Self-similarity Condition Research Articles

Comment on “Hydraulic characteristics of labyrinth sluice gates” by T. Hashem, A.Y. Mohammed, T.J. Alfatlawi

In this paper, the stage-discharge relationship of a labyrinth sluice gate is theoretically deduced by the Π-Theorem of dimensional analysis and the self-similarity condition. This approach was then compared with the classical approach using the discharge coefficient Cd proposed by Hashem et al. (2024). The classical stage-discharge equation (Eq. 4), with the discharge coefficient estimated by the empirical relationship (Eq. (1)) proposed by Hashem et al. (2024), resulted in errors E that are less than or equal to ±5 % for 74.3 % of the investigated cases and a mean square error MSE of 0.0078. The new stage-discharge relationship, deduced by dimensional analysis and self-similarity theory, was calibrated using the dataset by Hashem et al. (2024). The obtained stage-discharge equation (Eq. 20) is characterized by errors in the discharge estimate E that are less than or equal to ±5 % for 82.2 % of the investigated cases and a mean square error MSE of 0.0051. In conclusion, the stage-discharge relationship deduced by dimensional analysis allows for the best performance in the discharge estimate, which can be obtained without knowledge of the empirical discharge coefficient.

ПРИМЕНЕНИЕ МЕТОДОВ ИСКУССТВЕННОГО ИНТЕЛЛЕКТА ДЛЯ КОМПЬЮТЕРНОГО МОДЕЛИРОВАНИЯ ФРАКТАЛЬНЫХ ПОВЕРХНОСТЕЙ

The paper considers the possibility of using artificial intelligence methods for computer modeling of fractal surfaces. Fractals act as a mathematical model for creating a random surface relief. The random profile is constructed using the random displacement method, which is an algorithm for generating random functions with a spectrum. Surfaces are defined with the help of data arrays, which are checked by the self-similarity condition. Based on the defined arrays, models are built using the Weierstrass function. The algorithm for constructing surfaces has been refined and improved using machine learning neural network generative models. Thus, instead of simply creating a fractal surface using random functions, the generator creates fractal surfaces based on the distribution learned during training. The verification criterion is an algorithm based, in general, on a mathematical Monte Carlo method. The obtained results show the realism of the constructed fractal surfaces using neural networks. The models of the obtained surface reliefs can be used in modeling contact mechanics, mechanics of deformable solid bodies.

Numerical validation of novel scaling laws for air–water flows including compressibility and heat transfer

Air–water flows are among the most important flow types in hydraulic engineering. Their experimental modelling at reduced size using Froude scaling laws introduces scale effects. This study introduces novel scaling laws for compressible air–water flows in which the air is considered compressible. This is achieved by applying the one-parameter Lie group of point-scaling transformations to the governing equations of these flows. The scaling relationships between variables are derived for the fluid properties and the flow variables including temperature. The novel scaling laws are validated by computational fluid dynamics modelling of a Taylor bubble at different scales. The resulting velocity, density, temperature, pressure and volume of the bubble are shown to be self-similar at different scales, i.e. all these variables behave the same in dimensionless form. This study shows that the self-similar conditions of the derived novel scaling laws for compressible air–water flows have the potential to improve laboratory modelling.

Flow discharge measurement by a linear width contraction device

In this paper, the outflow process of a linear width contraction device for a free-flow condition is modeled using the dimensional analysis and the incomplete self-similarity condition. The proposed theoretical stage-discharge relationship is tested using measurements available in the literature. The proposed power stage-discharge equation is characterized by a value of the exponent close to 2 and a coefficient depending on the angle of the device sides with the channel bank. The proposed flume is characterized by simple construction, easy installation, low cost, and a good accuracy of the measured discharge (errors in the estimate ranging from − 3.84 to 1.9%). The deduced stage-discharge equation is characterized by errors in the estimate of discharge lower than or equal to ± 3% for 93.7% of the investigated cases. The main result of this paper is a stage-discharge relationship giving an accurate estimate of discharge but, at the same time, having the advantage of working regardless of the discharge coefficient estimate.

