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In recent years, the rapid advancement of deep learning has significantly impacted various fields, particularly in solving partial differential equations (PDEs) in the realm of solid mechanics, benefiting greatly from the remarkable approximation capabilities of neural networks. In solving PDEs, physics‐informed neural networks (PINNs) and the deep energy method (DEM) have garnered substantial attention. The principle of minimum potential energy and complementary energy are two important variational principles in solid mechanics. However, the well‐known DEM is based on the principle of minimum potential energy, but it lacks the important form of minimum complementary energy. To bridge this gap, we propose the deep complementary energy method (DCEM) based on the principle of minimum complementary energy. The output function of DCEM is the stress function, which inherently satisfies the equilibrium equation. We present numerical results of classical linear elasticity using the Prandtl and Airy stress functions, and compare DCEM with existing PINNs and DEM algorithms when modeling representative mechanical problems. The results demonstrate that DCEM outperforms DEM in terms of stress accuracy and efficiency and has an advantage in dealing with complex displacement boundary conditions, which is supported by theoretical analyses and numerical simulations. We extend DCEM to DCEM‐Plus (DCEM‐P), adding terms that satisfy PDEs. Furthermore, we propose a deep complementary energy operator method (DCEM‐O) by combining operator learning with physical equations. Initially, we train DCEM‐O using high‐fidelity numerical results and then incorporate complementary energy. DCEM‐P and DCEM‐O further enhance the accuracy and efficiency of DCEM. [ABSTRACT FROM AUTHOR]

In this paper, a series of ripple waves with decay modes for the (3+1)$$ \left(3+1\right) $$‐dimensions Kadomtsev–Petviashvili equation, always viewed as nonintegrable KP equation, are investigated. The decay mode wave solutions including ripplon, lump ripples, and lump chain ripples are described by Airy function, and all solutions are constructed from the Gram determinant form. Their propagation dynamics behavior is studied, and a comprehensive analysis of the asymptotic properties of these solutions has been diligently conducted. [ABSTRACT FROM AUTHOR]

In this paper, a new type of structured electron beam is introduced, which is the electron Airy tornado wave (eATW) with the characteristics of abrupt auto‐focusing and rotation property in free space. It is also found that the eATWs propagate periodically in a magnetic field, and their patterns of intensity distributions show the time‐like axial symmetry about a half‐period point in a period of time. Furthermore, the influence of the magnetic field's magnitude and direction on eATWs with different parameters has also been discussed. [ABSTRACT FROM AUTHOR]

It is known that random directed graphs D(n,p)$$ D\left(n,p\right) $$ undergo a phase transition around the point p=1/n$$ p=1/n $$. Earlier, Łuczak and Seierstad have established that as n→∞$$ n\to \infty $$ when p=(1+μn−1/3)/n$$ p=\left(1+\mu {n}^{-1/3}\right)/n $$, the asymptotic probability that the strongly connected components of a random directed graph are only cycles and single vertices decreases from 1 to 0 as μ$$ \mu $$ goes from −∞$$ -\infty $$ to ∞$$ \infty $$. By using techniques from analytic combinatorics, we establish the exact limiting value of this probability as a function of μ$$ \mu $$ and provide more statistical insights into the structure of a random digraph around, below and above its transition point. We obtain the limiting probability that a random digraph is acyclic and the probability that it has one strongly connected complex component with a given difference between the number of edges and vertices (called excess). Our result can be extended to the case of several complex components with given excesses as well in the whole range of sparse digraphs. Our study is based on a general symbolic method which can deal with a great variety of possible digraph families, and a version of the saddle point method which can be systematically applied to the complex contour integrals appearing from the symbolic method. While the technically easiest model is the model of random multidigraphs, in which multiple edges are allowed, and where edge multiplicities are sampled independently according to a Poisson distribution with a fixed parameter p$$ p $$, we also show how to systematically approach the family of simple digraphs, where multiple edges are forbidden, and where 2‐cycles are either allowed or not. Our theoretical predictions are supported by numerical simulations when the number of vertices is finite, and we provide tables of numerical values for the integrals of Airy functions that appear in this study. [ABSTRACT FROM AUTHOR]

