Numerical Integration (numerical + integration)

Distribution by Scientific Domains

Terms modified by Numerical Integration

  • numerical integration procedure
  • numerical integration scheme

  • Selected Abstracts

    Numerical integration of differential-algebraic equations with mixed holonomic and control constraints

    Mahmud Quasem
    The present work aims at the incorporation of control (or servo) constraints into finite,dimensional mechanical systems subject to holonomic constraints. In particular, we focus on underactuated systems, defined as systems in which the number of degrees of freedom exceeds the number of inputs. The corresponding equations of motion can be written in the form of differential,algebraic equations (DAEs) with a mixed set of holonomic and control constraints. Apart from closed,loop multibody systems, the present formulation accommodates the so,called rotationless formulation of multibody dynamics. To this end, we apply a specific projection method to the DAEs in terms of redundant coordinates. A similar projection approach has been previously developed in the framework of generalized coordinates by Blajer & Ko,odziejczyk [1]. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

    3-Fluoro-2,4-dioxa-3-phosphadecalins as Inhibitors of Acetylcholinesterase.

    A Reappraisal of Kinetic Mechanisms, Diagnostic Methods
    Abstract A systematic survey of the acetylcholine-mimetic 2,4-dioxa-3-phosphadecalins as irreversible inhibitors of acetylcholinesterase revealed hitherto overlooked properties as far as the kinetic mechanisms of interaction are concerned. As a support to past and future work in this field, we describe the kinetics of eight reaction schemes that may be found in irreversible enzyme modification and compare them with two mechanism of reversible, slow-binding inhibition. The relevant kinetic equations and their associated graphical representations are given for all mechanisms, and concrete examples illustrate their practical use. Since irreversible inhibition is a time-dependent phenomenon, kinetic analysis is greatly facilitated by fitting the appropriate integrated rate equations to reaction-progress curves by nonlinear regression. This primary scrutiny provides kinetic parameters that are indispensable tools for diagnosing the kinetic mechanism and for calculating inhibition constants. Numerical integration of sets of differential equations is an additional useful investigation tool in critical situations, e.g., when inhibitors are unstable and/or act as irreversible modifiers only temporarily. [source]

    Strategies for the numerical integration of DAE systems in multibody dynamics

    E. Pennestrě
    Abstract The number of multibody dynamics courses offered in the university is increasing. Often the instructor has the necessity to go through the steps of an algorithm by working out a simple example. This gives the student a better understand of the basic theory. This paper provides a tutorial on the numerical integration of differential-algebraic equations (DAE) arising from the dynamic modeling of multibody mechanical systems. In particular, some algorithms based on the orthogonalization of the Jacobian matrix are herein discussed. All the computational steps involved are explained in detail and by working out a simple example. It is also reported a brief description and an application of the multibody code NumDyn3D which uses the Singular Value Decomposition (SVD) approach. © 2004 Wiley Periodicals, Inc. Comput Appl Eng Educ 12: 106,116, 2004; Published online in Wiley InterScience (; DOI 10.1002/cae.20005 [source]

    Error estimation of closed-form solution for annual rate of structural collapse

    Brendon A. Bradley
    Abstract With the increasing emphasis of performance-based earthquake engineering in the engineering community, several investigations have been presented outlining simplified approaches suitable for performance-based seismic design (PBSD). Central to most of these PBSD approaches is the use of closed-form analytical solutions to the probabilistic integral equations representing the rate of exceedance of key performance measures. Situations where such closed-form solutions are not appropriate primarily relate to the problem of extrapolation outside of the region in which parameters of the closed-form solution are fit. This study presents a critical review of the closed-form solution for the annual rate of structural collapse. The closed-form solution requires the assumptions of lognormality of the collapse fragility and power model form of the ground motion hazard, of which the latter is more significant regarding the error of the closed-form solution. Via a parametric study, the key variables contributing to the error between the closed-form solution and solution via numerical integration are illustrated. As these key variables cannot be easily measured, it casts doubt on the use of such closed-form solutions in future PBSD, especially considering the simple and efficient nature of using direct numerical integration to obtain the solution. Copyright © 2008 John Wiley & Sons, Ltd. [source]

