High-order asymptotic field of tensile plane-strain nonlinear crack problems

In this paper, the governing equations and the analytical method of the secondorder asymptotic field for the plane-straln crack problems of mode I have been presented. The numerical calculation has been carried out. The amplitude coefficients k2 of the second term of the asymptotic field have been determined after comparing the present results with some fine results of the finite element calculation. The variation of coefficients k2 with changes of specimen geometry and developments of plastic zone have been analysed. It is shown that the second term of the asymptotic field has significant influence on the near-crack-tip field. Therefore, we may reasonably argue that both the J-integral and the coefficient k2 can beeome two characterizing parameters for crack initiation.

Quantifying chaos: practical estimation of the correlation dimension

Be it a physical object or a mathematical model, a nonlinear dynamical system can display complicated aperiodic behavior, or "chaos." In many cases, this chaos is associated with motion on a strange attractor in the system's phase space. And the dimension of the strange attractor indicates the effective number of degrees of freedom in the dynamical system.

In this thesis, we investigate numerical issues involved with estimating the dimension of a strange attractor from a finite time series of measurements on the dynamical system.

Of the various definitions of dimension, we argue that the correlation dimension is the most efficiently calculable and we remark further that it is the most commonly calculated. We are concerned with the practical problems that arise in attempting to compute the correlation dimension. We deal with geometrical effects (due to the inexact self-similarity of the attractor), dynamical effects (due to the nonindependence of points generated by the dynamical system that defines the attractor), and statistical effects (due to the finite number of points that sample the attractor). We propose a modification of the standard algorithm, which eliminates a specific effect due to autocorrelation, and a new implementation of the correlation algorithm, which is computationally efficient.

Finally, we apply the algorithm to chaotic data from the Caltech tokamak and the Texas tokamak (TEXT); we conclude that plasma turbulence is not a low- dimensional phenomenon.

I. Singular perturbation problems involving singular points and turning points. II. On the averaged Lagrangian technique for nonlinear dispersive waves

In Part I a class of linear boundary value problems is considered which is a simple model of boundary layer theory. The effect of zeros and singularities of the coefficients of the equations at the point where the boundary layer occurs is considered. The usual boundary layer techniques are still applicable in some cases and are used to derive uniform asymptotic expansions. In other cases it is shown that the inner and outer expansions do not overlap due to the presence of a turning point outside the boundary layer. The region near the turning point is described by a two-variable expansion. In these cases a related initial value problem is solved and then used to show formally that for the boundary value problem either a solution exists, except for a discrete set of eigenvalues, whose asymptotic behaviour is found, or the solution is non-unique. A proof is given of the validity of the two-variable expansion; in a special case this proof also demonstrates the validity of the inner and outer expansions.

Nonlinear dispersive wave equations which are governed by variational principles are considered in Part II. It is shown that the averaged Lagrangian variational principle is in fact exact. This result is used to construct perturbation schemes to enable higher order terms in the equations for the slowly varying quantities to be calculated. A simple scheme applicable to linear or near-linear equations is first derived. The specific form of the first order correction terms is derived for several examples. The stability of constant solutions to these equations is considered and it is shown that the correction terms lead to the instability cut-off found by Benjamin. A general stability criterion is given which explicitly demonstrates the conditions under which this cut-off occurs. The corrected set of equations are nonlinear dispersive equations and their stationary solutions are investigated. A more sophisticated scheme is developed for fully nonlinear equations by using an extension of the Hamiltonian formalism recently introduced by Whitham. Finally the averaged Lagrangian technique is extended to treat slowly varying multiply-periodic solutions. The adiabatic invariants for a separable mechanical system are derived by this method.

Asymptotic scaling in the two-dimensional O(3) Nonlinear sigma model: a Monte Carlo study on parallel computers

We investigate the 2d O(3) model with the standard action by Monte Carlo simulation at couplings β up to 2.05. We measure the energy density, mass gap and susceptibility of the model, and gather high statistics on lattices of size L ≤ 1024 using the Floating Point Systems T-series vector hypercube and the Thinking Machines Corp.'s Connection Machine 2. Asymptotic scaling does not appear to set in for this action, even at β = 2.10, where the correlation length is 420. We observe a 20% difference between our estimate m/Λ^─_(Ms) = 3.52(6) at this β and the recent exact analytical result . We use the overrelaxation algorithm interleaved with Metropolis updates and show that decorrelation time scales with the correlation length and the number of overrelaxation steps per sweep. We determine its effective dynamical critical exponent to be z' = 1.079(10); thus critical slowing down is reduced significantly for this local algorithm that is vectorizable and parallelizable.

