Simple Theory Research Articles

A simple theory for training response of deep neural networks

Abstract Deep neural networks give us a powerful method to model the training dataset’s relationship between input and output. We can regard that as a complex adaptive system consisting of many artificial neurons that work as an adaptive memory as a whole. The network’s behavior is training dynamics with a feedback loop from the evaluation of the loss function. We already know the training response can be constant or shows power law-like aging in some ideal situations. However, we still have gaps between those findings and other complex phenomena, like network fragility. To fill the gap, we introduce a very simple network and analyze it. We show the training response consists of some different factors based on training stages, activation functions, or training methods. In addition, we show feature space reduction as an effect of stochastic training dynamics, which can result in network fragility. Finally, we discuss some complex phenomena of deep networks.

Arctic amplification‐induced intensification of planetary wave modulational instability: A simplified theory of enhanced large‐scale waviness

AbstractIn the mid–high latitude atmosphere, the instability of planetary waves characterizes enhanced planetary wave activity or amplified large‐scale waviness leading to increased regional weather extremes. In this paper, a nonlinear Schrödinger equation is derived to describe the evolution of planetary waves. Then the consequences of Arctic amplification (AA)‐induced meridional background potential vorticity (PVy) changes on the modulational instability of planetary waves are examined. It is found that the modulational instability of uniform planetary wave trains mainly results from the presence of high‐order dispersion and nonlinearity, even though such an instability depends on the amplitude, vertical structure and zonal wavenumber of uniform planetary waves and the atmospheric stratification. Because the nonlinearity and high‐order dispersion depend on the magnitude of PVy, the modulational instability of planetary waves is significantly influenced by the variation of PVy associated with AA. It is also revealed that stronger modulational instability of planetary waves tends to occur in the smaller PVy region or in higher latitudes due to both stronger nonlinearity and weaker high‐order dispersion for fixed background and planetary wave parameters, which is conducive to more intense large‐scale waviness. However, because AA can reduce PVy in the mid–high latitudes mainly in the lower troposphere via reductions of winter zonal winds and meridional temperature gradients, the reduced PVy under AA can significantly enhance the modulational instability. Thus, the role of AA is to amplify planetary wave activity in mid–high latitudes through strengthening the modulational instability of planetary waves due to reduced PVy, which further enhances large‐scale waviness.

Time-dependent reliability analysis for structures under multilevel loads based on fatigue damage theory

This paper proposes an efficient method to evaluate the time-dependent fatigue reliability of structures under complex multilevel loads using a finite set of fatigue test data obtained under single-level loading conditions. The method preserves the simplicity of linear damage accumulation theory while considering the interactions among multiple load levels and the influence of load sequence on reliability. Leveraging the linear damage rule, a cumulative damage probability model was established under single-level constant amplitude loading. This model incorporates a fatigue life degradation function to characterize the cumulative damage effect of load cycles and corrects the fatigue reliability curve under single-level constant amplitude loading. By employing the principle of fatigue damage equivalence, the method determines the damage equivalence at the load switching point using the damage curve method, enabling the conversion of equivalent reliability under multilevel loads. The proposed method was validated by combining the test data obtained under three working conditions, including two-level loads loading, load blocks loading, and changing loads loading. The results demonstrate that the proposed method can accurately and efficiently evaluate the time-dependent fatigue reliability under different types of multilevel loads, using only a limited set of fatigue life data under single constant amplitude loads.

Development and Application of an FSI Model for Floating VAWT by Coupling CFD and FEA

The emerging floating vertical axis wind turbines (VAWTs) are regarded as a preferred solution to overcome the challenges faced by the traditional horizontal type in open-sea environments. Numerous numerical models have been advanced for assessing this novel object. However, current fully coupled models predominantly rely on simplified theories, assuming a linear fluid load and a one-dimensional slender beam structure. Despite computational fluid dynamic and finite element (CFD-FEA) coupling being qualified for high precision, this technology remains limited to the fixed VAWT field. To predict the load and structural response accurately and comprehensively, this study aims to extend CFD-FEA technology to floating VAWTs. First, an aero-hydro-moor-elastic fully coupled model is developed, and this model is validated by comparing it with several model experiments. Subsequently, a full-scale floating straight-bladed VAWT is simulated with the geometry and numerical models introduced. Furthermore, load and structural responses in a typical case are analyzed in both time and frequency domains. Finally, the sensitivity analysis of each structure part in floating VAWTs to environmental parameters is conducted and discussed. The discovery highlights the intricate nature of tower structural response, which incorporates 2-node, 3-node, wind frequency, and wave frequency components. Distinct from blades or floating foundations, which are primarily influenced by a single environmental parameter, the tower response is significantly amplified by the combined effects of wind and waves.

