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Quadrature Method (quadrature + method)

Distribution by Scientific Domains
Engineering42%

Kinds of Quadrature Method

  • differential quadrature method
    		•

    Selected Abstracts

    Poroelastodynamic Boundary Element Method in Time Domain: Numerical Aspects

    PROCEEDINGS IN APPLIED MATHEMATICS & MECHANICS, Issue 1 2005
    Martin Schanz
    Based on Biot's theory the governing equations for a poroelastic continuum are given as a coupled set of partial differential equations (PDEs) for the unknowns solid displacements and pore pressure. Using the Convolution Quadrature Method (CQM) proposed by Lubich a boundary time stepping procedure is established based only on the fundamental solutions in Laplace domain. To improve the numerical behavior of the CQM-based Boundary Element Method (BEM) dimensionless variables are introduced and different choices studied. This will be performed as a numerical study at the example of a poroelastic column. Summarizing the results, the normalization to time and spatial variable as well as on Young's modulus yields the best numerical behavior. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

    An adaptive direct quadrature method of moment for population balance equations

    AICHE JOURNAL, Issue 11 2008
    Junwei Su
    Abstract Quadrature method of moments (QMOM) and direct quadrature method of moments (DQMOM) for population balance equations (PBE) have been shown to be accurate and computationally efficient for isotropic systems or when used with computational fluid dynamics (CFD) codes. However, numerical difficulties can arise for cases where there is a large variation of moments or where two abscissas have similar values. Previous study has demonstrated that introducing an appropriate adjustable factor to the QMOM, the numerical difficulty can be alleviated in some cases with an additional benefit of improving numerical accuracy or significantly reducing computational time. However, no reliable method is available to determine the optimal adjustable factor that allows the highest possible accuracy to be obtained while maintaining computational efficiency. In this work, an adjustable factor is introduced to the DQMOM and a novel procedure is proposed that enables the optimal adjustable factor to be found for a given problem. A number of test cases including pure aggregation, pure breakage, pure growth, aggregation and breakage, aggregation and growth have been carried out. Our results show that the proposed method is capable of either improving numerical accuracy or reducing the computational time for a variety of problems. The novelty of this method is that the optimal adjustable factor is determined based on the actual particle size distribution at a given time, thereby reducing error accumulation. It also allows other factor-searching procedures to be incorporated in a straightforward manner without influencing the adaptive DQMOM (ADQMOM)itself. © 2008 American Institute of Chemical Engineers AIChE J, 2008 [source]

    DQ-based simulation of weakly nonlinear heat conduction processes

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 11 2008
    S. Tomasiello
    Abstract In this paper, an explicit form for the numerical solution of problems in the space,time domain by using quadrature rules is proposed. The compact form of the shape functions recently proposed by the author is useful to the scope. Numerical solutions for the time-dependent one-dimensional nonlinear heat conduction problem are calculated by means of the iterative differential quadrature method, a method proposed by the author and based on quadrature rules. The accuracy of the solution and stability analysis show good performance of the approach. Copyright © 2007 John Wiley & Sons, Ltd. [source]

    An interpolation-based local differential quadrature method to solve partial differential equations using irregularly distributed nodes

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 7 2008
    Hang Ma
    Abstract To circumvent the constraint in application of the conventional differential quadrature (DQ) method that the solution domain has to be a regular region, an interpolation-based local differential quadrature (LDQ) method is proposed in this paper. Instead of using regular nodes placed on mesh lines in the DQ method (DQM), irregularly distributed nodes are employed in the LDQ method. That is, any spatial derivative at a nodal point is approximated by a linear weighted sum of the functional values of irregularly distributed nodes in the local physical domain. The feature of the new approach lies in the fact that the weighting coefficients are determined by the quadrature rule over the irregularly distributed local supporting nodes with the aid of nodal interpolation techniques developed in the paper. Because of this distinctive feature, the LDQ method can be consistently applied to linear and nonlinear problems and is really a mesh-free method without the limitation in the solution domain of the conventional DQM. The effectiveness and efficiency of the method are validated by two simple numerical examples by solving boundary-value problems of a linear and a nonlinear partial differential equation. Copyright © 2007 John Wiley & Sons, Ltd. [source]

    New approaches in application of differential quadrature method to fourth-order differential equations

