Lagrange Inversion Research Articles

Construction method for generating functions of special numbers and polynomials arising from analysis of new operators

The aim of this paper is to construct a new method related to a family of operators to define generating functions for special numbers and polynomials. With the help of this method, we investigate various properties of these special numbers and polynomials with their generating functions, functional equations, and differential equations. We also give some algorithms to calculate the values of these numbers and polynomials. Moreover, we introduce some fundamental properties of this operator. By applying integral method to this operator, we obtain integral formulas together with combinatorial sums. Moreover, by applying this operator, the Lagrange inversion formula and convolution formula to these generating functions, we derive some novel identities and relations including combinatorial sums and combinatorial numbers including these new numbers and polynomials, ie, the Apostol‐Bernoulli numbers, the Apostol‐Euler numbers, the Stirling numbers, the central factorial numbers, and the other special numbers and polynomials. In addition, we give some examples derived from relations between composita and this operator on the set of formal power series. Finally, we give complex form of the Fourier series for these generating functions. By using these Fourier series, we not only derive some series relations and trigonometric sums including trigonometric function and hyperbolic functions but also give an example for a differential equation.

Accurate evaluation of failure indices of composite layered structures via various FE models

The objective of the current work is to perform a failure evaluation of fiber composite structures based on failure indices computed using the Hashin 3D failure criterion. The analysis employs 1D and 3D finite elements. 1D elements use higher-order structural theories from the Carrera Unified Formulation based on Lagrange expansions of the displacement field. The 3D model analysis exploits ABAQUS. Attention is paid to the free-edge effects, the mode of failure initiation - matrix or fiber tension, delamination -, and the loads at which first ply failure occurs. The results underline the paramount importance of out-of-plane stress components for accurate prediction and the computational efficiency of refined 1D models. In fact, 1D models lead from one to twofold reductions of the CPU time if compared to 3D models.

Computer-extended series solution for laminar flow between a fixed impermeable disk and a porous rotating disk

PurposeThe purpose of this paper is to study the laminar boundary layer flow between a stationary nonporous disk and a porous rotating disk, both being immersed in large amount of fluid.Design/methodology/approachThe governing nonlinear momentum equations in cylindrical polar coordinates together with relevant boundary conditions are reduced to a system of coupled nonlinear ordinary differential equations (NODEs) using similarity transformations. The resulting coupled NODEs are solved using computer-extended series solution and homotopy analysis method.FindingsThe analytical solutions are explicitly expressed in terms of recurrence relation for determining the universal coefficients. The nature and location of singularity which restricts the convergence of series is analyzed by using Domb–Sykes plot. Reversion of series is used for the improvement of series. The region of validity of series is extended for much larger values of Reynolds number (R), i.e. R = 6 to 15.Originality/valueThe resulting solutions are compared with earlier works in the literature and are found to be in good agreement.

A Comparison of Point Target Spectrum Derived for Multi-Aperture SAS

Synthetic aperture sonar has an important application prospect in ocean resources exploration. The SAS receiver antennas generally have multi-aperture configuration, which lead that it is more difficult to obtain the analytical solution of the two-dimensional spectrum. In this paper, the analytical solutions of the multi-aperture SAS with two methods of displaced phase centre antenna (DPCA) and four-order the method of series reversion (MSR) are given respectively, and the phase errors and the imaging results of the two methods under the same conditions are compared by computer simulation. The results of the computer simulation show that MSR has a smaller phase error than the DPCA at the same squint angle, and the DPCA and MSR methods have the same image quality in the small squint angle, but the quality of MSR is much better than that of DPCA in the medium squint angle.

Two‐dimensional frequency decoupling method for curved trajectory synthetic aperture radar imaging

Synthetic aperture radar (SAR) imaging with curved trajectory is difficult due to its severe two-dimensional (2D) space variance. When processing the echo with azimuth-variance, the range cell migration and Doppler modulation of the targets with different azimuth locations are different, which brings a big challenge to image formation. Here, a 2D frequency decoupling method for this case is proposed. Firstly, the range history of the curved trajectory is accurately modelled based on fourth-order Taylor series expansion. Then, the 2D point target reference spectrum of the echo signal is obtained by the method of series reversion. The 2D space variance characteristics of two typical cases with curved trajectory, i.e. geosynchronous SAR and missile-borne SAR, are analysed. Based on the space variance characteristics, a 2D frequency decoupling step is implemented to handle the space-variant phase. Simulations are presented to demonstrate the effectiveness of the proposed algorithm.