Deducing the stage-discharge relationship of rectangular broad and sharp-crested contraction devices

Flumes with a local contraction of the channel width are a common way for accurately measuring the flow discharge. In this paper, the outflow process of rectangular broad and sharp-crested contraction devices is modeled using the dimensional analysis and the incomplete self-similarity condition. The proposed theoretical stage-discharge relationship is tested using measurements available in the literature. The proposed power equation is characterized by a value of the exponent close to 1.5 and a coefficient depending on the contraction ratio. The proposed flow-measuring flumes are characterized by a good accuracy of the measured discharge (ranging from −2.71 to 3.28% for the broad-crested contraction device and in the range from −0.24 to 0.31% for the sharp-crested one). Even if both flume types have good accuracy of the measured discharge (less than the limit of ±5% suggested in the literature), the developed analysis demonstrated that the sharp-crested device is characterized by the lowest values of the errors in the estimate of discharge, and this result increases the interest of this device in practical applications. The advantage of the proposed approach is combining an accurate estimate of the flow discharge and working regardless of the discharge coefficient estimate.

Generation of High Power Ultrashort Pulses in Tapered Yb-Doped PCF Through Self-Similar Compression

We present a numerical approach for analysing simultaneous amplification and compression of a chirped hyperbolic secant soliton using a tapered Yb-doped photonic crystal fiber (PCF) based self-similar compressor. Theoretical studies reveal the conservation of input soliton during the amplification process due to self-similar property, which in turn helps for high quality compression of amplified soliton. In order to satisfy the self-similar condition, we note that the Yb-doped fiber should exhibit a decreasing group velocity dispersion (GVD). Such a constraint is met using a tapered PCF in our work, which is modeled using a fully vectorial effective index method (FVEIM). Influence of spatial and spectral dependent gain, saturation energy, amplified spontaneous emission (ASE), dispersion and nonlinearity on pulse propagation are analyzed using the combined solution of dynamic rate equations (DRE) and the generalized nonlinear Schrödinger equation (GNLSE). Numerical outcomes upon solving the aforementioned equations show that a 1 ps pulse is compressed down to a pulsewidth of 55.4 fs with a peak power of 1.03 kW in a total fiber length of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$L=2.5$ </tex-math></inline-formula> m and taper ratio of 2.34. Overall quality of the compression scheme is evidenced by a relatively high compression factor (F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>c</i></sub> ) = 18.12, and low percentage of pedestal energy ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$PE$ </tex-math></inline-formula> ) = 10.95 %

On some geometrical aspects of the potential structure of the equations of evolution: The case of Navier-Stokes

In this paper we discuss the potential structure of the evolution equations, in particular Navier-Stokes. To this end, the method of prolongation of Wahlquist H. D. and Estabrook F. B., J. Math. Phys., 16 (1975) 1 is introduced and the most general potential for the flow velocity is found, expressing everything in terms of the representative differential forms of the system of equations. Steady-flow and self-similar solutions and conditions are presented and briefly discussed, as well as the most general solution when a general transformation similar to the one given by Cole is introduced into the original system. In this theoretical context, the solution can be associated with a damped acoustic wave. Consequently, a useful application area for the present work is certainly in nonlinear acoustics, as we discuss briefly at the end of this letter.

Smoothness-Adaptive Contextual Bandits

In nonparametric contextual bandit formulations, a key complexity driver is the smoothness of payoff functions with respect to covariates. In many practical settings, the smoothness of payoffs is unknown, and misspecification of smoothness may severely deteriorate the performance of existing methods. In the paper “Smoothness-Adaptive Contextual Bandits,” Yonatan Gur, Ahmadreza Momeni, and Stefan Wager consider a framework where the smoothness of payoff functions is unknown and study when and how algorithms may adapt to unknown smoothness. First, they establish that designing algorithms that adapt to unknown smoothness is, in general, impossible. However, under a natural self-similarity condition, they establish that adapting to unknown smoothness is possible and devise a general policy for achieving smoothness-adaptive performance. The policy infers the smoothness of payoffs throughout the decision-making process while leveraging the structure of off-the-shelf nonadaptive policies. It matches (up to a logarithmic scale) the performance that is achievable when the smoothness of payoffs is known in advance.

