Higher-order Field Research Articles

Realizing the Calculation of a Fully Normalized Associated Legendre Function Based on an FPGA.

A large number of fully normalized associated Legendre function (fnALF) calculations are required to compute Earth's gravity field elements using ultra high-order gravity field coefficient models. In the surveying and mapping industry, researchers typically rely on CPU-based systems for these calculations, which leads to limitations in execution speed and power efficiency. Although modern CPUs improve instruction execution efficiency through instruction-level parallelism, the constraints of a shared memory architecture impose further limitations on the execution speed and power efficiency. This results in exponential increases in computation time as demand rises alongside high power consumption. In this article, we present a new computational implementation of an fnALF based on the ZYNQ platform. We design a task-parallel "pipeline" architecture which converts the original serial logic into a more efficient hardware implementation, and we utilize a redundant calculation layer to handle repetitive coefficient computations separately. The experimental results demonstrate that our system achieved accurate and rapid calculations. Under the only one geocentric residual latitude condition, we measured the computation times for spherical harmonic coefficient degrees of 360, 720, and 1080 to be 0.155922 s, 0.520950 s, and 1.401609 s, respectively. In the case of the multiple geocentric residual latitudes condition, our design generally yielded efficiency gains of over three times those of MATLAB R2020b implementation. Additionally, our calculated results were used to determine the geoid height in the field with an error of less than ±0.1m, confirming the reliability of our computations.

Unveiling non-monochromatic modes and nonlinearity in Piet Hein quantum semiconductor waveguides

We investigate the propagation characteristics of transverse electric (TE) and transverse magnetic (TM) modes in a semiconductor quantum plasma-filled coaxial waveguide with a Piet Hein cross-section. The unique geometry of the Piet Hein cross-section offers intriguing possibilities for tailoring modal properties and exploiting novel nonlinear phenomena. Using analytical and numerical methods, we unveil the non-monochromatic behaviour of TE and TM modes, characterized by distinct peaks and troughs in their field components. The Ez\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$E_{z}$$\\end{document} component of the TM mode exhibits a suppressed field within the central core and higher concentration in the cladding region, potentially minimizing energy loss within the waveguide. The Eρ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$E_{\\rho }$$\\end{document} component exhibits pronounced oscillations near the waveguide boundaries, suggesting constructive and destructive interference patterns. The Eξ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$E_{\\xi }$$\\end{document} component displays a non-monotonic behaviour with multiple pits and bumps, highlighting the interplay between the TE mode, the Piet Hein geometry, and the semiconductor quantum plasma properties. The Bρ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_{\\rho }$$\\end{document} component of the TM mode exhibits a non-monochromatic behaviour with multiple maxima and minima, attributed to the complex interactions between propagating waves with different phase shifts. The Bξ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_{\\xi }$$\\end{document} component exhibits a non-uniform distribution with distinct pits and bumps, with a resurgence towards the waveguide edges potentially due to higher-order mode contributions and local field enhancement effects. Our findings pave the way for the development of novel photonic devices with enhanced functionalities based on Piet Hein semiconductor quantum plasma waveguides.

Mixed variational formulation and finite-element implementation of second-order poro-elasticity

We present a variational formulation of second-order poro-elasticity that can be readily implemented into finite-element codes by using standard Lagrangian interpolation functions. Point of departure is a two-field minimization principle in terms of the displacement and the fluid flux as independent variables. That principle is taken as a basis for the derivation of continuous and incremental saddle-point formulations in terms of an extended set of independent variables. By static condensation this formulation is then reduced to a minimization principle in terms of the displacement and fluid flux as well as associated higher-order fields. Once implemented into a finite-element code, the resulting formulation can be applied to the numerical simulation of porous media in consideration of second-order effects. Here, we analyze the model response by means of several example problems including two standard tests in poro-elasticity, namely the consolidation problems of Terzaghi and Mandel, and compare the results with those of a corresponding first-order model. As becomes clear, the second-order formulation can unleash its full potential when applied to the study of porous media having spatial dimensions comparable to the size of their microstructure. In particular, it is capable to regularize steep field gradients at external as well as internal surfaces and to describe material dilatation effects known from experiments.

