Higher-order Terms Research Articles

A strain-based criterion for predicting size-dependent fracture resistance of quasi-brittle materials under mixed mode loading

In the present study, a strain-based approach called the modified maximum tangential strain criterion has been developed for evaluating size effect on the mixed-mode fracture resistance of quasi-brittle materials. This approach relies on the generalized maximum tangential strain criterion, which considers the singular (K) and the first non-singular (T) terms of Williams series expansion. Furthermore, as an important and size-dependent parameter in the proposed criterion, a new formulation is presented to calculate the critical distance (rc). The predictions of the new criterion are compared not only with the available experimental data, but also with the results estimated by another size effect criterion named the modified maximum tangential stress criterion. It is shown that the proposed strain-based criterion is highly capable of estimating the size-dependent mixed-mode fracture behavior of quasi-brittle materials without requiring to compute the other higher order terms.

High-Resolution Imaging of a Complex Moving Target for Higher-Order Phase Terms Using a Novel Technique

High-Resolution Imaging of a Complex Moving Target for Higher-Order Phase Terms Using a Novel Technique

Influence of Bed Variations on Linear Wave Propagation beyond the Mild Slope Condition

As water waves travel from deep to shallow waters, they experience increased nonlinearity and decreased dispersion due to the reduced water depth. While the impact of bed slope on wave propagation celerity is documented, it is often overlooked in commonly used depth-integrated wave models. This study uses the WKB approximation and solves higher-order slope-related terms to analyze the influence of varying depth, including the gradient and laplacian of water depth. One result is an extended longwave for linear wave reflection and transmission on a ramp to deeper waters. The main outcome and focus of this work, however, is a new, simple analytical expression for linear dispersion that includes bed variations. The results are applied to two cases: wind-generated water wave propagation in the nearshore, emphasizing corrections to bathymetry inversion methods, and tsunami propagation over the continental slope, highlighting the limitations of neglecting slope on dispersion and the significant role of the Laplacian of water depth.

Can AI Predict the Magnitude and Direction of Ortho-K Contact Lens Decentration to Limit Induced HOAs and Astigmatism?

Background: The aim is to investigate induced higher-order aberrations (HOA)s and astigmatism as a result of non-toric ortho-k lens decentration and utilise artificial intelligence (AI) to predict its magnitude and direction. Methods: Medmont E300 Video topographer was used to scan 249 corneas before and after ortho-k wear. Custom-built MATLAB codes extracted topography data and determined lens decentration from the boundary and midpoint of the central flattened treatment zone (TZ). An evaluation was carried out by conducting Zernike polynomial fittings via a computer-coded digital signal processing procedure. Finally, an AI-based machine learning neural network algorithm was developed to predict the direction and magnitude of TZ decentration. Results: Analysis of the first 21 Zernike polynomial coefficients indicate that the four low-order and four higher-order aberration terms were changed significantly by ortho-k wear. While baseline astigmatism was not correlated with lens decentration (R = 0.09), post-ortho-k astigmatism was moderately correlated with decentration (R = 0.38) and the difference in astigmatism (R = 0.3). Decentration was classified into three groups: ≤0.50 mm, reduced astigmatism by -0.9 ± 1 D; 0.5~1 mm, increased astigmatism by 0.8 ± 0.1 D; >1 mm, increased astigmatism by 2.7 ± 1.6 D and over 50% of lenses were decentred >0.5 mm. For lenses decentred >1 mm, 29.8% of right and 42.7% of left lenses decentred temporal-inferiorly and 13.7% of right and 9.4% of left lenses decentred temporal-superiorly. AI-based prediction successfully identified the decentration direction with accuracies of 70.2% for right and 71.8% for left lenses and predicted the magnitude of decentration with root-mean-square (RMS) of 0.31 mm and 0.25 mm for right and left eyes, respectively. Conclusions: Ortho-k lens decentration is common when fitting non-toric ortho-k lenses, resulting in induced HOAs and astigmatism, with the magnitude being related to the amount of decentration. AI-based algorithms can effectively predict decentration, potentially allowing for better control over ortho-k fitting and, thus, preferred clinical outcomes.

Linearised Fokker–Planck collision model for gyrokinetic simulations

Abstract We introduce a gyrokinetic, linearised Fokker–Planck collision model that satisfies conservation laws and is accurate at arbitrary collisionalities. The differential test-particle component of the operator is exact; the integral field-particle component is approximated using a spherical harmonic and a modified Laguerre polynomial expansion developed by Hirshman and Sigmar (1976 Phys. Fluids 19 1532). The numerical methods of the implementation in the δf-gyrokinetic code stella (Barnes et al 2019 J. Comput. Phys. 391 365–80) are discussed, and conservation properties of the operator are demonstrated. The collision model is then benchmarked against the collision model of the gyrokinetic solver GS2 in the limiting cases of a reduced test-particle collision operator and energy- and momentum-conserving operator. The accuracy of the full collision model is investigated by solving the parallel Spitzer-Härm problem for the transport coefficients. It is shown that retaining collisional energy flux and higher-order terms in the field-particle operator reduces errors in the transport coefficients from 10%–25% for a simple momentum- and energy-conserving model to under 1%.

