Legendre Polynomial Expansion Research Articles

On the Fourier coefficients of the derivative with respect to celebrated orthogonal systems

The main goal of this paper is to find the coefficients of the Jacobi polynomials and the integrals of Legendre polynomials expansion of the derivative of a function in terms of the coefficients in the expansion of the original function. More precisely, if {Q_{n}} is a sequence or orthogonal polynomials, and if p(x)=∑_{j=0}ⁿa_{j}Q_{n-j}(x) is such that p′(x)=∑_{j=0}ⁿ⁻¹d_{j}Q_{n-j-1}(x), we find an explicit relation for the coefficients d_{j}, as linear combinations of the coefficients a_{j}. This will be done for two celebrated classes of orthogonal functions, namely the Jacobi polynomials and the integrals of the Legendre polynomials.

A numerical scheme based on the Taylor expansion and Lie product formula for the second‐order acoustic wave equation and its application in seismic migration

AbstractWe have developed a numerical scheme for the second‐order acoustic wave equation based on the Lie product formula and Taylor‐series expansion. The scheme has been derived from the analytical solution of the wave equation and in the approximation of the time derivative for a wavefield. Through these two equations, we obtained the first‐order differential equation in time, where the time evolution operator of the analytic solution of this differential equation is written as a product of exponential matrices. The new numerical solution using a Lie product formula may be combined with Taylor‐series, Chebyshev, Hermite and Legendre polynomial expansion or any other expansion for the cosine function. We use the proposed scheme combined with the second‐ or fourth‐order Taylor approximations to propagate the wavefields in a recursive procedure, in a stable manner, accurately and efficiently with even larger time steps than the conventional finite‐difference method. Moreover, our numerical scheme has provided results with the same quality as the rapid expansion method but requiring fewer computations of the Laplacian operator per time step. The numerical results have shown that the proposed scheme is efficient and accurate in seismic modelling, reverse time migration and least‐squares reverse time migration.

Propagation of spherical wave in a three-layer sphere under cyclic pressure

In this paper, the dynamic characteristics of a three-layer sphere under a spherical wave induced by cyclic pressure are studied. Firstly, on the basis of the Lamb Decomposition and Variable Separation methods, the analytical expression of the spherical wave is conducted, which satisfies the stress equilibrium on the outer and inner surfaces of the sphere. Next, algebraic equations with respective boundary conditions are composed and solved by Legendre Polynomial expansion and effective truncation techniques. The obtained results enable us to reveal the influence on the dynamic stress concentration factor intensity under proper conditions. The conclusions of this article are verified by comparing the analytical solutions to the ones obtained by the finite element method. This paper can provide a theoretical method for the analysis of mechanical properties of multi-layered spherical structures under dynamic loading.

Component Analysis and Accuracy Improvement of Linear Power Flow Equation Based on Legendre Polynomial Expansion

The linearization of the power flow equation is widely used in power industry. Existing linearization approaches are generally regarded as nonlinear state variable space transformation, where the original power flow equation can be approximately reformulated as the linear forms in transformed state variable spaces. However, the optimality of linear power flow equations cannot be theoretically compared in different spaces and is only verified based on case studies. In this paper, based on Legendre polynomial expansion, power flow equations are uniformly transformed into a single state variable space, where dimensions correspond to certain orders of Legendre expansion. Through component analysis of power flow equations on each dimension, we find that the accuracy of the linear power flow equation can be improved by formulating an appropriate state variable space transformation to reduce Legendre component loss. Based on this, a linearization method minimizing Legendre component loss is formulated. Compared with the regular case-based linearization, the proposed method does not depend on empirical data, and can theoretically guide the selection of the linear power flow equation within the specified operating bound. The accuracy performance is verified in OPF calculation based on numerous IEEE and Polish test systems.

Velocity field and cavity dynamics in drop impact experiments.