This research examines the uniaxial behavior of core concrete, which is confined by a UHPC (ultra-high-performance concrete) shell, through experimental, numerical, and analytical methods. Various configurations, including different UHPC shell shapes, thicknesses, and core concrete compressive strengths, were tested until failure occurred. Results indicate that the UHPC shell altered the failure modes and enhanced the maximum stress and corresponding strain in uniaxial loading. Additionally, in square and rectangular specimens, debonding between the UHPC and NC (normal concrete) interfaces was observed. Furthermore, we constructed numerical models which integrate the concrete damage plasticity model for NC/UHPC and the coupled adhesive-frictional model implemented in cohesive elements. The model accurately reflects the crack and debonding evolution of the tested specimen. Subsequently, an analytical stress-strain model for uniaxial loading was created based on the experimental results. The confining pressure for square/rectangular specimens was determined using Airy's function. The validity of the analytical model was verified by comparing its predictions with the experimental results. [ABSTRACT FROM AUTHOR]

An unconstrained global ocean simulation for 2020 supports past observations of diurnal internal tides by acoustic tomography during the 1991–1992 Acoustic Mid‐Ocean Dynamics Experiment in the Western North Atlantic. Explicitly representing the tides, the simulation reproduces the functional form and resonant state of K1 and O1 internal‐tide standing waves, while providing a more realistic physical picture of them. The tomographic data were used to predict the tides in 2020. Not surprisingly, the characteristics of the barotropic and internal tides of the unconstrained simulation deviate from observations. The simulated barotropic tidal currents have excessive, irregular amplitude and lead the acoustic tidal predictions by about 2 hr. While internal‐tide phase coherence is apparent, the simulated internal‐tide variations were irregular in amplitude and phase, unlike the observations. The tomographic tidal measurements therefore provide a quantitative benchmark for improved model representation of tides, internal tides, and dissipation. Plain Language Summary: Oceanic internal waves, supported by stratification and the local rate of rotation of the earth (Coriolis frequency), are ubiquitous, generated by winds, tides, and uneven seafloor. Over the past 30 years, internal‐tide waves of the gravest vertical mode (half‐cosine vertically, 150–500‐km wavelength) were discovered to travel coherently across ocean basins, observed by their signals in temperature or sea‐surface height. Internal tides in a high‐resolution ocean model were compared to acoustical observations made in the western North Atlantic in 1991, with encouraging agreement. Internal waves are refracted back toward the equator at the latitude where their frequency equals twice the earth's local rotation. In the model, internal tides with 24‐hr periods (diurnal) are found to be standing waves with Airy function form, trapped between Puerto Rico and their turning latitude at 30°N, confirming an earlier interpretation of the acoustical observations. The acoustical tidal observations are an excellent benchmark for the development and adjustment of ocean models. Key Points: Past interpretations that signals in acoustic data were caused by standing‐wave, diurnal internal tides are supported by an ocean modelThe historical acoustic data can be used to predict model internal‐tide variability much like ordinary tide predictionThe acoustical observations provide a quantitative benchmark for improved model representation of tides, internal tides, and dissipation [ABSTRACT FROM AUTHOR]

Using a differential equation approach, asymptotic expansions are rigorously obtained for Lommel, Weber, Anger–Weber, and Struve functions, as well as Neumann polynomials, each of which is a solution of an inhomogeneous Bessel equation. The approximations involve Airy and Scorer functions, and are uniformly valid for large real order ν and unbounded complex argument z. An interesting complication is the identification of the Lommel functions with the new asymptotic solutions, and in order to do so, it is necessary to consider certain sectors of the complex plane, as well as introduce new forms of Lommel and Struve functions. [ABSTRACT FROM AUTHOR]

Most widely used special functions, such as orthogonal polynomials, Bessel functions, Airy functions, etc., are defined as solutions to differential equations with polynomial coefficients. This class of functions is referred to as D‐finite functions. There are many symbolic algorithms (and implementations thereof) to operate with these objects exactly. Recently, we have extended this notion to a more general class that also allows for good symbolic handling: differentially definable functions. In this paper, we give an overview on what is currently known about this new class. [ABSTRACT FROM AUTHOR]