First-principles calculations of dynamical Born effective charges, quadrupoles, and higher order terms from the charge response in large semiconducting and metallic systems

First-principles calculations of dynamical Born effective charges, quadrupoles, and higher order terms from the charge response in large semiconducting and metallic systems

A new strain-based approach to investigate the size and geometry effects on fracture resistance of rocks

In this paper, a new strain-based criterion is suggested for assessing the effects of size and geometry of specimen on the fracture resistance of rocks under mixed-mode (I/II) loading. The new approach named the modified maximum tangential strain (MMTSN) criterion is based on the classical maximum tangential strain (MTSN) criterion, in which the first non-singular term (T) of Williams series expansion is considered in addition to the singular terms (K). Furthermore, to provide more coherence, the critical distance (rc) from the crack tip is defined according to a new strain-based failure model. Unlike similar strain-based fracture models available in the literature, the critical distance rc in the MMTSN criterion is assumed to be size-dependent and a semi-empirical formulation is utilized for describing this size-dependency. To assess the ability of MMTSN for considering the size and geometry effects, the experimental data existing in the literature for a number of cracked Brazilian disk (CBD) and semi-circular bend (SCB) specimens manufactured from Guiting limestone are taken into account. It is demonstrated that the MMTSN criterion can predict the experimental data very well by taking into consideration the size and geometry effects without needing to calculate the other higher order terms.

Approximate Effective Interaction for Nuclear Matter and Finite Nuclei

In this paper, an approximate effective nucleon-nucleon interaction for nuclear matter and finite studies has been derived using the lowest order constrained variational (LOCV) approach. The LOCV method, a functional minimization procedure, uses a normalization constraint to keep higher-order terms as small as possible. As a first step, two-body matrix elements based on the Reid93 nucleon-nucleon potential were calculated for the nuclear system A = 16 in a harmonic oscillator basis, with the oscillator size parameter <i>ћω </i>= 14.0 MeV, and separated into the central, spin-orbit and tensor channels in conformity with the potentials for Inelastic scattering. Following this, a least squares fitting of the matrix elements to a sum of Yukawa functions was performed to determine the strengths of the effective interaction in the singlet-even, singlet-odd, triplet-even and triplet-odd (Central); tensor-even and tensor-odd (Tensor); spin-orbit-even and spin-orbit-odd (Spin-orbit) channels. Of all the matrix elements, only the triplet-even and tensor-even components, being attractive, are affected by the tensor correlations (<i>a </i>= 0.05); and are shown to exhibit the same trend of variation in conformity with past work, in terms of magnitude, as one goes from the lower-node quantum numbers (n’, n) = (0, 0) to higher ones (n’, n) = (2, 2). When compared with the G-matrix results of previous researchers, the results obtained herein have been found to be in good agreement. This, therefore, gives hope that the new effective interaction promises to be a reliable tool for nuclear matter and nuclear structure studies.

The endogenous growth and asset prices nexus revisited with closed-form solution

Endogenous growth models exhibit long-run risks, which are considered a potential explanation of the equity premium puzzle. Unlike previous literature, we use a closed-form solution of a simplified model to make the following contributions. First, we derive a set of conditions for a positive and large equity premium. Second, we match a key driver of endogenous growth, the R&D spending-to-GDP ratio. Third, we include a novel discussion on the role of patent obsolescence. Given that the literature concerns the accuracy of loglinear-lognormal solutions, we solve our model numerically with third-order perturbation. We find additional risk correction due to higher-order terms.

Optimization Method of Mine Ventilation Network Regulation Based on Mixed-Integer Nonlinear Programming

Mine ventilation is crucial for ensuring safe production in mines, as it is integral to the entire underground mining process. This study addresses the issues of high energy consumption, regulation difficulties, and unreasonable regulation schemes in mine ventilation systems. To this end, we construct an optimization model for mine ventilation network regulation using mixed-integer nonlinear programming (MINLP), focusing on objectives such as minimizing energy consumption, optimal regulation locations and modes, and minimizing the number of regulators. We analyze the construction methods of the mathematical optimization model for both selected and unselected fans. To handle high-order terms in the MINLP model, we propose a variable discretization strategy that introduces 0-1 binary variables to discretize fan branches’ air quantity and frequency regulation ratios. This transformation converts high-order terms in the constraints of fan frequency regulation into quadratic terms, making the model suitable for solvers based on globally accurate algorithms. Example analysis demonstrate that the proposed method can find the optimal solution in all cases, confirming its effectiveness. Finally, we apply the optimization method of ventilation network regulation based on MINLP to a coal mine ventilation network. The results indicate that the power of the main fan after frequency regulation is 71.84 kW, achieving a significant energy savings rate of 65.60% compared to before optimization power levels. Notably, ventilation network can be regulated without adding new regulators, thereby reducing management and maintenance costs. This optimization method provides a solid foundation for the implementation of intelligent ventilation systems.