Scattering of kinks in scalar-field models with higher-order self-interactions

Higher-order scalar field models in two dimensions, including the ϕ8 model, have been researched. It has been shown that for some special cases of the minima positions of the potential, the explicit kink solutions can be found. However, in physical applications, it is very important to know all the explicit solutions of a model for any minima position. In the present study, with the help of some deformation functions, we have shown that higher-order scalar field theories can be obtained with explicit kinks. In particular, we introduced two deformation functions that, when applied to the well known ϕ4 and ϕ6 models, produce modified ϕ8 and ϕ10 models, respectively, with all their explicit kink-like solutions which depend on a single parameter. Since this parameter controls the position of the minima of the potential, we have found interesting new solutions in many distinct cases. We have also studied the kink mass, the behavior of the excitation spectra and several kink-antikink collisions for these two new modified models. The collision outcome is determined by the initial configuration, specifically the sequence in which the kink-antikink and antikink-kink pairings emerge. Another interesting finding is the suppression of resonance windows, which may be explained by the presence of a set of internal modes in the model.

Framework of Physically Consistent Homogenization and Multiscale Modeling of Shell Structures**

A framework of physically consistent homogenization and multiscale modeling (PCHMM) for reduced-order analysis of plate/shell structures is developed in this paper. To address the inapplicability of conventional periodic boundary conditions and Hill’s condition involved in homogenization of shear-deformable shell structures, the paper proposes physically consistent boundary conditions and modified Hill’s condition for plate/shells. Unlike the PCHMM method for beams, considering the contradiction between high-order displacement fields induced by shear forces and low-order kinematic assumptions, additional constraints are applied to the plate/shell structure sectional strains during the solution of perturbation fields. The correctness and effectiveness of the proposed plate/shell PCHMM framework and method are verified by typical numerical examples. The proposed theory can also be conveniently embedded into commercial finite element software for homogenization and multiscale analysis of structures such as microscale metamaterials like lattice plates and large complex structures like aircraft fuselage sections.

Enhancing Temporal Knowledge Graph Representation with Curriculum Learning

Temporal knowledge graph representation approaches encounter significant challenges in handling the complex dynamic relations among entities, relations, and time. These challenges include the high difficulty of training and poor generalization performance, particularly with large datasets. To address these issues, this paper introduces curriculum learning strategies from machine learning, aiming to improve learning efficiency through effective curriculum planning. The proposed framework constructs a high-dimensional filtering model based on graph-based high-order receptive fields and employs a scoring model that uses a curriculum temperature strategy to evaluate the difficulty of temporal knowledge graph data quadruples at each stage. By progressively expanding the receptive field and dynamically adjusting the difficulty of learning samples, the model can better understand and capture multi-level information within the graph structure, thereby improving its generalization capabilities. Additionally, a temperature factor is introduced during model training to optimize parameter gradients, alongside a gradually increasing training strategy to reduce training difficulty. Experiments on the benchmark datasets ICEWS14 and ICEWS05-15 demonstrate that this framework not only significantly enhances model performance on these datasets but also substantially reduces training convergence time.

Growth of organized flow coherent motions within a single-stream shear layer: 4D-PTV measurements

This study investigates the evolution of a single-stream shear layer (SSSL) originating from a wall boundary layer past a backward-facing step. Utilizing a time-resolved 3D-Particle Tracking Velocimetry (4D-PTV) technique, we track the trajectories of fluorescent particles to gain insight into the flow characteristics of the SSSL. A compact water tunnel facility (Reτ=1240\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{Re}_\ au =1\\,240$$\\end{document}) is fabricated to obtain an SSSL with a perpendicular slow entrainment stream past the separation edge. A hybrid interpolation approach that combines ensemble binning and Gaussian weighting is implemented to derive minimally filtered mean and instantaneous lower- and higher-order flow field parameters. Spanwise-dominant coherent motion accompanied by finer flow scales is observed to grow due to flow entrainment through “nibbling” actions of small-scale vortices, “engulfing” by large-scale vortices, and vortex pairing events. Furthermore, the non-zero-speed stream edge grows relatively faster than the zero-speed stream edge, showing a strong asymmetry in mixing composition across a mixing layer. The SSSL reaches self-similarity at a streamwise distance of ≈55θ0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\approx 55\\,\ heta _{0}$$\\end{document}, where θ0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ heta _0$$\\end{document} is the initial momentum thickness from the separation edge, i.e., considerably shorter than reported in previous studies. A literature comparison of growth rate parameters raises intriguing questions regarding a potential inclusive growth scaling unifying the free shear layers. A turbulent kinetic energy (TKE) budget analysis reveals a negative production region immediately downstream of the separation edge attributed to a large positive streamwise gradient of streamwise velocity. In the self-similar region, the phase-averaged flow mapping demonstrates a larger concentration of turbulence production rate around the outer edges of spanwise vortices, specifically at the intersection of braids and vortices. Furthermore, a spatial separation exists in the regions of peak production and dissipation rates within the vortex core region favoring dissipation. The braids exhibit a larger concentration of turbulence diffusion rates, indicating their function as a conduit for exchanging turbulence between neighboring coherent motions.

