High-order Taylor Series Expansion Research Articles

Combining the Asymptotic Numerical Method with the Harmonic Balance Method to capture the nonlinear dynamics of spur gears

The present work proposes the Asymptotic Numerical Method (ANM) combined to the Harmonic Balance Method (HBM) as a valuable approach to solve the nonlinear dynamics of gear pairs. The ANM is a continuation method based on high-order Taylor series expansion of the computed solution branch. The HBM is a periodic solution representation method based on high-order Fourier series. Thanks to a quadratic recast of the equation of motion, the Taylor and Fourier series can be computed in a very efficient way and each step produces a continuous representation of the solution branch making the continuation very robust. By employing this method, the periodic solutions may be easily expressed with respect to both the shaft rotation frequency and the gear mesh frequency as the adoption of a high number of harmonics has negligible effects on the computational burden. Effectiveness and reliability of the method are proven by comparing the numerical results with that obtained from the Runge–Kutta time integration scheme and experimental data from literature. Afterwards, a comparison in terms of computational efficiency is performed. Finally, some considerations are drawn in order to highlights the main differences between the two methods within gear dynamics computation.

Local Convergence of Traub’s Method and Its Extensions

In this article, we examine the local convergence analysis of an extension of Newton’s method in a Banach space setting. Traub introduced the method (also known as the Arithmetic-Mean Newton’s Method and Weerakoon and Fernando method) with an order of convergence of three. All the previous works either used higher-order Taylor series expansion or could not derive the desired order of convergence. We studied the local convergence of Traub’s method and two of its modifications and obtained the convergence order for these methods without using Taylor series expansion. The radii of convergence, basins of attraction, comparison of iterations of similar iterative methods, approximate computational order of convergence (ACOC), and a representation of the number of iterations are provided.

Modified radius-weighted lattice Boltzmann model to address singularities in axisymmetric multiphase flows

The radius-weighted lattice Boltzmann model has achieved great success in the simulation of axisymmetric flows. However, severe spurious currents near the axis are observed when this model is extended to simulate axisymmetric multiphase flows. In this study, to determine the origin of this singularity, we conducted a truncation error analysis based on high-order Taylor series expansion and identified the leading error terms through dimensionless analysis. By neglecting the error terms in proportion to the radius, we obtained the final forms of the singular terms in the axisymmetric lattice Boltzmann model. We proposed a modified model by including an additional correction term, to remove the singularity at the third order. We validated the proposed model using numerical tests for flat and spherical interfaces. Results showed that the present modified model reduced the spurious currents near the axis by two orders of magnitude compared with the original model. This modified model also has been successfully applied to predict bubble dynamics in an air-water system. Our numerical results are in excellent agreement with available experimental observations in terms of bubble shapes and terminal velocities.

Higher-Order Approximations for Stabilizing Zero-Energy Modes in Non-Ordinary State-Based Peridynamics Models

The non-ordinary state-based peridynamics theory combines non-local dynamic techniques with a desirable correspondence material principle, allowing for the use of continuum mechanics constitutive models. Such an approach presents a unique capability for solving problems involving discontinuities (e.g., fracture and crack propagation). However, the correspondence-based peridynamics models often suffer from zero-energy mode instabilities in numerical implementation, primarily due to the weak integral formulation in non-local approximations of the deformation gradient tensor. This paper focuses on a computational scheme for eliminating the zero-energy mode oscillations using a choice of influence functions that improve the truncation error in a higher-order Taylor series expansion of the deformation gradient. The novelty here is a tensor-based derivation of the linear constraint equations, which can be used to systematically identify the particle interaction weight functions for various user-specified horizon radii. In this paper, the higher-order stabilization scheme is demonstrated for multi-dimensional examples involving polycrystalline and composite microstructures, along with comparisons against conventional finite element methods. The proposed stabilization scheme is shown to be highly effective in suppressing the spurious zero-energy mode oscillations in all of the numerical examples.