Fast magnetosonic (MS) waves have been found to propagate mainly in the radial direction inside the plasmasphere. One main topic about the radial propagation of MS waves is whether they would propagate down to the ground or be reflected at some regions. Since WKB assumption might become invalid near the reflection point (if the reflection happens), an analytic approach has been utilized to investigate the propagation behavior of MS waves in this study. It is found that as MS waves propagate inward into the Earth, they might reach the cut‐off regions when their frequencies become close to the cut‐off frequency. Moreover, as demonstrated by our theoretical model, the behavior of MS waves around these regions can be described by the Airy's equation. Our results find that MS waves would be fully reflected near the cut‐off point, since only evanescent waves are allowed in the cut‐off regions. Plain Language Summary: After generated, plasma waves would propagate away from the source region. During their propagation paths, these waves can interact with magnetospheric particles, which is thought as an important way to affect the dynamic evolution of the terrestrial magnetosphere. Fast magnetosonic (MS) waves are found to be available of propagating in both the radial and azimuthal directions. Particularly, they are propagating dominantly along the radial direction inside the plasmasphere. Thus, whether they could propagate down to the ground becomes an interesting topic. To investigate this topic, ray tracing, based on the WKB assumption, might cease to work when certain wave reflections occurs. Instead, an analytical approach is adopted in this study, which can give the detailed MS wave behaviors near the reflection region. Key Points: Dispersive cut‐off profile of magnetosonic (MS) waves has been reported with observations of the Van Allen ProbeAnalytical approach is utilized to obtain wave behaviors of MS waves around the cut‐off frequencyWave fields of perpendicular MS waves near the cut‐off point can be described via an Airy's equation [ABSTRACT FROM AUTHOR]

Conventional approaches for generating Airy beams rely upon bulky and costly systems, which impede the miniaturization of optical systems. Metasurfaces have provided a compact method to generate Airy optical beams, but the current schemes suffer from the issues of narrow bandwidths and/or low transmission efficiencies. Here, a design strategy of constructing broadband and high‐efficiency transmissive Airy optical beam generator by taking advantages of the high birefringence and transmission efficiency of dielectric metasurfaces is proposed. The local amplitude and phase of transmitted light can be simultaneously modulated to satisfy the Airy function by controlling the orientation angles of silicon nano‐pillars, regardless of the incidence wavelength. The resultant Airy beam generator is experimentally verified within an ultra‐wide operation bandwidth from 1.1 to 1.7 µm. The remarkable diffraction‐free, self‐accelerating, and self‐healing properties of Airy beams are also demonstrated. The results may power up viable opportunities for practical devices and applications of metasurface‐based Airy beams. [ABSTRACT FROM AUTHOR]

A theoretical study of the spontaneous and stimulated undulator radiation (UR) from electrons in undulators with multiple periods in both transversal directions is presented. Exact expressions are derived for the UR intensities in terms of the generalized Bessel and Airy functions, accounting for undulator field harmonics of arbitrary strength and for real parameters of the beams and installations. Theoretical results are verified with numerical and experimental data for SWISS‐XFEL, PAL‐XFEL, LEUTL, LCLS etc. The spectrum, UR line shape and width, and the harmonic evolution along the undulators are analyzed and compared with the available data for these experiments. Moreover, the effect of the field harmonics is elucidated. It is demonstrated that the third field harmonic can cause distinct odd UR harmonics. The asymmetric undulator field configuration is identified, which allows intense radiation of these harmonics. The power evolution in a free‐electron laser (FEL) with such an undulator is studied by means of an analytical FEL model. The latter is enhanced by a true description of the gradual power saturation of harmonics. A FEL with elliptic undulator and electron–photon phase‐shifting is proposed and modeled. It is shown that the resulting harmonic power from the phase‐shifted elliptic undulator can be significantly higher than from a planar undulator with the same phase‐shifting. [ABSTRACT FROM AUTHOR]

