Fractal Spectrum Research Articles

A Fractal Study on Random Distribution of Recycled Concrete and Its Influence on Failure Characteristics

In order to quantitatively describe the influence of aggregate distribution on crack development and peak stress of recycled aggregate concrete, a multifractal spectrum theory was proposed to quantitatively characterize aggregate distribution in specimens. A mesomechanical model of reclaimed aggregate concrete mixed with natural aggregate and artificial aggregate was constructed. Numerical simulation tests were conducted on the uniaxial compression mechanical behavior of 25 groups of sample models with the same proportion and different aggregate distribution forms. Based on the box dimension theory, the multiple fractal spectrum method was used to quantitatively characterize the aggregate distribution form, and the key factors affecting cracks were explored based on the gray correlation degree. The research results show that the aggregate distribution in recycled aggregate concrete has multifractal characteristics. The multifractal spectrum was used to effectively characterize the aggregate distribution pattern, which can enlarge local details and provide new ideas for the quantitative analysis of the damage mode of recycled concrete. Secondly, by establishing a statistical model of the correlation between the multifractal spectrum width of the aggregate distribution pattern and the crack distribution box dimension, it was found that there was a positive correlation between the two, that is, the greater the multifractal spectrum width of the aggregate distribution pattern, the greater the crack box dimension, and the more complex the crack distribution. The complexity of aggregate distribution is closely related to the irregularity and complexity of mesoscopic failure crack propagation in recycled concrete specimens. In addition, gray correlation theory was applied to analyze the key factors affecting the formation of cracks in the specimens. The results showed that aggregate distribution had a first-order correlation with crack formation, and changes in aggregate distribution were an important factor affecting the performance of recycled concrete. Secondly, the poor mechanical properties of NAITZ led to obvious material damage, while NCA and MZ had a significant impact on the skeleton effect in the stress–strain process due to their large areas. This study deepens people’s understanding of the damage characteristics and cracking failure modes of recycled concrete. The study verifies the feasibility of the application of recycled aggregates and provides a valuable reference for engineering practice.

Topological modes, vibration attenuation, and energy harvesting in electromechanical metastructures

The dynamics of topological boundary modes in both periodic and quasi-periodic electromechanical metastructures is investigated, with a focus on their applications to energy harvesting and vibration reduction. The metastructure analyzed in this study is based on a shunted array of piezoelectric patches, with electrical parameters modulated according to the 1D Aubry–André–Harper model. As a result of this modulation, a fractal spectrum is generated near the central frequency of the resonators, a hallmark of nontrivial topology that enables the emergence of digitally controllable edge states and ensuing localization phenomena at subwavelength frequencies. In this framework, a detailed analysis of the metastructure spectral characteristics is conducted to investigate the influence of the modulation parameters on mode localization, both at the boundaries and within the interior of the beam. Such localization effects are then studied in relation to the energy harvesting, attenuation, and wave transport capabilities of the system. These functionalities point toward the realization of self-powered structures with low frequency and digitally controllable vibration attenuation capabilities, and are considered of significant technological interest in applications involving elastic waves and vibrations, where the ability to precisely control and harness these phenomena could lead to innovative solutions in energy-efficient and adaptive systems.

Blood Plasma Film Multifractal Scanning in COVID-19 Consequences Diagnostics.

A 3D phase scanning method was applied to study blood plasma facies, generating layered polarization maps of the object field. The most sensitive parameters to changes in birefringence distribution were identified. Multifractal analysis using wavelet transforms and fractal dimension spectra provided specific insights into the scale self-similarity of the polarization maps. The multifractal spectra of ellipticity distributions were algorithmically derived, revealing that the third- and fourth-order statistical moments were most sensitive to changes in the supramolecular networks of the facies. These findings were successfully applied to differentiate post-COVID-19 effects with high accuracy.