Higher-order Taylor series expansion for uncertainty quantification with efficient local sensitivity

An efficient numerical approach for uncertainty quantification using a higher-order Taylor series expansion is presented. Moreover, local sensitivities in the Taylor series are evaluated using a high-accuracy and computationally efficient approach called modified forward finite difference (ModFFD). Once the sensitivities are estimated, with ModFFD, the stochastic response can be obtained for different realizations of the random inputs without incurring additional function evaluations. The main advantage of this approach is that it is applicable for any input distributions and is unrestricted by the nature of random input variables (correlated and uncorrelated). To test the presented approach, several analytical and engineering problems were considered with up to twenty-two random variables. For the engineering problem, a ten-bar truss problem with twenty-two random variables and buckling of a composite laminate with twenty random variables is considered. The comparison of the results with that of a large number of Monte Carlo Simulations demonstrated its high accuracy and computational efficiency for random inputs with non-standard random inputs and varying correlation.

A recursive system-free single-step temporal discretization method for finite difference methods

Single-stage or single-step high-order temporal discretizations of partial differential equations (PDEs) have shown great promise in delivering high-order accuracy in time with efficient use of computational resources. There has been much success in developing such methods for finite volume method (FVM) discretizations of PDEs. The Picard Integral formulation (PIF) has recently made such single-stage temporal methods accessible for finite difference method (FDM) discretizations. PIF methods rely on the so-called Lax-Wendroff procedures to tightly couple spatial and temporal derivatives through the governing PDE system to construct high-order Taylor series expansions in time. Going to higher than third order in time requires the calculation of Jacobian-like derivative tensor-vector contractions of an increasingly larger degree, greatly adding to the complexity of such schemes. To that end, we present in this paper a method for calculating these tensor contractions through a recursive application of a discrete Jacobian operator that readily and efficiently computes the needed contractions entirely agnostic of the system of partial differential equations (PDEs) being solved.

Sixth-order compact finite difference scheme with discrete sine transform for solving Poisson equations with Dirichlet boundary conditions

Compact finite difference methods are very popular for solving differential equations that arise in a wide variety of real-world applications. Despite their popularity, the efficiency of these methods is limited by the need for matrix inversion which is troubling when the size of the matrix is very large. However, there is a growing demand for methods with better accuracy and computational speed. To meet such needs, we design a new method which uses a sixth-order compact finite difference scheme and a discrete sine transform to solve Poisson equations with Dirichlet boundary conditions. The scheme is developed using the concept of higher-order Taylor series expansion which is then used to discretize Poisson equations that results in a system of linear algebraic equations. To solve this system, we propose a discrete sine transform designed based on the developed compact finite difference scheme. We proved analytically and numerically that the order of convergence of the proposed method is six. The efficiency of the new scheme is demonstrated by solving different test problems. The numerical results indicate that the proposed method outperforms an existing fourth-order scheme and provides solutions that are in excellent agreement with the exact solution.

Static Response Analysis of Uncertain Structures With Large-Scale Unknown-But-Bounded Parameters

This paper proposes an interval finite element method based on function decomposition for structural static response problems with large-scale unknown-but-bounded parameters. When there is a large number of uncertain parameters, it will lead to the curse of dimensionality. The existing Taylor expansion-based methods, which is often employed to deal with large-scale uncertainty problems, need the sensitivity information of response function to uncertain parameters. However, the gradient information may be difficult to obtain for some complicated structural problems. To overcome this drawback, univariate decomposition expression (UDE) and bivariate decomposition expression (BDE) are deduced by the higher-order Taylor series expansion. The original structure function with [Formula: see text]-dimensional interval parameters is decomposed into the sum of several low-dimensional response functions by UDE or BDE, each of which has only one or two interval parameters while the other interval parameters are replaced by their midpoint values. Therefore, solving the upper and lower bounds of the [Formula: see text]-dimensional function can be converted into solving those of the one- or two-dimensional functions, which savethe calculation costs and can be easily implemented. The accuracy and efficiency of the new method are verified by three numerical examples.

Large‐eddy simulation of subsonic turbulent jets using the compressible lattice Boltzmann method