We also use the cluster Monte Carlo algorithms, which are non-local Monte Carlo update schemes which can greatly increase the efficiency of computer simulations of spin models. The major computational task in these algorithms is connected component labeling, to identify clusters of connected sites on a lattice. We have devised some new SIMD component labeling algorithms, and implemented them on the Connection Machine. We investigate their performance when applied to the cluster update of the two dimensional Ising spin model.

Finally we use a Monte Carlo Renormalization Group method to directly measure the couplings of block Hamiltonians at different blocking levels. For the usual averaging block transformation we confirm the renormalized trajectory (RT) observed by Okawa. For another improved probabilistic block transformation we find the RT, showing that it is much closer to the Standard Action. We then use this block transformation to obtain the discrete β-function of the model which we compare to the perturbative result. We do not see convergence, except when using a rescaled coupling β_E to effectively resum the series. For the latter case we see agreement for m/ Λ^─_(Ms) at , β = 2.14, 2.26, 2.38 and 2.50. To three loops m/Λ^─_(Ms) = 3.047(35) at β = 2.50, which is very close to the exact value m/ Λ^─_(Ms) = 2.943. Our last point at β = 2.62 disagrees with this estimate however.

Mathematical study of complex networks : brain, internet, and power grid

The dissertation is concerned with the mathematical study of various network problems. First, three real-world networks are considered: (i) the human brain network (ii) communication networks, (iii) electric power networks. Although these networks perform very different tasks, they share similar mathematical foundations. The high-level goal is to analyze and/or synthesis each of these systems from a “control and optimization” point of view. After studying these three real-world networks, two abstract network problems are also explored, which are motivated by power systems. The first one is “flow optimization over a flow network” and the second one is “nonlinear optimization over a generalized weighted graph”. The results derived in this dissertation are summarized below.

Brain Networks: Neuroimaging data reveals the coordinated activity of spatially distinct brain regions, which may be represented mathematically as a network of nodes (brain regions) and links (interdependencies). To obtain the brain connectivity network, the graphs associated with the correlation matrix and the inverse covariance matrix—describing marginal and conditional dependencies between brain regions—have been proposed in the literature. A question arises as to whether any of these graphs provides useful information about the brain connectivity. Due to the electrical properties of the brain, this problem will be investigated in the context of electrical circuits. First, we consider an electric circuit model and show that the inverse covariance matrix of the node voltages reveals the topology of the circuit. Second, we study the problem of finding the topology of the circuit based on only measurement. In this case, by assuming that the circuit is hidden inside a black box and only the nodal signals are available for measurement, the aim is to find the topology of the circuit when a limited number of samples are available. For this purpose, we deploy the graphical lasso technique to estimate a sparse inverse covariance matrix. It is shown that the graphical lasso may find most of the circuit topology if the exact covariance matrix is well-conditioned. However, it may fail to work well when this matrix is ill-conditioned. To deal with ill-conditioned matrices, we propose a small modification to the graphical lasso algorithm and demonstrate its performance. Finally, the technique developed in this work will be applied to the resting-state fMRI data of a number of healthy subjects.

Communication Networks: Congestion control techniques aim to adjust the transmission rates of competing users in the Internet in such a way that the network resources are shared efficiently. Despite the progress in the analysis and synthesis of the Internet congestion control, almost all existing fluid models of congestion control assume that every link in the path of a flow observes the original source rate. To address this issue, a more accurate model is derived in this work for the behavior of the network under an arbitrary congestion controller, which takes into account of the effect of buffering (queueing) on data flows. Using this model, it is proved that the well-known Internet congestion control algorithms may no longer be stable for the common pricing schemes, unless a sufficient condition is satisfied. It is also shown that these algorithms are guaranteed to be stable if a new pricing mechanism is used.

Electrical Power Networks: Optimal power flow (OPF) has been one of the most studied problems for power systems since its introduction by Carpentier in 1962. This problem is concerned with finding an optimal operating point of a power network minimizing the total power generation cost subject to network and physical constraints. It is well known that OPF is computationally hard to solve due to the nonlinear interrelation among the optimization variables. The objective is to identify a large class of networks over which every OPF problem can be solved in polynomial time. To this end, a convex relaxation is proposed, which solves the OPF problem exactly for every radial network and every meshed network with a sufficient number of phase shifters, provided power over-delivery is allowed. The concept of “power over-delivery” is equivalent to relaxing the power balance equations to inequality constraints.

