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Time Derivatives (time + derivative)
Selected AbstractsA higher-order predictor,corrector scheme for two-dimensional advection,diffusion equationINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 4 2008Chuanjian Man Abstract A higher-order accurate numerical scheme is developed to solve the two-dimensional advection,diffusion equation in a staggered-grid system. The first-order spatial derivatives are approximated by the fourth-order accurate finite-difference scheme, thus all truncation errors are kept to a smaller order of magnitude than those of the diffusion terms. Therefore, there is no need to add an artificial diffusion term to balance the unwanted numerical diffusion. For the time derivative, the fourth-order accurate Adams,Bashforth predictor,corrector method is applied. The stability analysis of the proposed scheme is carried out using the Von Neumann method. It is shown that the proposed algorithm has good stability. This method also shows much less spurious oscillations than current lower-order accurate numerical schemes. As a result, the proposed numerical scheme can provide more accurate results for long-time simulations. The proposed numerical scheme is validated against available analytical and numerical solutions for one- and two-dimensional transport problems. One- and two-dimensional numerical examples are presented in this paper to demonstrate the accuracy and conservative properties of the proposed algorithm by comparing with other numerical schemes. The proposed method is demonstrated to be a useful and accurate modelling tool for a wide range of transport problems. Copyright © 2007 John Wiley & Sons, Ltd. [source] Numerical simulation of the miscible displacement of radionuclides in a heterogeneous porous mediumINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 10 2005C.-H. Bruneau Abstract The aim of this paper is to model and simulate the displacement of radioactive elements in a saturated heterogeneous porous medium. New schemes are proposed to solve accurately the convection,diffusion,reaction equations including nonlinear terms in the time derivative. Numerical tests show the stability and robustness of these schemes through strong heterogeneities of the medium. Finally the COUPLEX 1 benchmark concerning the far field simulation of a polluted flow by a leak of a nuclear waste disposal is performed and compared with the results available in the literature. Copyright © 2005 John Wiley & Sons, Ltd. [source] Towards very high-order accurate schemes for unsteady convection problems on unstructured meshesINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 8-9 2005R. Abgrall Abstract We construct several high-order residual-distribution methods for two-dimensional unsteady scalar advection on triangular unstructured meshes. For the first class of methods, we interpolate the solution in the space,time element. We start by calculating the first-order node residuals, then we calculate the high-order cell residual, and modify the first-order residuals to obtain high accuracy. For the second class of methods, we interpolate the solution in space only, and use high-order finite difference approximation for the time derivative. In doing so, we arrive at a multistep residual-distribution scheme. We illustrate the performance of both methods on several standard test problems. Copyright © 2005 John Wiley & Sons, Ltd. [source] LMI-based computation of optimal quadratic Lyapunov functions for odd polynomial systemsINTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL, Issue 1 2005G. Chesi Abstract The problem of estimating the domain of attraction (DA) of equilibria is considered for odd polynomial systems. Specifically, the computation of the optimal quadratic Lyapunov function (OQLF), i.e. the quadratic Lyapunov function (QLF) which maximizes the volume of the largest estimate of the DA (LEDA), is addressed. In order to tackle this double non-convex optimization problem, a relaxation approach based on homogeneous polynomial forms is proposed. The first contribution of the paper shows that a lower bound of the LEDA for a fixed QLF can be obtained via linear matrix inequalities (LMIs) based procedures. Also, condition for checking tightness of the lower bound are provided. The second contribution is a strategy for selecting a good starting point for the OQLF search, which is based on the volume maximization of the region where the time derivative of the QLFs is negative and is given in terms of LMIs. Several application examples are presented to illustrate the numerical behaviour of the proposed approach. Copyright © 2005 John Wiley & Sons, Ltd. [source] Shaping and timing gradient pulses to reduce MRI acoustic noise,MAGNETIC RESONANCE IN MEDICINE, Issue 2 2010Marcel Segbers MSc Abstract A method to reduce the acoustic noise generated by gradient systems in MRI has been recently proposed; such a method is based on the linear response theory. Since the physical cause of MRI acoustic noise is the time derivative of the gradient current, a common trapezoid current shape produces an acoustic gradient coil response mainly during the rising and falling edge. In the falling edge, the coil acoustic response presents a 180° phase difference compared to the rising edge. Therefore, by varying the width of the trapezoid and keeping the ramps constant, it is possible to suppress one selected frequency and its higher harmonics. This value is matched to one of the prominent resonance frequencies of the gradient coil system. The idea