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## Numerical Evaluation (numerical + evaluation)
## Selected Abstracts## Numerical evaluation of eigenvalues in notch problems using a region searching method INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 12 2006Y. Z. ChenAbstract This paper presents a method for finding the eigenvalues of some equations, or the zeros of analytic functions. There are two steps in the method. In the first step, integration along the edges of rectangle for an analytic function is performed. From the result of integration, one can know whether the zero exists in the rectangle or not. If the zero of an analytic function exists in the rectangle, we can perform the second step. In the second step, the zero is obtained by iteration. Therefore, the method is called a region searching method. Particular advantage of the suggested method is that the process for finding zero can be visualized. For example, one can clearly indicate the rectangles, which contain the zeros of an analytic function. Three numerical examples are presented. The obtained results are satisfactory even for a complicated case, for example, for finding eigenvalues of a composed wedge of dissimilar materials. Copyright © 2006 John Wiley & Sons, Ltd. [source] ## Asymptotic upper bounds for the errors of Richardson extrapolation with practical application in approximate computations INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 5 2009Aram SoroushianAbstract The results produced by Richardson extrapolation, though, in general, very accurate, are inexact. Numerical evaluation of this inexactness and implementation of the evaluation in practice are the objectives of this paper. First, considering linear changes of errors in the convergence plots, asymptotic upper bounds are proposed for the errors. Then, the achievement is extended to the results produced by Richardson extrapolation, and finally, an error-controlling procedure is proposed and successfully implemented in approximate computations originated in science and engineering. Copyright © 2009 John Wiley & Sons, Ltd. [source] ## Numerical evaluation of two discontinuous Galerkin methods for the compressible Navier,Stokes equations INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 1-2 2002F. BassiThis paper presents a critical comparison between two recently proposed discontinuous Galerkin methods for the space discretization of the viscous terms of the compressible Navier,Stokes equations. The robustness and accuracy of the two methods has been numerically evaluated by considering simple but well documented classical two-dimensional test cases, including the flow around the NACA0012 airfoil, the flow along a flat plate and the flow through a turbine nozzle. Copyright © 2002 John Wiley & Sons, Ltd. [source] ## A novel approach to scheduling multipurpose batch plants using unit-slots AICHE JOURNAL, Issue 7 2010Naresh SusarlaAbstract Several models for scheduling multipurpose batch plants exist in the literature. The models using unit-specific event points have shown better solution efficiency on various literature examples. This article presents a novel approach to scheduling multipurpose batch plants, which uses unit-slots instead of process-slots to manage shared resources such as material storage. We develop two slightly different models that are even more compact and simpler than that of Sundaramoorthy and Karimi, Chem Eng Sci. 2005;60:2679,2702. Although we focus on material as a shared resource, our multi-grid approach rationalizes, generalizes, and improves the current multi-grid approaches for scheduling with shared resources. Our models allow nonsimultaneous transfers of materials into and out of a batch. We show by an example that this flexibility can give better schedules than those from existing models in some cases. Furthermore, our approach uses fewer slots (event-points) on some examples than even those required by the most recent unit-specific event-based model. Numerical evaluation using literature examples shows significant gains in solution efficiency from the use of unit-slots except where the number of unit-slots required for the optimal solution equals that of process slots. We also highlight the importance of constraint sequencing in GAMS implementation for evaluating mixed-integer linear programming based scheduling models fairly. © 2009 American Institute of Chemical Engineers AIChE J, 2010 [source] ## A method for controlling the spins of atoms using optical near-fields JOURNAL OF MICROSCOPY, Issue 3 2003A. ShojiguchiSummary On the basis of the procedure for controlling the spins of atoms using circularly polarized evanescent light proposed by Hori et al.[(1996) Abstracts of the 1st Asia-Pacific Workshop on Near-field Optics] we discuss the influence of boundary conditions on the probability of spontaneous emission and thus on the spin polarization efficiency, which was not considered in the Hori et al. study. Using the Carniglia,Mandel mode expansion of electromagnetic fields, we derive the spontaneous emission and spin polarization probabilities of atoms near a dielectric surface, and show the atom,surface distance dependence and refractive index dependence. Numerical evaluation for the 6P1/2,6S1/2 transition of a Cs atom indicates an increase in the efficiency of spin polarization by 30%. [source] ## Numerical evaluation of pressure from experimentally measured film thickness in EHL point contact LUBRICATION SCIENCE, Issue 1 2008Michal VaverkaAbstract This paper is concerned with elastohydrodynamic lubrication, especially the determination of lubricant film thickness