Drop impact experiments allow to model a wide variety of natural processes, from raindrop impacts to planetary impact craters. In particular, interpreting the consequences of the planetary impacts requires an accurate description of the flow associated with the cratering process. In our experiments, we release a liquid drop above a deep liquid pool to investigate simultaneously the dynamics of the cavity and the velocity field produced around the air-liquid interface. Using particle image velocimetry, we analyse quantitatively the velocity field using a shifted Legendre polynomials decomposition. We show that the velocity field is more complex than considered in previous models, in relation to the non-hemispherical shape of the crater. In particular, the velocity field is dominated by the degrees 0 and 1, with contributions from the degree 2, and is independent of the Froude and the Weber number when these numbers are large enough. We then derive a semi-analytical model based on the Legendre polynomials expansion of an unsteady Bernoulli equation coupled with a kinematic boundary condition at the crater boundary. This model explains the experimental observations and can predict the time evolution of both the velocity field and the shape of the crater, including the initiation of the central jet.

Rotational state-changes in C5N− by collisions with He and H2

ABSTRACT The anion C5N− is one of the largest linear (C,N)-bearing chains detected in the interstellar medium. Here we present and discuss the general features of new ab initio potential energy surfaces describing the interaction of this linear anion with He and H2. We employ a Legendre Polynomials expansion representation for the former and an artificial neural network fit for the latter. We then carry out quantum scattering calculations to yield rotationally inelastic cross-sections for collisions with He and H2, using relative translational energy values in the range of 0.1–300 cm−1. We then obtained the corresponding inelastic rate coefficients as a function of temperature covering the range from 1 to 100 K. The results for these two systems are compared with each other, as well as with the earlier results on the C3N− colliding with the same partners. We found that the final inelastic rate coefficients for this anion are all fairly large, those from collisions with H2 being the largest. The consequences of such findings on their non-equilibrium rotational populations in interstellar environments are discussed in our conclusions.

Dynamic performance of bimaterial composite sphere under spherical wave

In this article, the dynamic characteristics of bimaterial composite spheres under spherical waves induced by cyclic pressure are studied. First, on the basis of the Lamb Decomposition and Variable Separation method, the analytical expression of spherical wave is conducted, which satisfies the stress equilibrium on the outer and inner surfaces of the sphere. Next, algebraic equations with respective boundary conditions are composed and solved by Legendre Polynomial expansion and effective truncation techniques. Obtained results enable to reveal the influence on the dynamic stress concentration factor intensity under proper conditions. The conclusions of this article are verified by comparing the analytical solutions to the ones obtained by the finite element method. This article can provide a theoretical method for the analysis of mechanical properties of multilayered spherical structure under dynamic loading.

An alternative expression for ad hoc field collision model

An alternative form of ad hoc collision model to the Fokker–Planck field collision operator is proposed. The model is so constructed that the results of exact Fokker–Planck field collision, applied to the low-order Legendre and associated Laguerre polynomials, are used to impose self-adjointness first and then collisional conservation laws. The model is shown to be actually identical to the conventional ad hoc model in the l = 0 and 1 parts in Legendre polynomial expansion in an isothermal case, while it is not limited to extend to the higher order, in contrast to the conventional ad hoc model, in which the collisional conservation laws are imposed primarily, using the exact test collision form of the Fokker–Planck operator. Furthermore, it is discussed that the proposed model is identified to the lowest order in an expansion of formal Hirshman–Sigmar operator, while there is a difference between the formal and conventional Hirshman–Sigmar operators. Numerical examples demonstrate that the proposed model can yield comparable results to those by the exact Fokker–Planck and conventional Hirshman–Sigmar operators.

Numerical simulations of spectroscopic properties in two-dimensional Mott insulator

We numerically investigate spectroscopic properties for a two-dimensional Mott insulator described by a half-filled Hubbard model on the square lattice. Firstly, we examine the spectral shape of the optical conductivity by using time-dependent density-matrix renormalization group implemented by the Legendre polynomial expansion. We find a shape peak resembling to an excitonic peak at absorption edge, whose origin is attributable to antiferromagnetic correlation in the spin background. We next investigate momentum-dependent transient charge and spin dynamics after photoirradiation by using time-dependent Lanczos-type exact diagonalization. We find a temporal oscillation in the intensity of the dynamical spin structure factor, which exhibits an antiphase behavior between, for example, two orthogonal momentum directions parallel and perpendicular to the electric field of a pump pulse. Observing such antiphase oscillations will be a big challenge for time-resolved resonant-inelastic x-ray scattering experiments.