of cancelling a single frequency is extended to a second frequency, using two successive trapezoid-shaped pulses presented at a selected interval. Overall sound pressure level reduction of 6 and 10 dB is found for the two trapezoid shapes and a single pulse shape, respectively. The acoustically optimized pulse shape proposed is additionally tested in a simulated echo planar imaging readout train, obtaining a sound pressure level reduction of 12 dB for the best case. Magn Reson Med, 2010. © 2010 Wiley-Liss, Inc. [source] Numerical solution of the free-surface viscous flow on a horizontal rotating elliptical cylinderNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 4 2008Roland Hunt Abstract The numerical solution of the free-surface fluid flow on a rotating elliptical cylinder is presented. Up to the present, research has concentrated on the circular cylinder for which steady solutions are the main interest. However, for noncircular cylinders, such as the ellipse, steady solutions are no longer possible, but there will be periodic solutions in which the solution is repeated after one full revolution of the cylinder. It is this new aspect that makes the investigation of noncircular cylinders novel. Here we consider both the time-dependent and periodic solutions for zero Reynolds number fluid flow. The numerical solution is expedited by first mapping the fluid film domain onto a rectangle such that the position of the free-surface is determined as part of the solution. For the time-dependent case a simple time-marching method of lines approach is adopted. For the periodic solution the discretised nonlinear equations have to be solved simultaneously over a time period. The resulting large system of equations is solved using Newton's method in which the form of the Jacobian enables a straightforward decomposition to be implemented, which makes matrix inversion manageable. In the periodic case all derivatives have been approximated pseudospectrally with the time derivative approximated by a differentiation matrix which has been specially derived so that the weight of fluid is algebraically conserved. Of interest is the solution for which the weight of fluid is at its maximum possible value, and this has been obtained by increasing the weight until a consistency break-down occurs. Time-dependent solutions do not produce the periodic solution after a long time-scale but have protuberances which are constantly appearing and disappearing. Periodic solutions exhibit spectral accuracy solutions and maximum supportable weight solutions have been obtained for ranges of eccentricity and angular velocity. The maximum weights are less than and approximately proportional to those obtained for the circular case. The shapes of maximum weight solutions is distinctly different from sub-maximum weight solutions. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008 [source] The Importance of dQ/dt on the Flow Field in a Turbodynamic Pump With Pulsatile FlowARTIFICIAL ORGANS, Issue 9 2009Fangjun Shu Abstract Fluid dynamic analysis of turbodynamic blood pumps (TBPs) is often conducted under steady flow conditions. However, the preponderance of clinical applications for ventricular assistance involves unsteady, pulsatile flow,due to the residual contractility of the native heart. This study was undertaken to demonstrate the importance of pulsatility and the associated time derivative of the flow rate (dQ/dt) on hemodynamics within a clinical-scale TBP. This was accomplished by performing flow visualization studies on a transparent model of a centrifugal TBP interposed within a cardiovascular simulator with controllable heart rate and stroke volume. Particle image velocimetry triggered to both the rotation angle of the impeller and phase of the cardiac cycle was used to quantify the velocity field in the outlet volute and in between the impeller blades for 16 phases of the cardiac cycle. Comparison of the unsteady flow fields to corresponding steady conditions at the same (instantaneous) flow rates revealed marked differences. In particular, deceleration of flow was found to promote separation within the outlet diffuser, while acceleration served to stabilize the velocity field. The notable differences between the acceleration and deceleration phases illustrated the prominence of inertial fluid forces. These studies emphasize the importance of dQ/dt as an independent variable for thorough preclinical validation of TBPs intended for use as a ventricular assist device. [source] The feasibility of electromagnetic gradiometer measurementsGEOPHYSICAL PROSPECTING, Issue 3 2001Daniel Sattel The quantities measured in transient electromagnetic (TEM) surveys are usually either magnetic field components or their time derivatives. Alternatively it might be advantageous to measure the spatial derivatives of these quantities. Such gradiometer measurements are expected to have lower noise levels due to the negative interference of ambient noise recorded by the two receiver coils. Error propagation models are used to compare quantitatively the noise sensitivities of conventional and gradiometer TEM data. To achieve this, eigenvalue decomposition is applied on synthetic data to derive the parameter uncertainties of layered-earth models. The results indicate that near-surface gradient measurements give a superior definition of the shallow conductivity structure, provided noise levels are 20,40 times smaller than those recorded by conventional EM instruments. For a fixed-wing towed-bird gradiometer system to be feasible, a noise reduction