and contact pressure within a point contact of friction surfaces of machine parts. A new solution technique for numerical determination of contact pressure is introduced. The direct measurement of contact pressure is very difficult. Hence, input data of lubricant film thickness obtained from the experiment based on colorimetric interferometry are used for the calculation of pressure using the inverse elasticity theory. The algorithm is enhanced by convolution in order to increase calculation speed. The approach described in this contribution gives reliable results on smooth contact and in the future, it will be extended to enable the study of contact of friction surfaces with asperities. Copyright © 2007 John Wiley & Sons, Ltd. [source] ## Axisymmetric interaction of a rigid disc with a transversely isotropic half-space INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, Issue 12 2010Amir Aabbas KatebiAbstract A theoretical formulation is presented for the determination of the interaction of a vertically loaded disc embedded in a transversely isotropic half-space. By means of a complete representation using a displacement potential, it is shown that the governing equations of motion for this class of problems can be uncoupled into a fourth-order partial differential equation. With the aid of Hankel transforms, a relaxed treatment of the mixed-boundary value problem is formulated as dual integral equations, which, in turn, are reduced to a Fredholm equation of the second kind. In addition to furnishing a unified view of existing solutions for zero and infinite embedments, the present treatment reveals a severe boundary-layer phenomenon, which is apt to be of interest to this class of problems in general. The present solutions are analytically in exact agreement with the existing solutions for a half-space with isotropic material properties. To confirm the accuracy of the numerical evaluation of the integrals involved, numerical results are included for cases of different degrees of the material anisotropy and compared with existing solutions. Further numerical examples are also presented to elucidate the influence of the degree of the material anisotropy on the response. Copyright © 2009 John Wiley & Sons, Ltd. [source] ## Steady infiltration from buried point source into heterogeneous cross-anisotropic unsaturated soil INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, Issue 10 2004G. J. ChenAbstract The paper presents the analytical solution for the steady-state infiltration from a buried point source into two types of heterogeneous cross-anisotropic unsaturated half-spaces. In the first case, the heterogeneity of the soil is modelled by an exponential relationship between the hydraulic conductivity and the soil depth. In the second case, the heterogeneous soil is represented by a multilayered half-space where each layer is homogeneous. The hydraulic conductivity varies exponentially with moisture potential and this leads to the linearization of the Richards equation governing unsaturated flow. The analytical solution is obtained by using the Hankel integral transform. For the multilayered case, the combination of a special forward and backward transfer matrix techniques makes the numerical evaluation of the solution very accurate and efficient. The correctness of both formulations is validated by comparison with alternative solutions for two different cases. The results from typical cases are presented to illustrate the influence on the flow field of the cross-anisotropic hydraulic conductivity, the soil heterogeneity and the depth of the source. Copyright © 2004 John Wiley & Sons, Ltd. [source] ## Robust and direct evaluation of J2 in linear elastic fracture mechanics with the X-FEM INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 10 2008G. LegrainAbstract The aim of the present paper is to study the accuracy and the robustness of the evaluation of Jk -integrals in linear elastic fracture mechanics using the extended finite element method (X-FEM) approach. X-FEM is a numerical method based on the partition of unity framework that allows the representation of discontinuity surfaces such as cracks, material inclusions or holes without meshing them explicitly. The main focus in this contribution is to compare various approaches for the numerical evaluation of the J2 -integral. These approaches have been proposed in the context of both classical and enriched finite elements. However, their convergence and the robustness have not yet been studied, which are the goals of this contribution. It is shown that the approaches that were used previously within the enriched finite element context do not converge numerically and that this convergence can be recovered with an improved strategy that is proposed in this paper. Copyright © 2008 John Wiley & Sons, Ltd. [source] ## Improvement of the asymptotic behaviour of the Euler,Maclaurin formula for Cauchy principal value and Hadamard finite-part integrals INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 4 2004U. Jin ChoiAbstract In the recent works (Commun. Numer. Meth. Engng 2001; 17: 881; to appear), the superiority of the non-linear transformations containing a real parameter b , 0 has been demonstrated in numerical evaluation of weakly singular integrals. Based on these transformations, we define a so-called parametric sigmoidal transformation and employ it to evaluate the Cauchy principal value and Hadamard finite-part integrals by using the Euler,Maclaurin formula. Better approximation is expected due to the prominent properties of the parametric sigmoidal transformation of whose local behaviour near