Accurate numerical, integral methods for computing drift-kinetic Trubnikov-Rosenbluth potentials

A novel numerical method is employed to compute the integral form of the axi-symmetric Trubnikov-Rosenbluth potentials. Two methods for quadrature in pitch-angle are described and their convergence properties are studied. Careful attention is given to quadrature over a singular Green's function. It is shown that an infinite series representation of the Green's function can be used more efficiently than its closed form involving complete elliptic integrals. Then a collocation method in speed, with its associated quadrature scheme, is laid out and its convergence properties are studied. Using the proposed scheme, accurate low-order moments of the field collision operator are obtained using relatively few velocity space degrees of freedom. The scheme is showcased by solving for the equilibrium, axi-symmetric bootstrap current in tokamaks. A C0 Gauss-Lobatto-Legendre finite element pitch-angle basis with vertex nodes at the trapped/passing boundary is shown, in the context of the integral methods used, to be much more efficient than the more common Legendre polynomial expansion.

An efficient nonlinear interval uncertain optimization method using Legendre polynomial chaos expansion

Many specialized numerical techniques have been developed on interval uncertainty optimization. However, the conventional nested optimization strategy is time-consuming. The improved Taylor-based interval optimization avoids the laborious inner optimization, nonetheless, it is intrusive and presents significant computational challenges in the large-level uncertain optimization problems. To end these, this paper proposes a Legendre polynomial expansion method combined with the subinterval technique to evaluate the range enclosure of an interval function, where the expansion coefficients are computed through the collocation method. The detailed comparisons demonstrate that such Legendre-based polynomial expansion provides much more confidence in interval bounds evaluation than the Taylor expansion, and is capable of maintaining very sharp solution enclosures even when the uncertainty level is increased. More exciting, it is highly efficient. Subsequent application of the Legendre polynomial expansion in the interval bound estimation of the uncertain objective function and constraints, and combining an appropriate outer optimizer, result in a novel nonlinear interval uncertain optimization method for complicated engineering systems, whereby the time-consuming optimization nesting is avoided. A beam design example and an electromagnetic buffer optimization example illustrate that the proposed interval optimization method has fine computational precision and high computational efficiency, especially is capable of attacking the increase of uncertainty level. Moreover, the new method does not require the derivative information of the uncertain objective and constraints, such that applies to any engineering uncertain optimization problems, including the complex engineering problems which are usually referred to as “black-box”.

Channel Estimation and Equalization for Alamouti SF-Coded OFDM-UWA Communications

In this paper, a new channel estimation and equalization algorithm for underwater acoustic (UWA) communications is presented. The proposed algorithm is developed to meet the requirements of underwater time-varying sparse channels that undergo Rayleigh fading. In addition, the algorithm takes into consideration a path-based channel model which describes each received path with significant power by an attenuation factor, a Doppler scale, and a delay. Transmit diversity enabled by Alamouti space-frequency block coding coupled with orthogonal frequency division multiplexing is employed in the form of two transmitters and multiple receivers. The proposed, non-data-aided, expectation-maximization (EM)-based maximum a posteriori probability sparse channel estimation first estimates the channel transfer functions from each transmit antenna to the receiver. Then, the estimation performance is greatly improved by taking into account the sparseness of the UWA channel and refining the estimation based on the sparse solution that best matches the frequency-domain channel estimates obtained during the first phase of the estimation process. Sparse channel path delays and Doppler scaling factors are estimated by a novel technique called delay focusing . After that, slow time-varying, complex-valued channel path gains are estimated using a basis expansion model based on the discrete Legendre polynomial expansion. Computer simulation results show that the resulting channel estimation algorithm can achieve excellent mean-square error and symbol error rate for both generated data and semi-experimental data taken at Sapanca Lake in Turkey and is capable of handling some mismatch due to different fading models.

