Numerical Integration Procedure Research Articles

Numerical Integration and Hilbert-Schmidt-Norms

We consider numerical integration procedures of the type on Hilbert spaces with a reproducing kernel over the polish space E . It turns out that the Hilbert-Schmidt norm of certain operators play an important role as an estimator for the efficiency of the underlying integration procedure. Applications of these general results for the important cases of integration over the s-dimensional unit cube E=[0,1) s , and several numerical experiments will be shown. All methods discussed in the present paper are inherently parallel.

Analysis of pulled axisymmetric membranes with wrinkling

The nonlinear behavior of an axisymmetric hyperelastic membrane subjected to pulling forces is analyzed. The membrane is considered to be ideal in the sense that it cannot carry compressive stress resultants. If the membrane has a positive initial Gaussian curvature, the pulling gives rise to wrinkles which form over parts of the surface. The full nonlinear equations governing the membrane behavior in the doubly tense and in the wrinkled regions are formulated, and then solved using a numerical integration procedure. Solutions for various examples are presented, with Hookean and neo-Hookean constitutive behavior. These include a few examples of wrinkled membranes with positive initial Gaussian curvatures, and one example of a membrane with a negative initial Gaussian curvature, where no wrinkles are formed.

Numerical analysis of dynamic stability under random excitation

It is well known that an elastic column subjected to a harmonically-varying axial force exhibits parametric resonance over a range of applied frequencies and amplitudes. In this paper, the effect of white noise on the dynamic stability properties of a model system is investigated. Because of the complex nature of the applied loads and the nonlinearity of the governing equilibrium equation, three different numerical integration procedures are considered for determining the response of the model structure, and the accuracy of each scheme is studied. Results from the numerical investigation show that the method of Successive Symmetric Quadratures provides greater accuracy than the conventional Newmark Method or Harmonic Acceleration Method for nonlinear dynamic systems under random excitation. Computation of the model response shows that band-limited low frequency noise can limit the maximum deformation of the model system relative to the unperturbed case. Wide-band white noise, however, was observed only to enhance the destabilizing effect of the applied loads.

Atomic and Molecular Calculations Using Quasirandom Numbers

We have calculated the total ground-state energy, kinetic energy, and potential energy of He, H2, Li, and Be using explicitly correlated wavefunctions and two numerical integration procedures. The first integration procedure uses pseudorandom numbers and converges as N−0.5. The second uses a set of quasirandom numbers generated by Halton's algorithm. Under some circumstances the convergence of this integration procedure can be as fast as N−1. When a small to intermediate number of quasirandom numbers are used, most of our expectation values converge faster than those computed using pseudorandom numbers. When a large number of quasirandom numbers are used, however, we find that most of the expectation values converge at roughly the same rate as those computed using pseudorandom numbers.

Approximations to directivity for linear, planar, and volumetric apertures and arrays

Even after decades of sonar design, approximations to the directivity factor (DF) or index of an array, are often used inappropriately. Many of the approximations commonly used provide accurate directivity approximations for only the simplest of array geometries. As the array's size, shape, weighting, and complexity increase, there is a renewed need for better directivity approximations. Directivity is defined as the ratio of the output signal-to-noise (SNR) of an array to the input SNR at an omnidirectional element in a spherically isotropic noise field. Calculation of directivity is obtained by integrating the magnitude-squared response of the array over all angles of incidence. In spherical coordinates, these arrival angles are denoted by an azimuthal angle /spl theta/ and a polar angle /spl phi/. Hence, calculation of the directivity requires a two-fold integration over the angular space defined by the azimuthal and polar angles. For complex, large-size arrays consisting of thousands of array elements, directivity calculations using numerical integration procedures can be time consuming, even on state-of-the-art computing systems. This report provides a number of accurate formulas for estimating the directivity of linear, planar, and volumetric apertures and arrays, which are allowed to have arbitrary shading coefficients, steering angles, and directional array element responses.

