Numerical Integration Procedure Research Articles

Evaluation of astrophysically useful parameters for HD molecule

The spectrum of HD molecule is the testing ground of both refined calculations and experimental techniques and its excited states are of considerable relevance to chemistry and astrophysics. In this study, the Franck–Condon (FC) factors and r-centroids are evaluated for the bands of d 3 Π u – a 3 Σ g + , e 3 Σ u + – a 3 Σ g + and I 1 Π g – B 1 Σ u + transitions of HD molecule. The evaluation is done by numerical integration procedure using a suitable potential. The physical and astrophysical significance of the estimated FC factors and r-centroids are also discussed.

An anisotropic visco-hyperelastic model for ligaments at finite strains. Formulation and computational aspects

In this paper we present a fully three-dimensional finite strain anisotropic visco-hyperelastic model for ligaments and tendons. The structural model is formulated within the framework of non-linear continuum mechanics and is well-suited for its finite element implementation. This model is based on a local additive decomposition of the stress tensor into initial and non-equilibrium parts as resulted from the assumed structure of the free-energy density function that generalizes Kelvin–Voigt linear viscous models. Also, we use a local multiplicative decomposition of the deformation gradient into volume-preserving and dilatational parts that permits to model the incompressible properties of soft biological tissues. To simulate the viscoelastic properties of this kind of tissues, we consider different viscoelastic behaviours for the matrix and the different families of fibers. A second-order accurate numerical integration procedure is used, established entirely in the reference configuration. Expressions for the stress and elasticity tensors in the spatial description are also presented. Of all soft tissues, we have focused in ligaments due to the importance of their viscoelastic properties in the clinical practise. In order to show clearly the performance of the constitutive model, we present 3D simulations of the behaviour of the anterior cruciate ligament and patellar tendon graft. The model was also tested for various multi-axial loading situations. The relaxation and creep responses and the strain rate dependent behaviour of anterior cruciate ligament and patellar tendon graft were accurately predicted.

Symplectic integrators for classical spin systems

We suggest a numerical integration procedure for solving the equations of motion of certain classical spin systems which preserves the underlying symplectic structure of the phase space. Such symplectic integrators have been successfully utilized for other Hamiltonian systems, e.g., for molecular dynamics or non-linear wave equations. Our procedure rests on a decomposition of the spin Hamiltonian into a sum of two completely integrable Hamiltonians and on the corresponding Lie–Trotter decomposition of the time evolution operator. In order to make this method widely applicable we provide a large class of integrable spin systems whose time evolution consists of a sequence of rotations about fixed axes. We test the proposed symplectic integrator for small spin systems, including the model of a recently synthesized magnetic molecule, and compare the results for variants of different order.

Nonlinear Response of Double-Wall Cylindrical Shell Vibrations under Random Excitation

An analytical model is presented to predict the nonlinear response of a double-wall sandwich cylindrical shell system subjected to random excitation. Nonlinear spring-dashpot models are integrated into the system to characterize the behavior of the soft core. Donnell’s thin shell theory is used to develop the governing nonlinear equations of motion. A Monte Carlo simulation of stationary random processes, multimode Galerkin-type approach, and numerical integration procedures are used to develop linear and nonlinear response solutions of simply supported cylindrical shells. Numerical results include time domain response histories, root-mean-square values and response spectral densities. Parametric studies are performed to investigate the effects of nonlinearity, shell thickness, core stiffness, and thickness.

Evaluation of astrophysically useful parameters for strontium monohydride and deuteride

The Franck-Condon factors and r-centroids which are very closely related to transition probabilities, have been evaluated by the more reliable numerical integration procedure for the band systems B2?-X2?, C2?-X2? and F2?-X2? of astrophysical molecules strontium monohydride and strontium deuteride using an adequate potential. The Franck-Condon factors are more intense, particularly for the ?? = 0 bands, for all the systems examined here. Thus the bands of the molecules are expected to be present in sunspot spectra, SC-stars, cool M-giant stars and other prominent astrophysical sources.

Discrete Time Transfer Matrix Method for Multibody System Dynamics

A new method for multibody system dynamics is proposed in this paper. This method, named as discrete time transfer matrix method of multibody system (MS-DT-TMM), combines and expands the advantages of the transfer matrix method (TMM), transfer matrix method of vibration of multibody system (MS-TMM), discrete time transfer matrix method (DT-TMM) and the numerical integration procedure. It does not need the global dynamics equations for the study of multibody system dynamics. It has the modeling flexibility and a small size of matrices, and can be applied to a wide range of problems including multi-rigid-body system dynamics and multi-flexible-body system dynamics. This method is simple, straightforward, practical, and provides a powerful tool for the study of multibody system dynamics. Formulations of the method as well as some numerical examples of multi-rigid-body system dynamics and multi-flexible-body system dynamics to validate the method are given.