    Effects of toroidal HVDC ground electrode on earth-return circuits

    W. Machczy
    This paper presents a method of evaluation of currents and potentials, excited by conductive effects of high-voltage direct-current (HVDC) transmission system, along two parallel earth-return circuits such as pipelines and cables buried in the vicinity of a toroidal ground electrode. It is assumed that the system considered is linear and that the earth is an isotropic, homogeneous medium of finite conductivity. Conductive coupling between earth-return circuits is taken into account, whereas the reaction of the conductors' currents on the electrode current is disregarded. The transmission line model of the conductor with earth-return, a segmental linear approximation of the curve of the primary earth potential distribution along the conductor and the concept of superposition have been used in the method. It should be pointed out, that the method does not require the time consuming numerical integration. The technical applications of the method are illustrated by examples. [source]

    Comparison of methods to model the gravitational gradients from topographic data bases

    Christopher Jekeli
    SUMMARY A number of methods have been developed over the last few decades to model the gravitational gradients using digital elevation data. All methods are based on second-order derivatives of the Newtonian mass integral for the gravitational potential. Foremost are algorithms that divide the topographic masses into prisms or more general polyhedra and sum the corresponding gradient contributions. Other methods are designed for computational speed and make use of the fast Fourier transform (FFT), require a regular rectangular grid of data, and yield gradients on the entire grid, but only at constant altitude. We add to these the ordinary numerical integration (in horizontal coordinates) of the gradient integrals. In total we compare two prism, two FFT and two ordinary numerical integration methods using 1, elevation data in two topographic regimes (rough and moderate terrain). Prism methods depend on the type of finite elements that are generated with the elevation data; in particular, alternative triangulations can yield significant differences in the gradients (up to tens of Eötvös). The FFT methods depend on a series development of the topographic heights, requiring terms up to 14th order in rough terrain; and, one popular method has significant bias errors (e.g. 13 Eötvös in the vertical,vertical gradient) embedded in its practical realization. The straightforward numerical integrations, whether on a rectangular or triangulated grid, yield sub-Eötvös differences in the gradients when compared to the other methods (except near the edges of the integration area) and they are as efficient computationally as the finite element methods. [source]

    Time-domain approach to linearized rotational response of a three-dimensional viscoelastic earth model induced by glacial-isostatic adjustment: I. Inertia-tensor perturbations

    k Martinec
    SUMMARY For a spherically symmetric viscoelastic earth model, the movement of the rotation vector due to surface and internal mass redistribution during the Pleistocene glaciation cycle has conventionally been computed in the Laplace-transform domain. The method involves multiplication of the Laplace transforms of the second-degree surface-load and tidal-load Love numbers with the time evolution of the surface load followed by inverse Laplace transformation into the time domain. The recently developed spectral finite-element method solves the field equations governing glacial-isostatic adjustment (GIA) directly in the time domain and, thus, eliminates the need of applying the Laplace-domain method. The new method offers the possibility to model the GIA-induced rotational response of the Earth by time integration of the linearized Liouville equation. The theory presented here derives the temporal perturbation of the inertia tensor, required to be specified in the Liouville equation, from time variations of the second-degree gravitational-potential coefficients by the MacCullagh's formulae. This extends the conventional approach based on the second-degree load Love numbers to general 3-D viscoelastic earth models. The verification of the theory of the GIA-induced rotational response of the Earth is performed by using two alternative approaches of computing the perturbation of the inertia tensor: a direct numerical integration and the Laplace-domain method. The time-domain solution of both the GIA and the induced rotational response of the Earth is readily combined with a time-domain solution of the sea level equation with a time-varying shoreline geometry. In a follow-up paper, we derive the theory for the case when GIA-induced perturbations in the centrifugal force affect not only the distribution of sea water, but also deformations and gravitational-potential perturbations of the Earth. [source]

    Comment on ,Yamada H, Nakamura F, Watanabe Y, Murakami M and Nogami T. 2005.