factor of at least 50,100 is required. One field test showed that noise reduction factors in excess of 60 are achievable with gradiometer measurements. However, other collected data indicate that the effectiveness of noise reduction can be hampered by the spatial variability of noise such as that encountered in built-up areas. Synthetic data calculated for a vertical plate model confirm the limited depth of detection of vertical gradient data but also indicate some spatial derivatives which offer better lateral resolution than conventional EM data. This high sensitivity to the near-surface conductivity structure suggests the application of EM gradiometers in areas such as environmental and archaeological mapping. [source] Adaptive superposition of finite element meshes in non-linear transient solid mechanics problemsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 9 2007Z. Yue Abstract An s-adaptive finite element procedure is developed for the transient analysis of 2-D solid mechanics problems with material non-linearity due to progressive damage. The resulting adaptive method simultaneously estimates and controls both the spatial error and temporal error within user-specified tolerances. The spatial error is quantified by the Zienkiewicz,Zhu error estimator and computed via superconvergent patch recovery, while the estimation of temporal error is based on the assumption of a linearly varying third-order time derivatives of the displacement field in conjunction with direct numerical time integration. The distinguishing characteristic of the s-adaptive procedure is the use of finite element mesh superposition (s-refinement) to provide spatial adaptivity. Mesh superposition proves to be particularly advantageous in computationally demanding non-linear transient problems since it is faster, simpler and more efficient than traditional h-refinement schemes. Numerical examples are provided to demonstrate the performance characteristics of the s-adaptive method for quasi-static and transient problems with material non-linearity. Copyright © 2007 John Wiley & Sons, Ltd. [source] State-space time integration with energy control and fourth-order accuracy for linear dynamic systemsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 5 2006Steen Krenk Abstract A fourth-order accurate time integration algorithm with exact energy conservation for linear structural dynamics is presented. It is derived by integrating the phase-space representation and evaluating the resulting displacement and velocity integrals via integration by parts, substituting the time derivatives from the original differential equations. The resulting algorithm has an exact energy equation, in which the change of energy is equal to the work of the external forces minus a quadratic form of the damping matrix. This implies unconditional stability of the algorithm, and the relative phase error is of fourth-order. An optional high-frequency algorithmic damping is constructed by optimal combination of three different damping matrices, each proportional to either the mass or the stiffness matrix. This leads to a modified form of the undamped algorithm with scalar weights on some of the matrices introducing damping of fourth-order in the frequency. Thus, the low-frequency response is virtually undamped, and the algorithm remains third-order accurate even when algorithmic damping is included. The accuracy of the algorithm is illustrated by an application to pulse propagation in an elastic medium, where the algorithmic damping is used to reduce dispersion due to the spatial discretization, leading to a smooth solution with a clearly defined wave front. Copyright © 2005 John Wiley & Sons, Ltd. [source] Numerical study of consistency of rate constitutive equations with elasticity at finite deformationINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 9 2002Ruocheng Lin Abstract The present work is concerned with the numerical study of the elasticity consistency of the spatial rate equations using the conventional Oldroyd, Truesdell, Cotter,Rivlin, Jaumann and Green,Naghdi rates and the three novel co-rotational ,E - and ,¯L -based, logarithmic rates, and of the rotated material rate equation describing the relationship between the material time derivatives of the rotated Kirchhoff stress and material logarithmic strain. To this end, three integration procedures for updating stress are presented. The stress responses of several typical deformation processes are simulated. According to the numerical results we know that among the spatial rate equations only the logarithmic rate equation is consistent with elasticity under constant material parameters. Integrating the other spatial rate equations will provide path-dependent stress response. These numerical conclusions support the arguments in H. Xiao et al. (Acta Mechanica 1999; 138:31,50). The reasons leading to elasticity inconsistency of spatial rate equations are analysed. If the material parameters are assumed to be strain-dependent, the logarithmic rate equation loses also its elasticity-consistent property. The numerical results prove also that the spatial logarithmic and rotated material rate equations are equivalent to each other. Copyright © 2002 John Wiley & Sons, Ltd. [source] Two-dimensional prediction of time dependent, turbulent flow around a square cylinder confined in a channelINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 11 2010M. Raisee Abstract This paper presents two-dimensional and unsteady RANS computations of time dependent, periodic, turbulent flow around