NUMERICAL SOLUTIONS OF THERMOELASTIC WAVE PROBLEMS BY THE METHOD OF CHARACTERISTICS

This article is concerned with the numerical treatment of thermal and thermal stress waves in thermoelastic solids. To keep the numerical treatment general, the development of the formulation is based on the generalized theory of thermoelasticity. A number of thermoelastic wave problems, which involve one or two space variables, are treated, in a uniform manner, by a system of first-order partial differential equations with stress, velocity, heat flow, and temperature as dependent variables. This system of equations is analyzed by the method of characteristics, yielding the characteristics and the characteristic equations. Procedures of numerical integration along the characteristics are established and carried out for several generalized and classical thermoelastic wave problems in homogeneous materials, composite materials, nonhomogeneous materials, and nonlinear elastic solids.

Crack edge element of three‐dimensional displacement discontinuity method with boundary division into triangular leaf elements

AbstractIn this paper, the existing Displacement Discontinuity Method (DDM) for three‐dimensional elastic analysis with boundary discretized into triangular elements, which is purely based on analytical integrals, is extended from the constant element to the square‐root crack edge element. In order to evaluate the singular integral when the receiver point falls into the remitter element, i.e., the observed point (x,y) ∈Δ, a part analytical and part numerical integration procedure is adopted effectively. The newly developed codes prove valid in estimating the Stress Intensity Factor (SIF) KI. Furthermore, for the sake of keeping the advantages of high‐speed and high‐accuracy in developing the numerical system, a novel method to realize pure analytical integration of influence function is found by the aid of symbolic computation technology of Mathematica. Copyright © 2001 John Wiley & Sons, Ltd.

Matrix approach to a numerical solution of the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi evolution equations

A matrix-based approach to numerical integration of the DGLAP evolution equations is presented. The method arises naturally on discretisation of the Bjorken x variable, a necessary procedure for numerical integration. Owing to peculiar properties of the matrices involved, the resulting equations take on a particularly simple form and may be solved in closed analytical form in the variable t=ln(alpha_0/alpha). Such an approach affords parametrisation via data x bins, rather than fixed functional forms. Thus, with the aid of the full correlation matrix, appraisal of the behaviour in different x regions is rendered more transparent and free of pollution from unphysical cross-correlations inherent to functional parametrisations. Computationally, the entire programme results in greater speed and stability; the matrix representation developed is extremely compact. Moreover, since the parameter dependence is linear, fitting is very stable and may be performed analytically in a single pass over the data values.

Prediction models for sound leakage through noise barriers.

Two numerical models are presented for the prediction of sound leakage through openings in thin hard barriers. The first numerical method is based on a simple procedure of numerical integration that can be implemented straightforwardly. This model is a more general approach, suitable for barriers with arbitrary gaps. The second model is a new method that permits prediction of sound leakage due to the presence of horizontal gaps in a long barrier. In the new method, effective barriers of appropriate heights represent the edges of the horizontal gaps. The sound diffracted by each effective barrier is calculated by a closed-form analytic expression. The total sound-pressure level is determined from a sum of these diffracted fields. Hence, the new method is fast, simple, and intuitive, allowing the leakage to be assessed accurately. The validity of these two numerical models is confirmed by precise experimental measurements.

Post-buckling of uniform cantilever column under a combined load

Post-buckling analysis of uniform column under a combined load consisting of a uniformly distributed axial load and concentrated load at the free end was investigated by using numerical integration procedure such as Butcher’s fifth-order Runge–Kutta method. The loads and deflections are obtained for various end slopes. The results obtained by the present analysis agree very well with published results. Linear buckling analysis under a combined load was investigated by using power-series method.

Towards an efficient meshless computational technique: the method of finite spheres

Computational efficiency and reliability are clearly the most important requirements for the success of a meshless numerical technique. While the basic ideas of meshless techniques are simple and well understood, an effective meshless method is very difficult to develop. The efficiency depends on the proper choice of the interpolation scheme, numerical integration procedures and techniques of imposing the boundary conditions. These issues in the context of the method of finite spheres are discussed.

TRANSITION PROBABILITIES AND DISSOCIATION ENERGY OF THE AsS+ MOLECULE

The Franck—Condon factors and r-centroids, which are very closely related to relative transition probabilities, have been computed by the more reliable numerical integration procedure for the bands of the A1Π–X1Σ+ system of the AsS+ molecule, using a suitable potential. The dissociation energy for the electronic ground state of AsS+ has been estimated by fitting the Steele and Lippincott potential function to the experimental potential energy curve, using the correlation coefficient.