A global-local approach for the elastoplastic analysis of compact and thin-walled structures via refined models

A computationally efficient framework has been developed for the elastoplastic analysis of compact and thin-walled structures using a combination of global-local techniques and refined beam models. The theory of the Carrera Unified Formulation (CUF) and its application to physically nonlinear problems are discussed. Higher-order models derived using Taylor and Lagrange expansions have been used to model the structure, and the elastoplastic behavior is described by a von Mises constitutive model with isotropic work hardening. Comparisons are made between classical and higher-order models regarding the deformations in the nonlinear regime, which highlight the capabilities of the latter in accurately predicting the elastoplastic behavior. The concept of global-local analysis is introduced, and two versions are presented - the first where physical nonlinearity is considered for both the global and local analyses, and the second where nonlinearity is considered only for the local analysis. The second version results in reasonably accurate results compared to a full 3D finite element analysis, with a twofold reduction in the number of degrees of freedom.

Δ-Cumulants in terms of moments

The [Formula: see text]-convolution of real probability measures, introduced by Bożejko, generalizes both free and Boolean convolutions. It is linearized by the [Formula: see text]-cumulants, and Yoshida gave a combinatorial formula for moments in terms of [Formula: see text]-cumulants, that implicitly defines the latter. It relies on the definition of an appropriate weight on noncrossing partitions. We give here two different expressions for the [Formula: see text]-cumulants: the first one is a simple variant of Lagrange inversion formula, and the second one is a combinatorial inversion of Yoshida’s formula involving Schröder trees.

Algebraic Conditions for Stability Analysis of Linear Time‐Invariant Distributed Order Dynamic Systems: A Lagrange Inversion Theorem Approach

AbstractBIBO stability of linear time‐invariant (LTI) distributed order dynamic systems with non‐negative weight functions is investigated in this paper by using Lagrange inversion theorem. New sufficient conditions of stability/instability are presented for these systems accordingly. These algebraically simple conditions are relatively tight and their conservatism is adjustable.

Wave propagation in compact, thin-walled, layered, and heterogeneous structures using variable kinematics finite elements

ABSTRACTThe article investigates wave propagation characteristics for a class of structures using higher-order one-dimensional (1D) models. 1D models are based on the Carrera Unified Formulation (CUF), a hierarchical formulation which provides a framework to obtain refined structural theories via a variable kinematics description. Theories are formulated by employing arbitrary expansions of the primary unknowns over the beam cross-section. Two classes of beam models are employed in the current work, namely Taylor Expansion (TE) and Lagrange Expansion (LE) models. Using the principle of virtual work and finite element method, the governing equations are formulated. The direct time integration of equation of motion is carried through an implicit scheme based on the Newmark method and a dissipative explicit method based on the Tchamwa–Wielgosz scheme. The framework is validated by comparing the response for the stress wave propagation in an isotropic beam to an analytical solution available in the literature. The capabilities of the proposed model are demonstrated by presenting results for wave propagation analysis of a sandwich beam and a layered annular cylinder structure. The ability of CUF models to detect 3D-like behavior with a reduced computational overhead is highlighted.

Experimental Testing and Analytical Solution by Means of Lambert W-Function of Inductor Air Gap Length

In this paper the exact analytical closed form solutions for inductor air gap length have been presented. To be specific, in a mathematical sense, the dependence between the air gap length, inductance and other constructional parameters of inductor is nonlinear, i.e. transcendental. In this paper that nonlinear equation is solved in two manners – by using the Lagrange inversion theorem of the Lambert W function or by using its asymptotic formula and by using Special Trans Functions Theory. The numerical results obtained by using these methods, are mutually compared in terms of the calculation error. In order to prove the validity of the mathematical dependence between the inductance, air gap length, and the other constructional parameters, two inductors have been created. The measured results of inductance and air gap length are compared with the corresponding results obtained by the calculations. Very good agreement is shown between the measured and the calculated value of both the air gap length and the inductance value.

Three-dimensional stress analysis for beam-like structures using Serendipity Lagrange shape functions

Simple1Both authors have equal contributions. analytical and finite element models are widely employed by practising engineers for the stress analysis of beam structures, because of their simplicity and acceptable levels of accuracy. However, the validity of these models is limited by assumptions of material heterogeneity, geometric dimensions and slenderness, and by Saint-Venant’s Principle, i.e. they are only applicable to regions remote from boundary constraints, discontinuities and points of load application. To predict accurate stress fields in these locations, computationally expensive three-dimensional (3D) finite element analyses are routinely performed. Alternatively, displacement based high-order beam models are often employed to capture localised three-dimensional stress fields analytically. Herein, a novel approach for the analysis of beam-like structures is presented. The approach is based on the Unified Formulation by Carrera and co-workers, and is able to recover complex, 3D stress fields in a computationally efficient manner. As a novelty, purposely adapted, hierarchical polynomials are used to define cross-sectional displacements. Due to the nature of their properties with respect to computational nodes, these functions are known as Serendipity Lagrange polynomials. This new cross-sectional expansion model is benchmarked against traditional finite elements and other implementations of the Unified Formulation by means of static analyses of beams with different complex cross-sections. It is shown that Serendipity Lagrange elements solve some of the shortcomings of the most commonly used Unified Formulation beam models based on Taylor and Lagrange expansion functions. Furthermore, significant computational efficiency gains over 3D finite elements are achieved for similar levels of accuracy.

