Coefficients Of Series Expansion Research Articles

Stress intensity factors, T-stresses and higher order coefficients of the Williams series expansion and their evaluation through molecular dynamics simulations

Molecular dynamics method and finite element analysis are applied for the analysis of the stress field in the neighborhood of the crack tip in a copper plate with a single edge notch. The molecular dynamics simulations implemented in a classical molecular dynamics code LAMMPS (Large-scale Atomic/Molecular Massively Parallel Simulator) are aimed at evaluating classical continuum linear elastic fracture mechanics parameters such as stress intensity factors, T-stresses and higher order coefficients of the Williams power series expansion of the near crack stress field for Mode I, Mode II and Mixed Mode (Mode I + Mode II) loadings of the cracked specimen in isotropic linear elastic materials. The key objective of the study is the comparison of continuum and atomistic approaches for the estimation of the near crack tip fields using the example of one of the most common cracked configurations. Stress intensity factors, T-stresses and higher order coefficients of the Williams series expansion for a copper plate with the single edge notch under Mode I and Mixed Mode loadings are evaluated by atomistic modeling and by finite element method. The wide class of the computations in LAMMPS is performed. The atomistic values of stress intensity factors and higher order terms of the Williams series expansion are compared with the values obtained from the numerical solutions given by finite element method. It is shown that the continuum fracture theory successfully describes fracture and the near crack tip fields even at extremely confined singular stress field of only several nanometers. The angular distributions of the stress components from atomistic modeling are retrieved and compared with the angular distributions of the stresses from continuum linear elastic fracture mechanics. The comparison shows good agreement between two approaches.

A prediction perspective on the Wiener–Hopf equations for time series

The Wiener–Hopf equations are a Toeplitz system of linear equations that naturally arise in several applications in time series. These include the update and prediction step of the stationary Kalman filter equations and the prediction of bivariate time series. The celebrated Wiener–Hopf technique is usually used for solving these equations and is based on a comparison of coefficients in a Fourier series expansion. However, a statistical interpretation of both the method and solution is opaque. The purpose of this note is to revisit the (discrete) Wiener–Hopf equations and obtain an alternative solution that is more aligned with classical techniques in time series analysis. Specifically, we propose a solution to the Wiener–Hopf equations that combines linear prediction with deconvolution. The Wiener–Hopf solution requires the spectral factorization of the underlying spectral density function. For ease of evaluation it is often assumed that the spectral density is rational. This allows one to obtain a computationally tractable solution. However, this leads to an approximation error when the underlying spectral density is not a rational function. We use the proposed solution with Baxter's inequality to derive an error bound for the rational spectral density approximation.

On approximation of helix by 3rd, 5th and 7th order Bezier curves in E3

Approximation of helices has been studied by using in many ways. In this study, it has been examined how a circular helix can be written as Bezier curve and written the 3th degree, 5th degree, and the 7th degree Maclaurin series expansions of helices for the polynomial forms. Hence, they can be written cubic, 5th order, and 7th order Bezier curves, based on the control points with matrix form we have already given in E3. Further we have given the control points of the Bezier curve based on the coefficients of the Maclaurin series expansion of the circular helix.

Experimental determination and finite element analysis of coefficients of the multi-parameter Williams series expansion in the vicinity of the crack tip in linear elastic materials. Part II

In this study coefficients of the multi-parameter Williams power series expansion for the stress field in the vicinity of the central crack in the rectangular plate and in the semi-circular notched disk under bending are obtained by the use of the finite element analysis. In SIMULIA Abaqus, the finite element analysis software, the numerical solutions for these two cracked geometries are found. The rectangular plate with the central crack has the geometry similar to the geometry used in the digital photoelasticity. Numerical simulations of the same cracked specimen as in the experimental photoelasticity method are performed. The numerical solutions obtained are utilized for the determination of the coefficients of the Williams series expansion. The higher-order coefficients are extracted from the finite element method calculations implemented in Simulia Abaqus software package and the outcomes are compared to experimental values. Determination of the coefficients of the terms of this series is performed using the least squares-based regression technique known as the over-deterministic method, for which stresses data obtained numerically in SIMULIA Abaqus software are taken as inputs. The plate with a small central crack has been considered either. This kind of the cracked specimen has been utilized for comparison of coefficients of the Williams series expansion obtained from the finite element analysis with the coefficients known from the theoretical solution based on the complex variable theory in plane elasticity. It is shown that the coefficients of the Williams series expansion match with good accuracy. The higher-order terms in the Williams series expansion for the semi-circular notch disk are found.

