Truncated Taylor Series Research Articles

A novel and effective way to impose boundary conditions and to mitigate the surface effect in state‐based Peridynamics

AbstractPeridynamics is a nonlocal continuum theory capable of modeling effectively crack initiation and propagation in solid bodies. However, the nonlocal nature of this theory is the cause of two main problems near the boundary of the body: an undesired stiffness fluctuation, the so‐called surface effect, and the difficulty of defining a rational method to properly impose the boundary conditions. The surface effect is analyzed analytically and numerically in the present paper in a state‐based peridynamic model. The authors propose a modified fictitious node method based on an extrapolation with a truncated Taylor series expansion. Furthermore, a rational procedure to impose the boundary conditions is defined with the aid of the fictitious nodes. In particular, Neumann boundary conditions are implemented via the peridynamic concept of force flux. The accuracy of the proposed method is assessed by means of several numerical examples for a state‐based peridynamic model: with respect to the peridynamic model adopting no corrections, the results are significantly improved even if low values of the truncation order for the Taylor expansion are chosen.

Derivation of the differential continuity equation in an introductory engineering fluid mechanics course

The differential continuity equation is elegantly derived in advanced fluid mechanics textbooks using the divergence theorem of Gauss, where the surface integral of the mass flux flowing out of a finite control volume is replaced by the volume integral of the divergence of the mass flux within the control volume. To avoid the need for introducing the Gauss divergence theorem in an introductory fluid mechanics course, introductory textbooks in fluid mechanics have opted to use a more simple approach, which depends on the consideration of an infinitesimal control volume and the use of Taylor series expansion. This approach, however, involves a first order truncation of the Taylor series expansion and the use of approximate equality signs which may imply to undergraduate students that the derived continuity equation is an approximate equation. The present study proposes an alternative derivation of the differential continuity equation using a finite control volume and is based on the simple concept of the antiderivative function and the fundamental theorem of calculus. The proposed derivation eliminates the need to formally introduce the Gauss divergence theorem in an introductory engineering fluid mechanics course while avoiding the use of truncated Taylor series expansion and approximate equality signs, hence providing a more simple and sound understanding of the derivation of the differential continuity equation to undergraduate engineering students.

Accurate Real-time CNC Curve Interpolators Based Upon Richardson Extrapolation

Real-time CNC interpolators achieving a constant or variable feedrate V along a parametric curve r(ξ) are usually based on truncated Taylor series expansions defining the time-dependence of the curve parameter ξ. Since the feedrate should be specified as a function of a physically meaningful variable (such as time t, path arc length s, or curvature κ) rather than ξ, successive applications of the differentiation chain rule are necessary to determine Taylor series coefficients beyond the linear term. The closed-form expressions for the higher-order coefficients are increasingly cumbersome to derive and implement, and consequently error-prone. To address this issue, the use of Richardson extrapolation as a simple means to compute rapidly convergent approximations to the higher-order coefficients is investigated herein. The methodology is demonstrated in the context of (1) an arc-length-dependent feedrate for cornering motions; (2) direct real-time offset curve interpolation; and (3) a curvature-dependent feedrate. All of these examples admit simple implementations that circumvent the need for tedious symbolic calculations of higher-order coefficients, and are compatible with real-time controllers with millisecond sampling intervals.

Resolution Learning in Deep Convolutional Networks Using Scale-Space Theory.

Resolution in deep convolutional neural networks (CNNs) is typically bounded by the receptive field size through filter sizes, and subsampling layers or strided convolutions on feature maps. The optimal resolution may vary significantly depending on the dataset. Modern CNNs hard-code their resolution hyper-parameters in the network architecture which makes tuning such hyper-parameters cumbersome. We propose to do away with hard-coded resolution hyper-parameters and aim to learn the appropriate resolution from data. We use scale-space theory to obtain a self-similar parametrization of filters and make use of the N-Jet: a truncated Taylor series to approximate a filter by a learned combination of Gaussian derivative filters. The parameter σ of the Gaussian basis controls both the amount of detail the filter encodes and the spatial extent of the filter. Since σ is a continuous parameter, we can optimize it with respect to the loss. The proposed N-Jet layer achieves comparable performance when used in state-of-the art architectures, while learning the correct resolution in each layer automatically. We evaluate our N-Jet layer on both classification and segmentation, and we show that learning σ is especially beneficial when dealing with inputs at multiple sizes.