Novel multifractal-based classification model for the quality grades of surrounding rock within tunnels

Understanding the variation patterns of tunnel boring machine (TBM) operational parameters is crucial for assessing engineering geological conditions and quality grades of surrounding rock within tunnels. Studying the multifractal characteristics of the TBM operational parameters can help identify the patterns, but the relevant research has not yet been explored. This paper proposed a novel classification model for quality grades of surrounding rock in TBM tunnels based on multifractal analysis theory. Initially, the statistical characteristics of eight TBM cycle data with different grades of surrounding rock were explored. Subsequently, the method of calculating and analyzing the multifractal characteristic parameters of the TBM operational data was deduced and summarized. The research results showed that the TBM operational parameters of cutterhead torque, total thrust, advance rate, and cutterhead rotation speed have significant multifractal characteristics. Its multifractal dimension, midpoint slope of the generalized fractal spectrum, and singularity strength range can be used to evaluate the surrounding rock grades of the tunnel. Finally, a novel classification model for the tunnel surrounding rocks based on the multifractal characteristic parameters was proposed using the multiple linear regression method, and the model was verified through four TBM cycle data containing different surrounding rock grades. The results showed that the proposed multifractal-based classification model for tunnel surrounding rocks has high accuracy and applicability. This study not only achieves multifractal feature representation and surrounding rock classification for TBM operational parameters but also holds the potential for adaptive adjustment of TBM operational parameters and automated tunneling applications.

Quantitative dispersion characterization of cement particles in hardened cement matrix

Optimizing the cement particle distribution in hardened cement matrix is one of the effective means to utilize cement efficiently, and obtaining information about cement distribution is a prerequisite for this. In view of this, this paper investigates the quantitative dispersion of cement particles in hardened cement paste from an experimental perspective. Cement pastes with different water-binder (W/B) ratios were prepared as the objects. BSE images of these samples were acquired for further analysis, including Delaunay triangulation algorithm, nearest neighbor distance function G(r) and fractal theory. Results showed that when the W/B ratio increases from 0.36 to 0.42, the average cement particle spacing rises from 32.7 μm to 34.9 μm and the proportion of particles with spacing between particle centroids <20 μm decreases from 41.96 % to 25.98 %. Similarly, the increasing W/B ratio causes the nearest neighbor distance function G(r) to shift towards larger distances. When the W/B ratio is 0.34, the median distance between particles (R50) exhibits the most significant change. Additionally, the intricate details of cement particle spacing information can be accurately described by fractal theory. Parameters such as R50 and the proportion of particles with spacing between centroids <20 μm demonstrate strong fits with various fractal parameters. Notably, they exhibit optimal fitting with the spectral width of the generalized fractal spectrum, with an R2 value of 0.99.

Non-Hermitian Delocalization in a Two-Dimensional Photonic Quasicrystal.

Theoretical and experimental studies suggest that both Hermitian and non-Hermitian quasicrystals show localization due to the fractal spectrum and to the transition to diffusive bands via exceptional points, respectively. Here, we present an experimental study of a dodecagonal photonic quasicrystal based on electromagnetically induced transparency in a Rb vapor cell. First, we observe the suppression of the wave packet expansion in the Hermitian case. We then discover a new regime, where increasing the non-Hermiticity leads to delocalization, demonstrating that the behavior in non-Hermitian quasicrystals is richer than previously thought.

Visual observation of photonic Floquet–Bloch oscillations

Bloch oscillations (BOs), an important transport phenomenon, have been studied extensively in static systems but remain mysterious in Floquet systems. Here, by harnessing notions from photonic analogy, we propose a generalization of the existing BOs in photonic Floquet lattices, namely the “photonic Floquet–Bloch oscillations”, which refer to rescaled photonic Bloch oscillations with a period of extended least common multiple of the modulation period and the Bloch oscillation period. Next, we report the first visual observation of such photonic Floquet–Bloch oscillations (FBOs) by employing waveguide fluorescence microscopy. Most significantly, the FBOs surpass the existing BOs in Floquet systems and exhibit exotic properties on their own, including fractal spectrum and fractional Floquet tunneling. This new transport mechanism offers an intriguing method of wave manipulation that may contribute to rapidly developing fields in photonics, condensed matter physics, and quantum physics.

Nonlinear Landau Fan Diagram for Graphene Electrons Exposed to a Moiré Potential.