Asymptotic solutions are derived for inhomogeneous differential equations having a large real or complex parameter and a simple turning point. They involve Scorer functions and three slowly varying analytic coefficient functions. The asymptotic approximations are uniformly valid for unbounded complex values of the argument, and are applied to inhomogeneous Airy equations having polynomial and exponential forcing terms. Error bounds are available for all approximations, including new simple ones for the well‐known asymptotic expansions of Scorer functions of large complex argument. [ABSTRACT FROM AUTHOR]

We consider the higher order asymptotics for the modified Korteweg‐de Vries equation in the Painlevé sector. We first show that the solution admits a uniform expansion to all orders in powers of t−1/3 with coefficients that are smooth functions of x(3t)−1/3. We then consider the special case when the reflection coefficient vanishes at the origin. In this case, the leading coefficient which satisfies the Painlevé II equation vanishes. We show that the leading asymptotics are instead described by the derivative of the Airy function. We are also able to express the subleading term explicitly in terms of the Airy function. [ABSTRACT FROM AUTHOR]

In this paper, we study polynomials orthogonal with respect to a Pollaczek–Jacobi type weight wpJ(x,t)=e−txxα(1−x)β,t≥0,α>0,β>0,x∈[0,1].The uniform asymptotic expansions for the monic orthogonal polynomials on the interval (0,1) and outside this interval are obtained. Moreover, near x=0, the uniform asymptotic expansion involves Airy function as ς=2n2t→∞,n→∞, and Bessel function of order α as ς=2n2t→0,n→∞; in the neighborhood of x=1, the uniform asymptotic expansion is associated with Bessel function of order β as n→∞. The recurrence coefficients and leading coefficient of the orthogonal polynomials are expressed in terms of a particular Painlevé III transcendent. We also obtain the limit of the kernel in the bulk of the spectrum. The double scaled logarithmic derivative of the Hankel determinant satisfies a σ‐form Painlevé III equation. The asymptotic analysis is based on the Deift and Zhou's steepest descent method. [ABSTRACT FROM AUTHOR]

A new approximate method for lightning-radiated extremely low-frequency (ELF) and very low-frequency (VLF) ground wave propagation over intermediate ranges is presented in this paper. In our approximate method, the original field attenuation function is divided into two factors in frequency domain representing the propagation effect of the ground conductivity and Earth’s curvature, and both of them have clearer formulations and can more easily be calculated rather than solving a complex differential equation related to Airy functions. The comparison results show that our new approximate method can predict the lightning-radiated field peak value over the intermediate range with a satisfactory accuracy within maximum errors of 0.0%, −3.3%, and −8.7% for the earth conductivity of 4 S/m, 0.01 S/m, and 0.001 S/m, respectively. We also find that Earth’s curvature has much more effect on the field propagation at the intermediate ranges than the finite ground conductivity, and the lightning-radiated ELF/VLF electric field peak value (V/m) at the intermediate ranges yields a propagation distance d (km) dependence of d−1.32. [ABSTRACT FROM AUTHOR]

ABSTRACT Surface waves are often used to estimate a near-surface shear-velocity profile. The inverse problem is solved for the locally one-dimensional problem of a set of homogeneous horizontal elastic layers. The result is a set of shear velocities, one for each layer. To obtain a P-wave velocity profile, the P-guided waves should be included in the inversion scheme. As an alternative to a multi-layered model, we consider a simple smooth acoustic constant-density velocity model, which has a negative constant vertical depth gradient of the squared P-wave slowness and is bounded by a free surface at the top and a homogeneous half-space at the bottom. The exact solution involves Airy functions and provides an analytical expression for the dispersion equation. If the [ABSTRACT FROM AUTHOR]