An Exact and Explicit Implied Volatility Inversion Formula

In this paper, we develop an exact and explicit (model-independent) Taylor series representation of the Black-Scholes implied volatility in terms of market-observed option prices. Based on an extended Faa di Bruno formula under the operator calculus setting, we first derive the Taylor series of the Black-Scholes formula with respect to the volatility around a pre-chosen positive initial value. Then we apply the Lagrange inversion theorem to explicitly invert the Taylor series to obtain the implied volatility formula. We rigorously establish that our formula converge to the true implied volatility as the truncation order increases, and choose the initial value as the model-independent upper bound of the true implied volatility. Numerical examples illustrate the remarkable accuracy of the formula. Our formula distinguishes from previous literature in that it converges to the true exact implied volatility, is a closed-form formula whose coefficients are explicitly determined and do not involve numerical iterations, and is extremely efficient, which makes it suitable for industrial implementation and adoption.

Modified Chirp Scaling Algorithm for Circular Trace Scanning Synthetic Aperture Radar

For circular trace scanning synthetic aperture radar (CTSSAR) with a circular track, the conventional hyperbolic equation becomes inadequate to express the range history of a point target accurately, and when it comes to the wide swath observation and imaging, the range variance makes it even harder to focus the target on the edge of the scene. Thus, an expression with high-order terms is needed to approximate the range history and the range variance should also be considered in the imaging algorithm. In this paper, based on the method of series reversion, a fourth-order approximated range model is established for the CTSSAR processing and the 2-D spectrum is derived for the echo signal in CTSSAR with circular trajectory. At the same time, in order to deal with the range-variant range cell migration problem in large-area CTSSAR imaging, a modified chirp scaling algorithm is proposed to realize precise wide swath CTSSAR focusing. Experiments and analyses are performed to validate the effectiveness of the proposed algorithm.

Elastoplastic analysis of compact and thin-walled structures using classical and refined beam finite element models

ABSTRACTThe paper presents results on the elastoplastic analysis of compact and thin-walled structures via refined beam models. The application of Carrera Unified Formulation (CUF) to perform elastoplastic analysis of isotropic beam structures is discussed. Particular attention is paid to the evaluation of local effects and cross-sectional distortions. CUF allows formulation of the kinematics of a one-dimensional (1D) structure by employing a generalized expansion of primary variables by arbitrary cross-section functions. Two types of cross-section expansion functions, TE (Taylor expansion) and LE (Lagrange expansion), are used to model the structure. The isotropically work-hardening von Mises constitutive model is incorporated to account for material nonlinearity. A Newton–Raphson iteration scheme is used to solve the system of nonlinear algebraic equations. Numerical results for compact and thin-walled beam members in plastic regime are presented with displacement profiles and beam deformed configurations along with stress contour plots. The results are compared against classical beam models such as Euler–Bernoulli beam theory and Timoshenko beam theory, reference solutions from literature, and three-dimensional (3D) solid finite element models. The results highlight: (1) the capability of the present refined beam models to describe the elastoplastic behavior of compact and thin-walled structures with 3D-like accuracy; (2) that local effects and severe cross-sectional distortions can be detected; (3) the computational cost of the present modeling approach is significantly lower than shell and solid model ones.

A Quillen Adjunction Between Algebras and Operads, Koszul Duality, and the Lagrange Inversion Formula

We define, for a somewhat standard forgetful functor from nonsymmetric operads to weight graded associative algebras, two functorial "enveloping operad" functors, the right inverse and the left adjoint of the forgetful functor. Those functors turn out to be related by operadic Koszul duality, and that relationship can be utilised to provide examples showing limitations of two standard tools of the Koszul duality theory. We also apply these functors to get a homotopical algebra proof of the Lagrange inversion formula.

Ramanujan’s Harmonic Number Expansion and Two Identities for Bernoulli Numbers

By the Lagrange–Burmann formula, we provide a new explicit formula for determining the coefficients of Ramanujan’s asymptotic expansion for the nth harmonic number. Based on the new explicit formula, we obtain two interesting identities for the Bernoulli numbers.

