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Approximate Representation (approximate + representation)
Selected AbstractsTangential-projection algorithm for manifold representation in unidentifiable model updating problemsEARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICS, Issue 4 2002Lambros S. Katafygiotis Abstract The problem of updating a structural model and its associated uncertainties by utilizing structural response data is addressed. In an identifiable case, the posterior probability density function (PDF) of the uncertain model parameters for given measured data can be approximated by a weighted sum of Gaussian distributions centered at a number of discrete optimal values of the parameters at which some positive measure-of-fit function is minimized. The present paper focuses on the problem of model updating in the general unidentifiable case for which certain simplifying assumptions available for identifiable cases are not valid. In this case, the PDF is distributed in the neighbourhood of an extended and usually highly complex manifold of the parameter space that cannot be calculated explicitly. The computational difficulties associated with calculating the highly complex posterior PDF are discussed and a new adaptive algorithm, referred to as the tangential-projection (TP) algorithm, allowing for an efficient approximate representation of the above manifold and the posterior PDF is presented. Using this approximation, expressions for calculating the uncertain predictive response are established. A numerical example involving noisy data is presented to demonstrate the proposed method. Copyright © 2002 John Wiley & Sons, Ltd. [source] The role of friction and secondary flaws on deflection and re-initiation of hydraulic fractures at orthogonal pre-existing fracturesGEOPHYSICAL JOURNAL INTERNATIONAL, Issue 3 2006Xi Zhang SUMMARY In this study, we explore the nature of plane-strain hydraulic fracture growth in the presence of pre-existing fractures such as joints without or with secondary flaws. The 2-D plane-strain fracture studied can be taken as a cross-section through the short dimensions of an elongated 3-D fracture or as an approximate representation of the leading edge of a 3-D fracture where the edge curvature is negligible. The fluid-driven fracture intersects a pre-existing fracture to which it is initially perpendicular and is assumed not to immediately cross, but is rather deflected into the pre-existing fracture. The intersection results in branching of the fracture and associated fluid flow into the pre-existing fracture. Further growth results in opening and frictional sliding along the pre-existing fracture. Fracture propagation in an impermeable homogeneous elastic medium and fluid invasion into a pre-existing fracture are both driven by an incompressible, Newtonian fluid injected at a constant rate. The frictional stress on the surfaces of pre-existing fractures is assumed to obey the Coulomb law. The governing equations for quasi-static fluid-driven fracture growth are given and a scaling is introduced to help identify important parameters. The displacement discontinuity method and the finite difference method are employed to deal with this coupling mechanism of rock fracture and fluid flow. In order to account for fluid lag, a method for separately tracking the crack tip and the fluid front is included in the numerical model. Numerical results are obtained for internal pressure, frictional contact stresses, opening and shear displacements, and fluid lag size, as well as for fracture re-initiation from secondary flaws. After fracture intersection, the hydraulic fracture growth mode changes from tensile to shearing. This contributes to increased injection pressure and to a reduction in fracture width. In the presence of pre-existing fractures, the fluid-driven cracks can be arrested or retarded in growth rate as a result of diversion of fluid flow into and frictional sliding along the pre-existing fractures. Frictional behaviour significantly affects the ability of the fluid to enter or penetrate the pre-existing fracture only for those situations where the fluid front is within a certain distance from the intersecting point. Importantly, fluid penetration requires higher injection pressure for frictionally weak pre-existing fractures. Fracture re-initiation from secondary flaws can reduce the injection pressure, but re-initiation is suppressed by large sliding on pre-existing fractures that are frictionally weak. [source] Do the Central Banks of Australia and New Zealand Behave Asymmetrically?THE ECONOMIC RECORD, Issue 261 2007Evidence from Monetary Policy Reaction Functions We test for evidence of asymmetric behaviour in the monetary policy reaction functions of the central banks of Australia and New Zealand. For the Reserve Bank of New Zealand, we found little evidence of asymmetric behaviour, whereas the Reserve Bank of Australia (RBA) appears to react more aggressively to negative output relative to positive output gaps of the same size. We impose additional structure on our model to help distinguish whether the asymmetric response originates from non-linearity in the inflation equation or from non-linearity in an approximate representation of the RBA's preferences over macroeconomic outcomes. We find that the preferences of the RBA may drive the asymmetry: the RBA appears to dislike negative output gaps more than positive output gaps of the same magnitude. We show this generates only a small increase in the conditional mean of inflation that is statistically indistinguishable from the target rate of inflation. [source] Linear random vibration by stochastic reduced-order modelsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 12 2010Mircea Grigoriu Abstract A practical method is developed for calculating statistics of the states of linear dynamic systems with deterministic properties subjected to non-Gaussian noise and systems with uncertain properties subjected to Gaussian and non-Gaussian noise. These classes of problems are relevant as most systems have uncertain properties, physical noise is rarely Gaussian, and the classical theory of linear random vibration applies to deterministic systems and can only deliver the first two moments of a system state if the noise is non-Gaussian. The method (1) is based on approximate representations of all or some of the random elements in the definition of linear random vibration problems by stochastic reduced-order models (SROMs), that is, simple random elements having a finite number of outcomes of unequal probabilities, (2) can be used to calculate statistics of a system state beyond its first two moments, and (3) establishes bounds on the discrepancy between exact and SROM-based solutions of linear random vibration problems. The implementation of the method has required to integrate existing and new numerical algorithms. Examples are presented to illustrate the application of the proposed method and assess its accuracy. Copyright © 2009 John Wiley & Sons, Ltd. [source] | ||||||||||