Multivariable adaptive decoupling control based on stochastic configuration networks with serial-parallel switching distribution

A multivariable adaptive decoupling control scheme is proposed based on stochastic configuration networks with serial-parallel switching distribution (SPSCN). Firstly, a linear controller is designed by combining a PID controller, feedback decoupling, and one-step optimal control. Secondly, a nonlinear controller is presented to deal with higher-order nonlinear terms and unknown external perturbations. SPSCN is used to improve the prediction accuracy of unmodeled dynamics. It combines uniform and normal search strategies in a serial-parallel fashion, aiming at improving the node quality and reducing the model complexity. The approximation performance of the SPSCN algorithm is demonstrated by performing approximation experiments with two functions and four benchmark datasets. Compared with the generalized minimum variance control (GMVC) algorithm in controlling the process of cement raw material decomposition, our proposed control scheme outperforms.

Coupled piecewise tri-stable stochastic resonance system driven by dual inputs and joint EMD/VMD analysis

A novel piecewise tri-stable stochastic resonance (NPTSR) system is proposed to address the issue of output saturation caused by high-order terms limitation in this paper. Building upon this, the exploration is extended to a coupled piecewise tri-stable stochastic resonance system driven by dual inputs (DCPTSR). First, we uncover the influence of dual input interaction on output quality, finding that when the fRequencies of the two input signals are consistent, varying the amplitude of the driving signal can effectively enhance the output performance of the target signal. Secondly, by utilizing the adiabatic approximation theory, the steady-state probability density (SPD) and signal-to-noise ratio (SNR) of the DCPTSR system are derived, which allows us to analyze the effects of various parameters on both SPD and SNR. Next, three combined denoising systems, namely EMD-DCPTSR, VMD-DCPTSR, and SDCPTSR, are constructed by utilizing empirical mode decomposition (EMD), variational mode decomposition (VMD), and the stochastic resonance (SR). Through numerical simulations, we demonstrate that the combined denoising system outperforms the stand-alone SR system, and we analyze the stochastic resonance phenomenon of the DCPTSR system using the spectral amplification (SA) coefficient as an evaluation index. Finally, to assess practical applicability, these systems are deployed for bearing fault detection. The experimental results exhibit notable signal-to-noise gain improvements for the DCPTSR system compared to standalone SR systems by 0.7699 ∼ 9.4541 dB. The EMD-DCPTSR system shows signal-to-noise gain improvements of 0.3245 ∼ 1.1709 dB compared to the VMD-DCPTSR and SDCPTSR systems. Moreover, all three combined denoising systems outperform the standalone SR system in terms of signal processing capabilities. In conclusion, this paper extensively investigates the interaction between the two input signals in a dual-input system and studies the output performance of using EMD, VMD, and SR as preprocessing methods for the SR system. Through numerical simulations and practical engineering applications, we highlight the substantial advantages of combined denoising systems. These findings offer essential theoretical insights and promising prospects for engineering applications.

Unconventional angular dependence of spin-orbit torque-induced harmonic Hall resistance in Pt/YIG bilayers

Spin-orbit torque (SOT), arising from spin-orbit coupling-induced spin currents, provides efficient control of magnetization. SOT characterization involving harmonic Hall resistances is typically done in low-current regimes, distinct from high-current regimes, where SOT-induced magnetization switching occurs. In this study, we investigate azimuthal angle (ϕ)-dependent harmonic Hall resistances of a Pt/yttrium iron garnet (YIG) layer across a wide range of measurement currents. Under low-current conditions, conventional ϕ-dependent Hall resistances are observed; the first harmonic Hall resistance exhibits sin 2ϕ behavior and the second harmonic Hall resistance comprises cos ϕ and cos 3ϕ terms. Interestingly, with increasing current, higher-order angular-dependent terms become non-negligible, referring to the sin 4ϕ and sin 6ϕ terms for the first harmonic and the cos 5ϕ and cos 7ϕ terms for the second harmonic Hall resistances. We attribute this unconventional angular dependence to the nonlinear response of magnetization direction to SOT, emphasizing its relevance to understanding the magnetization dynamics during SOT-induced switching under large currents.