Flow Networks: In this part of the dissertation, the minimum-cost flow problem over an arbitrary flow network is considered. In this problem, each node is associated with some possibly unknown injection, each line has two unknown flows at its ends related to each other via a nonlinear function, and all injections and flows need to satisfy certain box constraints. This problem, named generalized network flow (GNF), is highly non-convex due to its nonlinear equality constraints. Under the assumption of monotonicity and convexity of the flow and cost functions, a convex relaxation is proposed, which always finds the optimal injections. A primary application of this work is in the OPF problem. The results of this work on GNF prove that the relaxation on power balance equations (i.e., load over-delivery) is not needed in practice under a very mild angle assumption.

Generalized Weighted Graphs: Motivated by power optimizations, this part aims to find a global optimization technique for a nonlinear optimization defined over a generalized weighted graph. Every edge of this type of graph is associated with a weight set corresponding to the known parameters of the optimization (e.g., the coefficients). The motivation behind this problem is to investigate how the (hidden) structure of a given real/complex valued optimization makes the problem easy to solve, and indeed the generalized weighted graph is introduced to capture the structure of an optimization. Various sufficient conditions are derived, which relate the polynomial-time solvability of different classes of optimization problems to weak properties of the generalized weighted graph such as its topology and the sign definiteness of its weight sets. As an application, it is proved that a broad class of real and complex optimizations over power networks are polynomial-time solvable due to the passivity of transmission lines and transformers.

Statistical image modelling using interscale phase relationships of complex wavelet coefficients

A novel method for modelling the statistics of 2D photographic images useful in image restoration is defined. The new method is based on the Dual Tree Complex Wavelet Transform (DT-CWT) but a phase rotation is applied to the coefficients to create complex coefficients whose phase is shift-invariant at multiscale edge and ridge features. This is in addition to the magnitude shift invariance achieved by the DT-CWT. The increased correlation between coefficients adjacent in space and scale provides an improved mechanism for signal estimation. © 2006 IEEE.

Mixed nonlinear LES for DES suitable flows

It is shown that a new mixed nonlinear/eddy viscosity LES model reproduces profiles better than a number of competing nonlinear and mixed models for plane channel flow. The objective is an LES method that produces a fully resolved turbulent boundary layer and could be applied to a variety of aerospace problems that are currently studied with RANS, RANS-LES, or DES methods that lack a true turbulent boundary layer. There are two components to the new model. One an eddy viscosity based upon the advected subgrid scale energy and a relatively small coefficient. Second, filtered nonlinear terms based upon the Leray regularization. Coefficients for the eddy viscosity and nonlinear terms come from LES tests in decaying, isotropic turbulence. Using these coefficients, the velocity profile matches measurements data at Reτ ≈ 1000 exactly. Profiles of the components of kinetic energy have the same shape as in the experiment, but the magnitudes differ by about 25%. None of the competing LES gets the shape correct. This method does not require extra operations at the transition between the boundary layer and the interior flow.

Wave propagation in nonlinear one-dimensional soil model

The objective of the research conducted by the authors is to explore the feasibility of determining reliable in situ values of shear modulus as a function of strain. In this paper the meaning of the material stiffness obtained from impact and harmonic excitation tests on a surface slab is discussed. A one-dimensional discrete model with the nonlinear material stiffness is used for this purpose. When a static load is applied followed by an impact excitation, if the amplitude of the impact is very small, the measured wave velocity using the cross-correlation indicates the wave velocity calculated from the tangent modulus corresponding to the state of stress caused by the applied static load. The duration of the impact affects the magnitude of the displacement and the particle velocity but has very little effect on the estimation of the wave velocity for the magnitudes considered herein. When a harmonic excitation is applied, the cross-correlation of the time histories at different depths estimates a wave velocity close to the one calculated from the secant modulus in the stress-strain loop under steady-state condition. Copyright © 2008 John Wiley & Sons, Ltd.

Wave propagation due to sinusoidal excitation in nonlinear one-dimensional soil column

The objective of the author's on-going research is to explore the feasibility of determining reliable in situ curves of shear modulus as a function of strain using the dynamic test. The purpose of this paper is limited to investigating what material stiffness is measured from a dynamic test, focusing on the harmonic excitation test. A one-dimensional discrete model with nonlinear material properties is used for this purpose. When a sinusoidal load is applied, the cross-correlation of signals from different depths estimates a wave velocity close to the one calculated from the secant modulus in the stress-strain loops under steady-state conditions. The variables that contributed to changing the average slope of the stress-strain loop also influence the estimate of the wave velocity from cross-correlation. Copyright ASCE 2007.