    Measuring hydraulic permeability in a streambed using the packer test.
    Abstract In a recent paper, Yamada et al. (2005) derived an expression to calculate hydraulic permeability under non-Darcy flow conditions using the packer test; their results were obtained via numerical integration of the derived expression. Their findings are extended by providing a closed-form solution to the problem, and its dependence upon key parameters is illustrated. Copyright © 2008 John Wiley & Sons, Ltd. [source]

    Implicit integration of a mixed isotropic,kinematic hardening plasticity model for structured clays

    Angelo Amorosi
    Abstract In recent years, a number of constitutive models have been proposed to describe mathematically the mechanical response of natural clays. Some of these models are characterized by complex formulations, often leading to non-trivial problems in their numerical integration in finite elements codes. The paper describes a fully implicit stress-point algorithm for the numerical integration of a single-surface mixed isotropic,kinematic hardening plasticity model for structured clays. The formulation of the model stems from a compromise between its capability of reproducing the larger number of features characterizing the behaviour of structured clays and the possibility of developing a robust integration algorithm for its implementation in a finite elements code. The model is characterized by an ellipsoid-shaped yield function, inside which a stress-dependent reversible stiffness is accounted for by a non-linear hyperelastic formulation. The isotropic part of the hardening law extends the standard Cam-Clay one to include plastic strain-driven softening due to bond degradation, while the kinematic hardening part controls the evolution of the position of the yield surface in the stress space. The proposed algorithm allows the consistent linearization of the constitutive equations guaranteeing the quadratic rate of asymptotic convergence in the global-level Newton,Raphson iterative procedure. The accuracy and the convergence properties of the proposed algorithm are evaluated with reference to the numerical simulations of single element tests and the analysis of a typical geotechnical boundary value problem. Copyright © 2007 John Wiley & Sons, Ltd. [source]

    Generalized trapezoidal numerical integration of an advanced soil model

    Yunming Yang
    Abstract This paper investigates the numerical performance of the generalized trapezoidal integration rule by using an advanced soil model. The generalized trapezoidal integration rule can include many other integration algorithms by adjusting a single parameter , ranging from 1 to 0. The soil model used is the recently developed middle surface concept (MSC) sand model which simulates different soil response characteristics by using different pseudo-yield functions. The generalized trapezoidal rule and MSC sand model are used to simulate the responses of sand samples with different relative densities under various initial and loading conditions. Instead of a single step, multiple loading steps bring the sample to the vicinity of failure. These comprehensive investigations examine and compare the influences of various values of , on the numerical solution of integrated constitutive equations, the convergence of Newton's iterative scheme, and the integration accuracy. Copyright © 2007 John Wiley & Sons, Ltd. [source]

    The response of an elastic half-space under a momentary shear line impulse

    Moche Ziv
    Abstract The response of an ideal elastic half-space to a line-concentrated impulsive vector shear force applied momentarily is obtained by an analytical,numerical computational method based on the theory of characteristics in conjunction with kinematical relations derived across surfaces of strong discontinuities. The shear force is concentrated along an infinite line, drawn on the surface of the half-space, while being normal to that line as well as to the axis of symmetry of the half-space. An exact loading model is introduced and built into the computational method for this shear force. With this model, a compatibility exists among the prescribed applied force, the geometric decay of the shear stress component at the precursor shear wave, and the boundary conditions of the half-space; in this sense, the source configuration is exact. For the transient boundary-value problem described above, a wave characteristics formulation is presented, where its differential equations are extended to allow for strong discontinuities which occur in the material motion of the half-space. A numerical integration of these extended differential equations is then carried out in a three-dimensional spatiotemporal wavegrid formed by the Cartesian bicharacteristic curves of the wave characteristics formulation. This work is devoted to the construction of the computational method and to the concepts involved therein, whereas the interpretation of the resultant transient deformation of the half-space is presented in a subsequent paper. Copyright © 2003 John Wiley & Sons, Ltd. [source]