Development of Digital Signal Processing Techniques with BPM

In order to ensure the continuity of operation of the VEPP-2000 collider, accurate measurement of the betatron frequency is necessary. To do this, this work proposes to use methods that refine the Fourier transform, such as parabola interpolation (Gassior method), NAFF and window functions. The refined frequency is subsequently used in construction of phase portraits of the beam to control the interference of high-order magnetic fields. In addition, the work discusses methods for extracting a signal from a mixture for subsequent analysis – PCA and ICA. Finally, to improve the accuracy of frequency determination, the paper considers the simplest implementation of a Kalman filter to improve the accuracy of subsequent harmonic analysis. In addition to all of the above, the paper briefly discusses a method for monitoring the operation of the beam position monitors themselves.

Exact black holes in string-inspired Euler-Heisenberg theory

We consider higher-order derivative gauge field corrections that arise in the fundamental context of dimensional reduction of string theory and Lovelock-inspired gravities and obtain an exact and asymptotically flat black hole solution, in the presence of nontrivial dilaton configurations. Specifically, by considering the gravitational theory of Euler-Heisenberg nonlinear electrodynamics coupled to a dilaton field with specific coupling functions, we perform an extensive analysis of the characteristics of the black hole, including its geodesics for massive particles, the energy conditions, thermodynamical and stability analysis. The inclusion of a dilaton scalar potential in the action can also give rise to asymptotically (anti) de Sitter spacetimes and an effective cosmological constant. Moreover, we find that the black hole can be thermodynamically favored when compared to the Gibbons-Maeda-Garfinkle-Horowitz-Strominger black hole for those parameters of the model that lead to a larger black hole horizon for the same mass. Finally, it is observed that the energy conditions of the obtained black hole are indeed satisfied, further validating the robustness of the solution within the theoretical framework, but also implying that this self-gravitating dilaton-nonlinear-electrodynamics system constitutes another explicit example of bypassing modern versions of the no-hair theorem without any violation of the energy conditions. Published by the American Physical Society 2024

A unified theoretical framework of piezoelectric energy harvesters: Euler–Bernoulli, Timoshenko and Reddy beam models with the high-order electric field assumption

A unified theoretical framework of piezoelectric energy harvesters: Euler–Bernoulli, Timoshenko and Reddy beam models with the high-order electric field assumption

Assessment of new four variables 2D and quasi-3D higher order shear theories for bending analysis of different double curved FG shells

ABSRACT This study introduces a novel theory employing four unknowns to examine bending of a double-curvature functionally graded shells experiencing both uniform and sinusoidal pressure. This investigation utilizes 2D and quasi-3D higher order shear deformation theories, integrating a fresh model of a high-order displacement field. It focuses on exploring how geometric ratios, volume fraction and power law index influence the bending behavior, specifically examining their effects on the normal and shear stresses variation, along with their amplification-reduction impacts. The effectiveness of the developed model is evaluated by comparing its performance in simulating bending characteristics of plates, cylinders, spheres and ellipses with those presented in literature.

