High-order Taylor Series Expansion Research Articles

Effect of Higher Order Taylor Series Expansion Terms of the NI-RPIM on the Solution Accuracy of 2D Elastic Problems

Bu calismada, yuksek mertebeden Taylor serisi acilim terimlerinin radyal nokta interpolasyon yontemi dugum entegrasyon semasi (NI-RPIM) acisindan 2 boyutlu elastik problemlerin cozum dogruluguna etkileri incelenmistir. Dugum integrasyon semasi Liu ve dig. [1] tarafindan tasarlanmistir ve Taylor serisi acilimi uzerinedir. Bu calismada 4. mertebeye kadar terimleri arttirilarak kullanilmistir. 3 farkli alan calismasi yapilmis ve sonuclari analitik, sonlu elemanlar yontemi ve Gauss integrasyonlu RPIM sonuclari ile kisaylanmistir. Ayrica nokta sayisinin etkisi arastirilmistir. RPIM integrasyonu icerisinde kullanilan Taylor serisi acilimi ve Gauss metodu benzer cozum zamanlari verdigi kabul edilebilir. Bununla birlikte ozellikle yuksek sayida nokta iceren modellerin cozumunde, NI-RPIM ile yuksek mertebeden Taylor serisi acilimi terimleri Gauss integrasyonundan daha iyi cozum hizina sahiptir. 2. mertebeden dugum integrasyon terimlerinin yeterli sonuc verdigi belirlenmistir. Eger gerilme degerleri incelenmekteyse, cozum hassasiyeti icin 4. mertebe dugum integrasyon terimleri kullanilabilinir.

Research on Non-Linear Parameters Identification for Servo System of CNC Machine Tools

As phenomena of friction in computer numerical control(CNC) machine tools is inevitable, accurately modeling of friction is need for precise position control and performance evaluation of CNC machine tools. This paper proposed an identification method that non-linear friction model described as Stribeck are linearized by using high order Taylor series expansion and parameters of the model are obtained by least square method (LSM). Signals of servo motor current and rotating rate which are offered in many modern CNC machine tools are needed for the identification and experiments results show that parameters of Stribeck model can be obtained accurately by identification. The method is suited for industrial condition and has its practicality.

Optimal design of an automotive fan using the Turb’Opty meta-model

Optimal design techniques recently gained a wide popularity in the industry as relatively powerful computers have become broadly available and as attractive tools such as surrogate models and evolutionary optimization went through maturation. However, when dealing with complex geometries and difficult physical phenomena to be modeled, computing costs still remain high, due to the large number of required numerical simulations feeding the traditional surrogate models. Turb’Opty© is a meta-model which only requires a single CFD simulation at a reference configuration point, based on automatic differentiation of the discretized Reynolds-Averaged Navier–Stokes equations and high-order Taylor-series expansions. A flow database containing the derivatives of the physical variables with respect to the design variables is produced by this parameterization tool and thoroughly explored, in the post-processing step, by a multi-parameter and multi-objective genetic algorithm coupled to the associated extrapolation tool. In this paper, post-processing of the derivative database will be depicted through a 3D study of an automotive shrouded fan with casing treatment.

Sensitivities of effective properties computed using micromechanics differential schemes and high-order Taylor series: Application to piezo-polymer composites

This paper presents an approach using micromechanics differential schemes to compute the effective properties of composite materials and their first order sensitivities. The approach is based on a high-order Taylor series expansion of the initial value problem of the differential schemes and the usage of automatic differentiation tools. Numerical results are presented for electro-elastic properties of two-phase piezo-polymer composite materials. The comparison of the results to those obtained by CVODES which is one of the most advanced dynamic sensitivity solvers show the effectiveness of the method. This approach may be useful for microstructure sensitive design by a gradient-based optimization method or may be used to carry out sensitivity analysis around the optimal point after an optimal microstructure design.

High-order robust guidance of interplanetary trajectories based on differential algebra

Space trajectory design always requires the solution of an optimal control problem in order to maximize the payload launch-mass ratio while achieving the primary mission goals. A certain level of approximation always characterizes the dynamical models adopted to perform the design process. Furthermore the state identification is usually affected by navigation errors. Thus, after the nominal optimal solution is computed, a control strategy that assures the execution of mission goals in the real scenario must be implemented. In this frame differential algebraic techniques are here proposed as an effective alternative tool to design the guidance law. By using differential algebra the final state dependency on initial conditions, environmental and control parameters is represented by high order Taylor series expansions. The mission constraints can then be solved to high order using a so-called high order partial inversion of the polynomial relationship for every admissible uncertainty. The control strategy is eventually reduced to a simple function evaluation. The performances of the proposed methods are assessed by two examples of space mission trajectory design: a continuous propelled Earth-Mars transfer and an aerocapture maneuver at Mars.

