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Numerical Problems (numerical + problem)
Selected AbstractsAutomatic energy conserving space,time refinement for linear dynamic structural problemsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 3 2005P. Cavin Abstract In this paper a local space,time automatic refinement method (STAR method) is developed to efficiently solve time-dependent problems using FEM techniques. The automatic process is driven by an energy or a displacement error indicator which controls the precision of the result. The STAR method solves the numerical problem on grids with different mesh size. For the Newmark schemes, a general demonstration, using the energy method, gives the interface conditions between two successive grids which is compatible with the stability of the scheme. Finally, using a linear one-dimensional example, the convergence of the method and the precision of the results are discussed. Copyright © 2005 John Wiley & Sons, Ltd. [source] Efficient calculation of configurational entropy from molecular simulations by combining the mutual-information expansion and nearest-neighbor methods,,JOURNAL OF COMPUTATIONAL CHEMISTRY, Issue 10 2008Vladimir Hnizdo Abstract Changes in the configurational entropies of molecules make important contributions to the free energies of reaction for processes such as protein-folding, noncovalent association, and conformational change. However, obtaining entropy from molecular simulations represents a long-standing computational challenge. Here, two recently introduced approaches, the nearest-neighbor (NN) method and the mutual-information expansion (MIE), are combined to furnish an efficient and accurate method of extracting the configurational entropy from a molecular simulation to a given order of correlations among the internal degrees of freedom. The resulting method takes advantage of the strengths of each approach. The NN method is entirely nonparametric (i.e., it makes no assumptions about the underlying probability distribution), its estimates are asymptotically unbiased and consistent, and it makes optimum use of a limited number of available data samples. The MIE, a systematic expansion of entropy in mutual information terms of increasing order, provides a well-characterized approximation for lowering the dimensionality of the numerical problem of calculating the entropy of a high-dimensional system. The combination of these two methods enables obtaining well-converged estimations of the configurational entropy that capture many-body correlations of higher order than is possible with the simple histogramming that was used in the MIE method originally. The combined method is tested here on two simple systems: an idealized system represented by an analytical distribution of six circular variables, where the full joint entropy and all the MIE terms are exactly known, and the R,S stereoisomer of tartaric acid, a molecule with seven internal-rotation degrees of freedom for which the full entropy of internal rotation has been already estimated by the NN method. For these two systems, all the expansion terms of the full MIE of the entropy are estimated by the NN method and, for comparison, the MIE approximations up to third order are also estimated by simple histogramming. The results indicate that the truncation of the MIE at the two-body level can be an accurate, computationally nondemanding approximation to the configurational entropy of anharmonic internal degrees of freedom. If needed, higher-order correlations can be estimated reliably by the NN method without excessive demands on the molecular-simulation sample size and computing time. © 2008 Wiley Periodicals, Inc. J Comput Chem, 2008 [source] Approximation method for high-degree harmonics in normal mode modellingGEOPHYSICAL JOURNAL INTERNATIONAL, Issue 1 2002R. E. M. Riva Summary For some loading applications, the normal modes approach to the viscoelastic relaxation of a spherical earth requires the use of spherical harmonics up to a high degree. Examples include postseismic deformation (internal loading) and sea level variations due to glacial isostatic adjustment (external loading). In the case of postseismic modelling, the convergence of the solution, given as a spherical harmonic expansion series, is directly dependent on loading depth and requires several thousands of terms for shallow earthquake sources. The particular structure of the analytical fundamental solutions used in normal mode techniques usually does not allow a straightforward calculation, since numerical problems can readily occur due to the stiffness of the matrices used in the propagation routines. Here we show a way of removing this stiffness problem by approximating the fundamental matrix solutions, followed by a rescaling procedure, in this way we can virtually go up to whatever harmonic degree is required. [source] Simulating Seepage into Mine Shafts and Tunnels with MODFLOWGROUND WATER, Issue 3 2010Jacob Zaidel In cases when an equivalent porous medium assumption is suitable for simulating groundwater flow in bedrock aquifers, estimation of seepage into underground mine workings (UMWs) can be achieved by specifying MODFLOW drain nodes at the contact between water bearing rock and dewatered mine openings. However, this approach results in significant numerical problems when applied to simulate seepage into an extensive network of UMWs, which often exist at the mine sites. Numerical simulations conducted for individual UMWs, such as a vertical shaft or a horizontal drift, showed that accurate prediction of seepage rates can be achieved by either applying grid spacing that is much finer than the diameter/width of the simulated openings (explicit modeling) or using coarser grid with cell sizes exceeding the characteristic width of shafts or drifts