Advanced modeling of segmented Halbach cylinder for ironless brushless permanent-magnet two-phase motor

Compared with conventional permanent magnet (PM) cylinders, Halbach magnet cylinders have the advantage of strengthening the magnetic field on one side and canceling the field on the other side of the magnet structure. Outer-rotor radial-flux brushless ironless PM machines using a Halbach cylinder, therefore, can have a higher torque density compared with the ironless PM machines using a conventional magnet cylinder with the same magnet and coil volume and same air gap. For outer-rotor Halbach cylinder motors, existing designs usually take advantage of the fundamental harmonics of the magnetic field to obtain a linear toque model and to avoid the torque ripples from the higher-order field harmonics. To achieve this, the Lorentz coils must be placed where the fundamental harmonics of the field is dominant and the higher order harmonics can be negligible. With this, the coils should be sufficiently far from the magnet cylinder’s surface. However, how far the coils should be (from the magnet) is not clarified from the reported designs. To provide guidance on where to optimally place the Lorentz coils and to make the best use of the Halbach cylinders, there is a critical need for a simple but comprehensive analysis to find out all the field harmonics everywhere for both the ideal and segmented Halbach cylinders. This work is to provide that analysis.

The Herglotz variational principle for dissipative field theories

In the recent years, with the incorporation of contact geometry, there has been a renewed interest in the study of dissipative or non-conservative systems in physics and other areas of applied mathematics. The equations arising when studying contact Hamiltonian systems can also be obtained via the Herglotz variational principle. The contact Lagrangian and Hamiltonian formalisms for mechanical systems has also been generalized to field theories. The main goal of this paper is to develop a generalization of the Herglotz variational principle for first-order and higher-order field theories. In order to illustrate this, we study three examples: the damped vibrating string, the Korteweg–de Vries equation, and an academic example showing that the non-holonomic and the vakonomic variational principles are not fully equivalent.

Polynomial equations for additive functions II. The mixed parameter case

In this sequence of work we investigate polynomial equations of additive functions. This is the continuation of the paper [5] entitled Polynomial equations for additive functions I. We consider here the solutions of the equation ∑i=1nfi(xpi)gi(x)qi=0x∈F,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\sum _{i=1}^{n}f_{i}(x^{p_{i}})g_{i}(x)^{q_{i}}= 0 \\qquad \\left( x\\in \\mathbb {F}\\right) , \\end{aligned}$$\\end{document}where n is a positive integer, F⊂C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {F}\\subset \\mathbb {C}$$\\end{document} is a field, fi,gi:F→C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f_{i}, g_{i}:\\mathbb {F}\\rightarrow \\mathbb {C}$$\\end{document} are additive functions and pi,qi\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p_i, q_i$$\\end{document} are positive integers for all i=1,…,n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$i=1, \\ldots , n$$\\end{document}. Using the theory of decomposable functions we describe the solutions as compositions of higher-order derivations and field homomorphisms. In many cases, we also give a tight upper bound for the order of the involved derivations. Moreover, we present the full description of the solutions in some important special cases, too.

Nonlinear optimization DIC method inspired by unsupervised learning for high order displacement measurement

Digital Image Correlation (DIC) is the most widely used non-contact full-field strain measurement method, but due to the influence of subset calculations in principle, measuring complex deformation with high strain gradients within the subset remains a challenge. DIC based on neural networks performs well in complex deformation measurement scenarios, but its measurement accuracy and generalization ability depend on extensive training sets. This paper proposes a nonlinear optimization DIC method inspired by unsupervised learning that requires only a single pair of images. Unlike traditional unsupervised algorithms of optical flow estimation or Particle Image Velocimetry (PIV), the shape function in traditional DIC is incorporated into the loss function. This allows the use of only the reference image and the deformed image as input dataset, and the correct displacement field can be output after parameter optimization through training. This method does not require additional training sets to measure high-order strain fields like traditional neural networks-based DIC. The proposed method has demonstrated promising results in simulation experiments for high-order strain measurement and does not require additional datasets of the same type. In the measurement of the star-shaped displacement fields, its accuracy is better than the traditional DIC based on the subset, and it maintains convergence even when the strain is large, a situation where traditional DIC fails to converge. In addition, compared with the network obtained by supervised training as a pre-trained model, the method proposed in this paper can achieve superior results. The methods of the article can theoretically be applied to the result optimization of other networks beyond those discussed in this paper.

