Polynomial Expansion Research Articles

A new Legendre polynomial approach for computing the matrix exponential action on a vector

AbstractWe propose a new approach for computing the action of the matrix exponential over a vector. The approach is based on a Legendre polynomial expansion in the framework of the so‐called ‐product solution to ODEs. The new approach can be combined with Krylov subspace methods.

Fractional Wiener chaos: Part 1

AbstractIn this paper, we introduce a fractional analogue of the Wiener polynomial chaos expansion. It is important to highlight that the fractional order relates to the order of chaos decomposition elements, and not to the process itself, which remains the standard Wiener process. The central instrument in our fractional analogue of the Wiener chaos expansion is the function denoted as $${\mathcal {H}}_\alpha (x,y)$$ H α ( x , y ) , referred to herein as a power-normalised parabolic cylinder function. Through careful analysis of several fundamental deterministic and stochastic properties, we affirm that this function essentially serves as a fractional extension of the Hermite polynomial. In particular, the power-normalised parabolic cylinder function with the Wiener process and time as its arguments, $${\mathcal {H}}_\alpha (W_t,t)$$ H α ( W t , t ) , demonstrates martingale properties and can be interpreted as a fractional Itô integral with 1 as the integrand, thereby drawing parallels with its non-fractional counterpart. To build a fractional analogue of polynomial Wiener chaos on the real line, we introduce a new function, which we call the extended Hermite function, by smoothly joining two power-normalized parabolic cylinder functions at zero. We form an orthogonal set of extended Hermite functions as a one-parameter family and use tensor products of the extended Hermite functions as building blocks in the fractional Wiener chaos expansion, in the same way that tensor products of Hermite polynomials are used as building blocks in the Wiener chaos polynomial expansion.

Full field crack solutions in anti-plane flexoelectricity

In flexoelectric materials, strain gradients can induce electrical polarization. However, internal defects such as cracks profoundly affect the electromechanical coupling properties of flexoelectric solids. In particular, anti-plane cracks involve less physical fields, which are easier to study. In this study, we present a comprehensive and innovative investigation of the anti-plane crack problems in flexoelectric materials, including semi-infinite and finite-length anti-plane cracks. For the first time, we formulate a full-field solution for semi-infinite anti-plane cracks in flexoelectric media by applying the Wiener–Hopf technique. Furthermore, the collocation method and the Chebyshev polynomial expansion are used for the first time to derive the full-field hypersingular integral equation solution for finite-length anti-plane cracks in flexoelectric solids. In addition, a comparative analysis between the full-field and asymptotic solutions for semi-infinite cracks is performed, shedding light on the discrepancies in the representation of the electromechanical coupling behavior near the crack tip. The mixed finite element method is used to compare with the full-field solutions of finite-length cracks. The agreement between the numerical results and the full-field solutions demonstrates the rigor of our study. This research advances the knowledge of defects in flexoelectricity and provides significant insight into relevant failure mechanisms.

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%.

A note on the 1-D minimization problem related to solenoidal improvement of the uncertainty principle inequality

AbstractThis paper gives a second way to solve the one-dimensional minimization problem of the form : $$\begin{aligned} \min _{f\not \equiv 0}\frac{\displaystyle \int \limits _0^\infty \left( f''\right) ^2x^{\mu +1}dx\int \limits _0^\infty \left( {x}^2\left( f'\right) ^2 -\varepsilon f^2\right) {{x}}^{\mu -1}d{x}}{\displaystyle \left( \int \limits _0^\infty \left( f'\right) ^2 {{x}}^{\mu }d{x}\right) ^2} \end{aligned}$$ min f ≢ 0 ∫ 0 ∞ f ′ ′ 2 x μ + 1 d x ∫ 0 ∞ x 2 f ′ 2 - ε f 2 x μ - 1 d x ∫ 0 ∞ f ′ 2 x μ d x 2 for scalar-valued functions f on the half line, where $$f'$$ f ′ and $$f''$$ f ′ ′ are its derivatives and $$\varepsilon $$ ε and $$\mu $$ μ are positive parameters with $$\varepsilon < \frac{\mu ^2}{4}.$$ ε < μ 2 4 . This problem plays an essential part of the calculation of the best constant in Heisenberg’s uncertainty principle inequality for solenoidal vector fields. The above problem was originally solved by using (generalized) Laguerre polynomial expansion; however, the calculation was complicated and long. In the present paper, we give a simpler method to obtain the same solution, the essential part of which was communicated in Theorem 5.1 of the preprint (Hamamoto, arXiv:2104.02351v4, 2021).