a square block. Two turbulence models are used: the Launder,Sharma low-Reynolds number k,, model and a non-linear extension sensitive to the anisotropy of turbulence. The Reynolds number based on the free stream velocity and obstacle side is Re=2.2×104. The present numerical results have been obtained using a finite volume code that solves the governing equations in a vertical plane, located at the lateral mid-point of the channel. The pressure field is obtained with the SIMPLE algorithm. A bounded version of the third-order QUICK scheme is used for the convective terms. Comparisons of the numerical results with the experimental data indicate that a preliminary steady solution of the governing equations using the linear k,, does not lead to correct flow field predictions in the wake region downstream of the square cylinder. Consequently, the time derivatives of dependent variables are included in the transport equations and are discretized using the second-order Crank,Nicolson scheme. The unsteady computations using the linear and non-linear k,, models significantly improve the velocity field predictions. However, the linear k,, shows a number of predictive deficiencies, even in unsteady flow computations, especially in the prediction of the turbulence field. The introduction of a non-linear k,, model brings the two-dimensional unsteady predictions of the time-averaged velocity and turbulence fields and also the predicted values of the global parameters such as the Strouhal number and the drag coefficient to close agreement with the data. Copyright © 2009 John Wiley & Sons, Ltd. [source] Fifth-order Hermitian schemes for computational linear aeroacousticsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 9 2007Article first published online: 17 APR 200, G. Capdeville Abstract We develop a class of fifth-order methods to solve linear acoustics and/or aeroacoustics. Based on local Hermite polynomials, we investigate three competing strategies for solving hyperbolic linear problems with a fifth-order accuracy. A one-dimensional (1D) analysis in the Fourier series makes it possible to classify these possibilities. Then, numerical computations based on the 1D scalar advection equation support two possibilities in order to update the discrete variable and its first and second derivatives: the first one uses a procedure similar to that of Cauchy,Kovaleskaya (the ,,-P5 scheme'); the second one relies on a semi-discrete form and evolves in time the discrete unknowns by using a five-stage Runge,Kutta method (the ,RGK-P5 scheme'). Although the RGK-P5 scheme shares the same local spatial interpolator with the ,-P5 scheme, it is algebraically simpler. However, it is shown numerically that its loss of compactness reduces its domain of stability. Both schemes are then extended to bi-dimensional acoustics and aeroacoustics. Following the methodology validated in (J. Comput. Phys. 2005; 210:133,170; J. Comput. Phys. 2006; 217:530,562), we build an algorithm in three stages in order to optimize the procedure of discretization. In the ,reconstruction stage', we define a fifth-order local spatial interpolator based on an upwind stencil. In the ,decomposition stage', we decompose the time derivatives into simple wave contributions. In the ,evolution stage', we use these fluctuations to update either by a Cauchy,Kovaleskaya procedure or by a five-stage Runge,Kutta algorithm, the discrete variable and its derivatives. In this way, depending on the configuration of the ,evolution stage', two fifth-order upwind Hermitian schemes are constructed. The effectiveness and the exactitude of both schemes are checked by their applications to several 2D problems in acoustics and aeroacoustics. In this aim, we compare the computational cost and the computation memory requirement for each solution. The RGK-P5 appears as the best compromise between simplicity and accuracy, while the ,-P5 scheme is more accurate and less CPU time consuming, despite a greater algebraic complexity. Copyright © 2007 John Wiley & Sons, Ltd. [source] Development of a class of multiple time-stepping schemes for convection,diffusion equations in two dimensionsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 12 2006R. K. Lin Abstract In this paper we present a class of semi-discretization finite difference schemes for solving the transient convection,diffusion equation in two dimensions. The distinct feature of these scheme developments is to transform the unsteady convection,diffusion (CD) equation to the inhomogeneous steady convection,diffusion-reaction (CDR) equation after using different time-stepping schemes for the time derivative term. For the sake of saving memory, the alternating direction implicit scheme of Peaceman and Rachford is employed so that all calculations can be carried out within the one-dimensional framework. For the sake of increasing accuracy, the exact solution for the one-dimensional CDR equation is employed in the development of each scheme. Therefore, the numerical error is attributed primarily to the temporal approximation for the one-dimensional problem. Development of the proposed time-stepping schemes is rooted in the Taylor series expansion. All higher-order time derivatives are replaced with spatial derivatives through use of the model differential equation under investigation. Spatial derivatives with orders higher than two are not taken into account for retaining the linear production term in the convection,diffusion-reaction differential system. The proposed schemes with second, third and fourth temporal accuracy