Thermostatistics of extensive and non-extensive systems using generalized entropies

We describe in detail two numerical simulation methods valid to study systems whose thermostatistics is described by generalized entropies, such as Tsallis. The methods are useful for applications to non-trivial interacting systems with a large number of degrees of freedom, and both short- and long-range interactions. The first method is quite general and it is based on the numerical evaluation of the density of states with a given energy. The second method is more specific for Tsallis thermostatistics and it is based on a standard Monte Carlo Metropolis algorithm along with a numerical integration procedure. We show here that both methods are robust and efficient. We present results of the application of the methods to the one-dimensional Ising model both in a short-range and in a long-range (non-extensive) case. We show that the thermodynamic potentials for different values of the system size N and different values of the non-extensivity parameter q can be described by scaling relations which are an extension of the ones holding for the Boltzmann–Gibbs statistics ( q=1). Finally, we discuss the differences in using standard or non-standard mean value definitions in the Tsallis thermostatistics formalism and present a microcanonical ensemble calculation approach of the averages.

Approximate expressions for the directivity of a circular microstrip-patch antenna

The determination of directivity values for circular microstrip patches on thin substrates involves the use numerical integration procedures, which have to be repeated every time there is a change in patch-substrate permittivity, By a judicious use of certain approximations, it is shown here that it is possible to secure surprisingly accurate alternative descriptions of the patch directivity behavior (as a function of permittivity), in terms of either known tabulated functions, or functions that can be easily accommodated using a hand-held calculator.

Thermal analysis of solid-state devices and circuits: an analytical approach

An analytical model of the temperature distribution in integrated circuits is presented. The solution is based on a classic method of electrostatics: the integration of the temperature field produced by a point source. Using this approach, the temperature distribution is expressed by a closed-form analytical relation, while previous approaches involve the summation of a slowly convergent series (Fourier series method) or a numerical integration procedure (Fourier transform method). Therefore, the present approach is much more computationally effective, and is particularly suitable for being incorporated into electro-thermal simulation codes. Another significant advantage of the proposed analytical formulation is that a more clear understanding of the influence of geometric and layout parameters on the thermal behavior can be gained. The analysis applies to arbitrarily located surface or volume heat sources. Boundary conditions are properly taken into account using the method of images. Finally, an accurate analytical expression for the thermal resistance of an integrated device is derived, which accounts for all relevant geometric parameters. This result may be a useful guideline in the early stages of layout optimization.

Aryl substituent effects and solvent effects on the decarbonylation of phenacetyl radicals

Five aryl-substituted phenacetyl radicals (X = p-MeO, p-Me, H, p-Cl, p-CF3) were generated by laser photolysis of the corresponding dibenzyl ketones in n-hexane and acetonitrile. The decarbonylation reaction was monitored through the rise in time-resolved absorption of the benzyl radical chromophore at 317 nm. The decarbonylation rate constants were obtained by a numerical integration procedure, where second-order radical reactions were explicitly taken into account. Values of (2–3) × 106 s−1 in acetonitrile and (6–10) × 106 s−1 in n-hexane revealed a large solvent effect for all derivatives (by a factor of ∼3). The electronic substituent effect indicates that both electron-withdrawing and electron-donating para substituents accelerate the decarbonylation slightly. The rate constants followed the order MeO > Me, Cl, CF3 > H. The substituent effects are interpreted in terms of the ability of the para substituent to stabilize the benzyl radical resulting from decarbonylation. Evidence for a polar effect was not obtained. The underlying reasons for the observed solvent effect are discussed. Copyright © 2000 John Wiley & Sons, Ltd.

Computational implementation of stress integration in FE analysis of elasto-plastic large deformation problems

In this paper, we compare different numerical implementation algorithms for the rate type constitutive equation and present an integration scheme based on the physical meaning of the stress. Numerical implementation of various schemes is investigated in conjunction with the return mapping algorithm and the conditions to maintain plastic consistency. Jaumann and Truesdell rates are taken as the objective stress rates in the constitutive equation. An alternative numerical treatment for rate of deformation tensor D ij is presented and is shown to maintain incremental objectivity. Numerical examples included a single element under rigid body rotation, a necking bifurcation of a bar in tension and a punch indentation process. It is shown that the use of Truesdell stress rate with specific numerical integration procedure gives more accurate results than other procedures presented.