Multiple Floquet Chern insulator phases in the spin-charge coupled triangular-lattice ferrimagnet: Crucial role of higher-order terms in the high-frequency expansion

Multiple Floquet Chern insulator phases in the spin-charge coupled triangular-lattice ferrimagnet: Crucial role of higher-order terms in the high-frequency expansion

Stabilizing massless fields with fluxes in Landau-Ginzburg models

Recent work on flux compactifications suggests that the tadpole constraint generically allows only a limited number of complex structure moduli to become massive, i.e., be stabilized at quadratic order in the spacetime superpotential. We study the effects of higher-order terms systematically around the Fermat point in the 19 Landau-Ginzburg model. This model lives at strong coupling and features no Kähler moduli. We show that indeed massless fields can be stabilized in this fashion. We observe that, depending on the flux, this mechanism is more effective when the number of initially massless fields is large. These findings are compatible with both the tadpole conjecture and the massless Minkowski conjecture. Along the way, we complete the classification of integral flux vectors with small tadpole contribution. Thereby we are closing in on a future complete understanding of all possible flux configurations in the 19 Landau-Ginzburg model.

Intelligent cooperative control for flexible landing on small celestial bodies

In view of the potential landing risks in small celestial body landing missions, the concept of flexible landing has been proposed in recent years. By employing a flexible lander with structural compliance to complex surface topography, the risks of rebound or overturning upon touchdown can be reduced. During flexible landing, the overall motion of the flexible lander is controlled by several nodes embedded in the flexible structure. Due to the flexible deformation, the motion among the nodes is intricately coupled, posing challenges to the control algorithm design. To address this problem, an intelligent cooperative control method for flexible landing is proposed in this paper. By decomposing the flexible internal force into a low-order principal term with specific physical interpretation and an unmodeled high-order residual term, the coupling relationship among the nodes is characterized. On this basis, the low-order dynamics system is reconstructed into a piecewise affine system, upon which a multi-node cooperative optimal controller is designed. A long-short-term memory recurrent neural network is further established to compensate for the unmodeled high-order term. Through the combination of the low-order principal control and the high-order intelligent compensation, a feasible approach for achieving flexible landing is obtained. Finally, the effectiveness of the proposed control method is verified via 433 Eros-based flexible landing simulations.

Multi-stage trajectory tracking control design under comprehensive constraints based on Generalized Udwadia–Kalaba theory

A novel control design based on Generalized Udwadia–Kalaba (GUK) theory is proposed to achieve the trajectory tracking under comprehensive constraints in this paper. To describe the complex task requirements, a constraint framework including equality and inequality constraints is constructed. Thus, the complex trajectory tracking problem is converted into a constraint-following problem. Based on the framework above, GUK equation is introduced to consider the equality and inequality constraints individually. Specifically, the equality constraints are transformed into second-order matrix form, while the inequality constraints are addressed via a diffeomorphism. Then, a novel control method including servo control term and high-order control term is proposed. Through these two control terms, the constraint forces for equality and inequality can be solved separately. The effectiveness of this control method is verified by the stability analysis based on Lyapunov minimax apprach and illustrative application of unmanned surface vehicle (USV). Meanwhile, the superiority of this method is demonstrated through a comparative analysis with Udwadia–Kalaba (UK) method, Linear Quadratic Regulator (LQR) method and sliding mode control (SMC) method.

Average Entropy of Gaussian Mixtures.

We calculate the average differential entropy of a q-component Gaussian mixture in Rn. For simplicity, all components have covariance matrix σ21, while the means {Wi}i=1q are i.i.d. Gaussian vectors with zero mean and covariance s21. We obtain a series expansion in μ=s2/σ2 for the average differential entropy up to order O(μ2), and we provide a recipe to calculate higher-order terms. Our result provides an analytic approximation with a quantifiable order of magnitude for the error, which is not achieved in previous literature.

Multi-step Hermite-Birkhoff predictor-corrector schemes

In this study, we introduce a multi-step multi-derivative predictor-corrector time integration scheme analogous to the schemes in Schütz et al. (2022) [13], incorporating a multi-step quadrature rule. We conduct stability analysis up to order eight and optimize the schemes to achieve A(α)-stability for large α. Numerical experiments are performed on ordinary differential equations exhibiting diverse stiffness conditions, as well as on partial differential equations showcasing non-linearity and higher-order terms. Results demonstrate the convergence and flexibility of the proposed schemes across diverse situations.

Periodic wave solutions for a generalized reaction–convection–diffusion equation of high-order

In this paper, we investigate the uniqueness of periodic wave solutions for a generalized reaction–convection–diffusion equation with arbitrarily high-order reaction term or convection term. The main technique is to prove the monotonicity of the ratio of two Abelian integrals by a new criterion and Descartes’ rule of signs. A positive answer to a conjecture stated in Wei and Chen (2023) is given and some numeric simulations are carried out to illustrate the obtained theoretical results.