QSW_MPI: A framework for parallel simulation of quantum stochastic walks

QSW_MPI is a Python package developed for time-series simulation of continuous-time quantum stochastic walks. This model allows for the study of Markovian open quantum systems in the Lindblad formalism, including a generalisation of the continuous-time random walk and continuous-time quantum walk. Consisting of a Python interface accessing parallelised Fortran libraries utilising sparse data structures, QSW_MPI is scalable to massively parallel computers, which makes possible the simulation of a wide range of walk dynamics on directed and undirected graphs of arbitrary complexity. Program summaryProgram Title: QSW_MPICPC Library link to program files:http://dx.doi.org/10.17632/r8jsx4xxgv.1Licensing provisions: GPLv3Programming language: Python 3 + Fortran 2003External routines/libraries: NumPy [1], SciPy [2], MPI for Python [3], h5py[4]Nature of problem: QSW_MPI provides a framework for the simulation of quantum stochastic walks on arbitrary graphs (directed/undirected, weighted/unweighted).Solution method: A parallel distributed-memory implementation of the matrix exponential via a truncated Taylor series expansion with scaling and squaring [5].Additional comments including restrictions and unusual features: QSW_MPI will provide support for the simulation of multiple quantum walkers in a future version.

A note on Pad\xe9 approximants of tensor logarithm with application to Hencky-type hyperelasticity

We show that the logarithmic (Hencky) strain and its derivatives can be approximated, in a straightforward manner and with a high accuracy, using Padé approximants of the tensor (matrix) logarithm. Accuracy and computational efficiency of the Padé approximants are favourably compared to an alternative approximation method employing the truncated Taylor series. As an application, Hencky-type hyperelasticity models are considered, in which the elastic strain energy is expressed in terms of the Hencky strain, and of our particular interest is the anisotropic energy quadratic in the Hencky strain. Finite-element computations are carried out to examine performance of the Padé approximants of tensor logarithm in Hencky-type hyperelasticity problems. A discussion is also provided on computation of the stress tensor conjugate to the Hencky strain in a general anisotropic case.

New method for solving fractional partial integro-differential equations by combination of Laplace transform and resolvent kernel method

In this article, we obtained the approximate solution for a new class of Time-Fractional Partial Integro-Differential Equation (TFPIDE) of the Caputo-Volterra type in which the integral is not limited to the convolution type. This new class of TFPIDE is distinct from the common problem with the convolution integral kernel. The general expression of the analytical solution for this special type of TFPIDE was derived using a combination of Laplace transform and the resolvent kernel method. In the process, Laplace transform will transform the equation into a second kind Volterra integral equation in terms of the transformed function. Two main problems in deriving the approximate analytical solutions were identified as Case I and Case II problems. To obtain the approximate solutions for Case I and Case II problems, numerical methods were designed based on approximation of the resolvent kernel with truncated Neumann series as well as approximation of the Laplace transform based on truncated Taylor series. Several numerical examples are presented to indicate the plausibility, mechanism and performance of the proposed methods.

A Novel Approximation Methodology and Its Efficient VLSI Implementation for the Sigmoid Function

In this brief, a novel approximation method and its optimized hardware implementation are proposed for the sigmoid function used in Deep Neural Networks (DNNs). Based on piecewise approximation and truncated Taylor series expansion, the proposed method achieves very good approximation with low complexity while exploiting data representation with powers of two. In addition, by analyzing gradients of the sigmoid function, a small trick is introduced to improve the approximation precision. Furthermore, to reduce the hardware complexity and shorten the critical path, sampled values of the function are generated with simple logical-mapping. It is shown that the proposed approximation schemes can be implemented with purely combinational logic and the sigmoid function can be computed in one clock cycle. The experimental results demonstrate that the mean absolute errors are at the order of 1 × 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-3</sup> . Compared with prior arts, the new design can obtain significant improvement in critical path with comparable performance.