Existence conditions for functional ODE observer design of descriptor systems revisited

This paper is devoted to the problem of designing functional observers for linear time-invariant (LTI) descriptor systems. The observers are realized by using state–space systems governed by ordinary differential equations (ODEs). Available existence results for functional ODE observers in the literature are extended by introducing new and milder sufficient conditions. These conditions are purely algebraic and provided directly in terms of the system coefficient matrices. The proposed observer has an order less than or equal to the dimension of the functional vector to be estimated. The observer parameter matrices are obtained by using simple matrix theory, and the design algorithm is illustrated by numerical examples.

Large and small bubble breakup in gas–liquid centrifugal pumps

The present work studies bubble breakup in gas–liquid centrifugal pumps. Different inlet gas-fraction conditions are considered to provoke bubble-to-bubble, slug-to-bubble, and slug-to-slug pattern transitions. Results are presented for different mixture velocities and pump rotation frequencies. Data on mean equivalent bubble diameter and slug unit properties (length, passage frequency) are introduced. For small bubbles, a simple phenomenological theory is developed for the prediction of mean diameters. A discussion on the dynamics of small bubbles in the impeller region is also presented.

Investigating the different mechanisms in related neural activities: a focus on auditory perception and imagery.

Neuroimaging studies have shown that the neural representation of imagery is closely related to the perception modality; however, the undeniable different experiences between perception and imagery indicate that there are obvious neural mechanism differences between them, which cannot be explained by the simple theory that imagery is a form of weak perception. Considering the importance of functional integration of brain regions in neural activities, we conducted correlation analysis of neural activity in brain regions jointly activated by auditory imagery and perception, and then brain functional connectivity (FC) networks were obtained with a consistent structure. However, the connection values between the areas in the superior temporal gyrus and the right precentral cortex were significantly higher in auditory perception than in the imagery modality. In addition, the modality decoding based on FC patterns showed that the FC network of auditory imagery and perception can be significantly distinguishable. Subsequently, voxel-level FC analysis further verified the distribution regions of voxels with significant connectivity differences between the 2 modalities. This study complemented the correlation and difference between auditory imagery and perception in terms of brain information interaction, and it provided a new perspective for investigating the neural mechanisms of different modal information representations.

Theory construction and computation for polymer film melt stretching by modified Leonov model

Polymer film melt stretching exhibits a unique stress–strain behavior, featuring a stress plateau followed by a sudden increase with increasing the strain. This phenomenon poses a challenge to conventional melt-extension theories and methods. To address this, we derive a simplified flow theory according to the symmetric characteristics of thin melt-stretching, and propose a modified multi-mode Leonov model (m-Leonov) that accounts for molecular orientation effects, a crucial factor in accurately simulating the process. In order to improve the computational efficiency, an iterative algorithm was developed to decouple the interdependence between governing and constitutive equations. The computational results reveal that the viscoelastic nature of the film contributes to a delay in the transformation of stretching forces, leading to the observed stress plateau. Significantly, our m-Leonov model successfully predicts both the plateau and the subsequent abrupt increase in stress, across various temperatures and stretching rates. These simulations closely agree with experimental data, underlining the robustness of our approach. Our findings would enhance the understanding of rheological complexities in film stretching, offering potential applications in various materials and polymer processing.

Predictable Synthesis of 3D Polymer Networks Using Crystal Component-Linking.

Controlled synthesis of 3D polymer networks presents a significant challenge because of the complexity of the polymerization reaction in solution. In this study, a polymerization system that facilitates the prediction of a polymer network structure via percolation simulations is realized. The most significant difference between general percolation simulations and experimental polymerization systems is the mobility of the molecules during the reaction. A crystal component-linking method that connects the precisely arranged monomer as a supramolecular crystalline state to imitate the simple percolation theory is adopted. The percolation simulation based on the crystal structure of the arranged monomers is used to accurately calculate the gelation point, gel fraction, degree of swelling, and atomic formula, which correspond with the experimental results. This suggests that the network structures polymerized via the crystal component-linking method can be predicted precisely by a simple percolation simulation. Further, the percolation simulation predicts the structures of the loop, branched polymer, and crosslinking point, which are difficult to measure experimentally. The polymerization of precisely-arranged immobilized monomers in supramolecular structures is promising in synthesizing precisely controlled polymer networks.