Noise-Disruption-Inspired Neural Architecture Search with Spatial–Spectral Attention for Hyperspectral Image Classification

In view of the complexity and diversity of hyperspectral images (HSIs), the classification task has been a major challenge in the field of remote sensing image processing. Hyperspectral classification (HSIC) methods based on neural architecture search (NAS) is a current attractive frontier that not only automatically searches for neural network architectures best suited to the characteristics of HSI data, but also avoids the possible limitations of manual design of neural networks when dealing with new classification tasks. However, the existing NAS-based HSIC methods have the following limitations: (1) the search space lacks efficient convolution operators that can fully extract discriminative spatial–spectral features, and (2) NAS based on traditional differentiable architecture search (DARTS) has performance collapse caused by unfair competition. To overcome these limitations, we proposed a neural architecture search method with receptive field spatial–spectral attention (RFSS-NAS), which is specifically designed to automatically search the optimal architecture for HSIC. Considering the core needs of the model in extracting more discriminative spatial–spectral features, we designed a novel and efficient attention search space. The core component of this innovative space is the receptive field spatial–spectral attention convolution operator, which is capable of precisely focusing on the critical information in the image, thus greatly enhancing the quality of feature extraction. Meanwhile, for the purpose of solving the unfair competition issue in the traditional differentiable architecture search (DARTS) strategy, we skillfully introduce the Noisy-DARTS strategy. The strategy ensures the fairness and efficiency of the search process and effectively avoids the risk of performance crash. In addition, to further improve the robustness of the model and ability to recognize difficult-to-classify samples, we proposed a fusion loss function by combining the advantages of the label smoothing loss and the polynomial expansion perspective loss function, which not only smooths the label distribution and reduces the risk of overfitting, but also effectively handles those difficult-to-classify samples, thus improving the overall classification accuracy. Experiments on three public datasets fully validate the superior performance of RFSS-NAS.

A Double Legendre Polynomial Order N Benchmark Solution for the 1D Monoenergetic Neutron Transport Equation in Plane Geometry

As more and more numerical and analytical solutions to the linear neutron transport equation become available, verification of the numerical results becomes increasingly important. This presentation concerns the development of another benchmark for the linear neutron transport equation in a benchmark series, each employing a different method of solution. In 1D, there are numerous ways of analytically solving the monoenergetic transport equation, such as the Wiener–Hopf method, based on the analyticity of the solution, the method of singular eigenfunctions, inversion of the Laplace and Fourier transform solution, and analytical discrete ordinates in the limit, which is arguably one of the most straightforward, to name a few. Another potential method is the PN (Legendre polynomial order N) method, where one expands the solution in terms of full-range orthogonal Legendre polynomials, and with orthogonality and series truncation, the moments form an open set of first-order ODEs. Because of the half-range boundary conditions for incoming particles, however, full-range Legendre expansions are inaccurate near material discontinuities. For this reason, a double PN (DPN) expansion in half-range Legendre polynomials is more appropriate, where one separately expands incoming and exiting flux distributions to preserve the discontinuity at material interfaces. Here, we propose and demonstrate a new method of solution for the DPN equations for an isotropically scattering medium. In comparison to a well-established fully analytical response matrix/discrete ordinate solution (RM/DOM) benchmark using an entirely different method of solution for a non-absorbing 1 mfp thick slab with both isotropic and beam sources, the DPN algorithm achieves nearly 8- and 7-place precision, respectively.