Optimal Convergence of the Discrepancy Principle for Polynomially and Exponentially Ill-Posed Operators under White Noise

We consider a linear ill-posed equation in the Hilbert space setting under white noise. Known convergence results for the discrepancy principle are either restricted to Hilbert-Schmidt operators (and they require a self-similarity condition for the unknown solution additional to a classical source condition) or to polynomially ill-posed operators (excluding exponentially ill-posed problems). In this work, we show optimal convergence for a modified discrepancy principle for both polynomially and exponentially ill-posed operators (without further restrictions) solely under either Hölder-type or logarithmic source conditions. In particular, the method includes only a single simple hyper parameter, which does not need to be adapted to the type of ill-posedness.

Oscillations properties of the dynamic fractal structures

A wide class of systems in nature and engineering consists of self-similar fractal structures, where each cell repeats the structure of the previous one on a certain scale. However, the well-known B. Mandelbrot's fractals that use the scaling of geometric parameters do not reflect the dynamic properties of the system well enough. Therefore, when studying the system dynamics, it is suggested that structures which have a scaling of dynamic, parameters, should be considered as a special class of vibratory system. Such class of structures where stiffness and inertial parameters change with the same scale factor from cell to cell can be called dynamic fractals. Research has shown that such structures have a number of specific wave and vibrational properties. Thus, the dynamic fractal is equivalent in terms of frequency to a certain periodic structure, which is a band-pass filter. It has been also shown that the dynamic fractal having a scale factor greater than one with increasing magnitudes of stiffness and inertial parameters along its length has good vibroisolation properties. The main advantage of such a lattice is intensive vibration attenuation in all frequency ranges including in resonant modes of the lattice, which favorably compares with periodic lattices. On the contrary, the dynamic fractal having a scale factor less than one with decreasing magnitudes of stiffness and inertial parameters along its length has the properties of amplifying the incoming signal along the structure, which can be used in control systems.We have also studied the oscillation properties of a branched self-similar structure, a dichotomous lattice. It was found that its calculation model in the low-frequency range can be represented as a dynamically fractal structure. The natural frequencies and vibration modes at bending oscillations are determined. The conditions of dynamical self-similarity for such a lattice, its dispersion equation, and a favorable frequency range depending on the scaling factor are obtained.

Self-similar pulse compression by suspended core photonic crystal fiber with cubic-quintic nonlinearities

An enhanced self-similar pulse compression with suspended core photonic crystal fiber has been theoretically studied. The proposed model well support the essential criteria of having very low dispersion and increased nonlinearity with increasing suspension factor resulting high compression rate within short length of the fiber. Unlike the conventional tapering methods, the suspended core photonic crystal fiber can achieve the self-similar conditions more easily and reduces the practical difficulties to achieve the desired fiber parametric values. We have also studied the possible deformations of the fiber and temperature variations that can effect the pulse compression process and compression factor.

