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Closed Formulas (closed + formula)
Selected AbstractsA fast, one-equation integration algorithm for the Lemaitre ductile damage modelINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 8 2002E. A. de Souza NetoArticle first published online: 3 MAY 200 Abstract This paper introduces an elastic predictor/return mapping integration algorithm for a simplified version of the Lemaitre ductile damage model, whose return mapping stage requires the solution of only one scalar non-linear equation. The simplified damage model differs from its original counterpart only in that it excludes kinematic hardening. It can be used to predict ductile damage growth whenever load reversal is absent or negligible,a condition met in a vast number of practical engineering applications. The one-equation integration scheme proves highly efficient in the finite element solution of typical boundary value problems, requiring computation times comparable to those observed in classical von Mises implementations. This is in sharp contrast to the previously proposed implementations of the original model whose return mapping may require, in the most general case, the solution of a system of 14 coupled algebraic equations. For completeness, a closed formula for the corresponding consistent tangent operator is presented. The performance of the algorithm is illustrated by means of a numerical example. Copyright © 2002 John Wiley & Sons, Ltd. [source] Wiener,Kolmogorov Filtering and Smoothing for Multivariate Series With State,Space StructureJOURNAL OF TIME SERIES ANALYSIS, Issue 3 2007Víctor Gómez Abstract., Wiener,Kolmogorov filtering and smoothing usually deal with projection problems for stochastic processes that are observed over semi-infinite and doubly infinite intervals. For multivariate stationary series, there exist closed formulae based on covariance generating functions that were first given independently by N. Wiener and A.N. Kolmogorov around 1940. In this article, we consider multivariate series with a state,space structure and, using a new purely algebraic approach to the problem, we prove the equivalence between Wiener,Kolmogorov filtering and Kalman filtering. Up to now, this equivalence has only been partially shown. In addition, we get some new recursions for smoothing and some new recursions to compute the filter weights and the covariance generating functions of the errors. The results are extended to nonstationary series. [source] A note on formulas for localized failure of frictional materials in compression and biaxial loading modesINTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, Issue 10 2001Matthias Lambrecht Abstract The paper investigates aspects of the localization analysis of frictional materials. We derive closed formulas and diagrams for the inclination angle of critical discontinuity surfaces which develop in homogeneous compression and biaxial loading tests. The localization analysis is based on a Drucker,Prager-type elastoplastic hardening model for non-associated plastic flow at small strains, which we represent in spectral form. For this type of constitutive model, general analytical formulas for the so-called critical hardening modulus and the inclination angle of critical discontinuity surfaces are derived for the plane strain case. The subsequent treatment then specializes these formulas for the analysis of compression and biaxial loading modes. The key contribution here is a detailed analysis of plane strain deformation modes where the localized failure occurs after subsequent plastic flow. The derived formulas and diagrams can be applied to the checking of an accompanying localization analysis of frictional materials in finite-element computations. Copyright © 2001 John Wiley & Sons, Ltd. [source] Stochastic perturbation approach to the wavelet-based analysisNUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 4 2004M. Kami Abstract The wavelet-based decomposition of random variables and fields is proposed here in the context of application of the stochastic second order perturbation technique. A general methodology is employed for the first two probabilistic moments of a linear algebraic equations system solution, which are obtained instead of a single solution projection in the deterministic case. The perturbation approach application allows determination of the closed formulas for a wavelet decomposition of random fields. Next, these formulas are tested by symbolic projection of some elementary random field. Copyright © 2004 John Wiley & Sons, Ltd. [source] | |||||||||||||