Nonlinear analysis of cardiac rhythm fluctuations using DFA method

An approach by which the detrented fluctuation analysis (DFA) method can be used to help diagnose heart failure was demonstrated. DFA was applied to patients suffering from congestive heart failure (CHF) to check correlations between DFA indices and CHF, and determine a correlation between DFA indices and mortality, with a particular attention to the residue parameter, which is a measure of the departure of the DFA from its power law approximation. DFA parameters proved to be useful as a complement to the physiological parameters weber and FE to sort out the patients into three prognostic group.

Randomized nonlinear component analysis

Copyright © (2014) by the International Machine Learning Society (IMLS) All rights reserved. Classical methods such as Principal Component Analysis (PCA) and Canonical Correlation Analysis (CCA) are ubiquitous in statistics. However, these techniques are only able to reveal linear re-lationships in data. Although nonlinear variants of PCA and CCA have been proposed, these are computationally prohibitive in the large scale. In a separate strand of recent research, randomized methods have been proposed to construct features that help reveal nonlinear patterns in data. For basic tasks such as regression or classification, random features exhibit little or no loss in performance, while achieving drastic savings in computational requirements. In this paper we leverage randomness to design scalable new variants of nonlinear PCA and CCA; our ideas extend to key multivariate analysis tools such as spectral clustering or LDA. We demonstrate our algorithms through experiments on real- world data, on which we compare against the state-of-the-art. A simple R implementation of the presented algorithms is provided.

Optical properties of NAEC-PMMA nonlinear polymeric thin film

Novel guest nonlinear optical (NLO) chromophore molecules (4-nitrobenzene)-3-azo-9-ethylcarbazole (NAEC) were doped in poly (methyl methacrylate) (PMMA) host with a concentration of approximately 15% by weight. For a useful macroscopic electro-optic (EO) effect, these NLO molecules NAEC were arranged in a noncentrosymmetric structure in the host polymer by corona-onset poling at elevated temperature (COPET). For applying NAEC-PMMA polymer in optical devices such as EO switch, its optical properties have been investigated. The UV/Visible absorption spectra for the unpoled and poled polymer film were determined. The refractive index of the film was also determined from measurements of the coupling angles with the reflective intensity at 632.8 nm wavelength. Using the simple reflection technique, the EO coefficient 33 value was measured as 60 pm/V at 632.8 nm wavelength. The second-order nonlinear coefficient d(33) was characterized by the second-harmonic-generation (SHG) experimental setup and the calculated d(33) value reached 18.4 pm/V at 1064 nm wavelength. The relation between the second-order nonlinear coefficients d(33) and d(13) for the poled polymer film was also discussed in detail and the ratio d(33)/d(13) value was obtained as 3.3. (C) 2002 Kluwer Academic Publishers.

Effect of the electrostatic interactions of chromophores on the macroscopic second-order nonlinear optical properties of the polymer films

The polyetherketone (PEK-c) guest-host system thin films doped with 3-(1,1-dicyanothenyl)-1-phenyl-4,5-dihydro-1H-pryazole (DCNP) were prepared. Their second-order nonlinear optical (NLO) coefficients chi(33)((2)) were measured by using Maker fringe method for the polymer films doped with different weight percents of DCNP. Experimental results indicate that the second-order NLO properties of the poled polymer films could decrease with the chromophore loading increasing when the chromophore loading reaches a fairly high level. In this paper, the relationship between the macroscopic second-order NLO coefficient and the chromophore number density was modified under considering the role of the electrostatic interactions of chromophores in the polymer film. According to the modified relationship, the macroscopic second-order NLO coefficient is no longer in direct proportion with the chromophore number density in the polymer film. The effect of the electrostatic interactions of chromophores on second-order NLO properties was discussed. The attenuation of the macroscopic second-order NLO activity can be demonstrated by the role of the chromophore electrostatic interactions at high loading of chromophore in the polymer systems.

Are the available boiling heat transfer coefficients suitable for silicon microchannel heat sinks?

The typical MEMS fabrication of micro evaporators ensures the perfect smooth wall surface that is lack of nucleation sites, significantly decreasing the heat transfer coefficients compared with miniature evaporators fabricated using copper or stainless steel. In the present paper, we performed the boiling heat transfer experiment in silicon triangular microchannel heat sink over a wide parameter range for 102 runs. Acetone was used as the working fluid. The measured boiling heat transfer coefficients versus the local vapor mass qualities are compared with the classical Chen’s correlation and other correlations for macro and miniature capillary tubes. It is found that most of these correlations significantly over-predict the measured heat transfer coefficients. New correlations are given. There are many reasons for such deviations. The major reason is coming from the perfect smooth silicon surface that lowers the heat transfer performances. New theory is recommended for the silicon microchannel heat sink that should be different from metallic capillary tubes.