    Visual framework for development and use of constitutive models

    Youssef M. A. Hashash
    Abstract Advanced constitutive relations are used in geotechnical engineering to capture measured soil and rock behaviour in the laboratory, and in numerical models to represent the material response. These constitutive relations have traditionally been difficult to use, understand, and develop except by a limited number of specialists. This paper describes a framework for transforming the representation of constitutive relations, as well as stress and strain quantities from a series of mathematical equations and matrix quantities to multidimensional geometric/visual objects in a dynamic interactive colour-rich display environment. The paper proposes a shift in current approaches to the development of constitutive equations and their use in numerical simulations by taking advantage of rapid advancements in information technology and computer graphics. A novel interactive visualization development and learning environment for material constitutive relations referred to as VizCoRe is presented. Visualization examples of two constitutive relations, the linear elastic with von Mises failure criteria and the Modified Cam Clay (MCC) are shown. These include two- and three-dimensional renderings of stress states and paths and yield and failure surfaces. In addition, the environment allows for the visualization of the implicit integration algorithm used for the numerical integration of both constitutive models. Copyright © 2002 John Wiley & Sons, Ltd. [source]

    A simple robust numerical integration algorithm for a power-law visco-plastic model under both high and low rate-sensitivity

    E. A. de Souza Neto
    Abstract This note describes a simple and extremely robust algorithm for numerical integration of the power-law-type elasto-viscoplastic constitutive model discussed by Peri, (Int. J. Num. Meth. Eng. 1993; 36: 1365,1393). As the rate-independent limit is approached with increasing exponents, the evolution equations of power-law-type models are known to become stiff. Under such conditions, the solution of the implicitly discretized viscoplastic evolution equation cannot be easily obtained by standard root-finding algorithms. Here, a procedure which proves to be remarkably robust under stiff conditions is obtained by means of a simple logarithmic mapping of the basic backward Euler time-discrete equation for the incremental plastic multiplier. The logarithm-transformed equation is solved by the standard Newton,Raphson scheme combined with a simple bisection procedure which ensures that the iterative guesses for the equation unknown (the incremental equivalent plastic strain) remain within the domain where the transformed equation makes sense. The resulting implementation can handle small and large (up to order 106) power-law exponents equally. This allows its effective use under any situation of practical interest, ranging from high rate-sensitivity to virtually rate-independent conditions. The robustness of the proposed scheme is demonstrated by numerical examples. Copyright © 2003 John Wiley & Sons, Ltd. [source]

    Interface handling for three-dimensional higher-order XFEM-computations in fluid,structure interaction

    Ursula M. Mayer
    Abstract Three-dimensional higher-order eXtended finite element method (XFEM)-computations still pose challenging computational geometry problems especially for moving interfaces. This paper provides a method for the localization of a higher-order interface finite element (FE) mesh in an underlying three-dimensional higher-order FE mesh. Additionally, it demonstrates, how a subtetrahedralization of an intersected element can be obtained, which preserves the possibly curved interface and allows therefore exact numerical integration. The proposed interface algorithm collects initially a set of possibly intersecting elements by comparing their ,eXtended axis-aligned bounding boxes'. The intersection method is applied to a highly reduced number of intersection candidates. The resulting linearized interface is used as input for an elementwise constrained Delaunay tetrahedralization, which computes an appropriate subdivision for each intersected element. The curved interface is recovered from the linearized interface in the last step. The output comprises triangular integration cells representing the interface and tetrahedral integration cells for each intersected element. Application of the interface algorithm currently concentrates on fluid,structure interaction problems on low-order and higher-order FE meshes, which may be composed of any arbitrary element types such as hexahedra, tetrahedra, wedges, etc. Nevertheless, other XFEM-problems with explicitly given interfaces or discontinuities may be tackled in addition. Multiple structures and interfaces per intersected element can be handled without any additional difficulties. Several parallelization strategies exist depending on the desired domain decomposition approach. Numerical test cases including various geometrical exceptions demonstrate the accuracy, robustness and efficiency of the interface handling. Copyright © 2009 John Wiley & Sons, Ltd. [source]

    Rigid body dynamics in terms of quaternions: Hamiltonian formulation and conserving numerical integration