Non-singular straight dislocations in anisotropic crystals

A non-singular dislocation theory of straight dislocations in anisotropic crystals is derived using simplified anisotropic incompatible first strain gradient elasticity theory. Based on the non-singular theory of dislocations for anisotropic crystals, all dislocation key-formulas of straight dislocations are derived in generalized plane strain, for the first time. In this model, the singularity of the dislocation fields at the dislocation core is regularized owing to the nonlocal nature of strain gradient elasticity. The non-singular dislocation fields of straight dislocations are obtained in terms of two-dimensional anisotropic Green functions of simplified anisotropic strain gradient elasticity. All necessary Green functions, including the two-dimensional Green tensor of the twofold anisotropic Helmholtz-Navier operator and the two-dimensional F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{F}$$\\end{document}-tensor of generalized plane strain, are derived as sum of the classical part and a gradient part in terms of Meijer G-functions. Among others, we calculate the regularization of the Barnett solution for the elastic distortion of straight dislocations in an anisotropic crystal. In the framework of simplified anisotropic first strain gradient elasticity, the necessary material parameters are computed for cubic materials including aluminum (Al), copper (Cu), iron (Fe) and tungsten (W) using a second nearest-neighbour modified embedded-atom-method interatomic potential. The elastic distortion and stress fields of screw and edge dislocations of 12⟨111⟩\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\frac{1}{2} \\langle 111\\rangle$$\\end{document} Burgers vector in bcc iron and bcc tungsten and screw and edge dislocations of 12⟨110⟩\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\frac{1}{2} \\langle 110\\rangle$$\\end{document} Burgers vector in fcc copper and fcc aluminum have been computed and presented in contour plots.

Active vibration control of rotating smart bidirectional FGPTC subjected to internal mechanical shock

This study considers the active vibration control of a sandwich truncated cone, whose intermediate layer is made of bidirectional functionally graded (FG) porous material bonded with integrated piezoelectric actuator and sensor layers on its outer and inner surfaces. The material properties of the bidirectional functionally graded porous truncated cone (FGPTC), with even and uneven porosity distributions along the length and thickness directions, are changed using a power law function. The governing motion equations utilize the two-dimensional (2D) axisymmetric theory of elasticity rather than simple shell theories. The solving of the governing motion equations is accomplished through the utilization of graded finite element and Newmark methods. The study investigates the impact of dimensions, parameters, and porosity types on transient response behavior, including the semi-vertex angle and the ratio of truncated cone length to outer radius. Furthermore, the effects of power index, porosity volume fractions, rotational velocities, and boundary conditions are examined. Active vibration control (AVC) of the piezoelectric FGPTC is achieved by employing a closed-loop control algorithm based on velocity feedback. Several numerical simulations are conducted to evaluate the effectiveness of the proposed active vibration control system in reducing structural vibrations. The results demonstrate significant reductions in vibration amplitudes and improved structural stability achieved through the active control approach.

Pre-notch and pre-crack size effects on T-peel fracture behaviors of SAC305 solder joints

T-peel test of ductile adhesive is always an interesting and important topic in peeling mechanics and fracture mechanics. In this paper, the pre-notch and pre-crack effect on the T-peel test with a ductile adhesive layer is studied. Based on the experimental test of SAC 305 adhesive and copper adherent T-peel tests, it is found that the peak loads, fracture toughness, resistance curve, and adhesive toughness are sensitive to pre-crack or pre-notch lengths. Short pre-notch or pre-crack trigger unsteady state debonding frequently, and deep pre-notch or pre-crack conditions achieve steady state debonding more easily. Different data reduction schemes such as simple beam theory (SBT), corrected beam theory (CBT), and compliance-based beam method (CBBM) will lead to variations of fracture toughness extractions for T-peel tests. Trapezoidal shape cohesive zone model (CZM) is found to be easier matching T-peel tests conducted in this paper. However, T-peel samples with shorter pre-notch or pre-crack lengths are hard to simulate with the CZM method due to the unstable crack propagation. The reasons, which may lead to the discrepancy between pre-notch and pre-crack T-peel samples, are also presented and discussed. The selection of different pre-notch or pre-crack lengths is a very important factor that needs to be considered in the T-peel test.