One-dimensional finite element formulation with node-dependent kinematics

The present paper presents a refined one-dimensional finite element model with node-dependent kinematics. When this model is adopted, the beam theory can be different at each node of the same element. For instance, in the case of a 2-node beam element the Euler-Bernoulli theory could be used for node 1 and the Timoshenko beam theory could be used for node 2. Classical and higher-order refined models have been established with the Carrera Unified Formulation. Such a capability would allow the kinematic assumptions to be continuously varied along the beam axis, that is, no ad hoc mixing techniques such as the Arlequin method would be required. Different combinations of structural models have been proposed to account for different kinematic approximations of beams, and, beam models based on the Taylor and the Lagrange expansions have in particular been used. The numerical model has been assessed, and a number of applications to thin-walled structures have been proposed. The results have been compared with those obtained from uniform kinematic models and convergence analyses have been performed. The results show the efficiency of the proposed model. The high accuracy of refined one-dimensional models has been preserved while the computational costs have been reduced by using refined models only in those zones of the beam that require them.

Lagrange inversion formula, Laguerre polynomials and the free unitary Brownian motion

This paper is devoted to the computations of some relevant quantities associated with the free unitary Brownian motion. Using the Lagrange inversion formula, we first derive an explicit expression for its alternating star cumulants of even lengths and relate them to those having odd lengths by means of a summation formula for the free cumulants with product as entries. Next, we use again this formula together with a generating series for Laguerre polynomials in order to compute the Taylor coefficients of the reciprocal of the $R$-transform of the free Jacobi process associated with a single projection of rank $1/2$ and those of the $S$-transform as well. This generating series lead also to the Taylor expansions of the Schur function of the spectral distribution of the free unitary Brownian motion and of its first iterate.

An Indirect Range-Doppler Algorithm for Multireceiver Synthetic Aperture Sonar Based on Lagrange Inversion Theorem

This paper presents a novel imaging algorithm which has the capacity of processing multireceiver synthetic aperture sonar (SAS) data based on Vu’s spectrum. There are two important issues when developing the fast Fourier-based imaging algorithms. One is the point target reference spectrum (PTRS) with analytical expression in the 2-D frequency domain; the other is the space variance of the PTRS. Vu has solved the first issue based on the Lagrange inversion theorem. However, the second issue has not been solved yet. Our approach in this paper is to apply Vu’s spectral result in an indirect range-Doppler (R-D) imaging algorithm. For each transmitter/receiver pair, the close-form azimuth modulation is derived first, and then, range/azimuth coupling can be obtained via the difference between the PTRS and the azimuth modulation. Unlike traditional R-D algorithm, which utilizes the interpolation operation to eliminate the range cell migration (RCM), the RCM correction (RCMC) in this paper is carried out through two steps, i.e., bulk RCMC and differential RCMC. Bulk RCMC accounts for the RCMC in one reference range. Differential RCMC accomplished with a range-dependent subblock postprocessing method is used to compensate for the space variant deviation phase error. Via the proposed method, the echoed signal corresponding to each transmitter/receiver pair is processed first. Following which, a high-resolution SAS complex image is obtained by coherent superposition of all the coarse subimages. Numerical simulations and real data processing results show that the proposed method has the similar focusing capabilities with the accurate back projection algorithm across the whole swath. Although the presented method has the disadvantage of latency, it is very easy to make a complete phase center approximation correction, allowing for the tandem nature of the collected data. Besides, the proposed algorithm can handle general case with explicit point of stationary phase (PSP) no matter how complicated the PSP is, and it is also suitable for the imagery of tandem synthetic aperture radar.

Exact solutions for free vibration analysis of laminated, box and sandwich beams by refined layer-wise theory

The present work addresses a closed-form solution for the free vibration analysis of simply supported composite laminated beams via a refined one-dimensional (1D) model, which employs the Carrera Unified Formulation (CUF). In the framework of CUF, the 3D displacement field can be expanded as any order of generic unknown variables over the cross section, in the case of beam theories. Particularly, Lagrange expansions of cross-sectional displacement variables in conjunction with layer-wise (LW) theory are adopted in this analysis, which makes it possible to refine the kinematic fields of complex cross section by arbitrary order and accuracy. As a consequence, the governing equations can be derived using the principle of virtual work in a unified form and can be solved by a Navier-type, closed-form solution. Numerical investigations are carried out to test the performance of this novel method, including composite and sandwich beams ranging from simple to complex configurations of the cross section. The results are compared with those available in the literature as well as the 3D finite element method (FEM) solutions computed by commercial codes. The present CUF model is proved to be able of achieving high accurate results with less computational costs. Besides, they may serve as benchmarks for future assessments in this field.