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 2 2005
    Xinwei Wang
    Abstract Various methods to apply multiple boundary conditions in the differential quadrature method are summarized and discussed. Two of them are new approaches appearing for the first time. Numerical examples demonstrate the accuracy of the new methods in applying multiple boundary conditions. Copyright © 2004 John Wiley & Sons, Ltd. [source]

    Stability and accuracy of the iterative differential quadrature method

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 9 2003
    Stefania Tomasiello
    Abstract In this paper the stability and accuracy of an iterative method based on differential quadrature rules will be discussed. The method has already been proposed by the author in a previous work, where its good performance has been shown. Nevertheless, discussion about stability and accuracy remained open. An answer to this question will be provided in this paper, where the conditional stability of the method will be pointed out, in addition to an examination of the possible errors which arise under certain conditions. The discussion will be preceded by an overview of the method and its foundations, i.e. the differential quadrature rules, and followed by a numerical case which shows how the method behaves when applied to reduce continuous systems to two-degree-of-freedom systems in the non-linear range. In particular, here the case of oscillators coupled in non-linear terms will be treated. The numerical results, used to draw Poincaré maps, will be compared with those obtained by using the Runge,Kutta method with a high precision goal. Copyright © 2003 John Wiley & Sons, Ltd. [source]

    A multi-QMOM framework to describe multi-component agglomerates in liquid steel

    AICHE JOURNAL, Issue 9 2010
    L. Claudotte
    Abstract A variant of the quadrature method of moments (QMOM) for solving multiple population balance equations (PBE) is developed with the objective of application to steel industry processing. During the process of oxygen removal in a steel ladle, a large panel of oxide inclusions may be observed depending on the type of oxygen removal and addition elements. The final quality of the steel can be improved by accurate numerical simulation of the multi-component precipitation. The model proposed in this article takes into account the interactions between three major aspects of steelmaking modeling, namely fluid dynamics, thermo-kinetics and population balance. A commercial CFD code is used to predict the liquid steel hydrodynamics, whereas a home-made thermo-kinetic code adjusts chemical composition with nucleation and diffusion growth, and finally a set of PBE tracks the evolution of inclusion size with emphasis on particle aggregation. Each PBE is solved by QMOM, the first PBE/QMOM system describing the clusters and each remaining PBE/QMOM system being dedicated to the elementary particles of each inclusion species. It is shown how this coupled model can be used to investigate the cluster size and composition of a particular grade of steel (i.e., Fe-Al-Ti-O). © 2010 American Institute of Chemical Engineers AIChE J, 2010 [source]

    Bubble size distribution modeling in stirred gas,liquid reactors with QMOM augmented by a new correction algorithm

    AICHE JOURNAL, Issue 1 2010
    Miriam Petitti
    Abstract Local gas hold-up and bubbles size distributions have been modeled and validated against experimental data in a stirred gas,liquid reactor, considering two different spargers. An Eulerian multifluid approach coupled with a population balance model (PBM) has been employed to describe the evolution of the bubble size distribution due to break-up and coalescence. The PBM has been solved by resorting to the quadrature method of moments, implemented through user defined functions in the commercial computational fluid dynamics code Fluent v. 6.2. To overcome divergence issues caused by moments corruption, due to numerical problems, a correction scheme for the moments has been implemented; simulation results prove that it plays a crucial role for the stability and the accuracy of the overall approach. Very good agreements between experimental data and simulations predictions are obtained, for a unique set of break-up and coalescence kinetic constants, in a wide range of operating conditions. © 2009 American Institute of Chemical Engineers AIChE J, 2010 [source]