Comparison of the Solution of the Van der Pol Equation Using the Modified Adomian Decomposition Method and Truncated Taylor Series Method

In this paper, we compare the solution of the van der Pol equation obtained by using the truncated Taylor series method and the modified Adomian decomposition method with the solution obtained by the Poincare-Lindstedt (P-L) method. The approximating 4-component modified Adomian decomposition method behaves more like an approximate P-L analytic method than the tenth-order Taylor series. Also, with the addition of just one term, the approximating 5-component modified Adomian decomposition method produces a more convergent solution to that of P-L analytic method than the twenty second-order Taylor series approximation as the independent variable t representing time progressively increases. A general comparison of the two solutions revealed that the absolute errors generated by the approximating polynomial from the Taylor series are greater than the ones generated from the modified Adomian decomposition method. It was further revealed that very few components of the modified Adomian decomposition could yield a series of about 3 times the order of the one obtained by using the Taylor series method. Hence, we recommend the inclusion of the modified Adomian Decomposition Method in modern mathematical tools.

Industry Tip: Quick and Easy Matrix Exponentials

Computation of matrix exponentials is commonly encountered in aerospace and electronic systems. Computing matrix exponentials via a reverse summation of the truncated Taylor series is discussed.

Some practical regards on the application of the harmonic balance method for hysteresis models

Describing hysteretic systems with a closed-form solution is a challenging task due to some pitfalls regarding the non-smooth and memory effect mechanisms that do not permit, for example, to apply conventional frequency domain methods. Consequently, it is necessary to use some previous smoothing scheme to approximate the hysteresis loop. Thus, this work proposes a new way for approximating the hysteresis loops analytically using a truncated Taylor series as a simple and effective smoothing procedure to enable the use of the harmonic balance method. Two benchmark hysteretic systems, which were not addressed yet by closed-form solutions obtained by the harmonic balance method, are simulated to demonstrate the benefits of the proposed strategy. The first one is a Bouc-Wen oscillator and the second one is a LuGre model. The comparison with numerical integrations and other literature methods have shown that the obtained analytical solutions of the suggested smoothed hysteresis loops are adequate to describe the fundamental dynamics in both models using a feasible frequency domain approximation.

Approximate Nonlinear Discrete-Time Models Based on B-Spline Functions

We consider the discretization of continuous-time nonlinear systems described by normal forms. In particular, we consider the case when the input to the system is generated by a B-spline hold device to obtain an approximate discrete-time model. It is shown that the corresponding sampled-data model and its accuracy (measured in terms of the local truncation error) depend on the smoothness of the input and on the applied integration strategy, namely, the truncated Taylor series expansion. Moreover, the sampling zero dynamics of the discrete-time model are asymptotically characterized as the sampling period goes to zero, and it is shown that these zero dynamics converge to the asymptotic sampling zeros of the linear case.

Numerical solution of the Kirchhoff-transformed Richards equation for simulating variably saturated flow in heterogeneous layered porous media

A new numerical method was developed to accurately and efficiently compute a solution of the nonlinear Richards equation with a layered soil. In the proposed method, the Kirchhoff integral transformation was applied. However, in the Kirchhoff integral transformation approach, the transformed Kirchhoff head has dyadic characteristics at the material interface between different soil types. To avoid the dyadic characteristics at the material interface, a truncated Taylor series expansion was applied to the Kirchhoff head at the material interface and so the Kirchhoff head was replaced with a single pressure head value at the material interface. Accordingly, through the Taylor series expansion, a set of algebraic equations in the one-dimensional control volume finite difference discretized system formed a tridiagonal matrix system. Through a series of numerical experiments, the new method was compared to other numerical methods to determine its superiority. The results clearly demonstrated that the approach was not only more computationally efficient, but also more accurate and robust than other numerical methods. Computational performance was greatly enhanced with the proposed method, and which could be used to simulate complicated heterogeneous flow at a large-scale watershed or regional scale.