A non-perturbative mixed anomaly and fractional hydrodynamic transport

We present a new non-perturbative ’t Hooft anomaly afflicting a quantum field theory with symmetry group G = U(1) × ℤ2 in four dimensions. We use the Adams spectral sequence to compute that the bordism group \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\Omega }_{5}^{{\ ext{Spin}}}$$\\end{document}(BG), which classifies anomalies that remain when perturbative anomalies cancel, is ℤ4. By constructing a mapping torus and evaluating the Atiyah-Patodi-Singer η-invariant, we show that the mod 4 anomaly is generated by a pair of Weyl fermions that are vector-like under U(1), but with only one component charged under ℤ2. We construct a simple microscopic field theory that realises the anomaly, before investigating its impact in the hydrodynamic limit. We find that the anomaly dictates transport phenomena in the U(1) current and energy-momentum tensor akin to the chiral vortical and magnetic effects (even though the perturbative anomalies here vanish), but with the conductivities being fractionally quantised in units of a quarter, reflecting the mod 4 nature of the bordism group. Along the way, we compute the (relevant) bordism groups \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\Omega }_{d}^{{\ ext{Spin}}}$$\\end{document}(Bℤ2 × BU(1)) and \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\Omega }_{d}^{{{\ ext{Pin}}}^{-}}$$\\end{document} (BU(1)) in all degrees d = 0 through 5.

Model-based linear control of nonlinear pneumatic soft bending actuators

Advanced model-based control techniques hold great promise for the precise control of pneumatic soft bending actuators (PSBAs) with strong nonlinearities. However, most previous controllers were designed in a cumbersome nonlinear form. Considering the simplicity of linear system theory, this paper presents a novel perspective on using model-based linear control to handle nonlinear PSBAs, and for the first time, summarizes two methodologies, global linearization and pseudo-linear construction. Derived from them, Koopman-based and hysteresis-based linear control approaches are proposed, respectively. For the former, a novel fusion prediction equation is developed to build a high-fidelity Koopman model, realizing global linearization, and then the linear model predictive control (MPC) is deployed. For the latter, the inverse of the generalized Prandtl–Ishlinskii (GPI) model cascades with the PSBA to construct a pseudo-linear system, eliminating the asymmetric hysteresis, which activates the linear proportional-integral-derivative (PID) control. It is worth noting that the above two are based on data-driven models adapted to various PSBAs with material and structural customization. Finally, the two model-based linear control approaches are verified and compared through a series of experiments. The results show that the proposed linear controls, with more concise designs, achieve comparable or even superior performance than nonlinear controls.

Crushing performance evaluation of gradient Sierpinski triangular fractal column

Thin-Walled Structures with excellent mechanical properties are the key to the protective system of the vehicle. This paper proposes a novel Sierpinski triangular fractal column with graded thickness (STFC-GT) to improve the crushing stability and energy absorption of Thin-Walled Structures. The crashworthiness of the STFC-GT is investigated through the experimental test and numerical simulation under axial crushing load. The results show that the Sierpinski triangular fractal column contributes to the energy absorption, and the graded thickness design can decrease the initial peak force. Specifically, the energy absorption of STFC-GT improves by 144.22 % from 0th to 2nd. And the initial peak force of 2nd STFC-GT is 55.32 % lower than that of 2nd STFC with uniform thickness. In addition, a theoretical relationship between the parameter configurations and the mean crushing force of STFC-GT are developed using the simplified super folding element theory (SSFE). The maximum error is less than 7.89 % between the theoretical and numerical results. Finally, the influence of gradient types on crushing performance is investigated to determine the optimal gradient configuration and maximize the mechanical performance of the STFC. The research result provides reliable guidance for the design of a new vehicle safety protection device.