A sensitivity equation method for fast evaluation of nearby flows and uncertainty analysis for shape parameters†

This paper presents the application of the continuous sensitivity equation method (CSEM) to fast evaluation of nearby flows and to uncertainty analysis for shape parameters. The flow and sensitivity fields are solved using an adaptive finite-element method. A new approach is presented to extract accurate flow derivatives at the boundary, which are needed in the shape sensitivity boundary conditions. Boundary derivatives are evaluated via high order Taylor series expansions used in a constrained least-squares procedure. The proposed method is first applied to fast evaluation of nearby flows: the baseline flow and sensitivity fields around a NACA 0012 airfoil are used to predict the flow around airfoils with nearby shapes obtained by modifications of the thickness (NACA 0015), the angle of attack and the camber (NACA 4512). The method is then applied to evaluate the influence of geometrical uncertainties on the flow around a NACA 0012 airfoil.

On accurate boundary conditions for a shape sensitivity equation method

AbstractThis paper studies the application of the continuous sensitivity equation method (CSEM) for the Navier–Stokes equations in the particular case of shape parameters. Boundary conditions for shape parameters involve flow derivatives at the boundary. Thus, accurate flow gradients are critical to the success of the CSEM. A new approach is presented to extract accurate flow derivatives at the boundary. High order Taylor series expansions are used on layered patches in conjunction with a constrained least‐squares procedure to evaluate accurate first and second derivatives of the flow variables at the boundary, required for Dirichlet and Neumann sensitivity boundary conditions. The flow and sensitivity fields are solved using an adaptive finite‐element method. The proposed methodology is first verified on a problem with a closed form solution obtained by the Method of Manufactured Solutions. The ability of the proposed method to provide accurate sensitivity fields for realistic problems is then demonstrated. The flow and sensitivity fields for a NACA 0012 airfoil are used for fast evaluation of the nearby flow over an airfoil of different thickness (NACA 0015). Copyright © 2005 John Wiley & Sons, Ltd.

Deriving derivatives of derivative securities

Various techniques are used to simplify the derivations of "greeks" of path-independent claims in the Black-Merton-Scholes model. First, delta, gamma, speed, and other higher-order spatial derivatives of these claims are interpreted as the values of certain quantoed contingent claims. It is then shown that all partial derivatives of such claims can be represented in terms of these spatial derivatives. These observations permit the rapid deployment of high-order Taylor series expansions, and this is illustrated for the case of European options.

Padé approximant algorithm for solving nonlinear ordinary differential equation boundary value problems on an unbounded domain

We describe a four-step algorithm for solving ordinary differential equation nonlinear boundary-value problems on infinite or semi-infinite intervals. The first step is to compute high-order Taylor series expansions using an algebraic manipulation language such as Maple or Mathematica. These expansions will contain one or more unknown parameters z which will be determined by the boundary condition at infinity. The second step is to convert the Taylor expansions into diagonal Padé approximants. The boundary condition that u(x) decays to zero at infinity becomes the condition that the coefficient of the highest power of x in the numerator polynomial must be zero. The third step is to solve this equation for the free parameter z. The final step is to evaluate each of the multiple solutions of this equation for physical plausibility and convergence (as N increases). This algorithm can be implemented in as few as seven lines of Maple (sample program provided!). We illustrate the method with three examples: the Flierl–Petviashvili vortex of geophysical fluid dynamics, the quartic oscillator of quantum mechanics, and the Blasius function for the boundary layer above a semi-infinite plate in fluid mechanics. Methods for nonlinear problems are almost always iterative and need a first guess to initialize the iteration. The Padé algorithm is unusual in that it is a direct method that requires no a priori information about the solution. © 1997 American Institute of Physics.

Applicability of a Higher-Order Taylor Series Expansion Method for Random Wave Simulations

Applicability of a Higher-Order Taylor Series Expansion Method for Random Wave Simulations

Applicability of a Higher Order Taylor Series Expansion Method for Random Wave Simulations.

Applicability of a Higher Order Taylor Series Expansion Method for Random Wave Simulations.

Numerical Simulation of Turbulent Shear Flow Using Higher Order Taylor Series Expansion Method

Numerical Simulation of Turbulent Shear Flow Using Higher Order Taylor Series Expansion Method

Numerical Simulation of Turbulent Shear Flow Using Higher Order Taylor Series Expansion Method.

A higher order Taylor series expansion method is applied to highly accurate numerical simulation of turbulent flow. In the present study, central difference scheme-like methods are adopted for an even expansion order, and upwind difference scheme-like methods are adopted for an odd order. The latter methods provide reduced numerical errors, but also numerical instability, arising from the convection terms in the governing equations. The numerical simulation of turbulent shear flow in a rectangular channel (Reynolds number = 1.7 × 10 4 ) is carried out. The numerical velocity fields are compared with experimental results measured by LDV, and it is confirmed that the higher and odd order methods can simulate mean velocity distributions, root-mean-square velocity fluctuations and Reynolds stress with good agreement