Chlorophyll a fluorescence induction kinetics in leaves predicted from a model describing each discrete step of excitation energy and electron transfer associated with Photosystem II

Chlorophyll a fluorescence induction (FI) is widely used as a probe for studying photosynthesis. On illumination, fluorescence emission rises from an initial level O to a maximum P through transient steps, termed J and I. FI kinetics reflect the overall performance of photosystem II (PSII). Although FI kinetics are commonly and easily measured, there is a lack of consensus as to what controls the characteristic series of transients, partially because most of the current models of FI focus on subsets of reactions of PSII, but not the whole. Here we present a model of fluorescence induction, which includes all discrete energy and electron transfer steps in and around PSII, avoiding any assumptions about what is critical to obtaining O J I P kinetics. This model successfully simulates the observed kinetics of fluorescence induction including O J I P transients. The fluorescence emission in this model was calculated directly from the amount of excited singlet-state chlorophyll in the core and peripheral antennae of PSII. Electron and energy transfer were simulated by a series of linked differential equations. A variable step numerical integration procedure (ode15s) from MATLAB provided a computationally efficient method of solving these linked equations. This in silico representation of the complete molecular system provides an experimental workbench for testing hypotheses as to the underlying mechanism controlling the O J I P kinetics and fluorescence emission at these points. Simulations based on this model showed that J corresponds to the peak concentrations of Q(-)AQB (QA and QB are the first and second quinone electron acceptor of PSII respectively) and Q(-)AQ(-)B and I to the first shoulder in the increase in concentration of Q(-)AQ(2-)B. The P peak coincides with maximum concentrations of both Q(-)AQ(2-)B and PQH2. In addition, simulations using this model suggest that different ratios of the peripheral antenna and core antenna lead to differences in fluorescence emission at O without affecting fluorescence emission at J, I and P. An increase in the concentration of QB-nonreducing PSII centers leads to higher fluorescence emission at O and correspondingly decreases the variable to maximum fluorescence ratio (F v/F m).

Hiemenz flow of a micropolar viscoelastic fluid in hydromagnetics

Boundary-layer equations are solved for the hydromagnetic problem of two-dimensional Hiemenz flow, for a micropolar, viscoelastic, incompressible, viscous, electrically conducting fluid, impinging perpendicularly onto a plane in the presence of a transverse magnetic field. The governing system of equations is first transformed into a dimensionless form. The resulting equations then are solved by using the Runge–Kutta numerical integration procedure in conjunction with shooting technique. Numerical solutions are presented for the governing momentum and angular-momentum equations. The proposed approximate solution, although simple, is nevertheless sufficiently accurate for the entire investigated range of values of the Hartman number. The effect of micropolar and viscoelastic parameters on Hiemenz flow in hydromagnetics is discussed.PACS No.: 46.35

CALCULATION OF IMMERSION DOSES FROM EXTERNAL EXPOSURE TO A PLUME OF RADIOACTIVE MATERIAL

The immersion doses from external exposure to a Gaussian plume of noble gases accidentally released into the atmosphere have been calculated. A numerical integration procedure employing Gauss-Legendre of 64th order has been used. The numerical procedure allows calculating the dose rate at any downwind horizontal or vertical distance. The dose rates were calculated using various forms of gamma dose build-up factors, including Linear, Berger and Geometric Progression (GP). The GP form, having an extraordinarily precise formulation, is a favored choice because the build-up factor levels off for large distances and does not increase exponentially as does the Berger form. The Linear form much under predicts the build-up and subsequently the dose rates for large distances from the source. The dose predictions using a simple uniform cloud model (that does not use any form of build-up factor) is also presented for comparison purposes. The comparison of dose rates with the already reported results indicated that the numerical procedure could be used for dose calculations from a Gaussian plume for all downwind and crosswind distances. The comparison of dose rates obtained using different forms of the build-up factors indicated that the Geometric Progression form was a favored choice and has a wider range of applicability as compared to the Linear or Berger form. The simple uniform cloud model for dose calculations is only suitable for plume centerline doses and should be used with caution for off-center distances.