Cooperative properties of multiple quantum scattering: I quantum nutation

The cooperative models of the bimodal field in the multiple quantum scattering nutations are discussed and proposed for possible detections in open cavities. We proposed two types of cooperation between the converted photon processes in these multiple steps of scattering nutation in the cavity. One of them takes into consideration the cooperative process between the photons of each step of the multiple steps of Raman conversion. The second cooperative process takes place between the photons belonging to different steps of multiple scattering conversions. The proposed novel bimodal entangled sources take into consideration both the coherence and collective phenomena between the photons belonging to the system of the bimodal field obtained in multiple scattering emissions. The application of higher-order multiple Raman bimodal coherent field in quantum information is proposed.

Dynamic stiffness method and CUF-based component-wise theories applied to free vibration analysis of solid beams, thin-walled structures and reinforced panels

In this paper, an exact dynamic stiffness formulation using higher-order theories with displacement variables only is presented and subsequently used to investigate the free vibration characteristics of solid beams, thin-walled structures and reinforced panel structures. In essence, higher-order displacement fields are developed by using the Carrera unified formulation (CUF), and by discretizing the cross-section kinematics with bilinear, cubic and fourth-order Lagrange polynomials. In particular, the component-wise (CW) approach based on Lagrange expansion is applied in which the solid part and thin-walled part are considered as two independent components that can be assembled. The principle of virtual displacements is used to derive the governing differential equations and the associated natural boundary conditions. An exact dynamic stiffness matrix is then developed by relating the amplitudes of harmonically varying loads to those of the responses. The explicit terms of the dynamic stiffness matrices are also presented. The Wittrick–Williams algorithm of the dynamic stiffness method (DSM) is applied with the explicit expressions of the J0 count for beam elements under all above support conditions. In return, there is no need to refine the element in the DSM, and thus, it becomes immensely efficient. The accuracy and efficiency of the proposed methodology are extensively assessed for different solid beams, thin-walled structures and reinforced panels and the results are compared with those appearing in published literature and also checked by 3D finite element (FE) solutions.

Higher-order asymptotic crack-tip fields in simplified strain gradient elasticity

Explicit representation for the higher-order in-plane crack-tip fields is derived using Papkovich–Neuber stress functions within the framework of simplified strain gradient elasticity (SGE). In the considered theory, strain energy density depends on both the infinitesimal strain tensor and its spatial gradient. The constitutive equations of simplified SGE contain an additional length scale parameter that can be correlated with the characteristic size of the material microstructure. Presented asymptotic solution has separable form and reduces to classical Williams’ series when the gradient effects are negligible. The leading terms in the derived solution coincide with the previously known asymptotic solutions for crack problems in SGE. The higher-order terms have coupled amplitude factors and modified definitions for the angular distribution as compared to the classical solution. Derived asymptotic fields are compared to the full-field numerical solution for the Mode I crack problem to quantify the amplitude factors and the zones of dominance for up to eight terms. The presented solution has practical applications in the development of advanced numerical methods, particularly concerning enriched finite elements for crack problems in SGE.

Topological high-harmonic spectroscopy

Linearly polarized vector beams are structured lasers whose topology is characterized by a well-defined Poincaré index, which is a topological invariant during high-order harmonic generation. As such, harmonics are produced as extreme-ultraviolet vector beams that inherit the topology of the driver. This holds for isotropic targets such as noble gases, but analogous behaviour in crystalline solids is still open to discussion. Here, we demonstrate that this conservation rule breaks in crystalline solids, in virtue of their anisotropic non-linear susceptibility. We identify the topological properties of the harmonic field as unique probes, sensitive to both the microscopic and macroscopic features of the target’s complex non-linear response. Our simulations, performed in single-layer graphene, show that the harmonic field is split into a multi-beam structure whose topology encodes information about laser-driven electronic dynamics. Our work promotes the topological analysis of the high-order harmonic field as a spectroscopic tool to reveal the nonlinearities in the coupling of light and target symmetries.

A novel beam model suitable at the microscale

A novel beam model is proposed to address the limitations of existing microscale beam models by introducing a higher-order displacement field assumption for the modified couple stress theory. The governing equations and boundary conditions are derived using the variational principle. Validation through finite element analysis for a simply supported beam problem demonstrates the beam model successfully captures the width-dependent behavior of microbeams, which existing beam models cannot explain. Additionally, the beam model accurately represents the plane strain behavior as the beam width increases and the classical beam behavior at a small length scale parameter.