Implicit characteristic Galerkin method for convection-diffusion equations

This paper presents a characteristic Galerkin finite element method with an implicit algorithm for solving multidimensional, time-dependent convection–diffusion equations. The method is formulated on the basis of the combination of both the precise and the implicit numerical integration procedures aiming to reference particles. The precise integration procedure with a 2N algorithm is taken as a tool to determine the material (Lagrangian) derivative of the convective function in the operator splitting procedure. The stability analysis of the algorithm and numerical results illustrate good performance of the present method in stability and accuracy. Copyright © 2000 John Wiley & Sons, Ltd.

Contaminant transport with non-equilibrium processes in unsaturated soils and implicit characteristic Galerkin scheme

A non-equilibrium sorption—advection—diffusion model to simulate miscible pollutant transport in saturated–unsaturated soils is presented. The governing phenomena modelled in the present simulation are: convection, molecular diffusion, mechanical dispersion, sorption, immobile water effect and degradation, including both physical and chemical non-equilibrium processes. A finite element procedure, based on the characteristic Galerkin method with an implicit algorithm is developed to numerically solve the model equations. The implicit algorithm is formulated by means of a combination of both the precise and the traditional numerical integration procedures. The stability analysis of the algorithm shows that the unconditional stability of the present implicit algorithm is enhanced as compared with that of the traditional implicit numerical integration procedure. The numerical results illustrate good performance of the present algorithm in stability and accuracy, and in simulating the effects of all the mentioned phenomena governing the contaminant transport and the concentration distribution. Copyright © 2000 John Wiley & Sons, Ltd.

Applications of Artificial Wind Numerical Scheme for Relativistic Hydrodynamics in Astrophysics

We apply a new way for the construction of efficient non-oscillation shock-capturing schemes, namely artificial wind (AW) scheme, for relativistic hydrodynamics (RHD) simulations in astrophysics. We also discuss physical foundations for computational RHD. An appropriate choice for a realistic equation of state of the matter at relativistic temperatures drastically facilitates the procedure for numerical integration of hydrodynamics equations. The time-consuming iteration approach to solve the coupling equation is avoided. We have developed a 2-D relativistic hydrodynamics code based on the AW scheme and the new RHD. Here we present two applications of the code in astrophysics, namely, numerical simulations of ultra-relativistic jet propagation andrelativistic Richtmyer-Meshkov instability. Due to numerical difficulties arising from strong relativistic shocks, motions with large value ofthe Lorentz f as well as due to the complicated structure ofthe relativistic equations, relativistic fluid simulations are traditionally believed to be much more difficult as compared with usual hydrodynamic simulations. Furthermore, the coupling equation between the primitive variables and the conservative ones through Lorentz factor γ is a quartic equation which should be solved at each cell several times ofiteration using Newton Raphson procedure to update the numerical solution for one time step. As a result, the relativistic codes are drastically time-consuming. Here, we apply a new way for the construction of efficient non-oscillation shockcapturing scheme, i.e., artificial wind (AW) scheme, for relativistic hydrodynamics simulation. The basic idea ofAW scheme is to solve the governing equations in different steadily moving frames of reference chosen in such a way that the flow is supersonic there, resulting in simple upwind formulas. The concept and principle of the AW scheme is given in an accompanying paper. 1) Recently, we proposed a new form of computational relativistic hydrodynamics (RHD). The key point is a special equation ofstate (EOS) f or the matter at relativistic temperature. 2) The new EOS is much more realistic, and can drastically facilitate the procedure for explicitly expressing the conservative variables via the primitive ones, and reversely, expressing the primitive variables via the conservative ones. In fact, in the new form of RHD, conservative variables are related to primitive variables through a simple second-order algebraic equation, the time-consuming iteration to solve the coupling equation is avoided. Here, we present two astrophysical applications. The first one is ultra-relativistic jet propagating through a homogeneous medium. The simulation has been run on