Smoothness-Adaptive Contextual Bandits

We study a non-parametric multi-armed bandit problem with stochastic covariates, where a key complexity driver is the smoothness of payoff functions with respect to covariates. Previous studies have focused on deriving minimax-optimal algorithms in cases where it is a priori known how smooth the payoff functions are. In practice, however, the smoothness of payoff functions is typically not known in advance, and misspecification of smoothness may severely deteriorate the performance of existing methods. In this work, we consider a framework where the smoothness of payoff functions is not known, and study when and how algorithms may adapt to unknown smoothness. First, we establish that designing algorithms that adapt to unknown smoothness of payoff functions is, in general, impossible. However, under a self-similarity condition (which does not reduce the minimax complexity of the dynamic optimization problem at hand), we establish that adapting to unknown smoothness is possible, and further devise a general policy for achieving smoothness-adaptive performance. Our policy infers the smoothness of payoffs throughout the decision-making process, while leveraging the structure of off-the-shelf non-adaptive policies. We establish that for problem settings with either differentiable or non-differentiable payoff functions, this policy matches (up to a logarithmic scale) the regret rate that is achievable when the smoothness of payoffs is known a priori.

Adaptation bounds for confidence bands under self-similarity

We derive bounds on the scope for a confidence band to adapt to the unknown regularity of a nonparametric function that is observed with noise, such as a regression function or density, under the self-similarity condition proposed by Giné and Nickl (Ann. Statist. 38 (2010) 1122–1170). We find that adaptation can only be achieved up to a term that depends on the choice of the constant used to define self-similarity, and that this term becomes arbitrarily large for conservative choices of the self-similarity constant. We construct a confidence band that achieves this bound, up to a constant term that does not depend on the self-similarity constant. Our results suggest that care must be taken in choosing and interpreting the constant that defines self-similarity, since the dependence of adaptive confidence bands on this constant cannot be made to disappear asymptotically.

Flow resistance law under suspended sediment laden conditions

The uniform flow resistance equation, in the form due to Manning or Darcy-Weisbach, is widely applied to establish the stage-discharge relationship of a river cross-section. The application of this equation, namely the slope-area method, allows to indirectly measure the corresponding river discharge by measurements of bed slope, water level, cross-section area, wetted perimeter and an estimate of channel roughness. In this paper, a recently deduced flow resistance equation for open channel flow was tested during conditions of suspended sediment-laden flow. First, the flow resistance equation was determined by dimensional analysis and by applying the condition of incomplete self-similarity for the flow velocity profile. Then the analysis was developed by the following steps: (i) for sediment-laden flows characterized by known values of mean diameter and concentration of suspended sediments, a relationship (Eq. (28)) between the Γ function of the velocity profile, the channel slope and the Froude number was calibrated by the available measurements; and (ii) a relationship for estimating the Γ function (Eq. (29)) which also takes into account the mean concentration of suspended particles was also established. The theoretical flow resistance law (Eq. (26)) coupled with the relationship for estimating the Γ function (Eq. (28) or Eq. (29)), which is characterized by the applicability of a wide range of flow conditions, allowed to estimate the Darcy-Weisbach friction factor for flows with suspended-load. The analysis showed that for large-size mixtures the Darcy-Weisbach friction factor can be accurately estimated neglecting the effect of mean concentration of suspended sediments while for small-size mixtures the friction factor decreases when the mean sediment concentration increases.

Self-similar grooving solutions to the Mullins’ equation

In 1957, Mullins proposed surface diffusion motion as a model for thermal grooving. By adopting a small slope approximation, he reduced the model to the Mullins’ linear surface diffusion equation, ( M E ) y t + B y x x x x = 0 , \begin{equation} \nonumber ({\mathrm {ME}})\quad \quad y_t + B y_{xxxx}=0, \end{equation} known also more simply as the Mullins’ equation. Mullins sought self-similar solutions to (ME) for planar initial conditions, prescribing boundary conditions at the thermal groove, as well as far field decay. He found explicit series solutions which are routinely used in analyzing thermal grooving to this day. While (ME) and the small slope approximation are physically reasonable, Mullins’ choice of boundary conditions is not always appropriate. Here we present an in depth study of self-similar solutions to the Mullins’ equation for general self-similar boundary conditions, explicitly identifying four linearly independent solutions defined on R ∖ { 0 } \mathbb {R}\setminus \{0\} ; among these four solutions, two exhibit unbounded growth and two exhibit asymptotic decay, far from the origin. We indicate how the full set of solutions can be used in analyzing the effective boundary conditions from experimental profiles and in evaluating the governing physical parameters.

