Efficient Numerical Integration Algorithm Research Articles

Efficient Numerical Integration Algorithm of Probabilistic Risk Assessment for Aero-Engine Rotors Considering In-Service Inspection Uncertainties

Numerical integration methods have the characteristics of high efficiency and precision, making them attractive for aero-engine probabilistic risk assessment and design optimization of an inspection plan. One factor that makes the numerical integration method a suitable approach to in-service inspection uncertainties is the explicit derivation of the integration formula and integration domains. This explicit derivation ensures accurate characterization of a multivariable system’s failure risk evolution mechanism. This study develops an efficient numerical integration algorithm for probabilistic risk assessment considering in-service inspection uncertainties. The principle of probability conservation is applied to the transformation of the integration domain from the current flight cycle to the initial (N = 0) computational space. Consequently, the integration formula of failure probability is deduced, and a detailed mathematical demonstration of the proposed method is provided. An actual compressor disk is evaluated using the efficient numerical integration algorithm and the Monte Carlo simulation to validate the accuracy and efficiency of the proposed method. Results show that the time cost of the proposed algorithm is dozens of times lower than that of the Monte Carlo simulation, with a maximum relative error of 5%. Thus, the efficient numerical integration algorithm can be applied to failure analysis in the airworthiness design of commercial aero-engine components.

On the representation and implicit integration of general isotropic elastoplasticity based on a set of mutually orthogonal unit basis tensors

SUMMARYA simple and compact representation framework and the corresponding efficient numerical integration algorithm are developed for constitutive equations of isotropic elastoplasticity. Central to this work is the utilization of a set of mutually orthogonal unit tensor bases and the corresponding invariants. The set of bases can be regarded equivalently as a local cylindrical coordinate system in the three‐dimensional coaxial tensor subspace, namely, the principal space. The base tensors are given in the global coordinate system. Similar to the principal space approach, the proposed method reduces the problem dimension from six to three. In contrast to the conventional approach, the transformation procedure between the principal space and the general space is avoided and explicit computation of the principal axes is bypassed. With the proposed technique, the matrices, which need to be inverted during iteration, take a simple form for the great majority of constitutive equations in use. The tangent operator consistent with the proposed algorithm can be decomposed into the direct sum of two linear maps over the coaxial tensor subspace and the subspace orthogonal to it. Consequently, its closed form is derived in an extremely simple manner. Finally, numerical examples demonstrate the high quality performances of the proposed scheme. Copyright © 2014 John Wiley & Sons, Ltd.

An efficient numerical integration algorithm for single-layer Raster cellular neural networks simulator

An efficient numerical integration algorithm for single layer Raster cellular neural networks(CNN) simulator is presented in this paper. The simulator is capable of performing CNN simulations for any size of input image, thus a powerful tool for researchers investigating potential applications of CNN. This paper reports an efficient algorithm exploiting the latency properties of CNN along with numerical integration techniques; these numerical integration techniques are Euler, Runge Kutta fourth-second order 4(2), and Runge Kutta fourth-third order 4(3). Simulation results and comparisons are also presented. Comparisons include CPU time, quality measures of the pictures used in comparison, step size used, the format of input pictures. The CPU time used here is the simulation time. Key words: Single-layer cellular neural networks, numerical integration algorithms, RK 4(2), RK 4(3).

System of second order robot arm problem by an efficient numerical integration algorithm

A robot arm problem solved by adapting RK-sixth order algorithm is presented in this paper. Results and comparison show the efficiency of the RK-sixth order algorithm based on the absolute error between the exact and approximate solutions at different time intervals. In the present paper, a corrective measure has been taken to obtain the stability polynomial, the ranges for the real part of λh and graphical representation of the stability regions for the case of RK-Butcher algorithm, obtained by Sekar et al. (2004), Murugesan et al. (2004, 2005), Murugesh and Murugesan (2009) and Park et al. (2004a, 2004b). It is found that the error involved in the numerical solution obtained by RK-sixth order algorithm is less in comparison with that obtained by the RK-fifth order and RK-fourth order algorithms respectively.

Fast Fourier-Galerkin methods for first-kind logarithmic-kernel integral equations on open arcs

We propose a fully discrete fast Fourier-Galerkin method for solving an integral equation of the first kind with a logarithmic kernel on a smooth open arc, which is a reformulation of the Dirichlet problem of the Laplace equation in the plane. The optimal convergence order and quasi-linear complexity order of the proposed method are established. A precondition is introduced. Combining this method with an efficient numerical integration algorithm for computing the single-layer potential defined on an open arc, we obtain the solution of the Dirichlet problem on a smooth open arc in the plane. Numerical examples are presented to confirm the theoretical estimates and to demonstrate the efficiency and accuracy of the proposed method.