Due to Landau quantization, the conductance of two-dimensional electrons exposed to a perpendicular magnetic field exhibits oscillations that generate a fan of linear trajectories when plotted in the parameter space spanned by density and field. This fan looks identical, irrespective of the dispersion and field dependence of the Landau level energy. This is no surprise because the position of conductance minima depends solely on the level degeneracy that is linear in flux. The fractal energy spectrum that emerges within each Landau band when electrons are also exposed to a two-dimensional superlattice potential produces numerous additional oscillations, but they also create just linear fans for identical reasons. Here, we report conductance oscillations of graphene electrons exposed to a moiré potential that defy this general rule and form nonlinear trajectories in the density-field plane. We attribute this anomalous behavior to the simultaneous occupation of multiple minibands and magnetic breakdown-induced open orbits.

Injection spectroscopy of momentum state lattices

The energy spectrum of quantum systems contain a wealth of information about their underlying properties. Spectroscopic techniques, especially those with access to spatially resolved measurements, can be challenging to implement in real-space systems of cold atoms in optical lattices. Here we explore a technique for probing energy spectra in synthetic lattices that is analogous to scanning tunneling microscopy. Using one-dimensional synthetic lattices of coupled atomic momentum states, we explore this spectroscopic technique and observe qualitative agreement between the measured and simulated energy spectra for small two- and three-site lattices as well as a uniform many-site lattice. Finally, through simulations, we show that this technique should allow for the exploration of the topological bands and the fractal energy spectrum of the Hofstadter model as realized in synthetic lattices.

Onset of Lagrangian chaos: From fractal power spectrum to the absolutely continuous one

Onset of Lagrangian chaos: From fractal power spectrum to the absolutely continuous one

Linear or non-linear stability monitor in BWRs? Introducing a new non-linear monitor based on the fractal spectrum

The stability phenomenon in nuclear boiling water reactors (BWRs) continues to be of scientific and technological interest contributing to the development of advanced stability monitors. The Mexican scientific community over the course of more than two decades (in the world more than four decades) has developed innovative techniques for stability analysis in BWRs, both online and offline; inspired in the first instance by the Laguna Verde NPP instability event occurred on January 24, 1995. Under this perspective, an extensive discussion of linear and non-linear techniques for stability analysis of BWRs is included in this work as well a novel method of stability analysis in BWR based on a Higuchi Fractal Spectrum for the design of a non-linear monitor. This methodology, based on a generalization of the Higuchi Fractal Dimension (HFD), permits to determine in real-time, the stability of a BWR. Hence, real challenging cases in which the instability of the reactor is not evident, were used to test this proposed stability indicator. In general, this proposal gives a better forecast than that based on the simple HFD.

Hofstadter butterfly in optical multilayers

This research focuses on the theoretical examination of the propagation of electromagnetic waves in a multilayer structure in which the refractive index in each medium is modulated by a similar expression of Harper’s potential for the electronic 1D model, or, equivalently, to on-site energies in the Aubry–André model. We applied the transfer matrix approach to different multilayers, with a refractive index modulated by a similar expression of Harper’s potential to obtain the optical transmittance spectra for normal incidence. We calculate the transmittance spectrum as a function of frequency and the Harper model phase for two scenarios in order to search for probable edge states that might arise in this system: first, when the quarter wavelength ratio relates to layer thicknesses through njdj=λ0/4, with nj being the refractive index of the jth layer of the 1D photonic crystal and λ0 is the central wavelength; second, we consider the effects when the thicknesses of the layers are related by d2j=2d2j+1. Besides, to obtain the photonic equivalent for Hofstadter butterfly fractal spectra, we calculate the transmittance spectrum as a function of the periodicity control parameter of the Harper model at the critical point.

Probing miniband structure and Hofstadter butterfly in gated graphene superlattices via magnetotransport

The presence of periodic modulation in graphene leads to a reconstruction of the band structure and formation of minibands. In an external uniform magnetic field, a fractal energy spectrum called Hofstadter butterfly is formed. Particularly interesting in this regard are superlattices with tunable modulation strength, such as electrostatically induced ones in graphene. We perform quantum transport modeling in gate-induced square two-dimensional superlattice in graphene and investigate the relation to the details of the band structure. At low magnetic field the dynamics of carriers reflects the semi-classical orbits which depend on the mini band structure. We theoretically model transverse magnetic focusing, a ballistic transport technique by means of which we investigate the minibands, their extent and carrier type. We find a good agreement between the focusing spectra and the mini band structures obtained from the continuum model, proving usefulness of this technique. At high magnetic field the calculated four-probe resistance fit the Hofstadter butterfly spectrum obtained for our superlattice. Our quantum transport modeling provides an insight into the mini band structures, and can be applied to other superlattice geometries.