SummaryThe lattice Boltzmann method (LBM) is a powerful technique for the computational modeling of a wide variety of single‐s and multiphase flows involving complex geometries. Although the LBM has been demonstrated to be effective for the solution of incompressible flow problems, there are limitations when this methodology is applied to the solution of compressible flows, especially for flows at high Mach numbers. In this article, we investigate strategies to overcome some of the limitations associated with the application of LBM to compressible flows. To this purpose, one of the key contributions of this study is the synthesis and integration of previous efforts concerning the formulation of LBM for the large‐eddy simulation (LES) of compressible turbulent flows in the subsonic flow regime. It is shown how certain limitations of applying the LBM to compressible flows can be addressed by using either a higher order Taylor series expansion of the Maxwell–Boltzmann equilibrium distribution function or using the Kataoka and Tsutahara (KT) LBM model formulation for compressible flows. The proposed LBM/LES methodology for compressible flows has been combined with the Kirchhoff integral formulation for computational aeroacoustics and used to simulate the flow and acoustic fields of compressible jet flows at high subsonic speeds with practical relevance for providing a better understanding of problems associated with jet noise. In this context, simulations of the physics associated with the jet flow and concomitant noise in the near‐ and far‐field regimes were conducted using the proposed framework of a compressible LBM/LES and Kirchhoff integral method. The results of the subsonic isothermal and nonisothermal jet flow simulations for the flow and acoustic fields have been compared with available numerical and experimental results with generally good to excellent agreement.

Nonlinear bending analysis of functionally graded porous beams using the multiquadric radial basis functions and a Taylor series-based continuation procedure

In this paper, the nonlinear bending analysis of Functionally Graded Porous (FGP) beams is investigated using an efficient numerical algorithm associating a meshless collocation technique uses the Multiquadric Radial Basis Function (MQRBF) approximation method and a higher-order Taylor series-based continuation procedure. Material properties of the FGP beams are described by adopting a modified power-law function taking into account the effect of porosities. Based on the First Order Shear Deformation Theory (FSDT) of beams with the von-Kármán kinematic hypothesis, the strong form of nonlinear equations is established. For an efficient application of the proposed numerical approach, a quadratic matrix strong form of the problem is presented. The resulting nonlinear equations are solved numerically with the proposed algorithm which leaned on the following three steps: a higher-order Taylor series expansion to transform the quadratic nonlinear equations into a sequence of linear ones, a meshless collocation technique based on MQRBF approximation method to solve numerically the resulting linear equations and a continuation procedure to get the whole solution branch. To demonstrate the robustness of the developed algorithm, convergence and validation studies have been carried out. Furthermore, the effects of power-law index, porosity volume fraction, Young’s modulus ratio, loads and boundary conditions are investigated.

A purely frequency based Floquet-Hill formulation for the efficient stability computation of periodic solutions of ordinary differential systems

Since the founding theory established by G. Floquet more than a hundred years ago, computing the stability of periodic solutions has given rise to various numerical methods, mostly depending on the way the periodic solutions are themselves determined, either in the time domain or in the frequency domain. In this paper, we address the stability analysis of branches of periodic solutions that are computed by combining a pure Harmonic Balance Method (HBM) with an Asymptotic Numerical Method (ANM). HBM is a frequency domain method for determining periodic solutions under the form of Fourier series and ANM is continuation technique that relies on high order Taylor series expansion of the solutions branches with respect to a path parameter. It is well established now that this HBM-ANM combination is efficient and reliable, provided that the system of ODE is first of all recasted with quadratic nonlinearities, allowing an easy manipulation of both the Taylor and the Fourier series. In this context, Hill's method, a frequency domain version of Floquet theory, is revisited so as to become a by-product of the HBM applied to a quadratic system, allowing the stability analysis to be implemented in an elegant way and with good computing performances. The different types of stability changes of periodic solutions are all explored and illustrated through several academic examples, including systems that are autonomous or not, conservative or not, free or forced.

Real-time estimation of state-of-charge in lithium-ion batteries using improved central difference transform method

Abstract The accurate and real-time estimation of state-of-charge (SOC) in lithium-ion batteries (LIBs) are crucial for battery management system (BMS) in electric vehicles. The SOC estimation is affected by several factors like temperature, aging and many other battery characteristics, making it challenging. In this study, improved central difference transform Kalman filter method based on square root second-order central difference transform (SRCDKF) was utilized for real-time estimation of SOC in LIBs. The hybrid pulse power characterization (HPPC) tests were combined with recursive least squares (RLS) method to identify the second-order equivalent circuit model parameters. To avoid high order Taylor series expansion and complicated multi-parameter adjustment in other Kalman filters, the CDKF with square root second-order difference transform is developed to generate Sigma point. The effectiveness of the proposed SRCDKF was then verified by pulse discharge and urban dynamometer driving schedule (UDDS) testing, and the results were compared with those obtained from extended Kalman filter (EKF) and unscented Kalman filter (UKF). In particular, the proposed algorithm provided small error within 2% under UDDS testing, and the convergence time was earlier than those obtained with the other two algorithms. The proposed SRCDKF can also guarantee the non-negative covariance and reduce the computational complexity. Overall, these findings look promising for future BMS SOC estimation in practice.