Energy release rate and mode partition for interlaminar crack in circular laminated beams

This work describes an improved technique for the analysis of interlaminar crack growth in laminated circular beams subjected to mixed-mode loading conditions. By adopting a multi-layer curved beam model and curved interface elements leads to model the structure as a succession of first-order shear deformable beams joined by adhesive interfaces. The interface containing the crack is simulated by means of a cracking linear interfacial constitutive law which provides normal and shear interlaminar stresses. At the perfectly bonded interfaces the connection between beam elements is imposed by the Lagrange multiplier method. The total energy release rate and its individual mode components are obtained when the stiffnesses, considered as penalty parameters, tend to infinity thus reproducing the results of the linear elastic fracture mechanics theory. Results are obtained by adopting an accurate numerical integration procedure able to deal with stress singularities at the crack tip. Comparisons with results available from literature obtained by using the elasticity theory and solid finite elements show that the model is able to capture accurately both the total and the individual mode components of energy release rate. Numerical examples demonstrate the capability of the proposed method to provide a complete description of the problem of mixed-mode crack growth in curved interfaces, which takes into account the influence of curvature and of transverse shear deformability.

High-order difference schemes based on new Marchuk integral identities for one-dimensional interface problems

High-order finite difference approximations of the solution and the flux to model interface problems in one-dimension are constructed and analyzed. Explicit formulas based on new Marchuk integral identities that give O(h2), O(h4),… accuracy are derived. Numerical integration procedures using Lobatto quadratures for computing three-point schemes of any prescribed order of accuracy are developed. A rigorous rate of convergence analysis is presented. Numerical experiments confirm the theoretical results.

Eigenanalysis for Membranes with Stringers Using the Methods of Fundamental Solutions and Domain Decomposition

We use a meshless numerical method to analyze the eigenanalysis of thin circular membranes with degenerate boundary conditions, composed by different orientations and structures of stringers. The membrane eigenproblem is studied by solving the two-dimensional Helmholtz equation utilizing the method of fundamental solutions and domain decomposition technique as well. The method of singular value decomposition is adopted to obtain eigenvalues and eigenvectors of the resulting system of global linear equation. The proposed novel numerical scheme was first validated by three circular membranes which are structured with a single edge stringer, two opposite edge stringers and an internal stringer. Present results for those three cases match very well with the solutions obtained by the analytical approach as well as by methods of dual boundary element, and finite element. The analysis is then extended to solve a completely new problem of a circular membrane with a cross stringer at the center of the membrane. We illustrate the proposed innovative numerical scheme which is simpler and more efficient to solve Helmholtz problems with degenerate boundary conditions. The good features of this scheme are not depending upon the treatments of mesh, singularity, hypersingularity, numerical integration and iterative procedure, which are generally required by other conventional mesh-dependent methods. keyword: eigenanalysis, Helmholtz equation, method of fundamental solutions, domain decomposition method, degenerate boundary condition 1 Department of Civil Engineering & Hydrotech Research Institute, National Taiwan University, Taipei, Taiwan 2 Correspond to: D.L. Young, Fax: +886-2-23626114, E-mail: dlyoung@ntu.edu.tw 3 Department of Information Technology, Toko University, Chia-Yi County, Taiwan

Interaction integral procedures for 3-D curved cracks including surface tractions

This study examines a two-state interaction integral for the direct computation of mixed-mode stress intensity factors along curved cracks under remote mechanical loads and applied crack-face tractions. We investigate the accuracy of stress intensity factors computed along planar, curved cracks in homogeneous materials using a simplified interaction integral that omits terms to reflect specifically the effects of local crack-front curvature. We examine the significance of the crack-face traction term in the interaction integral, and demonstrate the benefit of a simple, exact numerical integration procedure to evaluate the integral for one class of three-dimensional elements. The work also discusses two approaches to compute auxiliary, interaction integral quantities along cracks discretized by linear and curved elements. Comparisons of numerical results with analytical solutions for stress intensity factors verify the accuracy of the proposed interaction integral procedures.

Franck-Condon factors and r-centroids for certain band systems of astrophysical molecules SrF and ScF

The Franck-Condon factors and r-centroids, which are very closely related to vibrational transition probabilities, have been evaluated by the more reliable numerical integration procedure for the bands of B2∑+ − X2∑+, F2∑+ − X2∑+ systems of SrF and C1∑+ − X1∑+, G1Π − X1∑+ systems of ScF molecules of astrophysical interest, using a suitable potential.

High-order difference schemes based on new Marchuk integral identities for one-dimensional interface problems

High-order finite difference approximations of the solution and the flux to model interface problems in one-dimension are constructed and analyzed. Explicit formulas based on new Marchuk integral identities that give O ( h 2 ), O ( h 4 ),… accuracy are derived. Numerical integration procedures using Lobatto quadratures for computing three-point schemes of any prescribed order of accuracy are developed. A rigorous rate of convergence analysis is presented. Numerical experiments confirm the theoretical results.