A New Attractive Method in Solving Families of Fractional Differential Equations by a New Transform

In this paper, we use the ARA transform to solve families of fractional differential equations. New formulas about the ARA transform are presented and implemented in solving some applications. New results related to the ARA integral transform of the Riemann-Liouville fractional integral and the Caputo fractional derivative are obtained and the last one is implemented to create series solutions for the target equations. The procedure proposed in this article is mainly based on some theorems of particular solutions and the expansion coefficients of binomial series. In order to achieve the accuracy and simplicity of the new method, some numerical examples are considered and solved. We obtain the solutions of some families of fractional differential equations in a series form and we show how these solutions lead to some important results that include generalizations of some classical methods.

Calculation of multipole moments of axistationary electrovacuum spacetimes

The multipole moments of stationary axially symmetric vacuum or electrovacuum spacetimes can be expressed in terms of the power series expansion coefficients of the Ernst potential on the axis. In this paper we present a simpler, more efficient calculation of the multipole moments, applying methods introduced by B\"ackdahl and Herberthson. For the nonvacuum electromagnetic case, our results for the octupole and higher moments differ from the results already published in the literature. The reason for this difference is that we correct an earlier unnoticed mistake in the power series solution of the Ernst equations. We also apply the presented method to directly calculate the multipole moments of a five-parameter charged magnetized generalization of the Kerr and Tomimatsu-Sato exact solutions.

Legendre polynomial-based robust Fourier transformation and its use in reduction to the pole of magnetic data

A new inversion based Fourier transformation technique named as Legendre-Polynomials Least-Squares Fourier Transformation (L-LSQ-FT) and Legendre-Polynomials Iteratively Reweighted Least-Squares Fourier Transformation (L-IRLS-FT) are presented. The introduced L-LSQ-FT algorithm establishes an overdetermined inverse problem from the Fourier transform. The spectrum was approximated by a series expansion limited to a finite number of terms, and the solution of inverse problem, which gives the values of series expansion coefficients, was obtained by the LSQ method. Practically, results from the least square method are responsive to data outliers, thus scattered large errors and the estimated model values may be far from reality. A definitely better option is attained by introducing Steiner’s Most Frequent Value method. By combining the IRLS algorithm with Cauchy-Steiner weights, the Fourier transformation process was robustified to give the L-IRLS-FT method. In both cases Legendre polynomials were applied as basis functions. Thus the approximation of the continuous Fourier spectra is given by a finite series of Legendre polynomials and their coefficients. The series expansion coefficients were obtained as a solution to an overdetermined non-linear inverse problem. The traditional DFT and the L-IRLS-FT were tested numerically using synthetic datasets as well as field magnetic data. The resulting images clearly show the reduced sensitivity of the newly developed L-IRLS-FT methods to outliers and scattered noise compared to the traditional DFT. Conclusively, the newly developed L-IRLS-FT can be considered to be a better alternative to the traditional DFT.

Low-frequency scattering defined by the Helmholtz equation in one dimension

The Helmholtz equation in one dimension, which describes the propagation of electromagnetic waves in effectively one-dimensional systems, is equivalent to the time-independent Schrödinger equation. The fact that the potential term entering the latter is energy-dependent obstructs the application of the results on low-energy quantum scattering in the study of the low-frequency waves satisfying the Helmholtz equation. We use a recently developed dynamical formulation of stationary scattering to offer a comprehensive treatment of the low-frequency scattering of these waves for a general finite-range scatterer. In particular, we give explicit formulas for the coefficients of the low-frequency series expansion of the transfer matrix of the system which in turn allow for determining the low-frequency expansions of its reflection, transmission, and absorption coefficients. Our general results reveal a number of interesting physical aspects of low-frequency scattering particularly in relation to permittivity profiles having balanced gain and loss.