    Peter Betsch
    Abstract In the present paper unit quaternions are used to describe the rotational motion of a rigid body. The unit-length constraint is enforced explicitly by means of an algebraic constraint. Correspondingly, the equations of motion assume the form of differential-algebraic equations (DAEs). A new route to the derivation of the mass matrix associated with the quaternion formulation is presented. In contrast to previous works, the newly proposed approach yields a non-singular mass matrix. Consequently, the passage to the Hamiltonian framework is made possible without the need to introduce undetermined inertia terms. The Hamiltonian form of the DAEs along with the notion of a discrete derivative make possible the design of a new quaternion-based energy,momentum scheme. Two numerical examples demonstrate the performance of the newly developed method. In this connection, comparison is made with a quaternion-based variational integrator, a director-based energy,momentum scheme, and a momentum conserving scheme relying on the discretization of the classical Euler's equations. Copyright © 2009 John Wiley & Sons, Ltd. [source]

    Adaptive through-thickness integration for accurate springback prediction

    I. A. Burchitz
    Abstract Accurate numerical prediction of springback in sheet metal forming is essential for the automotive industry. Numerous factors influence the accuracy of prediction of this complex phenomenon by using the finite element method. One of them is the numerical integration through the thickness of shell elements. It is known that the traditional numerical schemes are very inefficient in elastic,plastic analysis and even for simple problems they require up to 50 integration points for an accurate springback prediction. An adaptive through-thickness integration strategy can be a good alternative. The main characteristic feature of the strategy is that it defines abscissas and weights depending on the integrand's properties and, thus, can adapt itself to improve the accuracy of integration. A concept of an adaptive through-thickness integration strategy for shell elements is presented in this paper. Its potential is demonstrated using two examples. Calculations of a simple test,bending a beam under tension,show that for a similar set of material and process parameters the adaptive rule with seven integration points performs significantly better than the traditional trapezoidal rule with 50 points. Simulations of an unconstrained cylindrical bending problem demonstrate that the adaptive through-thickness integration strategy for shell elements can guarantee an accurate springback prediction at minimal costs. Copyright © 2007 John Wiley & Sons, Ltd. [source]

    Development of a genetic algorithm-based lookup table approach for efficient numerical integration in the method of finite spheres with application to the solution of thin beam and plate problems

    Suleiman BaniHani
    Abstract It is observed that for the solution of thin beam and plate problems using the meshfree method of finite spheres, Gaussian and adaptive quadrature schemes are computationally inefficient. In this paper, we develop a novel technique in which the integration points and weights are generated using genetic algorithms and stored in a lookup table using normalized coordinates as part of an offline computational step. During online computations, this lookup table is used much like a table of Gaussian integration points and weights in the finite element computations. This technique offers significant reduction of computational time without sacrificing accuracy. Example problems are solved which demonstrate the effectiveness of the procedure. Copyright © 2006 John Wiley & Sons, Ltd. [source]

    An integrated procedure for three-dimensional structural analysis with the finite cover method

    Kenjiro Terada
    Abstract In this paper an integrated procedure for three-dimensional (3D) structural analyses with the finite cover method (FCM) is introduced. In the pre-process of this procedure, the geometry of a structure is modelled by 3D-CAD, followed by digitization to have the corresponding voxel model, and then the structure is covered by a union of mathematical covers, namely a mathematical mesh independently generated for approximation purposes. Since the mesh topology in the FCM does not need to conform to the physical boundaries of the structure, the mesh can be regular and structured. Thus, the numerical analysis procedure is free from the difficulties mesh generation typically poses and, in this sense, enables us to realize the mesh-free analysis. After formulating the FCM with interface elements for the static equilibrium state of a structure, we detail the procedure of the finite cover modelling, including the geometry modelling with 3D-CAD and the identification of the geometry covered by a regular mesh for numerical integration. Prior to full 3D modelling and analysis, we present a simple numerical example to confirm the equivalence of the performance of the FCM and that of the standard finite element method (FEM). Finally, representative numerical examples are presented to demonstrate the capabilities of the proposed analysis procedure. Copyright © 2005 John Wiley & Sons, Ltd. [source]

    F-bar-based linear triangles and tetrahedra for finite strain analysis of nearly incompressible solids.