Spectral properties of two coupled Fibonacci chains

The Fibonacci chain, i.e. a tight-binding model where couplings and/or on-site potentials can take only two different values distributed according to the Fibonacci word, is a classical example of a one-dimensional quasicrystal. With its many intriguing properties, such as a fractal eigenvalue spectrum, the Fibonacci chain offers a rich platform to investigate many of the effects that occur in three-dimensional quasicrystals. In this work, we study the eigenvalues and eigenstates of two identical Fibonacci chains coupled to each other in different ways. We find that this setup allows for a rich variety of effects. Depending on the coupling scheme used, the resulting system (i) possesses an eigenvalue spectrum featuring a richer hierarchical structure compared to the spectrum of a single Fibonacci chain, (ii) shows a coexistence of Bloch and critical eigenstates, or (iii) possesses a large number of degenerate eigenstates, each of which is perfectly localized on only four sites of the system. If additionally, the system is infinitely extended, the macroscopic number of perfectly localized eigenstates induces a perfectly flat quasi band. Especially the second case is interesting from an application perspective, since eigenstates that are of Bloch or of critical character feature largely different transport properties. At the same time, the proposed setup allows for an experimental realization, e.g. with evanescently coupled waveguides, electric circuits, or by patterning an anti-lattice with adatoms on a metallic substrate.

Predicting irreducible water saturation of unconventional reservoirs by using NMR T2 spectra: Methods of morphological division and fractal models

AbstractIrreducible water saturation restricts an efficient petroleum drainage in unconventional reservoirs. T2 cutoff value identification by using NMR T2 spectrum is the key to clarifying irreducible water saturation. In this paper, saturation and centrifugal T2 spectra of sandstone and coal samples were carried out to calculate T2 cutoff value, with methods of single fractal dimension, multi‐fractal dimension, and spectrum morphological discrimination; In addition, the applicability of these three methods in characterizing T2 cutoff was discussed. The parameters related to T2 cutoff can be divided into two types: (1) The first type is morphological parameters including main peak position ™ and smaller pore volume percentage (SPVP); with an increase of T2 cutoff, TM increases linearly and SPVP decreases exponentially, and the correlation between SPVP and T2 cutoff is stronger than that of TM; (2) Second type is fractal parameters of D2 (larger pore), D−10‐ D10 and D−10/D10; with the increase of T2 cutoff, all single and multi‐fractal dimensions increase linearly, and correlation between D2 and T2 cutoff is stronger than that of multi‐fractal dimension. When calculating the T2cutoff of samples with developed macro‐pores, spectrum morphological methods should be preferentially used, while the fractal dimension discrimination methods need to be used for T2cutoff of samples with developed micro‐pores.

Multifractal analysis of the heterogeneity of nanopores in tight reservoirs based on boosting machine learning algorithms

Exploring the geological factors that affect fluid flow has always been a hot topic. For tight reservoirs, the pore structure and characteristics of different lithofacies reveal the storage status of fluids in different reservoir environments. The size, connectivity, and distribution of fillers in different sedimentary environments have always posed a challenge in studying the microscopic heterogeneity. In this paper, six logging curves (gamma-ray, density, acoustic, compensated neutron, shallow resistivity, and deep resistivity) in two marker wells, namely, J1 and J2, of the Permian Lucaogou Formation in the Jimsar Basin are tested by using four reinforcement learning algorithms: LogitBoost, GBM, XGBoost, and KNN. The total percent correct of training well J2 is 96%, 96%, 96%, and 96%, and the total percent correct of validation well J1 is 75%, 68%, 72%, and 75%, respectively. Based on the lithofacies classification obtained by using reinforcement learning algorithm, micropores, mesopores, and macropores are comprehensively described by high-pressure mercury injection and low-pressure nitrogen gas adsorption tests. The multifractal theory servers for the quantitative characterization of the pore distribution heterogeneity regarding different lithofacies samples, and as observed, the higher probability measure area of the generalized fractal spectrum affects the heterogeneity of the local interval of mesopores and macropores of the estuary dam. In the micropore and mesopore, the heterogeneity of the evaporation lake showed a large variation due to the influence of the higher probability measure area, and in the mesopore and macropore, the heterogeneity of the evaporation lake was controlled by the lower probability measure area. According to the correlation analysis, the single-fractal dimension is well related to the multifractal parameters, and the individual fitting degree reaches up to 99%, which can serve for characterizing the pore size distribution uniformity. The combination of boosting machine learning and multifractal can help to better characterize the micro-heterogeneity under different sedimentary environments and different pore size distribution ranges, which is helpful in the exploration and development of oil fields.