Photon fluence and dose estimation in computed tomography using a discrete ordinates Boltzmann solver

In this study, cone-beam single projection and axial CT scans are modeled with a software package—DOCTORS, which solves the linear Boltzmann equation using the discrete ordinates method. Phantoms include a uniform 35 cm diameter water cylinder and a non-uniform abdomen phantom. Series simulations were performed with different simulation parameters, including the number of quadrature angles, the order of Legendre polynomial expansions, and coarse and fine mesh grid. Monte Carlo simulations were also performed to benchmark DOCTORS simulations. A quantitative comparison was made between the simulation results obtained using DOCTORS and Monte Carlo methods. The deterministic simulation was in good agreement with the Monte Carlo simulation on dose estimation, with a root-mean-square-deviation difference of around 2.87%. It was found that the contribution of uncollided photon fluence directly from the source dominates the local absorbed dose in the diagnostic X-ray energy range. The uncollided photon fluence can be calculated accurately using a ‘ray-tracing’ algorithm. The accuracy of collided photon fluence estimation is largely affected by the pre-calculated multigroup cross-sections. The primary benefit of DOCTORS lies in its rapid computation speed. Using DOCTORS, parallel computing with GPU enables the cone-beam CT dose estimation nearly in real-time.

Exclusive π0p electroproduction off protons in the resonance region at photon virtualities 0.4GeV2≤Q2≤1GeV2

The exclusive electroproduction process ep→e′p′π0 was measured in the range of photon virtualities Q2=0.4–1.0GeV2 and the invariant mass range of the pπ0 system of W=1.1–1.8 GeV. These kinematics are covered in exclusive π0 electroproduction off the proton with nearly complete angular coverage in the pπ0 center-of-mass system and with high statistical accuracy. Nearly 36 000 cross-section points were measured, and the structure functions σT+εσL,σLT, and σTT, were extracted via fitting the ϕπ0 dependence of the cross section. A Legendre polynomial expansion analysis demonstrates the sensitivity of our data to high-lying N* and Δ* resonances with M>1.6 GeV. As part of a broad effort to determine the electrocouplings of the N* and Δ* resonances using both single- and double-pion electroproduction, this dataset is crucial for the reliable extraction of the high-lying resonance electrocouplings from the combined isospin analysis of the Nπ and π+π−p channels.17 MoreReceived 6 August 2019DOI:https://doi.org/10.1103/PhysRevC.101.015208©2020 American Physical SocietyPhysics Subject Headings (PhySH)Research AreasLepton induced nuclear reactionsResonance reactionsStrong interactionPhysical SystemsBaryonsNuclear Physics

Discrete unified gas kinetic scheme for multiscale anisotropic radiative heat transfer

In this work, a discrete unified gas kinetic scheme (DUGKS) is developed for radiative transfer in anisotropic scattering media. The method is an extension of a previous one for isotropic radiation problems [1]. The present scheme is a finite-volume discretization of the anisotropic gray radiation equation, where the anisotropic scattering phase function is approximated by the Legendre polynomial expansion. With the coupling of free transport and scattering processes in the reconstruction of the flux at cell interfaces, the present DUGKS has the nice unified preserving properties such that the cell size is not limited by the photon mean free path even in the optical thick regime. Several one- and two-dimensional numerical tests are conducted to validate the performance of the present DUGKS, and the numerical results demonstrate that the scheme is a reliable method for anisotropic radiative heat transfer problems.

Dynamics of vibrating beams using first-order theory based on Legendre polynomial expansion

First-order models are used in the analysis of the tension–compression and transverse bending modes of beam vibration. The equation of motion for each mode and the expressions for boundary conditions are obtained using the generalized variational principle. Systems of partial differential equations for the longitudinal and bending modes of vibrating beams are reduced to a single fourth-order equation, and frequency equations are obtained. The problem of free and forced vibrations of beams that are simply supported at both ends is presented. An analysis and comparison with well-known theories is performed using computer algebra system Mathematica.