x = 0 is governed by parameter b. Through the asymptotic error analysis of the Euler,Maclaurin formula using the parametric sigmoidal transformation, we can observe that it provides a distinct improvement on its predecessors using traditional sigmoidal transformations. Numerical results of some examples show the availability of the present method. Copyright © 2004 John Wiley & Sons, Ltd. [source] ## The S and G transformations for computing three-center nuclear attraction integrals INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, Issue 8 2009Richard Mikael SlevinskyAbstract It is now well established that nonlinear transformations can be extremely useful in the case of oscillatory integrals. In previous work, we could show that the G transformation, which is not so well known among those interested in the numerical evaluation of highly oscillatory integrals, works very well for the extremely challenging integral called Twisted Tail. In this work, we demonstrate that these techniques also apply to three-center nuclear attraction integrals over exponential type functions. The accurate and rapid evaluation of these integrals is required in ab initio molecular structure calculations and density functional theory. Using a basis set of B functions and profiting from their relatively simple Fourier representation, these integrals are formulated as analytical expressions involving highly oscillatory spherical Bessel integral functions. In the present work, we implement two highly accurate algorithms for three-center nuclear attraction integrals. The first algorithm is based on the G transformation and the second is based on a combination of the S and G transformations. The application of these transformations is largely due to the properties of special functions that allow the computation of higher order derivatives of the integrands with exceptional simplicity. The numerical results illustrate the accuracy of these algorithms applied to three-center nuclear attraction integrals over exponential type functions with a miscellany of different parameters. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2009 [source] ## Extrapolation methods for improving convergence of spherical Bessel integrals for the two-center Coulomb integrals INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, Issue 11 2006Hassan SafouhiAbstract Multi-center two-electron Coulomb integrals over Slater-type functions are required for any accurate molecular electronic structure calculations. These integrals, which are numerous, are to be evaluated rapidly and accurately. Slater-type functions are expressed in terms of the so-called B functions, which are best suited to apply the Fourier transform method. The Fourier transform method allowed analytic expressions for these integrals to be developed. Unfortunately, the analytic expressions obtained turned out to be extremely difficult to evaluate accurately due to the presence of highly oscillatory spherical Bessel integrals. In this work, we used techniques based on nonlinear transformation and extrapolation methods for improving convergence of these oscillatory spherical Bessel integrals. These techniques, which led to highly efficient and rapid algorithms for the numerical evaluation of three- and four-center two-electron Coulomb and exchange integrals, are now shown to be applicable to the two-center two-electron Coulomb integrals. The numerical results obtained for the molecular integrals under consideration illustrate the efficiency of the algorithm described in the present work compared with algorithms using the epsilon (,) algorithm of Wynn and Levin's u transform. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2006 [source] ## Notes on quantitative structure-properties relationships (QSPR) (1): A discussion on a QSPR dimensionality paradox (QSPR DP) and its quantum resolution JOURNAL OF COMPUTATIONAL CHEMISTRY, Issue 7 2009Ramon Carbó-DorcaAbstract Classical quantitative structure-properties relationship (QSPR) statistical techniques unavoidably present an inherent paradoxical computational context. They rely on the definition of a Gram matrix in descriptor spaces, which is used afterwards to reduce the original dimension via several possible kinds of algebraic manipulations. From there, effective models for the computation of unknown properties of known molecular structures are obtained. However, the reduced descriptor dimension causes linear dependence within the set of discrete vector molecular representations, leading to positive semi-definite Gram matrices in molecular spaces. To resolve this QSPR dimensionality paradox (QSPR DP) here is proposed to adopt as starting point the quantum QSPR (QQSPR) computational framework perspective, where density functions act as infinite dimensional descriptors. The fundamental QQSPR equation, deduced from employing quantum expectation value numerical evaluation, can be approximately solved in order to obtain models exempt of the QSPR DP. The substitution of the quantum similarity matrix by an empirical Gram matrix in molecular spaces, build up with the original non manipulated discrete molecular descriptor vectors, permits to obtain classical QSPR models with the same characteristics as in QQSPR, that is: possessing a certain degree of causality and explicitly independent of the descriptor dimension. © 2008 Wiley Periodicals, Inc. J Comput Chem, 2009 [source] ## Transient Behavior and Gelation of Free Radical Polymerizations in Continuous Stirred Tank Reactors MACROMOLECULAR THEORY AND SIMULATIONS, Issue 4 2005Rolando C. S. DiasAbstract Summary: Using the authors' previously developed