Comparison of two algorithms for Harmonic Balance and path continuation

In this work we apply and compare two algorithms for setting up Harmonic Balance equations and numerical continuation of the solution path for harmonically driven mechanical systems. The first algorithm relies on a predictor-corrector scheme and an Alternating Frequency-Time approach (AFT-PreCo). The second algorithm relies on a high-order Taylor series expansion of the solution path (asymptotic numerical method) and classical Harmonic Balance formulated entirely in the frequency domain (cHB-ANM). We conclude that the cHB-ANM is suited for smooth nonlinearities, for instance geometrically nonlinear finite element models. Here, cHB-ANM avoids aliasing errors and convinces with a numerically robust adjustment of the continuation step length and a continuous representation of the solution path. For non-smooth nonlinearities such as stick-slip friction or unilateral constraints, AFT-PreCo is better suited, and convinces with high numerical robustness and efficiency.

Estimation of Non-smooth Functionals in Hilbert Sample Space Using the Edgeworth Expansions

An arbitrary non-smooth functional is estimated using a nonparametric set-up. Exploratory data analysis methods are relied on to come up with the functional form for the sample to allow both robustness and optimality to be achieved. An infinite number of parameters are involved and thus the Hilbert sample space is a natural choice. An important step in understanding this problem is the normal means problem, . The basic difficulty of estimating as defined can be traced back to the non differentiability of the absolute value function, at the origin. Accordingly, constructing an optimal estimator is not easy partly due to the nonexistence of an unbiased estimate of the absolute value function. Therefore, best polynomial approximation was used to smooth the singularity at the origin and then an unbiased estimator for every term in the expansion constructed by use of Hermite polynomials when the averages are bounded by a given constant M > 0 say. The expansion of the Gaussian density function in terms of Hermite polynomials gives a clear and almost accurate estimate that admits cumulant generating function; the Saddle point approximation. Additional precision is obtained by using a higher order Taylor series expansion about the mean resulting in Edgeworth expansion techniques.

A Fast High-Resolution Imaging Algorithm for Helicopter-Borne Rotating Array SAR Based on 2-D Chirp-Z Transform

In view of the azimuth resolution of the helicopter-borne rotating array synthetic aperture radar (RoASAR) depending on the azimuth reconstruction angle and the sector distortion caused by the azimuth Deramp processing, this paper proposes an efficient helicopter-borne RoASAR high-resolution imaging algorithm based on two-dimensional (2-D) chirp-z transform (CZT). First, the high-order Taylor series expansion is performed on the slant range, and the accurate 2-D spectral expression of the point target is obtained by using the method of series reversion (MSR). Based on that, the space-variant characteristics of the range cell migration (RCM) terms are analyzed. After that, the space-variant RCM and the sector distortion effect caused by the azimuth Deramp processing are removed by using efficient 2-D CZT, thereby increasing the azimuth reconstruction angle and improving the azimuth resolution. The proposed algorithm is efficient without the interpolation operation, and it is easy to implement in real-time. Finally, the simulations are provided to verify the effectiveness of the proposed algorithm.

A Screw Approach to the Approximation of the Local Geometry of the Configuration Space and of the Set of Configurations of Certain Rank of Lower Pair Linkages

Abstract A motion of a mechanism is a curve in its configuration space (c-space). Singularities of the c-space are kinematic singularities of the mechanism. Any mobility analysis of a particular mechanism amounts to investigating the c-space geometry at a given configuration. A higher-order analysis is necessary to determine the finite mobility. To this end, past research leads to approaches using higher-order time derivatives of loop closure constraints assuming (implicitly) that all possible motions are smooth. This continuity assumption limits the generality of these methods. In this paper, an approach to the higher-order local mobility analysis of lower pair multiloop linkages is presented. This is based on a higher-order Taylor series expansion of the geometric constraint mapping, for which a recursive algebraic expression in terms of joint screws is presented. An exhaustive local analysis includes analysis of the set of constraint singularities (configurations where the constraint Jacobian has certain corank). A local approximation of the set of configurations with certain rank is presented, along with an explicit expression for the differentials of Jacobian minors in terms of instantaneous joint screws. The c-space and the set of points of certain corank are therewith locally approximated by an algebraic variety determined algebraically from the mechanism's screw system. The results are shown for a simple planar 4-bar linkage, which exhibits a bifurcation singularity and for a planar three-loop linkage exhibiting a cusp in c-space. The latter cannot be treated by the higher-order local analysis methods proposed in the literature.