Forced vibrations of an elastic triangular plate supported around its perimeter by a unilateral support via the Chebyshev polynomial expansion

The present study introduces static and dynamic analysis of an elastic triangular plate on unilateral edge supports, including forced vibrations due to loading and unloading excitation. The plate is assumed to be subjected to a uniformly distributed load and an eccentrically applied concentrated load. Furthermore, the loads are assumed to display dynamic variations. The governing equation of the problem is derived by considering the static and the dynamic responses of the plate by including inertia forces. The analytical solution is conducted by employing a series of Chebyshev polynomials for the admissible displacement functions and using Lagrange’s equations of motion. Having found the governing equation of the problem, an approximate numerical solution is accomplished using an iterative process due to the non-linear properties of the unilateral edge supports. The static behavior of the plate under a concentrated load is investigated in detail numerically by considering a wide range of parameters of the plate geometry and the stiffness of the support. In deriving the governing equation of the problem and in the numerical solution, special care is taken to use nondimensional parameters. This approach ensures that the analysis and conclusions remain valid across a wide range of parameter values. Dynamic treatment of the problem is carried out in the time domain by assuming a stepwise time variation of the concentrated load and by employing the constant acceleration procedure in the numerical solution of the system of governing differential equations derived from the equation of motion. Time variations of the displacements, the contact points, and the reactions of the plate’s support are presented in figures for various values of the parameters of the plate, as well as those of the supports, particularly focusing on the non-linearity of the problem due to the plate liftoff from the unilateral support. The effects of the parameters and the loading are investigated in detail. The results reveal that the unilateral property of the support stiffness significantly affects the static and dynamic behavior of the triangular plate. The analysis can be extended easily to cover a general type of loading as well. The main issues addressed in this study include: the analysis of a triangular plate with unilateral perimeter support that does not resist tensile forces; the use of Chebyshev polynomials as admissible functions in both directions; the consideration of static and dynamic loads; the determination of lifting areas of the support in both cases and the assessment of the global vertical force balance in the plate, including inertia forces in the dynamic case.

Design of Robust Ballistic Landings on the Secondary of a Binary Asteroid

ESA’s Hera mission aims to visit binary asteroid Didymos in late 2026, investigating its physical characteristics and the result of NASA’s impact by the DART spacecraft in more detail. Two CubeSats onboard Hera plan to perform a ballistic landing on the secondary of the system, called Dimorphos. For these types of landings the translational state during descent is not controlled, reducing the spacecraft’s complexity but also increasing its sensitivity to deployment maneuver errors and dynamic uncertainties. This paper introduces a novel methodology to analyze the effect of these uncertainties on the dynamics of the lander and design a trajectory that is robust against them. This methodology consists of propagating the uncertain state of the lander using the nonintrusive Chebyshev interpolation (NCI) technique, which approximates the uncertain dynamics using a polynomial expansion. The results are then analyzed using the pseudo-diffusion indicator. This indicator is derived from the coefficients of the polynomial expansion, which quantifies the rate of growth of the set of possible states of the spacecraft over time. The indicator is used here to constrain the impact velocity and angle to values that allow for successful settling on the surface. This information is then used to optimize the landing trajectory by applying the NCI technique inside the transcription of the problem. The resulting trajectory increases the robustness of the trajectory compared to a conventional method, improving landing success by 20% and significantly reducing the landing footprint.