Generating function for tensor network diagrammatic summation

The understanding of complex quantum many-body systems has been vastly boosted by tensor network (TN) methods. Among others, excitation spectrum and long-range interacting systems can be studied using TNs, where one however confronts the intricate summation over an extensive number of tensor diagrams. Here, we introduce a set of generating functions, which encode the diagrammatic summations as leading-order series expansion coefficients. Combined with automatic differentiation, the generating function allows us to solve the problem of TN diagrammatic summation. We illustrate this scheme by computing variational excited states and the dynamical structure factor of a quantum spin chain, and further investigating entanglement properties of excited states. Extensions to infinite-size systems and higher dimension are outlined.

Algorithms for Construction of Recurrence Relations for the Coefficients of the Fourier Series Expansions with Respect to Classical Discrete Orthogonal Polynomials

A new formula expressing explicitly the integrals, antidifference, of discrete orthogonal polynomials $$\{P_{n}(x):$$ Hahn, Meixner, Kravchuk, and Charlier $$\}$$ of any degree in terms of $$P_{n}(x)$$ themselves are proved. Other formulae for the expansion coefficients of general-order difference integrations $$\nabla ^{-s}f(x),\,\Delta ^{-s}f(x),$$ $$\nabla ^{-s}[x^{\ell }\nabla ^{q}f(x)]\,$$ and $$\Delta ^{-s}[x^{\ell }\Delta ^{q}f(x)],$$ of an arbitrary function f(x) of a discrete variable in terms of its original expansion coefficients are also obtained. Application of these formulae for solving ordinary difference equations with varying coefficients, by reducing them to recurrence relations in the expansion coefficients of the solution, is explained.

Coefficient Bounds and Fekete-Szeg¨o inequality for a Certain Families of Bi-Prestarlike Functions Defined by (M,N)-Lucas Polynomials

In the current work, we use the (M,N)-Lucas Polynomials to introduce a new families of holomorphic and bi-Prestarlike functions defined in the unit disk O and establish upper bounds for the second and third coefficients of the Taylor-Maclaurin series expansions of functions belonging to these families. Also, we debate Fekete-Szeg¨o problem for thesefamilies. Further, we point out several certain special cases for our results.

Free Vibration Analysis of Laminated Composite Double-Plate Structure System with Elastic Constraints Based on Improved Fourier Series Method

An analytical model of laminated composite double-plate system (LCDPS) is established, which efficiently analyzes the common 3D plate structure in engineering applications. The proposed model combines the first-order shear deformation theory (FSDT) and the classical delamination theory, and then the LCDPS’s vibration characteristics are investigated. In the process of analysis, the improved Fourier series method (IFSM) is used to describe the displacement admissible function of the LCDPS, which can remove the potential discontinuities at the boundaries. Five sets of artificial springs are introduced to simulate the elastic boundary constraints, and the restraints of the Winkler elastic layer can be adjustable. The improved Fourier series is substituted into the governing equations and boundary conditions; then, applying the Rayleigh–Ritz method, we take all the series expansion coefficients as the generalized coordinates. After that, a set of standard linear algebraic equations was obtained. On this basis, the natural frequency and mode shapes of the LCDPS can be obtained by solving the standard eigenvalue problem. By the discussion of numerical examples and the comparison with those of the reports in the literature, the convergence and the reliability of the present approach are validated. Finally, the parametric investigations of the free vibration with complex boundary conditions are carried out, including the influence of boundary conditions, lamination scheme, plate geometric parameters, and elastic coefficient between two plates.

Confined orbitals in fullerenes and quantum dots calculated by analytic continuation method

The Schrodinger equation with a Gaussian potential to model a confined system as a quantum dot or a hydrogen atom inside a fullerene is solved by using the analytic continuation method. The use of the Rodrigues formula allows us to obtain in an easy way the coefficients of the power series expansion of the Gaussian potential in terms of the Hermite polynomials. Recurrence formulas have been obtained for the series of the states of one electron confined by that potential. This method is simpler and computationally more efficient than others employed to model quantum dots using Gaussian potentials.