    Part I: formulation, benchmarking
    Abstract This paper proposes a new technique which allows the use of simplex finite elements (linear triangles in 2D and linear tetrahedra in 3D) in the large strain analysis of nearly incompressible solids. The new technique extends the F-bar method proposed by de Souza Neto et al. (Int. J. Solids and Struct. 1996; 33: 3277,3296) and is conceptually very simple: It relies on the enforcement of (near-) incompressibility over a patch of simplex elements (rather than the point-wise enforcement of conventional displacement-based finite elements). Within the framework of the F-bar method, this is achieved by assuming, for each element of a mesh, a modified (F-bar) deformation gradient whose volumetric component is defined as the volume change ratio of a pre-defined patch of elements. The resulting constraint relaxation effectively overcomes volumetric locking and allows the successful use of simplex elements under finite strain near-incompressibility. As the original F-bar procedure, the present methodology preserves the displacement-based structure of the finite element equations as well as the strain-driven format of standard algorithms for numerical integration of path-dependent constitutive equations and can be used regardless of the constitutive model adopted. The new elements are implemented within an implicit quasi-static environment. In this context, a closed form expression for the exact tangent stiffness of the new elements is derived. This allows the use of the full Newton,Raphson scheme for equilibrium iterations. The performance of the proposed elements is assessed by means of a comprehensive set of benchmarking two- and three-dimensional numerical examples. Copyright © 2005 John Wiley & Sons, Ltd. [source]

    A generalized dimension-reduction method for multidimensional integration in stochastic mechanics

    H. Xu
    Abstract A new, generalized, multivariate dimension-reduction method is presented for calculating statistical moments of the response of mechanical systems subject to uncertainties in loads, material properties, and geometry. The method involves an additive decomposition of an N -dimensional response function into at most S -dimensional functions, where S,N; an approximation of response moments by moments of input random variables; and a moment-based quadrature rule for numerical integration. A new theorem is presented, which provides a convenient means to represent the Taylor series up to a specific dimension without involving any partial derivatives. A complete proof of the theorem is given using two lemmas, also proved in this paper. The proposed method requires neither the calculation of partial derivatives of response, as in commonly used Taylor expansion/perturbation methods, nor the inversion of random matrices, as in the Neumann expansion method. Eight numerical examples involving elementary mathematical functions and solid-mechanics problems illustrate the proposed method. Results indicate that the multivariate dimension-reduction method generates convergent solutions and provides more accurate estimates of statistical moments or multidimensional integration than existing methods, such as first- and second-order Taylor expansion methods, statistically equivalent solutions, quasi-Monte Carlo simulation, and the fully symmetric interpolatory rule. While the accuracy of the dimension-reduction method is comparable to that of the fourth-order Neumann expansion method, a comparison of CPU time suggests that the former is computationally far more efficient than the latter. Copyright © 2004 John Wiley & Sons, Ltd. [source]

    A 3D mortar method for solid mechanics,

    Michael A. Puso
    Abstract A version of the mortar method is developed for tying arbitrary dissimilar 3D meshes with a focus on issues related to large deformation solid mechanics. Issues regarding momentum conservation, large deformations, computational efficiency and bending are considered. In particular, a mortar method formulation that is invariant to rigid body rotations is introduced. A scheme is presented for the numerical integration of the mortar surface projection integrals applicable to arbitrary 3D curved dissimilar interfaces. Here, integration need only be performed at problem initialization such that coefficients can be stored and used throughout a quasi-static time stepping process even for large deformation problems. A degree of freedom reduction scheme exploiting the dual space interpolation method such that direct linear solution techniques can be applied without Lagrange multipliers is proposed. This provided a significant reduction in factorization times. Example problems which touch on the aforementioned solid mechanics related issues are presented. Published in 2003 by John Wiley & Sons, Ltd. [source]