On Some Sufficiency-Type Global Stability Results for Time-Varying Dynamic Systems with State-Dependent Parameterizations

This paper formulates sufficiency-type global stability and asymptotic stability results for, in general, nonlinear time-varying dynamic systems with state-trajectory solution-dependent parameterizations. The stability proofs are based on obtaining sufficiency-type conditions which guarantee that either the norms of the solution trajectory or alternative interval-type integrals of the matrix of dynamics of the higher-order than linear terms do not grow faster than their available supremum on the preceding time intervals. Some extensions are also given based on the use of a truncated Taylor series expansion of chosen truncation order with multiargument integral remainder for the dynamics of the differential system.

Algorithmic Pirogov–Sinai theory

We develop an efficient algorithmic approach for approximate counting and sampling in the low-temperature regime of a broad class of statistical physics models on finite subsets of the lattice $\mathbb Z^d$ and on the torus $(\mathbb Z/n \mathbb Z)^d$. Our approach is based on combining contour representations from Pirogov-Sinai theory with Barvinok's approach to approximate counting using truncated Taylor series. Some consequences of our main results include an FPTAS for approximating the partition function of the hard-core model at sufficiently high fugacity on subsets of $\mathbb Z^d$ with appropriate boundary conditions and an efficient sampling algorithm for the ferromagnetic Potts model on the discrete torus $(\mathbb Z/n \mathbb Z)^d$ at sufficiently low temperature.

A numerical method for solving continuous population models for single and interacting species

In this study, a numerical approach is presented to obtain the approximate solutions of continuous population models for single and interacting species. This method is essentially based on the truncated Taylor series and its matrix representations with collocation points. By using Taylor polynomials and collocation points, this method transforms population models into a matrix equation. The matrix equation corresponds to a system of nonlinear equations with the unknown Taylor coefficients. To illustrate reliability and efficiency of the method, numerical examples are presented and results are compared with the other numerical methods. Additionally, residual correction procedure is applied to estimate the absolute errors. All numerical computations have been performed on the computer algebraic system Maple 15.

Optimal Design of Double Pipe Heat Exchanger Structures

This paper investigates the design optimization of double pipe heat exchanger using mathematical programming. The heat exchanger area is minimized and the thermo-fluid dynamic conditions are considered for the use of the right transport correlations, together with design specifications, such as, maximum pressure drops and minimum excess area. The modular nature of this kind of heat exchanger and the allocation of the streams (inside the inner tube or in the annulus) are also contemplated. Two mixed-integer nonlinear programming (MINLP) approaches are proposed. One approach relates the binary variables to the nonlinear constraints directly. In the second, the resulting nonlinearities involving binary variables are formally linearized, without loss of rigor (e.g., no use of truncated Taylor series). The proposed methodology can get better solutions than traditional trial and error procedures. The flexibility of the model is illustrated, together with a comparison between the performances of both MINLP formulations. Additionally, computational time and local optimality issues are discussed.

Series solution of fractional Pantograph equations via Taylor series

This article is devoted to develop a numerical approximation called Taylor minimization method for initial and boundary value fractional Pantograph equations, which governs the modelling of the train system, with neutral and multi-term delays. Taylor optimization technique is basically composed of truncated Taylor series approximation of unknown function while employment of procedure is accompanied by an optimization strategy that is simulated annealing for carrying out the learning phase of unknown Taylor series coefficients. The proposed technique is implemented on various models of Pantograph equations to study the applicability and effectiveness of the planned scheme while error analysis and comparison with previous methods are performed to validate the results. To measure the capability of convergence the data for 100 numbers of independent runs is demonstrated in the form of pictorial presentation.