Analysis of planar arbitrarily curved microbeams with simplified strain gradient theory and Timoshenko–Ehrenfest beam model

Analysis of planar arbitrarily curved microbeams with simplified strain gradient theory and Timoshenko–Ehrenfest beam model

Power control in LTE based on heuristic game theory for interference management

In the conventional LTE homogeneous network, sufficient transmit power of user equipment (UE) is determined by open-loop power control (OL-PC) and closed-loop power control (CL-PC) schemes. However, in a Het-Net environment, setting the UE’s transmit power requires delicate responsiveness to handle the severe and complicated uplink interference. In this paper, an interference-aware uplink power control mechanism based on Heuristic game theory approach is proposed for devices coexisting in a heterogeneous wireless network. Various wireless constraints like channel response, path loss, fading, shadowing, interference and metrics like SNR, SINR, throughput and bit rates are considered. Uplink power is controlled by suitably selecting the penalization factor (β) based on a simple Heuristic game theory approach considering the possible wireless constraints of each user depending on its location in the cell under consideration. The algorithm is framed in such a way to reduce inter-cell interference, limit transmit power, enhance bit rates and throughput of users. A significant improvement of 5.2% in the user coverage/distribution is achieved as a result of interference management compared to conventional power control scheme and power control with convex pricing.

Interference effect and Raman depolarization behavior in MoO2 bubbles

Here, MoO2 bubbles are firstly found in MoO2 nanosheets (NSs) synthesized by chemical vapor deposition and exposed to air for 2–3 years. Interestingly, the Newton’s rings are clearly observed in Raman map of MoO2 bubbles. Meanwhile, it’s found that the height of bright rings close to the 1/2, 2/2, 3/2 of the laser wavelength, confirming that the Newton’s rings in Raman map are caused by the laser interference on the surface of bubbles. At the same time, based on simple membrane theory, the adhesion energy 0.15 and 1.08 J/m2 between MoO2 NSs and different SiO2/Si substrates are extracted. Moreover, Raman depolarization effect is observed on the surface of MoO2 bubbles. As the aspect ratio of bubbles increases from 0.16 to 0.21, the relative-depolarization ratios of 205 cm−1 and 750 cm−1 phonon modes are gradually increased from 18% to 46% and 10–36%, respectively. The Raman depolarization behavior should be ascribed to the surface strain of MoO2 bubbles leading to the asymmetric changes of photoelastic tensor in all directions of the crystal. This finding maybe provides a reference for regulating the polarization characteristics of MoO2 by adjusting its surface strain and ultimately achieving application in the field of photonics.

Zero shear viscosity of two polyatomic glass-forming liquids calculated by Vogel–Fulcher law and schematic mode-coupling theory

A theoretical method based on the simple Vogel–Fulcher law for estimating the zero shear viscosity of two glass-forming liquids, 1,3-bis(1-nathtyl)-5–(2-naphthyl) benzene and polystyrene, over a range of temperatures from that close to their glass transition temperatures to high temperatures was reported. One parameter of this model, i.e., the critical number density, at which particles are trapped in a cage that only segmental vibration is possible was identified using the intermolecular radial distribution function derived from the Percus–Yevick integral equation and a simplified schematic mode-coupling theory.

Numerical and theoretical analysis of the out-of-plane crushing behavior of a sinusoidal-shaped honeycomb structure with tunable mechanical properties

Honeycomb structures are widely utilized for energy absorption in engineering, showing superior crushing strength and energy absorption in the vertical direction, yet their potential as energy absorbers is limited due to a high initial peak force (IPF) in that direction. In this study, a novel honeycomb structure with tunable out-of-plane mechanical properties is developed by incorporating sinusoidal-shaped features into traditional vertically straight-walled cells. Crushing behavior and energy absorption performance of sinusoidal-shaped honeycomb structure (SSHS) under quasi-static out-of-plane compression are investigated numerically and theoretically. Results demonstrate that the IPF and the fluctuation of crushing response of SSHS can be effectively reduced for the introduction of sinusoidal-curved configuration. The predictable collapse mode of SSHS could be achieved for the buckling-induced features of trough and peak of sinusoidal waves. Parametric analyses were carried out to explore the influence of the geometrical topology of sinusoidal waves, namely, wave amplitude (A = 0.6–1.6 mm), wave number (N = 1–6) and the relative density (ρ = 0.05–0.25) on the out-of-plane mechanical properties of SSHS, including IPF, mean crushing force (MCF), specific energy absorption (SEA) and crushing force efficiency (CFE). The results imply properly adjusting topology parameters can determine the collapse patterns and the stress distribution, thereby substantially improving the mechanical properties. Moreover, theoretical investigation was also performed based on the simplified super folding element theory. The calculation of MCF shows a good agreement between the theoretical predictions and the numerical results. The sinusoidally curved-walled topology design strategy provides insights into the development of lightweight structures with controllable and improved crushing performance.