by a factor of 3. Theoretical insight into this phenomenon is presented, based on the so-called well-index theory. It is demonstrated that applying this theory allows to minimize numerical errors associated with MODFLOW simulation of seepage into UMWs on a relatively coarse Cartesian grid. Presented examples include simulated steady-state groundwater flow from homogeneous, heterogeneous, and/or anisotropic rock into a vertical shaft, a horizontal drift/cross-cut, a ramp, two parallel drifts, and a combined system of a vertical shaft connected to a horizontal drift. [source] Non-local damage model based on displacement averagingINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 1 2005M. Jirásek Abstract Continuum damage models describe the changes of material stiffness and strength, caused by the evolution of defects, in the framework of continuum mechanics. In many materials, a fast evolution of defects leads to stress,strain laws with softening, which creates serious mathematical and numerical problems. To regularize the model behaviour, various generalized continuum theories have been proposed. Integral-type non-local damage models are often based on weighted spatial averaging of a strain-like quantity. This paper explores an alternative formulation with averaging of the displacement field. Damage is assumed to be driven by the symmetric gradient of the non-local displacements. It is demonstrated that an exact equivalence between strain and displacement averaging can be achieved only in an unbounded medium. Around physical boundaries of the analysed body, both formulations differ and the non-local displacement model generates spurious damage in the boundary layers. The paper shows that this undesirable effect can be suppressed by an appropriate adjustment of the non-local weight function. Alternatively, an implicit gradient formulation could be used. Issues of algorithmic implementation, computational efficiency and smoothness of the resolved stress fields are discussed. Copyright © 2005 John Wiley & Sons, Ltd. [source] A robust methodology for RANS simulations of highly underexpanded jetsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 12 2008G. Lehnasch Abstract This work aims at developing/combining numerical tools adapted to the simulation of the near field of highly underexpanded jets. An overview of the challenging numerical problems related to the complex shock/expansion structure encountered in these flows is given and an efficient and low-cost numerical strategy is proposed to overcome these, even on short computational domains. Based on common upwinding algorithms used on unstructured meshes in a mixed finite-volume/finite-element approach, it relies on an appropriate utilization of zonal anisotropic remeshing algorithms. This methodology is validated for the whole near field of cold air jets issuing from axisymmetric convergent nozzles and yielding various underexpansion ratios. In addition, the most usual corrections of the k,, model used to take into account the compressibility effects on turbulence are precisely assessed. Copyright © 2007 John Wiley & Sons, Ltd. [source] Vortex surface method: some numerical problems of the potential calculationINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 8 2001T. Belamri Abstract The singularities method is used to analyze the flow around an isolated profile or through a plane cascade. In this paper, a numerical study has been developed in order to discuss the accuracy of solutions. The aims of this study are summarized as follows: (1) to expose the elements that influence the method,precision in the geometrical profile definition, trailing-edge geometry, smoothing problems, number of discretization points, precision of calculation, etc.; (2) to provide an accurate solution for these different problems. For example, some profiles, obtained by the Joukowski transformation, present, in spite of an analytical definition, a crossing of the suction and pressure sides at the trailing edge. This crossing causes a serious error in the velocity field computation. A new procedure to solve this problem is presented. Copyright © 2001 John Wiley & Sons, Ltd. [source] Energy transfer in master equation simulations: A new approachINTERNATIONAL JOURNAL OF CHEMICAL KINETICS, Issue 12 2009John R. BarkerArticle first published online: 8 OCT 200 Collisional energy transfer plays a key role in recombination, unimolecular, and chemical activation reactions. For master equation simulations of such reaction systems, it is conventionally assumed that the rate constant for inelastic energy transfer collisions is independent of the excitation energy. However, numerical instabilities and nonphysical results are encountered when normalizing the collision step-size distribution in the sparse density of states regime at low energies. It is argued here that the conventional assumption is not correct, and it is shown that the numerical problems and nonphysical results are eliminated by making a plausible assumption about the energy dependence of the rate coefficient for inelastic collisions. The new assumption produces a model that is more physically realistic for any reasonable choice of collision step-size distribution, but more work remains to be done. The resulting numerical algorithm is stable and noniterative. Testing shows that overall accuracy in master equation simulations is better with this new approach than with the conventional one. This new approach is appropriate for all energy-grained master equation formulations. © 2009 Wiley Periodicals, Inc. Int J Chem Kinet 41: 748,763, 2009 [source] Bubble size distribution modeling in stirred gas,liquid reactors with QMOM augmented by a new correction algorithmAICHE JOURNAL, Issue 1 2010Miriam 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] | |||||||||||||