The digital photoelasticity method and finite element analysis for determination of the multi-point crack-tip field series expansions for notched semi-disks

The digital photoelastic method for the construction of the multi-point crack-tip series expansions in isotropic elastic semi-disks is employed. The series of cracked specimens in the form of the semi-disk with vertical and inclined notches from optically active materials (polycarbonates) were produced and tested under three-point bending. The principal intent of this experimental procedure of the phototelasticity technique was estimation of the amplitude coefficients in the Williams expansion in which the non-singular high-order terms are taken into account. The first ten coefficients in the multi-point series expansion have been ascertained. To retrieve the points belonging to the isochromatic fringe patterns and having the minimum light intensity the digital image processing scheme was utilized. The scheme of the over-deterministic method for evaluation of the amplitude coefficients (fracture mechanics parameters) is applied. Additionally, the problem was reduced to the problem of minimizing the objective function which is the quadratic form for the stress field in terms of unknown fracture mechanics parameters. The finite element analysis for the same series of the notched specimens was performed. The demeanour of the stress field in the neighbourhood of the notch tip for different types of mixed mode loading was elucidated from finite element analysis. The coefficients of the asymptotic presentation of the stress and displacement fields are recovered from finite element results. It is shown that the multi-point coefficients are in good agreement with experimental observations.

Finding both, the continued fraction and the Laurent series expansion of golden ratio analogs in the field of formal power series

The focus of this paper is on formal power series analogs of the golden ratio. We are interested in both their continued fractions expansions as well as their Laurent series expansions. Knowing both of them is for instance important for the study of Kronecker type sequences in the theory of uniform distribution.Our approach studies the Hankel matrices that are determined using the coefficients of the Laurent series expansions. We discover that both matrices in the LU decomposition of these Hankel matrices can be described by a simple recursive algorithm based on the continued fraction coefficients of the golden ratio analogs.The upper triangular factor U possesses several nice properties. First, its entries in the special case where the golden ratio analog is [0;X‾] can be given by using the Catalan's triangular numbers. Nicely, this relation together with our findings on the Hankel matrices are used to derive combinatorial identities involving Catalan's triangular numbers. As a side product we obtain a description of the distribution of the Catalan numbers modulo a prime number.Second, the upper triangular factor U is the columnwise composition of the Zeckendorf-type representations of the powers of X. This Zeckendorf representation for polynomials is introduced in the style of the Zeckendorf representation of the natural numbers and it is based on the Fibonacci polynomials instead of Fibonacci numbers.Third, in the case of positive characteristic, for each purely periodic golden ratio analog we detect a self-similar pattern in the upper triangular factor U of its Hankel matrix. We derive this pattern from a binomial type formula for two non-commutative matrices.Finally, we exemplary show, how this self-similar pattern in the upper triangular factor U can be used to describe the Laurent series expansion of the golden ratio analog [0;X‾].

Experimental determination and finite element analysis of coefficients of the multi-parameter Williams series expansion in the vicinity of the crack tip in linear elastic materials. Part I

This study aims at obtaining coefficients of the multi-parameter Williams series expansion for the stress field in the vicinity of the central crack in the rectangular plate and in the semi-circular notched disk under bending by the use of the digital photoelasticity method. The higher-order terms in the Williams asymptotic expansion are retained. It allows us to give a more accurate estimation of the near-crack-tip stress, strain and displacement fields and extend the domain of validity for the Williams power series expansion. The program is specially developed for the interpretation and processing of experimental data from the phototelasticity experiments. By means of the developed tool, the fringe patterns that contain the whole field stress information in terms of the difference in principal stresses (isochromatics) are captured as a digital image, which is processed for quantitative evaluations. The developed tool allows us to find points that belong to isochromatic fringes with the minimal light intensity. The digital image processing with the aid of the developed tool is performed. The points determined with the adopted tool are used further for the calculations of the stress intensity factor, T-stresses and coefficients of higher-order terms in the Williams series expansion. The iterative procedure of the over-deterministic method is utilized to find the higher order terms of the Williams series expansion. The procedure is based on the consistent correction of the coefficients of the Williams series expansion. The first fifteen coefficients are obtained. The experimentally obtained coefficients are used for the reconstruction of the isochromatic fringe pattern in the vicinity of the crack tip. The comparison of the theoretically reconstructed and experimental isochromatic fringe patterns shows that the coefficients of the Williams series expansion have a good match.