Multifractal spin-glass chaos projection and interrelation of multicultural music and brain signals

A complexity classification scheme is developed from the fractal spectra of spin-glass chaos and demonstrated with multigeographic multicultural music and brain electroencephalogram signals. Systematic patterns are found to emerge. Chaos under scale change is the essence of spin-glass ordering and can be obtained, continuously tailor-made, from the exact renormalization-group solution of Ising models on frustrated hierarchical lattices. The music pieces are from genres of Turkish music, namely Arabesque, Rap, Pop, Classical, and genres of Western music, namely Blues, Jazz, Pop, Classical. A surprising group defection occurs.

Perpendicular electric field drives Chern transitions and layer polarization changes in Hofstadter bands

Moiré superlattices engineer band properties and enable observation of fractal energy spectra of Hofstadter butterfly. Recently, correlated-electron physics hosted by flat bands in small-angle moiré systems has been at the foreground. However, the implications of moiré band topology within the single-particle framework are little explored experimentally. An outstanding problem is understanding the effect of band topology on Hofstadter physics, which does not require electron correlations. Our work experimentally studies Chern state switching in the Hofstadter regime using twisted double bilayer graphene (TDBG), which offers electric field tunable topological bands, unlike twisted bilayer graphene. Here we show that the nontrivial topology reflects in the Hofstadter spectra, in particular, by displaying a cascade of Hofstadter gaps that switch their Chern numbers sequentially while varying the perpendicular electric field. Our experiments together with theoretical calculations suggest a crucial role of charge polarization changing concomitantly with topological transitions in this system. Layer polarization is likely to play an important role in the topological states in few-layer twisted systems. Moreover, our work establishes TDBG as a novel Hofstadter platform with nontrivial magnetoelectric coupling.

Higher-dimensional Hofstadter butterfly on the Penrose lattice

It is now possible to use quasicrystals to search for novel topological phenomena enhanced by their peculiar structure characterized by an irrational number and high-dimensional primitive vectors. Here, we extend the concept of a topological insulator with an emerging staggered local magnetic flux (i.e., without external fields), similar to Haldane's honeycomb model, to the Penrose lattice as a quasicrystal. The Penrose lattice consists of two different tiles, where the ratio of the numbers of tiles corresponds to an irrational number. Contrary to periodic lattices, the periodicity of the energy spectrum with respect to the magnetic flux no longer exists, reflecting the irrational number in the Penrose lattice. Calculating the Bott index as a topological invariant, we find topological phases appearing in a fractal energy spectrum similar to the Hofstadter butterfly. More intriguingly, by folding the one-dimensional aperiodic magnetic flux into a two-dimensional periodic flux space, the fractal structure of the energy spectrum is extended to a higher dimension, whose section corresponds to the Hofstadter butterfly.

Texture direction analysis of micro-topographies using fractal geometry

In recent years fractal geometry has been repeatedly shown to help describe and characterize micro-topographies. Important properties of micro-topographies include roughness or texture direction. Consequently, in this work, fractal geometry is investigated to determine texture direction. For this purpose, synthetic data and real height maps of different micro-topographies are evaluated using the fractal power spectrum density method. The results are compared with a manual determination of the texture direction and determining the texture direction according to ISO 25 178 using the S td parameter. The results show that the fractal method is more accurate than the currently standardized method. Another advantage is that secondary texture directions can be detected. Thus, the fractal method is well suited for characterizing micro-topographies and can complement existing parameters from ISO 25 178.