    A numerical model for the cyclic deterioration of railway tracks

    Akke S. J. Suiker
    Abstract An elasto-plastic material model is proposed that can be used to simulate the cyclic deterioration of railway tracks. The model describes the envelope of the irreversible, plastic material response generated during a cyclic loading process, thereby distinguishing the mechanisms of frictional sliding and volumetric compaction. The reversible response is represented by a pressure-dependent, hypo-elastic material law. After the numerical integration of the model is specified, the model is calibrated on laboratory experiments and employed in a finite-element case study of the long-term settlement behaviour of a railway track. The main features of the model are illustrated by comparing the computed response with the response obtained by in situ track measurements. Copyright © 2003 John Wiley & Sons, Ltd. [source]

    Self-regular boundary integral equation formulations for Laplace's equation in 2-D

    A. B. Jorge
    Abstract The purpose of this work is to demonstrate the application of the self-regular formulation strategy using Green's identity (potential-BIE) and its gradient form (flux-BIE) for Laplace's equation. Self-regular formulations lead to highly effective BEM algorithms that utilize standard conforming boundary elements and low-order Gaussian integrations. Both formulations are discussed and implemented for two-dimensional potential problems, and numerical results are presented. Potential results show that the use of quartic interpolations is required for the flux-BIE to show comparable accuracy to the potential-BIE using quadratic interpolations. On the other hand, flux error results in the potential-BIE implementation can be dominated by the numerical integration of the logarithmic kernel of the remaining weakly singular integral. Accuracy of these flux results does not improve beyond a certain level when using standard quadrature together with a special transformation, but when an alternative logarithmic quadrature scheme is used these errors are shown to reduce abruptly, and the flux results converge monotonically to the exact answer. In the flux-BIE implementation, where all integrals are regularized, flux results accuracy improves systematically, even with some oscillations, when refining the mesh or increasing the order of the interpolating function. The flux-BIE approach presents a great numerical sensitivity to the mesh generation scheme and refinement. Accurate results for the potential and the flux were obtained for coarse-graded meshes in which the rate of change of the tangential derivative of the potential was better approximated. This numerical sensitivity and the need for graded meshes were not found in the elasticity problem for which self-regular formulations have also been developed using a similar approach. Logarithmic quadrature to evaluate the weakly singular integral is implemented in the self-regular potential-BIE, showing that the magnitude of the error is dependent only on the standard Gauss integration of the regularized integral, but not on this logarithmic quadrature of the weakly singular integral. The self-regular potential-BIE is compared with the standard (CPV) formulation, showing the equivalence between these formulations. The self-regular BIE formulations and computational algorithms are established as robust alternatives to singular BIE formulations for potential problems. Copyright © 2001 John Wiley & Sons, Ltd. [source]

    Defining and optimizing algorithms for neighbouring particle identification in SPH fluid simulations

    G. Viccione
    Abstract Lagrangian particle methods such as smoothed particle hydrodynamics (SPH) are very demanding in terms of computing time for large domains. Since the numerical integration of the governing equations is only carried out for each particle on a restricted number of neighbouring ones located inside a cut-off radius rc, a substantial part of the computational burden depends on the actual search procedure; it is therefore vital that efficient methods are adopted for such a search. The cut-off radius is indeed much lower than the typical domain's size; hence, the number of neighbouring particles is only a little fraction of the total number. Straightforward determination of which particles are inside the interaction range requires the computation of all pair-wise distances, a procedure whose computational time would be unpractical or totally impossible for large problems. Two main strategies have been developed in the past in order to reduce the unnecessary computation of distances: the first based on dynamically storing each particle's neighbourhood list (Verlet list) and the second based on a framework of fixed cells. The paper presents the results of a numerical sensitivity study on the efficiency of the two procedures as a function of such parameters as the Verlet size and the cell dimensions. An insight is given into the relative computational burden; a discussion of the relative merits of the different approaches is also given and some suggestions are provided on the computational and data structure of the neighbourhood search part of SPH codes. Copyright © 2008 John Wiley & Sons, Ltd. [source]