Analysis of thick-wire antennas using a novel and simple kernel treatment

This paper presents novel aspects concerning the analysis of thick-wire antennas. The analysis is carried out using an integral equation (IE) technique, in which the kernel is treated with a novel numerical-integration procedure. A comparison with other techniques are included, showing the advantages of the newly proposed method. The novel treatment of the kernel requires the efficient extraction of the singularities. For this purpose, two different techniques are presented, and their advantages and drawbacks are discussed. Using the procedure proposed, complex thick wire antennas with rotational symmetry are analyzed. A combined antenna composed of cylindrical and conical sections is studied. The results show that the new techniques derived are efficient and lead to accurate results. © 2005 Wiley Periodicals, Inc. Microwave Opt Technol Lett 46: 410–417, 2005; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.21002

Fully tensorial nodal and modal shape functions for triangles and tetrahedra

This paper presents nodal and modal shape functions for triangle and tetrahedron finite elements. The functions are constructed based on the fully tensorial expansions of one-dimensional polynomials expressed in barycentric co-ordinates. The nodal functions obtained from the application of the tensorial procedure are the standard h-Lagrange shape functions presented in the literature. The modal shape functions use Jacobi polynomials and have a natural global C0 inter-element continuity. An efficient Gauss–Jacobi numerical integration procedure is also presented to decrease the number of points for the consistent integration of the element matrices. An example illustrates the approximation properties of the modal functions. Copyright © 2005 John Wiley & Sons, Ltd.

Closed-form solutions for analysis of cylindrically conformal microstrip antennas with arbitrary radii

An efficient and accurate closed-form solution, based on the spectral-domain method of moments (MoM), is presented to analyze the characteristics of cylindrically conformal microstrip antennas with arbitrary radii. First, an algorithm that combines the recursive and the uniform asymptotic expansion method in different regions is developed to calculate the spectral-domain Green's functions for arbitrary dielectric coated cylinders accurately. Next, the integrands of the spectral integrals involved in MoM equation are partitioned into two parts, i.e., the fast oscillatory, slowly convergent part and the well-behaved part. Consequently, all of the integrals in MoM analysis can be categorized into two types. Through the Sommerfeld identity and its extension, these integrals are cast into closed-forms by using the nonlinear approximation method (generalized pencil-of-function method). In this method, any time-consuming numerical integration procedure is eliminated, which improves the computational efficiency drastically. Compared with the results from published references, the validity and accuracy of the method are demonstrated.

An internally consistent integration method for critical state models

A new procedure to integrate critical state models including Cam–Clay and modified Cam–Clay is proposed here. The proposed procedure makes use of the linearity of the virgin isotropic compression curve and the parallel anisotropic consolidation lines in e–ln p space which are basic features of the formulation of critical state models. Using this algorithm, a unique final stress state may be found as a function of a single unknown for elastoplastic loading. The key equations are given in this article for the Cam–Clay and modified Cam–Clay models. The use of the Newton–Raphson iterative method to minimize residuals and obtain a converged solution is described here. This new algorithm may be applied using the assumptions of linear elasticity or non-linear elasticity within a given loading step. The new algorithm proposed here is internally consistent and has computational advantages over the current numerical integration procedures. Numerical examples are presented to show the performance of the algorithm as compared to other integration algorithms. Published in 2005 by John Wiley & Sons, Ltd.

Numerically integrated modified virtual crack closure integral technique for 2-D crack problems

Modified virtual crack closure integral (MVCCI) technique has become very popular for computation of strain energy release rate (SERR) and stress intensity factor (SIF) for 2-D crack problems. The objective of this paper is to propose a numerical integration procedure for MVCCI so as to generalize the technique and make its application much wider. This new procedure called as numerically integrated MVCCI (NI-MVCCI) will remove the dependence of MVCCI equations on the type of finite element employed in the basic stress analysis. Numerical studies on fracture analysis of 2-D crack (mode I and II) problems have been conducted by employing 4-noded, 8-noded (regular & quarter-point), 9-noded and 12-noded finite elements. For non-singular (regular) elements at crack tip, NI-MVCCI technique generates the same results as MVCCI, but the advantage for higher order regular and singular elements is that complex equations for MVCCI need not be derived. Gauss numerical integration rule to be employed for 8-noded singular (quarter-point) element for accurate computation of SERR and SIF has been recommended based on the numerical studies.