Time-stepping wave-equation solution for seismic modeling using a multiple-angle formula and the Taylor expansion

We have developed an analytical solution for wave equations using a multiple-angle formula. The new solution based on the multiple-angle expansion allows us to generate a family of solutions for the acoustic-wave equation, which may be combined with Taylor-series, Chebyshev, Hermite, and Legendre polynomial expansions or any other expansion for the cosine function and used for seismic modeling, reverse time migration, and inverse problems. Extension of this method to the solution of elastic and anisotropic wave equations is also straightforward. We also derive a criterion using the stability and dispersion relations to determine the order of the solution for a given time step and, thus, obtaining stable wavefields free of numerical dispersion. Afterward, numerical tests are performed using complex 2D velocity models to evaluate the effectiveness and robustness of our method, combined with second- or fourth-order Taylor approximations. Our multiple-angle approach is stable and provides reliable seismic modeling results for larger times steps than those usually used by conventional finite-difference methods. Moreover, multiple-angle schemes using a second-order Taylor approximation for each cosine term have a lower computational cost than the mixed wavenumber-space rapid expansion method.

The eigenvalues of one-speed neutrons in a slab with forward and backward scattering

The eigenvalue spectrum is studied for one-speed neutrons in a slab with forward and backward scattering. First, the transport equation describing the interaction of neutrons in a system with general geometry is given. Then, the scattering function in transport equation is chosen as the forward-backward-isotropic (FBI) scattering model. The resultant transport equation is solved using the Legendre polynomials expansion (PN method) and the Chebyshev polynomials of second kind expansion (UN method) in neutron angular flux. Then, the PN and UN moments of the equations are obtained using the properties of the Legendre and the Chebyshev polynomials of the second kind. Finally, the eigenvalues for various values of the collision and scattering parameters are calculated using different orders of the presented methods and they are given in the tables for comparison.

Optimized Nonuniform FFTs and Their Application to Array Factor Computation

We deal with developing an optimized approach for implementing nonuniform fast Fourier transform (NUFFT) algorithms under a general and new perspective for 1-D transformations. The computations of nonequispaced results, nonequispaced data, and Type-3 nonuniform discrete Fourier transforms are tackled in a unified way. They exploit “uniformly sampled” exponentials to interpolate the “nonuniformly sampled” ones involved in the nonuniform discrete Fourier transforms (NUFDTs), so as to enable the use of standard fast Fourier transforms, and an optimized window. The computational costs and the memory requirements are analyzed, and their convenient performance is assessed also by comparing them with other approaches in the literature. Numerical results demonstrate that the method is more accurate and does not introduce any additional computational or memory burden. The computation of the window functions amounts to that of a Legendre polynomial expansion, i.e., a simple polynomial evaluation. This is convenient in terms of computational burden and of the proper arrangement of the calculations. A case study of electromagnetic interest has been carried out by applying the developed NUFFTs to the radiation of linear regular or irregular arrays onto a set of regular or irregular spectral points. Guidelines for multidimensional extension of the proposed approach are also presented.

Evaluation of shear and membrane locking in refined hierarchical shell finite elements for laminated structures

Shear and membrane locking phenomena are fundamental issues of shell finite element models. A family of refined shell elements for laminated structures has been developed in the framework of Carrera Unified Formulation, including hierarchical elements based on higher-order Legendre polynomial expansions. These hierarchical elements were reported to be relatively less prone to locking phenomena, yet an exhaustive evaluation of them regarding the mitigation of shear and membrane locking on laminated shells is still essential. In the present article, numerically efficient integration schemes for hierarchical elements, including also reduced and selective integration procedures, are discussed and evaluated through single-element p-version finite element models. Both shear and membrane locking are assessed quantitatively through the estimation of strain energy components. The numerical results show that the fully integrated hierarchical shell elements can overcome the shear and membrane locking effectively when a sufficiently high polynomial degree is reached. Reduced and selective integration schemes can help with the mitigation of locking on lower-order hierarchical shell elements.