Self-similarity in fate and transport of contaminants in groundwater

Similarity within non-linear geophysical processes under different space and time has been investigated in literature to understand and model the sophisticated nature of these processes. This study deals with the self-similarity of the governing processes of fate and transport of contaminants in groundwater, which include advection, dispersion, sorption, and degradation, either by chemical reaction or microbiological interaction. As such, self-similarity conditions of three-dimensional advective-dispersive-reactive transport (ADR) equation with various initial and boundary conditions are obtained by employing one-parameter Lie group of point scaling transformations. The strength of the proposed scaling approach lies in its ability to handle initial and boundary conditions of the process together with the governing equation. Then, utilizing reported conditions in Borden Field experiment domain, numerical simulations are performed to demonstrate examples of self-similarity when first-order degradation rate constant is zero and nonzero under three sorption conditions. The results of the numerical examples showed the effectiveness of the derived self-similarity conditions and the strength of the proposed approach in handling initial and boundary conditions of the governing transport process. It is possible to obtain both smaller and larger size self-similar (scaled-up or scaled-down) domains, which result in faster and slower transport processes, respectively. The obtained self-similarity criteria can offer further understanding of ADR transport phenomena and can provide spatial, temporal, and economical flexibility in constructing scaled models of field conditions governed by three-dimensional ADR transport process.

Self-similar Elastic Condition of Filtration Through the Moving Boundary

We consider the one-dimensional problem of the elastic filtration of a fluid though the moving boundary. The boundary conditions are introduced so that the problem be invariant. The invariant problem is reduced to a overdetermine boundary task for the Weber equation. Exact solutions are found. The asymptotic of a solution in infinite point determines the invariant law of a filtration according to the given boundary conditions. There is a connection between overdetermine invariant boundary conditions for any invariant law of a filtration.

Autoregressive Models and Non-Local Self Similarity in Sparse Representation for Image Deblurring

Local area within a normal natural image can be thought as a stationary process. This can be modelled well using autoregressive models. In this paper, a set of autoregressive models will be learned from a collection of high quality image patches. Out of these models, one will be selected adaptively and will be used to regularize the input image patches. In addition to the autoregressive models, a non-local self-similarity condition was proposed. The autoregressive models will exploit local correlation of individual image, but a natural will have many repetitive structures. These structures, which are basically redundant, are very much useful in image deblurring. The performance of these schemes is verified by applying to image deblurring.

Static and Dynamic Properties of Branched Self-Similar Structures: The Trichotomous Lattice

The static and dynamic properties of self-similar branched structures (lattices) in which each subsequent cell replicates the structure of the previous one on some scale are investigated. The trichotomous lattices in which each element in each subsequent line divides into three self-similar ones are analyzed. The analytical conditions of static and dynamic self-similarity are found. The conditions of the same static deformation of each element are obtained. It is proved that the lattice is a band-pass filter, and its pass band depends on the similarity factor. It is shown that in the lattice there are multiple natural frequencies equal to the partial frequency of the generating element.

On the formation of expanding crack tip precipitates

The stress driven growth of an expanding precipitate at a crack tip is studied. The material is assumed to be linearly elastic, and the expansion is considered to be isotropic or transversely isotropic. The extent of the precipitate is expected to be small as compared with the crack length and distance to boundaries. The problem has only a single length scale given by the squared ratio of the stress intensity factor and a critical hydrostatic stress that initiates the growth of the precipitate. Therefore, the growth occurs under self-similar conditions. The equations on non-dimensional form show that the free parameters are expansion strain, degree of anisotropy and Poisson’s ratio. It is found that the precipitate, once initiated, grows without remote load for expansion strains above a critical value. The anisotropy of the expansion strongly affects the shape of the precipitate but does not have a large effect on the crack tip shielding.