    Orthogonality of modal bases in hp finite element models

    V. Prabhakar
    Abstract In this paper, we exploit orthogonality of modal bases (SIAM J. Sci. Comput. 1999; 20:1671,1695) used in hp finite element models. We calculate entries of coefficient matrix analytically without using any numerical integration, which can be computationally very expensive. We use properties of Jacobi polynomials and recast the entries of the coefficient matrix so that they can be evaluated analytically. We implement this in the context of the least-squares finite element model although this procedure can be used in other finite element formulations. In this paper, we only develop analytical expressions for rectangular elements. Spectral convergence of the L2 least-squares functional is verified using exact solution of Kovasznay flow. Numerical results for transient flow over a backward-facing step are also presented. We also solve steady flow past a circular cylinder and show the reduction in computational cost using expressions developed herein. Copyright © 2007 John Wiley & Sons, Ltd. [source]

    The analytical resolution of parallel first- and second-order reaction mechanisms

    N. B. Caballero
    Given the species A1 and A2, the competition among the three different elementary processes (1) (2) (3) is frequently found in thermal and photochemical reaction systems. In the present paper, an analytical resolution of the system (1),(3), performed under plausible contour conditions, namely, finite initial molar concentrations for both reactants, [A2]0 and [A1]0, and nonzero reaction rate coefficients k1, k2, and k3, leads to the equation [A1] = ((,[A2], , [A2])/,) , ,, where , = k1/2k3, , = , + 1 = 2k3/k2, and , = ([A2]0 + ,[A1]0 + , ,))/[A2]0,. The comparison with a numerical integration employing the fourth-order Runge,Kutta algorithm for the well-known case of the oxidation of organic compounds by ferrate ion is performed. © 2010 Wiley Periodicals, Inc. Int J Chem Kinet 42: 562,566, 2010 [source]

    An accurate few-parameter ground state wave function for the lithium atom

    Nicolais L. Guevara
    Abstract A simple, seven-parameter trial function is proposed for a description of the ground state of the Lithium atom. It includes both spin functions. Inter-electronic distances appear in exponential form as well as in a pre-exponential factor, and the necessary energy matrix elements are evaluated by numerical integration in the space of the relative coordinates. Encouragingly accurate values of the energy and the cusp parameters as well as for some expectation values are obtained. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2009 [source]

    Recent topics in numerical integration

    Ronald Cools
    Abstract In this article, we describe some recent topics in the approximation of univariate and multivariate integrals. We especially pay attention to progress in the area of oscillatory integrals and in quasi-Monte Carlo methods using lattice rules. Many pointers to the relevant literature are provided. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2009 [source]

    Model density approach to the Kohn,Sham problem: Efficient extension of the density fitting technique

    Uwe Birkenheuer
    Abstract We present a novel procedure for treating the exchange-correlation contributions in the Kohn,Sham procedure. The approach proposed is fully variational and closely related to the so-called "fitting functions" method for the Coulomb Hartree problem; in fact, the method consistently uses this auxiliary representation of the electron density to determine the exchange-correlation contributions. The exchange-correlation potential and its matrix elements in a basis set of localized (atomic) orbitals can be evaluated by reusing the three-center Coulomb integrals involving fitting functions, while the computational cost of the remaining numerical integration is significantly reduced and scales only linearly with the size of the auxiliary basis. We tested the approach extensively for a large set of atoms and small molecules as well as for transition-metal carbonyls and clusters, by comparing total energies, atomization energies, structure parameters, and vibrational frequencies at the local density approximation and generalized gradient approximation levels of theory. The method requires a sufficiently flexible auxiliary basis set. We propose a minimal extension of the conventional auxiliary basis set, which yields essentially the same accuracy for the quantities just mentioned as the standard approach. The new method allows one to achieve substantial savings compared with a fully numerical integration of the exchange-correlation contributions. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005 [source]

    Efficiency in the calculation of absorption corrections for cylinders

    Takashi Ida
    Efficiency in the numerical calculation of absorption corrections for cylinders has been examined. Two mathematical expressions for the correction factors have been evaluated by two methods for numerical integration. It has been found that the Gauss,Legendre quadrature applied to the formula proposed by Thorkildsen & Larsen [Acta Cryst. (1998), A54, 172,185] gives results with relative errors ,10,6, using 12,×,12 terms in the numerical integration. The conventional approach, using Simpson's method in conjunction with the formula given by Dwiggins [Acta Cryst. (1975), A31, 146,148] for the absorption correction, is far less efficient. [source]