Arbitrary-order time-accurate semi-Lagrangian spectral approximations of the Vlasov–Poisson system

The Vlasov–Poisson system, modeling the evolution of non-collisional plasmas in the electrostatic limit, is approximated by a semi-Lagrangian technique. Spectral methods of periodic type are implemented through a collocation approach. Groups of particles are represented by the Fourier Lagrangian basis and evolve, for a single timestep, along a high-order accurate representation of the local characteristic lines. The time-advancing technique is based on truncated Taylor series that can be, in principle, of any order of accuracy. A variant is obtained by coupling the phase space discretization with high-order accurate Backward Differentiation Formulas (BDF). At each timestep, particle displacements are reinterpolated and expressed in the original basis to guarantee the order of accuracy in all the variables at relatively low costs. Thus, these techniques combine the excellent features of spectral approximations with high-order time integration. The resulting method has excellent conservation properties. Indeed, it can be proven that the total number of particles, proportional to the total mass and charge, is conserved up to the machine precision. Series of numerical experiments are performed in order to assess the real performance. In particular, comparisons with standard benchmarks are examined.

A generalised and low-dissipative multi-directional characteristics-based scheme with inclusion of the local Riemann problem investigating incompressible flows without free-surfaces

In the present study, we develop a generalised Godunov-type multi-directional characteristics-based (MCB) scheme which is applicable to any hyperbolic system for modelling incompressible flows. We further extend the MCB scheme to include the solution of the local Riemann problem which leads to a hybrid mathematical treatment of the system of equations. We employ the proposed scheme to hyperbolic-type incompressible flow solvers and apply it to the Artificial Compressibility (AC) and Fractional-Step, Artificial Compressibility with Pressure Projection (FSAC-PP) method. In this work, we show that the MCB scheme may improve the accuracy and convergence properties over the classical single-directional characteristics-based (SCB) and non-characteristic treatments. The inclusion of a Riemann solver in conjunction with the MCB scheme is capable of reducing the number of iterations up to a factor of 4.7 times compared to a solution when a Riemann solver is not included. Furthermore, we found that both the AC and FSAC-PP method showed similar levels of accuracy while the FSAC-PP method converged up to 5.8 times faster than the AC method for steady state flows. Independent of the characteristics- and Riemann solver-based treatment of all primitive variables, we found that the FSAC-PP method is 7–200 times faster than the AC method per pseudo-time step for unsteady flows. We investigate low- and high-Reynolds number problems for well-established validation benchmark test cases focusing on a flow inside of a lid driven cavity, evolution of the Taylor–Green vortex and forced separated flow over a backward-facing step. In addition to this, comparisons between a central difference scheme with artificial dissipation and a low-dissipative interpolation scheme have been performed. The results show that the latter approach may not provide enough numerical dissipation to develop the flow at high-Reynolds numbers. We found that the inclusion of a Riemann solver is able to overcome this shortcoming. Overall, the proposed generalised Godunov-type MCB scheme provides an accurate numerical treatment with improved convergence properties for hyperbolic-type incompressible flow solvers. Program summaryProgram Title: unified2D-CProgram Files doi:http://dx.doi.org/10.17632/8m3dw6zkgc.1Licensing provisions: CC BY NC 3.0Programming language: C++Nature of problem: Incompressible flow solver have generally a non-hyperbolic type and thus the method of characteristics and Riemann solvers cannot be used without modifications for low speed flows. In the framework of compressible flows, the Riemann problem – and the method of characteristics which is closely related to it – is an essential part of the solution procedure. The Riemann solver is able to preserve the conservativeness and, through the evaluation of the eigenstructure of the system, introduces transportiveness into the spatial reconstruction schemes. The characteristics-based scheme allows to couple the pressure and velocity in a physical manner which, together with the Riemann solver, presents a new multi-directional Godunov framework for incompressible flows.Solution method: We show a generalised description of a multi-directional characteristics-based schemes which may be used with any incompressible and hyperbolic system of equations. The Finite Volume approach is used where inter-cell fluxes are reconstructed through a simple but higher-order polynomial interpolation scheme which only adds numerical dissipation proportional to its Taylor-series truncation error. We use the Rusanov Riemann solver which provides the needed conservativeness and transportiveness. It also adds just enough numerical dissipation for cases where the dissipation of the numerical scheme is not sufficient while retaining a high level of accuracy.