Fast Minimum Error Entropy for Linear Regression

The minimum error entropy (MEE) criterion finds extensive utility across diverse applications, particularly in contexts characterized by non-Gaussian noise. However, its computational demands are notable, and are primarily attributable to the double summation operation involved in calculating the probability density function (PDF) of the error. To address this, our study introduces a novel approach, termed the fast minimum error entropy (FMEE) algorithm, aimed at mitigating computational complexity through the utilization of polynomial expansions of the error PDF. Initially, the PDF approximation of a random variable is derived via the Gram–Charlier expansion. Subsequently, we proceed to ascertain and streamline the entropy of the random variable. Following this, the error entropy inherent to the linear regression model is delineated and expressed as a function of the regression coefficient vector. Lastly, leveraging the gradient descent algorithm, we compute the regression coefficient vector corresponding to the minimum error entropy. Theoretical scrutiny reveals that the time complexity of FMEE stands at O(n), in stark contrast to the O(n2) complexity associated with MEE. Experimentally, our findings underscore the remarkable efficiency gains afforded by FMEE, with time consumption registering less than 1‰ of that observed with MEE. Encouragingly, this efficiency leap is achieved without compromising accuracy, as evidenced by negligible differentials observed between the accuracies of FMEE and MEE. Furthermore, comprehensive regression experiments on real-world electric datasets in northwest China demonstrate that our FMEE outperforms baseline methods by a clear margin.

Kinematics of nonlinear waves over variable bathymetry. Part I: Numerical modelling, verification and validation

Fluid particle kinematics due to wave motion (i.e. orbital velocities and accelerations) at and beneath the free surface is involved in many coastal and ocean engineering applications, e.g. estimation of wave-induced forces on structures, sediment transport, etc. This work presents the formulations of these kinematics fields within a fully nonlinear potential flow (FNPF) approach. In this model, the velocity potential is approximated with a high-order polynomial expansion over the water column using an orthogonal basis of Chebyshev polynomials of the first kind. Using the same basis, original analytical expressions of the components of velocity and acceleration are derived in this work. The estimation of particle accelerations in the course of the simulation involves the time derivatives of the decomposition coefficients, which are computed with a high-order backward finite-difference scheme in time. The capability of the numerical model in computing the particle kinematics is first validated for regular nonlinear waves propagating over a flat bottom. The model is shown to be able to predict both the velocity and acceleration of highly nonlinear and nearly breaking waves with negligible error compared to the corresponding stream function wave solution. Then, for regular waves propagating over an uneven bottom (bar-type bottom profile), the simulated results are confronted with existing experimental data, and very good agreement is achieved up to the sixth-order harmonics for free surface elevation, velocity and acceleration.

Coefficient Estimates for New Subclasses of Bi-Univalent Functions with Bounded Boundary Rotation by Using Faber Polynomial Technique

In this article, the authors use the Faber polynomial expansions to find the general coefficient estimates for a few new subclasses of bi-univalent functions with bounded boundary rotation and bounded radius rotation. Some of the results improve the existing coefficient bounds in the literature.

Q-boson model and relations with integrable hierarchies

This work investigates the intricate relationship between the q-boson model, a quantum integrable system, and classical integrable systems such as the Toda and KP hierarchies. Initially, we analyze scalar products of off-shell Bethe states and explore their connections to tau functions of integrable hierarchies. Furthermore, we discuss correlation functions within this formalism, examining their representations in terms of tau functions, as well as their Schur polynomial expansions.

Statistical distribution of static stress resolved onto geometrically-rough faults

The in-situ stress state within fault zones is technically challenging to characterize, requiring the use of indirect methods to estimate. Most work to date has focused on understanding average properties of resolved stress on faults, but fault non-planarity should induce spatial variations in resolved static stress on a single fault. Assuming a particular stochastic model for fault geometry (band-limited fractal) gives an approximate analytic solution for the probability density function (PDF) on fault stress that depends on the mean fault orientation, mean stress ratio, and roughness level. The mean stress is shown to be equal to the planar fault value, while deviations are described by substituting a second-order polynomial expansion of the stress ratio into the inverse distribution on fault slope. The result is an analytical expression for the PDF of shear-to-normal stress ratio on 2-D rough faults in a uniform background stress field. Two end-member distributions exist, one approximately Gaussian when all points on the fault are well away from failure, and one reverse exponential, which occurs when the mean stress ratio approaches the peak. For the range of roughness values expected to apply to crustal faults, stress deviations due to geometry can reach nearly 100% of the background stress level. Consideration of such a distribution of stress on faults suggests that geometric roughness and the resulting stress deviations may play a key role in controlling earthquake behavior.