Numerical integration algorithm of a new model for metal plasticity and fracture including pressure and lode angle dependence

This paper describes an efficient numerical integration algorithm of a new model for metal plasticity and fracture [2]. The constitutive equations of the material model critically include both the effect of pressure through the triaxiality ratio and the effect of third deviatoric stress invariant through the lode angle in the description of material. The necessary steps required to implement the model within an implicit quasi-static finite element environment are discussed. In particular, the stress update procedure, which is based on the so-called operator split concept resulting in the standard elastic predictor/return mapping algorithm, and the computation of tangent matrix consistent with the stress update are described. Finally, the convergence for large increments is illustrated and iso-error maps are used to analyse the accuracy and stability of the integration algorithm. The simulation of a cylindrical notched bar specimen subjected to tension [1] is presented to illustrate the robustness of the proposed algorithm.

Efficient Lie-Poisson Integrator for Secular Spin Dynamics of Rigid Bodies

A fast and efficient numerical integration algorithm is presented for the problem of the secular evolution of the spin axis. Under the assumption that a celestial body rotates around its maximum moment of inertia, the equations of motion are reduced to the Hamiltonian form with a Lie-Poisson bracket. The integration method is based on the splitting of the Hamiltonian function, and so it conserves the Lie-Poisson structure. Two alternative partitions of the Hamiltonian are investigated, and second-order leapfrog integrators are provided for both cases. Non-Hamiltonian torques can be incorporated into the integrators with a combination of Euler and Lie-Euler approximations. Numerical tests of the methods confirm their useful properties of short computation time and reliability on long integration intervals.

Computational Analysis of Quality Reduction during Drying of Lumber due to Irrecoverable Deformation. II: Algorithmic Aspects and Practical Application

This paper describes the development of an efficient numerical integration algorithm for an orthotropic viscoelastic-mechanosorptive-plastic material model for wood, which was presented in Part I of this serial. This development utilizes both a semianalytical algorithm for the viscoelastic parts, and the return-mapping algorithm known from plasticity for the mechanosorptive-plastic part of the model. This yields an easily computed, closed form expression for the consistent tangent operator, and, as a consequence, a quadratic rate of asymptotic convergence on the global level of the incremental-iterative analysis by means of the FEM. To prove the practical applicability of the presented formulation, a numerical study of the so-called case-hardening phenomenon as observed on fast dried pine wood was performed. Selected results are presented and discussed.

Peak profile function for synchrotron X-ray diffractometry

A formula of the instrumental function for a high-resolution synchrotron X-ray diffractometer, equipped with a flat crystal analyser and a set of Soller slits for limiting the axial divergence of the diffracted beam, has been derived. The formula incorporates the effects of (i) the axial divergence of the diffracted beam limited by the Soller slits, (ii) the Bragg angle of the flat crystal analyser, and (iii) the tilt angle defined as the deviation of the normal direction of the analyser face from the goniometer plane. The model profile function given by the convolution of a Lorentzian function with the instrumental function has been applied to fit the experimental diffraction peak profiles of standard Si powder (NIST SRM640b) measured with a high-resolution synchrotron X-ray diffractometer, MDS, on beamline BL4B2 at the Photon Factory in Tsukuba. The convolution has been calculated by applying an efficient algorithm for numerical integration. The profile function reproduces not only the experimental profiles measured with a well aligned crystal analyser, but also significantly distorted profiles arising from misalignment of the analyser, withRpvalues within 1.4%, by varying only the instrumental parameter for the tilt angle. It is suggested that further convolution with a Gaussian distribution is practically not necessary for the model instrumental function to fit the data collected with MDS. More rapid computation can be achieved by applying an analytical formula of the profile function, when the tilt angle of the crystal analyser is within about 0.2°.

Sommerfeld Integrals in Modeling Interconnects and Microstrip Elements in Multi-Layered Media

ABSTRACT An efficient numerical integration algorithm is developed to evaluate the spatial-domain Green's function of vector and scalar potentials. A rigorous formulation for the Green's function in multi-layered media is derived in the spectral domain involving Sommerfeld-type integrals. This approach requires little analytic derivation for singular terms, and reduces the computation effort by using the weighted-average method to evaluate the oscillatory tail integrals. It is found that this algorithm is very efficient and suitable to model interconnects and microstrip elements for printed circuits/antennas with arbitrary size and any number of layers.

Non-muffin-tin atomic scattering-matrices for semiconductor LEED-calculations

Non-muffin-tin potentials for LEED-calculations can be treated by the integration of the Lippmann-Schwinger equation as described by Nagano and Tong. We have solved this coupled system of equations numerically with a code for ordinary differential equations. The general scattering potential is generated by superimposing atomic Clementi-Roetti densities and Coulomb potentials for a cluster of crystal atoms. The exchange potential is modeled as a function of the superimposed electron density. Both, energy-dependent and energy-independent exchange potentials are investigated. The spherical wave decomposition of the potential entering the angular momentum representation of the Lippmann-Schwinger equation is performed with an efficient numerical integration algorithm. Spherical wave components of the superposition potential are shown to emphasize their influence on the LEED-calculation. Scattering amplitudes of this potential model are computed and compared with those of standard muffin-tin phase shifts of GaAs. Especially, surface configurations are considered within this procedure.