method for the general kinetic analysis of non-linear irreversible polymerizations, the simulation of free radical homogeneous polymerization systems with terminal branching and chain transfer to polymer has been carried out for continuous stirred tank reactors. Its improved accuracy on the numerical evaluation of generating functions has been exploited in order to perform their numerical inversion and chain length distributions could also be estimated with or without the presence of gel. A comparison with alternative techniques emphasizing the effect of their simplifying assumptions on the accuracy of calculations is also presented. Predicted CLD before gelation (t,=,1 h), after gelation (t,=,15 h, steady state), and close to gel point for a free radical polymerization with transfer to polymer in a CSTR with ,,=,60 min. [source] ## A numerical evaluation of chamber methodologies used in measuring the ,13C of soil respiration RAPID COMMUNICATIONS IN MASS SPECTROMETRY, Issue 17 2009Nick NickersonMeasurement of the ,13C value of soil-respired CO2 (,r) has become a commonplace method through which ecosystem function and C dynamics can be better understood. Despite its proven utility there is currently no consensus on the most robust method with which to measure ,r. Static and dynamic chamber systems are both commonly used for this purpose; however, the literature on these methods provides evidence suggesting that measurements of ,r made with these chamber systems are neither repeatable (self-consistent) nor comparable across methodologies. Here we use a three-dimensional (3-D) numerical soil-atmosphere-chamber model to test these chamber systems in a ,surrogate reality'. Our simulations show that each chamber methodology is inherently biased and that no chamber methodology can accurately predict the true ,r signature under field conditions. If researchers intend to use ,r to study insitu ecosystem processes, the issues with these chamber systems need to be corrected either by using diffusive theory or by designing a new, unbiased ,r measurement system. Copyright © 2009 John Wiley & Sons, Ltd. [source] ## Heart Rate Variability Fraction,A New Reportable Measure of 24-Hour R-R Interval Variation ANNALS OF NONINVASIVE ELECTROCARDIOLOGY, Issue 1 2005Maciej Sosnowski M.D.Background: The scatterplot of R-R intervals has several unique features. Its numerical evaluation may produce a new useful index of global heart rate variability (HRV) from Holter recordings. Methods: Two-hundred and ten middle-aged healthy subjects were enrolled in this study. The study was repeated the next day in 165 subjects. Each subject had a 24-hour ECG recording taken. Preprocessed data were transferred into a personal computer and the standard HRV time-domain indices: standard deviation of total normal R-R intervals (SDNN), standard deviation of averaged means of normal R-R intervals over 5-minute periods (SDANN), triangular index (TI), and pNN50 were determined. The scatterplot area (0.2,1.8 second) was divided into 256 boxes, each of 0.1-second interval, and the number of paired R-R intervals was counted. The heart rate variability fraction (HRVF) was calculated as the two highest counts divided by the number of total beats differing from the consecutive beat by <50 ms. The HRVF was obtained by subtracting this fraction from 1, and converting the result to a percentage. Results: The normal value of the HRVF was 52.7 ± 8.6%. The 2,98% range calculated from the normal probability plot was 35.1,70.3%. The HRVF varied significantly with gender (female 48.7 ± 8.4% vs male 53.6 ± 8.6%, P = 0.002). The HRVF correlated with RRI (r = 0.525) and showed a similar or better relationship with SDNN (0.851), SDANN (0.653), and TI (0.845) than did the standard HRV measures with each other. Bland-Altman plot showed a good day-by-day reproducibility of the HRVF, with the intraclass correlation coefficient of 0.839 and a low relative standard error difference (1.8%). Conclusion: We introduced a new index of HRV, which is easy for computation, robust, reproducible, easy to understand, and may overcome the limitations that belong to the standard HRV measures. This index, named HRV fraction, by combining magnitude, distribution, and heart-rate influences, might become a clinically useful index of global HRV. [source] ## UPPER BOUNDS ON THE MINIMUM COVERAGE PROBABILITY OF CONFIDENCE INTERVALS IN REGRESSION AFTER MODEL SELECTION AUSTRALIAN & NEW ZEALAND JOURNAL OF STATISTICS, Issue 3 2009Paul KabailaSummary We consider a linear regression model, with the parameter of interest a specified linear combination of the components of the regression parameter vector. We suppose that, as a first step, a data-based model selection (e.g. by preliminary hypothesis tests or minimizing the Akaike information criterion , AIC) is used to select a model. It is common statistical practice to then construct a confidence interval for the parameter of interest, based on the assumption that the selected model had been given to us,a priori. This assumption is false, and it can lead to a confidence interval with poor coverage properties. We provide an easily computed finite-sample upper bound (calculated by repeated numerical evaluation of a double integral) to the minimum coverage probability of this confidence interval. This bound applies for model selection by any of the following methods: minimum AIC, minimum Bayesian information criterion (BIC), maximum adjusted,R2, minimum Mallows' CP and,t -tests. The importance of this upper bound is that it delineates general categories of design matrices and model selection procedures for which this