathematical background appertaining to solitons and discusses in particular the numerical solution of three types of dispersive nonlinear partial differential equations exhibiting soliton-type solutions, namely the Korteweg-de Vries equation, the Nonlinear Schrodinger equation, and the Sine-Gordon equation. Throughout this study a semidiscrete Galerkin method is applied using a finite element discretization in space and a step-by-step time integration of the resulting system of nonlinear ordinary differential equations. Depending upon the special type of the evolutionary equation the application of a Petrov-Galerkin procedure may increase significantly the numerical stability. Accuracy and effectivity of the different approaches are demonstrated on a series of computer plots.

Die Erforschung des Chaos

17. Febr. 2011 Schon vor 16 Jahren erschien die erste Auflage von „Die Erforschung des Chaos“. Nachdem John Argyris 2004 im hohen Alter von 91 Jahren Die Erforschung des Chaos. Studienbuch für Naturwissenschaftler und Ingenieure. 1995. Vieweg+Teubner 790 S., 254 SW-Abb., 86 Farbabb., 24 Tabellen, 170 Seit Ihrem Aufkommen wurde die Chaosforschung als eine von wenigen mathematisch-physikalischen Gebieten von der Öffentlichkeit mit großem Interesse Die Erforschung des Chaos: Dynamische Systeme John Argyris, Gunter Faust, Maria Haase, Rudolf Friedrich ISBN: 9783662545454 Kostenloser Versand

Damage identification based on ridges and maxima lines of the wavelet transform

Abstract The paper analyses the transient vibration behaviour of structures using the continuous wavelet transform (CWT), which provides effective tools for detecting changes in the structure of the material. The advantage of the CWT over commonly used time-frequency methods like the Wigner–Ville and the Gabor transform is its ability to decompose signals simultaneously both in time (or space) and frequency (or scale) with adaptive windows. The essential information is contained in the maxima of the wavelet transform. From the ridges, the modal parameters of the decoupled modes can be extracted and the signal can be reconstructed. From the maxima lines, defects can be localized. This paper presents a new approach for the calculation of wavelet transform ridges and maxima lines, which is based on a direct integration of differential equations. The potential of the method is demonstrated by the analysis of the impact vibration response of different bars.

On an unconventional but natural formation of a stiffness matrix

Abstract The paper describes a procedure of deriving stiffness matrices for finite elements based on the patch-test using the natural method. The latter is found once more to greatly simplify the formulation and to result in compact natural stiffness matrices. The procedure is tested on the construction of a triangular plate-bending element TRUNC, the natural stiffness of which appears as a simple hyperdiagonal matrix. Sample problems using the TRUNC element show that the element converges quickly and is therefore well suited for engineering purposes.

Reconstruction of Complex Dynamical Systems Affected by Strong Measurement Noise

This Letter reports on a new approach to properly analyze time series of dynamical systems which are spoilt by the simultaneous presence of dynamical noise and measurement noise. It is shown that even strong external measurement noise as well as dynamical noise which is an intrinsic part of the dynamical process can be quantified correctly, solely on the basis of measured time series and proper data analysis. Finally, real world data sets are presented pointing out the relevance of the new approach.

An Exploration of Dynamical Systems and Chaos

Descriptive synopsis of the text.- Mathematical introduction to dynamical systems.- Dynamical systems without dissipation.- Dynamical systems with dissipation.- Local bifurcation theory.- Convective flow: Benard problem.- Routes to chaos.- Turbulence.- Computer experiments.

Extracting strong measurement noise from stochastic time series: applications to empirical data.

It is a big challenge in the analysis of experimental data to disentangle the unavoidable measurement noise from the intrinsic dynamical noise. Here we present a general operational method to extract measurement noise from stochastic time series even in the case when the amplitudes of measurement noise and uncontaminated signal are of the same order of magnitude. Our approach is based on a recently developed method for a nonparametric reconstruction of Langevin processes. Minimizing a proper non-negative function, the procedure is able to correctly extract strong measurement noise and to estimate drift and diffusion coefficients in the Langevin equation describing the evolution of the original uncorrupted signal. As input, the algorithm uses only the two first conditional moments extracted directly from the stochastic series and is therefore suitable for a broad panoply of different signals. To demonstrate the power of the method, we apply the algorithm to synthetic as well as climatological measurement data, namely, the daily North Atlantic Oscillation index, shedding light on the discussion of the nature of its underlying physical processes.

Discrete model for laser driven etching and microstructuring of metallic surfaces.

We present a unidimensional discrete solid-on-solid model evolving in time using a kinetic Monte Carlo method to simulate microstructuring of kerfs on metallic surfaces by means of laser-induced jet-chemical etching. The precise control of the passivation layer achieved by this technique is responsible for the high resolution of the structures. However, within a certain range of experimental parameters, the microstructuring of kerfs on stainless steel surfaces with a solution of H3PO4 shows periodic ripples, which are considered to originate from intrinsic dynamics. The model mimics a few of the various physical and chemical processes involved and within certain parameter ranges reproduces some morphological aspects of the structures, in particular ripple regimes. We analyze the range of values of laser beam power for the appearance of ripples in both experimental and simulated kerfs. The discrete model is an extension of one that has been used previously in the context of ion sputtering and is related to a noisy version of the Kuramoto-Sivashinsky equation used extensively in the field of pattern formation.

Some considerations on the natural approach

Abstract In this note we present some pertinent comments on the natural mode technique. In the first part we demonstrate that for arbitrary finite elements the choice of urelements may exclusively be based on simplex elements. In fact, the corresponding infinitesimal building blocks, the so-called subelements, may be used also for quadrilateral and hexahedronal elements or for that matter for any one-, two- or three-dimensional element. The second part develops some new considerations on unconventional models associated with the patch test.