An effective mesh-free fission source convergence indicator for Monte Carlo k-Eigenvalue problems

In Monte Carlo k-eigenvalue problems, the entropy of the fission site distribution is calculated after each fission cycle and its evolution over the entire fission cycles is examined. Here, the entropy evolution equilibrates as soon as the fission source distribution converges. Unfortunately, the standard entropy calculation requires space discretization with an additional effort to assign the fission sites into the space bins. Plus, the decision on choosing the number of spatial bins is subjective and based on code users’ experience. Therefore, a consistent mesh-free convergence indicator is proposed as an alternative to the conventional entropy indicator. The indicator utilizes the fundamental mode coefficient of the Fourier series expansion of the fission site distribution. Also, it eliminates the need for user specification on spatial meshing for the purpose of entropy calculation. The numerical tests conducted in this paper indicates that the new indicator is as reliable as the conventional entropy indicator and able to indicate the instance where the source distribution has converged.

Construction of Оne-Range Addition Theorems for Noninteger Slater Functions Using Self-Friction Exponential Type Orbitals and Polynomials

The addition theorems of Slater type orbitals (STOs) presented in literature are generally complicated to theoretically examine the electronic structure of atoms and molecules. The computational deficiencies in use of these theorems arise from the separation of integral variables. In this work, to eliminate these calculation efforts, a large number of independent one-range addition theorems for $$\chi $$ -noninteger Slater type orbitals ( $$\chi $$ -NISTOs) in terms of $$\chi $$ -integer STOs ( $$\chi $$ -ISTOs) is presented by using complete orthogonal basis sets of $${{L}^{{(p_{l}^{*})}}}$$ -self-friction (SF) polynomials ( $${{L}^{{(p_{l}^{*})}}}$$ -SFPs), $${{\psi }^{{(p_{l}^{*})}}}$$ -SF exponential type orbitals ( $${{\psi }^{{(p_{l}^{*})}}}$$ -SFETOs), $${{L}^{{(\alpha \text{*})}}}$$ -modified SFPs ( $${{L}^{{(\alpha \text{*})}}}$$ -MSFPs), and $${{\psi }^{{(\alpha \text{*})}}}$$ -modified SFETOs ( $${{\psi }^{{(\alpha \text{*})}}}$$ -MSFETOs) introduced by one of the authors. Here, $$p_{l}^{*} = 2l + 2 - \alpha \text{*}$$ and $$\alpha \text{*}$$ are the integer $$(\alpha \text{*} = \alpha ,\; - {\kern 1pt} \infty < \alpha \leqslant 2)$$ or noninteger ( $$\alpha \text{*} \ne \alpha ,$$ $$ - \infty < \alpha < 3$$ ) SF quantum numbers based on the Lorentz damping theory. The expansion coefficients of series for the one-range addition theorems are expressed through the analytical relations for the overlap integrals of $$\chi $$ -NISTOs with the same screening parameters. As an application, the calculations of overlap integrals with the different screening constants of $$\chi $$ -NISTOs are performed.

Synthesis of the Reflective Surface on an Elastic Shell

We consider the method for obtaining a surface that asymptotically approaches the necessary for converging convergent beams into parallel ones by directed action on the elastic shell, in particular, on a flat flexible disk. A condition is formulated for the coefficients of the power series expansion of the solution of the differential equation of bending, under which the resulting surface will (from the center to the edges) asymptotically approach the surface of a paraboloid of revolution, which gives an exact solution. We formulate this condition by the theorem about the limit of a complex-valued function given by the vector of coefficients of its analytic element. The physical basis of the method is the effect of pulling or pushing elements, the selection of which ensures the necessary transformation of the existing shell.

Crack tip asymptotic fields in anisotropic planes: Importance of higher order terms

This paper introduces the series expansion of the crack tip asymptotic fields in an anisotropic plane. Using stress-based definitions of the fracture loading modes, we present the series solutions of stresses, strains and displacements associated with the modes I and II in a cracked anisotropic plane. Special care is given to the first non-singular terms of the stress field, deriving expressions of the T-stress and the crack rigid body rotation. We then present the elasticity solution of an infinite anisotropic plane containing an inclined central crack subjected to uniaxial or biaxial loading. Under such conditions, exact analytical solutions are obtained for all coefficients of the crack tip series expansion. Our results show that the T-term is the only stress parameter that depends on the elastic constants of the anisotropic material. Finally, we evaluate the dependency of the T-stress on the material properties, and show that how significantly the higher order terms, particularly the T-stress, can contribute to the stresses near the crack tip.