Construction of some new traveling wave solutions to the space-time fractional modified equal width equation in modern physics

Nonlinear fractional evolution equations are important for determining various complex nonlinear problems that occur in various scientific fields, such as nonlinear optics, molecular biology, quantum mechanics, plasma physics, nonlinear dynamics, water surface waves, elastic media and others. The space-time fractional modified equal width (MEW) equation is investigated in this paper utilizing a variety of solitary wave solutions, with a particular emphasis on their implications for wave propagation characteristics in plasma and optical fibre systems. The fractional-order problem is transformed into an ordinary differential equation using a fractional wave transformation approach. In this article, the polynomial expansion approach and the sardar sub-equation method are successfully used to evaluate the exact solutions of space-time fractional MEW equation. Additionally, in order to graphically represent the physical significance of created solutions, the acquired solutions are shown on contour, 3D and 2D graphs. Based on the results, the employed methods show their efficacy in solving diverse fractional nonlinear evolution equations generated across applied and natural sciences. The findings obtained demonstrate that the two approaches are more effective and suited for resolving various nonlinear fractional differential equations.

FCHG: Fuzzy Cognitive Hypergraph for interpretable fault detection

While deep learning-based fault detection methods for rolling body mechanical components have achieved good results, their application in real-world scenarios is limited, and their reliability is not satisfactory due to the non-interpretability of deep learning. Meanwhile, automated component vibration signals are usually affected by irregularities, which make it challenging to obtain advanced features by conventional feature extraction methods. To solve the above problems, we develop a new fuzzy cognitive graph model that combines the interpretability of fuzzy cognitive graphs with the high-precision categorization of hypergraphs, called fuzzy cognitive hypergraph (FCHG). First, frequency domain analysis is combined with graph analysis to help the network extract high-level features. Second, causal learning is used to construct the FCHG weight matrix to make the FCHG interpretable and enhance the reliability of fault diagnosis. Finally, the FCHG network is built using spectrogram theory and Chebyshev polynomial expansion to capture multiple feature relationships and improve fault diagnosis accuracy. The practicality and effectiveness of the FCHG model for monitoring the health and stability status of rolling body mechanical components are verified on three datasets.

A Preconditioner-Based Data-Driven Polynomial Expansion Method: Application to Compressor Blade With Leading Edge Uncertainty

Abstract In engineering practice, the amount of measured data is often scarce and limited, posing a challenge in uncertainty quantification (UQ) and propagation. Data-driven polynomial chaos (DDPC) is an effective way to tackle this challenge. However, the DDPC method faces problems from the lack of robustness and convergence difficulty. In this paper, a preconditioner-based data-driven polynomial chaos (PDDPC) method is developed to deal with UQ problems with scarce measured data. Two numerical experiments are used to validate the computational robustness, convergence property, and application potential in case of scarce data. Then, the PDDPC is first applied to evaluate the uncertain impacts of real leading edge (LE) errors on the aerodynamic performance of a two-dimensional compressor blade. Results show that the overall performance of compressor blade is degraded and there is a large performance dispersion at off-design incidence conditions. The actual blade performance has a high probability of deviating from the nominal performance. Under the influence of uncertain LE geometry, the probability distributions of the total pressure loss coefficient and static pressure ratio have obvious skewness characteristics. Compared with the PDDPC method, the UQ results obtained by the fitted Gaussian and Beta probability distributions seriously underestimate the performance dispersion of compressor blade. The mechanism analysis illustrates that the large flow variation around the leading edge is the main reason for the overall performance degradation and the fluctuations of the entire flow field.