confidence interval has poor coverage properties. This upper bound is shown to be a finite-sample analogue of an earlier large-sample upper bound due to Kabaila and Leeb. [source] ## Residual Pattern Based Test for Interactions in Two-Way ANOVA BIOMETRICAL JOURNAL, Issue 3 2008Wei NingAbstract This article proposes a new test to detect interactions in replicated two-way ANOVA models, more powerful than the classical F -test and more general than the test of Terbeck and Davies (1998, Annals of Statistics26, 1279,1305) developed for the case with unconditionally identifiable interaction pattern. We use the parameterization without the conventional restrictions on the interaction terms and base our test on the maximum of the standardized disturbance estimates. We show that our test is unbiased and consistent, and discuss how to estimate the p -value of the test. In a 3 × 3 case, which is our main focus in this article, the exact p -value can be computed by using four-dimensional integrations. For a general I × J case which requires an (I , 1) × (J , 1) dimensional integration for a numerical evaluation of the exact p -value, we propose to use an improved Bonferroni inequality to estimate an upperbound of the p -value and simulations indicate a reasonable accuracy of the upperbound. Via simulations, we show that our test is more powerful than the classical F -test and also that it can deal with both situations: unconditionally identifiable and non-unconditionally identifiable cases. An application to genetic data is presented in which the new test is significant, while the classical F -test failed to detect interactions. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## Strategies for computing goal-oriented a posteriori error measures in non-linear elasticity INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 8 2002Fredrik LarssonAbstract We investigate the characteristics and performance of goal-oriented a posteriori error measures for a class of non-linear elasticity models, while restriction is made to small strain theory. The chosen error measure of the displacement field can be global or local (probing the chosen quantity in a specific spatial point). The error is computable with the aid of the solution of a dual problem whose data depend on the error measure. The main thrust of the paper is to evaluate the performance of a few different approximation strategies for computing the dual solution. The chosen strategies are compared in terms of accuracy, ease of implementation, reliability and cost-efficiency. A well-known numerical example, the Cook's membrane, is used for the numerical evaluations. Copyright © 2002 John Wiley & Sons, Ltd. [source] ## Classical and advanced multilayered plate elements based upon PVD and RMVT. INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 3 2002Part 2: Numerical implementationsAbstract This paper presents numerical evaluations related to the multilayered plate elements which were proposed in the companion paper (Part 1). Two-dimensional modellings with linear and higher-order (up to fourth order) expansion in the z -plate/layer thickness direction have been implemented for both displacements and transverse stresses. Layer-wise as well as equivalent single-layer modellings are considered on both frameworks of the principle of virtual displacements and Reissner mixed variational theorem. Such a variety has led to the implementation of 22 plate theories. As far as finite element approximation is concerned, three quadrilaters have been considered (four-, eight- and nine-noded plate elements). As a result, 22×3 different finite plate elements have been compared in the present analysis. The automatic procedure described in Part 1, which made extensive use of indicial notations, has herein been referred to in the considered computer implementations. An assessment has been made as far as convergence rates, numerical integrations and comparison to correspondent closed-form solutions are concerned. Extensive comparison to early and recently available results has been made for sample problems related to laminated and sandwich structures. Classical formulations, full mixed, hybrid, as well as three-dimensional solutions have been considered in such a comparison. Numerical substantiation of the importance of the fulfilment of zig-zag effects and interlaminar equilibria is given. The superiority of RMVT formulated finite elements over those related to PVD has been concluded. Two test cases are proposed as ,desk-beds' to establish the accuracy of the several theories. Results related to all the developed theories are presented for the first test case. The second test case, which is related to sandwich plates, restricts the comparison to the most significant implemented finite elements. It is proposed to refer to these test cases to establish the accuracy of existing or new higher-order, refined or improved finite elements for multilayered plate analyses. Copyright © 2002 John Wiley & Sons, Ltd. [source] ## Symbolic analytical solutions for the abundances differential equations of the Helium burning phase ASTRONOMISCHE NACHRICHTEN, Issue 5 2003M.I. NouhAbstract In this paper, a literal analytical solution is developed for the abundances differential equations of the helium burning phase in hot massive stars. The abundance for each of the basic elements 4He,12C,16O and 20Ne is obtained as a recurrent power series in time, which facilitates its symbolic and numerical evaluations. Numerical comparison between the present solution and the numerical integration of the differential equations for the abundances show good agreement. [source] |