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Exact Integration (exact + integration)
Selected AbstractsExact integration of polynomial,exponential products with application to wave-based numerical methodsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 3 2009G. Gabard Abstract Wave-based numerical methods often require to integrate products of polynomials and exponentials. With quadrature methods, this task can be particularly expensive at high frequencies as large numbers of integration points are required. This paper presents a set of closed-form solutions for the integrals of polynomial,exponential products in two and three dimensions. These results apply to arbitrary polygons in two dimensions, and for arbitrary polygonal surfaces or polyhedral volumes in three dimensions. Quadrature methods are therefore not required for this class of integrals that can be evaluated quickly and exactly. Copyright © 2008 John Wiley & Sons, Ltd. [source] Exact integration of the stiffness matrix of an 8-node plane elastic finite element by symbolic computationNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 1 2008L. Videla Abstract Computer algebra systems (CAS) are powerful tools for obtaining analytical expressions for many engineering applications in both academic and industrial environments. CAS have been used in this paper to generate exact expressions for the stiffness matrix of an 8-node plane elastic finite element. The Maple software system was used to identify six basic formulas from which all the terms of the stiffness matrix could be obtained. The formulas are functions of the Cartesian coordinates of the corner nodes of the element, and elastic parameters Young's modulus and Poisson's ratio. Many algebraic manipulations were performed on the formulas to optimize their efficiency. The redaction in CPU time using the exact expressions as opposed to the classical Gauss,Legendre numerical integration approach was over 50%. In an additional study of accuracy, it was shown that the numerical approach could lead to quite significant errors as compared with the exact approach, especially as element distortion was increased.© 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2007 [source] Error estimates in 2-node shear-flexible beam elementsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 7 2003Gajbir Singh Abstract The objective of the paper is to report the investigation of error estimates/or convergence characteristics of shear-flexible beam elements. The order and magnitude of principal discretization error in the usage of various types beam elements such as: (a) 2-node standard isoparametric element, (b) 2-node field-consistent/reduced integration element and (c) 2-node coupled-displacement field element, is assessed herein. The method employs classical order of error analyses that is commonly used to evaluate the discretization error of finite difference methods. The finite element equilibrium equations at any node are expressed in terms of differential equations through the use of Taylor series. These differential equations are compared with the governing equations and error terms are identified. It is shown that the discretization error in coupled-field elements is the least compared to the field-consistent and standard finite elements (based on exact integration). Copyright © 2003 John Wiley & Sons, Ltd. [source] Numerically exact integration of a family of axisymmetric finite elementsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 4 2003T. E. Price Abstract Axisymmetric finite element stress analysis involves repeated integration of a rational polynomial integrand. For elements near the axis of symmetry, such integrals are quasi-singular, are difficult to integrate numerically, and can lead to significant computational errors. This paper describes a Gaussian quadrature procedure to integrate exactly, within computational limits, a class of rational polynomials over undistorted triangular and quadrilateral finite elements. The procedure's accuracy and efficiency are illustrated through a numerical example. Copyright © 2003 John Wiley & Sons, Ltd. [source] On the spectrum of the electric field integral equation and the convergence of the moment methodINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 1 2001Karl F. Warnick Abstract Existing convergence estimates for numerical scattering methods based on boundary integral equations are asymptotic in the limit of vanishing discretization length, and break down as the electrical size of the problem grows. In order to analyse the efficiency and accuracy of numerical methods for the large scattering problems of interest in computational electromagnetics, we study the spectrum of the electric field integral equation (EFIE) for an infinite, conducting strip for both the TM (weakly singular kernel) and TE polarizations (hypersingular kernel). Due to the self-coupling of surface wave modes, the condition number of the discretized integral equation increases as the square root of the electrical size of the strip for both polarizations. From the spectrum of the EFIE, the solution error introduced by discretization of the integral equation can also be estimated. Away from the edge singularities of the solution, the error is second order in the discretization length for low-order bases with exact integration of matrix elements, and is first order if an approximate quadrature rule is employed. Comparison with numerical results demonstrates the validity of these condition number and solution error estimates. The spectral theory offers insights into the behaviour of numerical methods commonly observed in computational electromagnetics. Copyright © 2001 John Wiley & Sons, Ltd. [source] A precise boundary element method for macromolecular transport propertiesJOURNAL OF COMPUTATIONAL CHEMISTRY, Issue 9 2004Sergio Aragon Abstract A very precise boundary element numerical solution of the exact formulation of the hydrodynamic resistance problem with stick boundary conditions is presented. BEST, the Fortran 77 program developed for this purpose, computes the full transport tensors in the center of resistance or the center of diffusion for an arbitrarily shaped rigid body, including rotation-translation coupling. The input for this program is a triangulation of the solvent-defined surface of the molecule of interest, given by Connolly's MSROLL or other suitable triangulator. The triangulation is prepared for BEST by COALESCE, a program that allows user control over the quality and number of triangles to describe the surface. High numerical precision is assured by effectively exact integration of the Oseen tensor over triangular surface elements, and by scaling the hydrodynamic computation to the precise surface area of the molecule. Efficiency of computation is achieved by the use of public domain LAPACK routines that call BLAS Level 3 hardware-optimized subroutines available for most processors. A protein computation can be done in less than 10 min of CPU time in a modern Pentium IV processor. The present work includes a complete analysis of the sources of error in the numerical work and techniques to eliminate these errors. The operation of BEST is illustrated with applications to ellipsoids of revolution, and Lysozyme, a small protein. The typical numerical accuracy achieved is 0.05% compared to analytical theory. The numerical precision for a protein is better than 1%, much better than experimental errors in these quantities, and more than 10 times better than traditional bead-based methods. © 2004 Wiley Periodicals, Inc. J Comput Chem 9: 1191,1205, 2004 [source] Direct assessment of structural resistance against pressurized fractureINTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, Issue 5 2003G. Bolzon Abstract The determination of the load bearing capacity of hydraulic structures such as dams, reservoirs and retaining walls requires the consideration of mixed-mode fracture, possibly driven by the fluid pressure, in correspondence to artificial and natural joints (or cracks, in the latter case). A friction-cohesive softening interface model with coupled degradation of normal and tangential strength is introduced here to account for the essential features of the joint behaviour; its predictive capability is assessed through extensive calculations. Alternative numerical techniques resting on the discrete-crack approach are considered, focusing on simplified approaches for the direct appraisal of the structural resistance. Comparison is made with the results of evolutionary analyses, based on a priori piecewise linearization of the interface model and on ,exact integration'. Copyright © 2003 John Wiley & Sons, Ltd. [source] | ||||||||||