Angular Motion Decoupling and Attitude Determination Based on High Dynamic Gyro

Aiming at the problem of attitude angular motion coupling of a high-spinning rigid body, the model of the compound regular precession and the random angular motion is constructed by studying the angular motion characteristics of the high-spinning rigid body. This paper presents a new decoupling method with least squares angular rate zero-crossing detection. This method can estimate the coupling parameters in the model by estimating the angular motion period, and then, it extracts the real angular velocity information. In order to improve the attitude solution accuracy, a new four-subsample quaternion attitude algorithm is proposed by combining the angular velocity polynomial fitting and the attitude quaternion high-order Taylor series expansion. The self-developed high dynamic gyro and Missile-borne computer and simulation software are used to verify the attitude angle resolution accuracy between the traditional method and the new method proposed in this paper. The results show that the method is superior to the traditional algorithm, and increase by an order of magnitude. Finally, the flight test with the high dynamic gyro and Missile-borne computer is carried out to verify the real-time decoupling and attitude angle real-time solving ability of the proposed algorithm in the flight test.

An accurate and efficient gridless method based on implicit, fast, and constrained weights optimization schemes for compressible flows

An accurate and efficient gridless method is presented for calculation of inviscid compressible flows in subsonic to supersonic flows. The Taylor series least squares is used for discretization of spatial derivatives at each point. Fast gridless method using the second- and fourth-order artificial dissipation terms is applied for solving the Euler equations. Two methods are proposed for increasing the accuracy; implementing the constrained weights optimization and using high-order Taylor series expansion on the fast gridless method. The explicit and implicit dual-time schemes are applied for temporal discretization. To speed up convergence, local time stepping and residual smoothing techniques are used. The capability and accuracy of the methods are compared with other gridless methods, finite volume method and AGARD data for some test cases in subsonic, transonic and supersonic flows. Results show that the use of the constraint weights optimization increases the accuracy especially with fewer point distributions in comparison with conventional gridless methods and it is more efficient than other gridless methods.

An Improved RD Algorithm for Maneuvering Bistatic Forward-Looking SAR Imaging with a Fixed Transmitter.

In order to improve the azimuth resolution beyond what monostatic synthetic aperture radar (SAR) can achieve in the forward-looking area, an asymmetric configuration bistatic SAR system and its imaging algorithm are proposed in this paper. The transmitter is mounted on a fixed platform in side-looking mode while the receiver moves along a nonlinear trajectory in forward-looking mode. Due to the high velocity and acceleration of the maneuvering platform in both along-track and height direction, the traditional algorithms are no longer applicable. In this paper, a new algorithm based on the high precise 2-D frequency spectrum is proposed, which takes high-order Taylor series expansion terms of the slant range into consideration. The proposed algorithm compensates high-order range-azimuth coupling terms to guarantee the focus accuracy in SAR imaging. The simulation results and error analysis validate the effectiveness of the proposed algorithm and the correctness of our analysis.

High-order Taylor series expansion methods for error propagation in geographic information systems

The quality of modeling results in GIS operations depends on how well we can track error propagating from inputs to outputs. Monte Carlo simulation, moment design and Taylor series expansion have been employed to study error propagation over the years. Among them, first-order Taylor series expansion is popular because error propagation can be analytically studied. Because most operations in GIS are nonlinear, first-order Taylor series expansion generally cannot meet practical needs, and higher-order approximation is thus necessary. In this paper, we employ Taylor series expansion methods of different orders to investigate error propagation when the random error vectors are normally and independently or dependently distributed. We also extend these methods to situations involving multi-dimensional output vectors. We employ these methods to examine length measurement of linear segments, perimeter of polygons and intersections of two line segments basic in GIS operations. Simulation experiments indicate that the fifth-order Taylor series expansion method is most accurate compared with the first-order and third-order method. Compared with the third-order expansion; however, it can only slightly improve the accuracy, but on the expense of substantially increasing the number of partial derivatives that need to be calculated. Striking a balance between accuracy and complexity, the third-order Taylor series expansion method appears to be a more appropriate choice for practical applications.