orders have been theoretically explored by conducting Fourier and dispersion analyses and numerically validated by solving three test problems with analytic solutions. Copyright © 2006 John Wiley & Sons, Ltd. [source] On the geometric conservation law in transient flow calculations on deforming domainsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 12 2006Ch. Förster Abstract This note revisits the derivation of the ALE form of the incompressible Navier,Stokes equations in order to retain insight into the nature of geometric conservation. It is shown that the flow equations can be written such that time derivatives of integrals over moving domains are avoided prior to discretization. The geometric conservation law is introduced into the equations and the resulting formulation is discretized in time and space without loss of stability and accuracy compared to the fixed grid version. There is no need for temporal averaging remaining. The formulation applies equally to different time integration schemes within a finite element context. Copyright © 2005 John Wiley & Sons, Ltd. [source] Model reference adaptive iterative learning control for linear systemsINTERNATIONAL JOURNAL OF ADAPTIVE CONTROL AND SIGNAL PROCESSING, Issue 9 2006A. Tayebi Abstract In this paper, we propose a model reference adaptive control (MRAC) strategy for continuous-time single-input single-output (SISO) linear time-invariant (LTI) systems with unknown parameters, performing repetitive tasks. This is achieved through the introduction of a discrete-type parametric adaptation law in the ,iteration domain', which is directly obtained from the continuous-time parametric adaptation law used in standard MRAC schemes. In fact, at the first iteration, we apply a standard MRAC to the system under consideration, while for the subsequent iterations, the parameters are appropriately updated along the iteration-axis, in order to enhance the tracking performance from iteration to iteration. This approach is referred to as the model reference adaptive iterative learning control (MRAILC). In the case of systems with relative degree one, we obtain a pointwise convergence of the tracking error to zero, over the whole finite time interval, when the number of iterations tends to infinity. In the general case, i.e. systems with arbitrary relative degree, we show that the tracking error converges to a prescribed small domain around zero, over the whole finite time interval, when the number of iterations tends to infinity. It is worth noting that this approach allows: (1) to extend existing MRAC schemes, in a straightforward manner, to repetitive systems; (2) to avoid the use of the output time derivatives, which are generally required in traditional iterative learning control (ILC) strategies dealing with systems with high relative degree; (3) to handle systems with multiple tracking objectives (i.e. the desired trajectory can be iteration-varying). Finally, simulation results are carried out to support the theoretical development. Copyright © 2006 John Wiley & Sons, Ltd. [source] Approximation of time-dependent, viscoelastic fluid flow: Crank-Nicolson, finite element approximation,NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 2 2004Vincent J. Ervin Abstract In this article we analyze a fully discrete approximation to the time dependent viscoelasticity equations with an Oldroyd B constitutive equation in ,, = 2, 3. We use a Crank-Nicolson discretization for the time derivatives. At each time level a linear system of equations is solved. To resolve the nonlinearities we use a three-step extrapolation for the prediction of the velocity and stress at the new time level. The approximation is stabilized by using a discontinuous Galerkin approximation for the constitutive equation. For the mesh parameter, h, and the temporal step size, ,t, sufficiently small and satisfying ,t , Ch, existence of the approximate solution is proven. A priori error estimates for the approximation in terms of ,t and h are also derived. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 248,283, 2004 [source] Dynamics and control of underactuated mechanical systems: analysis and simple experimental verificationPROCEEDINGS IN APPLIED MATHEMATICS & MECHANICS, Issue 1 2009Wojciech Blajer Underactuated mechanical systems are systems with fewer control inputs than the degrees of freedom, m < n, the relevant technical examples being e.g. cranes, aircrafts and flexible manipulators. The determination of an input control strategy that forces an underactuated system to complete a set of m specified motion tasks (servo-constraints) is a demanding problem. The solution is conditioned to differential flatness of the problem, denoted that all 2n state variables and m control inputs can algebraically be expressed, at least theoretically, in terms of the desired m outputs and their time derivatives up to a certain order. A more practical formulation, motivated hereafter, is to pose the problem as a set of differential-algebraic equations, and then obtain the solution numerically. The theoretical considerations are illustrated by a simple two-degree-of-freedom underactuated system composed of two rotating discs connected by a flexible rod (torsional spring), in which the pre-specified motion of the first disc is actuated by the torque applied to the second disc, n = 2 and m = 1. The determined control strategy is then verified experimentally on a laboratory stand representing the two-disc system. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] | |||||||||||||||||||||||||