    An adaptive direct quadrature method of moment for population balance equations

    AICHE JOURNAL, Issue 11 2008
    Junwei Su
    Abstract Quadrature method of moments (QMOM) and direct quadrature method of moments (DQMOM) for population balance equations (PBE) have been shown to be accurate and computationally efficient for isotropic systems or when used with computational fluid dynamics (CFD) codes. However, numerical difficulties can arise for cases where there is a large variation of moments or where two abscissas have similar values. Previous study has demonstrated that introducing an appropriate adjustable factor to the QMOM, the numerical difficulty can be alleviated in some cases with an additional benefit of improving numerical accuracy or significantly reducing computational time. However, no reliable method is available to determine the optimal adjustable factor that allows the highest possible accuracy to be obtained while maintaining computational efficiency. In this work, an adjustable factor is introduced to the DQMOM and a novel procedure is proposed that enables the optimal adjustable factor to be found for a given problem. A number of test cases including pure aggregation, pure breakage, pure growth, aggregation and breakage, aggregation and growth have been carried out. Our results show that the proposed method is capable of either improving numerical accuracy or reducing the computational time for a variety of problems. The novelty of this method is that the optimal adjustable factor is determined based on the actual particle size distribution at a given time, thereby reducing error accumulation. It also allows other factor-searching procedures to be incorporated in a straightforward manner without influencing the adaptive DQMOM (ADQMOM)itself. © 2008 American Institute of Chemical Engineers AIChE J, 2008 [source]

    Predicting the phase equilibria of petroleum fluids with the SAFT-VR approach

    AICHE JOURNAL, Issue 3 2007
    Lixin Sun
    Abstract The SAFT-VR equation of state is combined with a semi-continuous thermodynamic approach to model several synthetic and crude oil systems. In our approach, the oil fractions are defined by a continuous distribution that is then represented as discrete pseudo-components using the Gaussian quadrature method. The SAFT-VR parameters for the pseudo-components are obtained from simple linear relationships that were defined in earlier work, which allows the approach to be easily applied to undefined oil systems. Good agreement between the theoretical predictions and experimental data is obtained for bubble point pressure calculations of several gas condensates and the solubility of gases such as methane, ethane, and carbon dioxide in several crude oils. © 2007 American Institute of Chemical Engineers AIChE J, 2007 [source]

    Generalization and numerical investigation of QMOM

    AICHE JOURNAL, Issue 1 2007
    R. Grosch
    Abstract A generalized framework is developed for the quadrature method of moments (QMOM), which is a solution method for population balance models. It further evaluates the applicability of this method to industrial suspension crystallization processes. The framework is based on the concepts of generalized moments and coordinate transformations, which have been used already in earlier solution approaches. It is shown how existing approaches to QMOM are derived from the suggested unified framework. Thus, similarities and differences between the various QMOM methods are uncovered. Further, potential error sources involved in the different approaches to QMOM are discussed and assessed by means of a series of test cases. The test cases are selected to be challenging. The error in the QMOM solution is evaluated by comparison to an adaptive, error controlled solution of the population balance. The behavior of a range of different QMOM formulations is analyzed by means of numerical quadrature, dynamic simulation, as well as numerical continuation and bifurcation analysis. As a result of this detailed analysis, some general limitations of the method are detected and guidelines for its application are developed. This article is limited to lumped population balance models with one internal coordinate. © 2006 American Institute of Chemical Engineers AIChE J, 2007 [source]

    Performance and numerical behavior of the second-order scheme of precise time-step integration for transient dynamic analysis

    NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 6 2007
    Hang Ma
    Abstract Spurious high-frequency responses resulting from spatial discretization in time-step algorithms for structural dynamic analysis have long been an issue of concern in the framework of traditional finite difference methods. Such algorithms should be not only numerically dissipative in a controllable manner, but also unconditionally stable so that the time-step size can be governed solely by the accuracy requirement. In this article, the issue is considered in the framework of the second-order scheme of the precise integration method (PIM). Taking the Newmark-, method as a reference, the performance and numerical behavior of the second-order PIM for elasto-dynamic impact-response problems are studied in detail. In this analysis, the differential quadrature method is used for spatial discretization. The effects of spatial discretization, numerical damping, and time step on solution accuracy are explored by analyzing longitudinal vibrations of a shock-excited rod with rectangular, half-triangular, and Heaviside step impact. Both the analysis and numerical tests show that under the framework of the PIM, the spatial discretization used here can provide a reasonable number of model types for any given error tolerance. In the analysis of dynamic response, an appropriate spatial discretization scheme for a given structure is usually required in order to obtain an accurate and meaningful numerical solution, especially for describing the fine details of traction responses with sharp changes. Under the framework of the PIM, the numerical damping that is often required in traditional integration schemes is found to be unnecessary, and there is no restriction on the size of time steps, because the PIM can usually produce results with machine-like precision and is an unconditionally stable explicit method. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007 [source]