Sensitivity analysis of turbine stage aerothermal characteristics and blade cooling performance considering combustor swirl and hot spot

This paper investigates the global sensitivity of high-pressure-turbine (HPT) first stage cascades aerothermal characteristics and blade cooling performance under the influence of combustor swirl and hot spot. The combustor outflow conditions impact on turbine working performance and robustness. This paper is focused on the influence of uncertain combustor swirling flow and hot spot, rather than deterministic working conditions. A simplified combustor outflow simulation method is firstly proposed, and the effects of combustor outflow are quantified by combining chaos polynomial expansion with Sobol’ decomposition method. The results show that different rotating directions of the swirl lead to restricted influences on the averaged aerothermal performance of the turbine stage, but significantly affects its robustness. The positive swirl increases the total pressure loss generated by the lower passage vortex of the stator cascade, and the high uncertainty region of the stator cascade total pressure loss profile is the combined products of the residual swirl and the interaction between the passage vortex and mainstream. The total pressure loss of the rotor cascade due to leakage vortex has low sensitivity to the hot spot and swirl strength, while the total pressure loss generated by the passage vortex is affected by hot spot temperature. Temperature distribution affected by the passage vortex is sensitive to swirl strength. Generally, uncertainty variance of stage aerothermal parameters with positive swirl is about 30% larger than that of the negative case, since the positive swirl has stronger interactions with the blades, leading to more significant uncertainty. However, the averaged heat transfer coefficients with positive swirl are 37% less sensitive, compared to the negative swirl case. The results give better understanding on the influence mechanisms of turbine inflow conditions, emphasize the meaning of investigating uncertain characteristics of the turbine cascades. Investigations of this article provide reference for robust turbine blade cooling design.

Conforming Capacitive Load Cells for Conical Pick Cutters.

In underground coal mining, machine operators put themselves at risk when getting close to the machine or cutting face to observe the process. To improve the safety and efficiency of machine operators, a cutting force sensor is proposed. A linear cutting machine is used to cut two separate coal samples cast in concrete with conical pick cutters to simulate mining with a continuous miner. Linear and neural network regression models are fit using 100 random 70:30 test/train splits. The normal force exceeds 60 kN during the rock-cutting tests, and it is averaged using a low pass filter with a 10 Hertz cutoff frequency. The sensor uses measurements of the resonant frequency of capacitive cells in a steel case to determine cutting forces. When used in the rock-cutting experiments, the sensor conforms to the tooling and the stiffness and sensitivity are increased compared to the initial configuration. The sensor is able to track the normal force on the conical picks with a mean absolute error less than 6 kN and an R2 score greater than 0.60 using linear regression. A small neural network with a second-order polynomial expansion is able to improve this to a mean absolute error of less than 4 kN and an R2 score of around 0.80. Filtering measurements before regression fitting is explored. This type of sensor could allow operators to assess tool wear and material type using objective force measurements while maintaining a greater distance from the cutting interface.

Hydrodynamic performance of multi-chamber oscillating water columns in a caisson array

A theoretical model based on the linear potential flow theory and eigenfunction expansion-matching method is developed to analyze the interaction between water waves and multi-chamber oscillating water columns (OWCs) embedded in a caisson array. The semi-analytical solution consists of the diffraction and radiation components of periodic OWCs. The velocity singularity at the tip of the OWC chamber walls is resolved by implementing a Chebyshev polynomial expansion. The unknown expansion coefficients are determined by the continuity equations of velocity and pressure at the interface of subdomains. The semi-analytical solution is verified using the Haskind relation and the conservation law of wave energy flux. It is found that multi-chamber OWCs have higher hydrodynamic efficiency over a broader frequency bandwidth than traditional single- and dual-chamber OWCs. The maximum hydrodynamic efficiency η increases with increasing of chamber number (J), from 0.5 (J = 1) to nearly 1.0 (J = 8 and 12). It is also found that a larger incident wave angle leads to a narrower frequency bandwidth for energy capturing. Increasing the incident wave angle from π/3 to 5π/12 results in a decrease of the first cutoff frequency kc from 2.41 to 2.02, and therefore results in the energy capture bandwidth narrowing accordingly. Both constructive and destructive hydrodynamic interactions between caissons array and OWCs are observed over the tested frequency range. Interestingly, when configuration of the OWC chambers is reversed, the transmission coefficient remains unchanged.