A representation theorem is obtained for an arbitrarily loaded elastic bimaterial solid consisting of an infinite plane containing a circular inhomogeneity. The elastic image method is used for the analysis. The theorem expresses the Airy stress functions that generate the elastic fields for the composite solid explicitly in terms of the Airy stress function for the corresponding homogeneous infinite solid. It shows that if the solution for the homogeneous infinite solid is available, then the solutions for the corresponding bimaterial solid can be deduced by the process of differentiation and integration. The result could provide the important advantage of economy of effort in the determination of the elastic fields for composite planes with circular interfaces. [ABSTRACT FROM AUTHOR]

Asymptotic properties of nonlinear dispersion equations [ABSTRACT FROM AUTHOR]

Open holes are essential design features in many technical structures. For a lightweight optimal design precise stress analysis tools are vital. Focus of this contribution is the stress field determination of an open hole in a rectangular plate of finite width by analytical means, particularly the Airy stress function. This is beneficial in terms of computational effort. To obtain results of high precision, the stress boundary conditions at the straight edges as well as along the hole boundary must be fulfilled. This is reached by supplementing the field modelling the corresponding infinite dimensions problem – the Kirsch solution – with auxiliary functions. [ABSTRACT FROM AUTHOR]

Recently Eirola and Nevanlinna have proposed an iterative solution method for unsymmetric linear systems, in which the preconditioner is updated from step to step. Following their ideas we suggest variants of GMRES, in which a preconditioner is constructed per iteration step by a suitable approximation process, e.g., by GMRES itself. Our numerical experiments indicate that this may lead to considerable savings in CPU-time and memory requirements in typical CFD applications. [ABSTRACT FROM AUTHOR]

Recently, a new method has been presented for the discrete simulation of multidimensional systems, which are described by linear partial differential equations with constant coefficients. It is based on methods customary in linear systems theory and digital signal processing and uses a frequency-domain representation of the continuous system to be simulated, A proper choice of functional transformations for each independent variable allows us to treat the influence of initial conditions, boundary conditions and excitation functions separately by suitable transfer functions. From these, corresponding discrete transfer functions and the structure of a discrete system for the simulation of the continuous problem are derived. The application of this method to wave propagation problems on uniform transmission lines is presented here. At first, the continuous problem is characterized by transfer functions; then the derivation of a discrete system is shown, and finally, some simulation results and a comparison with other methods are given. [ABSTRACT FROM AUTHOR]

The present study deals with the nonlinear dynamic response and vibration of functionally graded carbon nanotubes (FG‐CNT) reinforced composite elliptical cylindrical shells resting on elastic foundations in thermal environments using an analytical solution. The nanocomposite elliptical cylindrical shells are reinforced by single‐walled carbon nanotubes. Based on the classical shell theory with the geometrical nonlinearity in von Kármán and the nonlinear dynamic response and vibration results are obtained by using the Airy stress function, Galerkin method, and Runge‐Kutta method. In numerical results, the effect of geometrical parameters, initial geometric imperfection, Winkler and Pasternak elastic foundations and volume fractions and types of distribution of CNTs on the nonlinear dynamic response and vibration of FG‐CNT reinforced composite elliptical cylindrical shells are investigated. [ABSTRACT FROM AUTHOR]

The Airy beam has attracted considerable research interest owing to the intriguing diffraction‐free, self‐accelerating, and self‐healing properties, and hence has found numerous applications in photonics. Metasurfaces provide a compact method to generate Airy beams, but the available Airy beam generators suffer from the issue of narrow bandwidth since they typically rely on resonance principles. Here, hyperbolic metamaterials (HMMs) are proposed to address this issue by taking advantage of the broadband high birefringence. By changing the orientation angle of the HMM unit, the local amplitude and phase distributions of the transmitted electromagnetic waves passing through such HMM unit can be well adjusted to follow the Airy function within a wide spectral range. The resultant microwave Airy beam generator promises an operation bandwidth from 7.2 to 9.6 GHz, while the near‐infrared Airy beam generator has a working bandwidth from 1.25 to 2.40 µm over 10% transmission efficiency. The proposed Airy beam overcomes the limitation of the customary Airy beams with the issue of narrow bandwidth and can stimulate the design of complicated optical devices with the requirement of maintaining the amplitude and phase distributions over a wide spectral band. [ABSTRACT FROM AUTHOR]