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Stagnation point heat flux characterization under numerical error and boundary conditions uncertainty

EUROPEAN JOURNAL OF MECHANICS B-FLUIDS 2022年第0期95卷 221-230页

作者： Capriati, Michele Cortesi, Andrea Magin, Thierry E. Congedo, Pietro M. IPP Ecole Polytech Ctr Math Appl Inria F-91120 Palaiseau France Karman Inst Fluid Dynam Chaussee Waterloo 72 B-1640 Rhode Saint Genese Belgium

The numerical simulation of hypersonic atmospheric entry flows is a challenging problem. Prediction of quantities of interest, such as surface heat flux and pressure, is strongly influenced by the mesh quality using conventional second-order spatial accuracy schemes while depending on boundary conditions, which may generally suffer from uncertainty. This paper illustrates one of the first systematic quantification of the numerical error and the uncertainty-induced variability for the simulation of hypersonic flows. Specifically, a mesh-convergence study using grid adaptation tools is coupled with surrogate-based approaches to Uncertainty Quantification. The illustrative example is the simulation of the EXPERT vehicle of the European Space Agency employing the US3D solver. First, we show the benefits in using mesh adaptation to simulate hypersonic flows under uncertainty. On the one hand, this practice reduces the numerical uncertainty associated with each prediction and, on the other hand, allows us to obtain a more reliable surrogate model for Uncertainty Quantification by preventing non-physical heat flux values. Secondly, we perform a sensitivity analysis to compare the numerical uncertainty associated with a given mesh with the UQ-induced variability for a specific quantity of interest. In the case considered, the impact of the numerical uncertainty turned out to be at least one order of magnitude less than the quantity of interest variability. This result indicates the possibility of using coarse and adapted meshes for future UQ studies.

关键词： Atmospheric entry flows Uncertainty quantification Surrogate model numerical error

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A Priori Assessment of Subgrid-Scale Models and numerical error in Forced Convective Flow at High Prandtl Numbers

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FLOW TURBULENCE AND COMBUSTION 2020年第2期105卷 377-392页

作者： Sufra, Lorenzo Steiner, Helfried Graz Univ Technol Inst Fluid Mech & Heat Transfer Graz Austria

An extensive a priori analysis has been carried out on data from Direct numerical simulation of fully developed heated turbulent pipe flow at high molecular Prandtl numbers Pr=10/20, testing three popular modelling candidates for subgrid-scale closure in Large-Eddy simulation (LES). Aside from assessing the models' capabilities to describe quantitatively the unresolved turbulent fluxes, a special focus is also put on the role of the numerical error, which arises from the discretization of the filtered advective fluxes on a coarse LES grid. The present analysis extends here previous studies on subgrid-scale momentum transport in a isothermal mixing layer and channel flow carried out by Brandt (J Numer Methods Fluids 51: 635-657, 2006) and Vreman et al. (J Eng Math 29: 299-327, 1995), respectively, to the subgrid-scale transport of heat at high Prandtl numbers. The statistical dependence between the individual contributions (resolved, subgrid-scale, numerical discretization error) constituting the filtered advective flux divergence in the LES formulation is investigated as well, in terms of corresponding cross-correlations. The sensitivity of the tested sgs-models to a grid refinement is further examined performing also a posteriori LES, where the basically more sophisticated candidates turn out to be more demanding in terms of required grid resolution.

关键词： A priori LES Subgrid-scale closure numerical error Grid sensitivity

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Study of numerical error of a Eulerian-Lagrangian scheme in the presence of particle source

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COMPUTER PHYSICS COMMUNICATIONS 2021年 264卷 107960-107960页

作者： Tatsumi, Ryoko Hoshino, Kazuo Hatayama, Akiyoshi Keio Univ Grad Sch Sci & Technol Yokohama Kanagawa 2238522 Japan

Eulerian-Lagrangian (EL-LG) scheme is a numerical scheme that tracks pseudo particles in Eulerian cells. It is widely used in the computational fluid dynamics, however, numerical errors associated with a particle source term has not yet been investigated much. Hence this study focuses on numerical errors of EL-LG caused by particle sources. The purposes are: (i) to clarify causes and situations that bring larger numerical errors by source terms and (ii) to suggest an idea to reduce them. For those purposes, we focus on the particle continuity equation and carry out systematic analysis of the numerical error by setting the following three simple cases: Case (A) No source, Case (B) Constant source, and Case (C) source with arbitrary spatial profile. For each case, we have obtained a theoretical expression of the numerical error. It has been clarified that the errors become relatively large when (i) the spatial profile of the particle source has a large gradient and (ii) the source is localized in the region with high flow-velocity. These were caused by the treatment of the particle source: If pseudo particles due to the source are added in a simple way at the start or the end of the time step, this can lead to larger numerical errors. To reduce those errors, a time-averaging scheme has been suggested. Although the analyzed cases are simple, the results obtained in this study would give important knowledge and insight into numerical errors associated with particle sources in EL-LG schemes. (C) 2021 Published by Elsevier B.V.

关键词： CFD Eulerian-Lagrangian scheme numerical error Particle source Plasma fluid simulation

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The numerical error Analysis of Trace Ripple Simulation in Strong Bunching in TWT 18

The Numerical Error Analysis of Trace Ripple Simulation in S...

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18th International Vacuum Electronics Conference (IVEC)

作者： Hu, YuLu Wang, YanMei Hu, Quan Zhu, XiaoFang Li, Bin Univ Elect Sci & Technol China Sch Phys Elect Natl Key Lab Sci & Technol Vacuum Elect 4Sect 2North Jianshe Rd Chengdu 610054 Sichuan Peoples R China Beijing Vacuum Elect Res Inst POB 749 Beijing 100016 Peoples R China

ISBN: (纸本)9781509059164

In this paper the numerical error of trace ripple simulation in TWT is first presented, which is happened in nonlinear interaction region especially in strong bunching and will ultimately lead the error calculation of the beam-wave interaction. Then the reason is discussed, and some efforts to suppress the numerical error has been shown in this paper.

关键词： numerical error Trace Ripple Beam Wave Interaction MTSS

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Optimized Vertical Layers for the Hybrid Terrain-Following Coordinate Minimizing numerical errors in a 2D Rising Bubble Experiment near Steep Terrain

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Journal of Meteorological Research 2023年第6期37卷 896-907页

作者： Hao YANG Yiyuan LI Bin WANG State Key Laboratory of Numerical Modeling for Atmospheric Sciences and Geophysical Fluid Dynamics Institute of Atmospheric PhysicsChinese Academy of SciencesBeijing100029China College of Earth and Planetary Sciences University of Chinese Academy of SciencesBeijing100049China

The basic terrain-following(BTF)coordinate simplifies the lower boundary conditions of a numerical model but leads to numerical error and instability on steep *** terrain-following(HTF)coordinates with smooth slopes of vertical layers(slopeVL)generally overcome this ***,the HTF coordinate becomes very desirable for atmospheric and oceanic numerical ***,improper vertical layering in HTF coordinates may also increase the incidence of *** for the slopeVL of an HTF coordinate,this study further optimizes the HTF coordinate focusing on the thickness of vertical layers(thickVL).Four HTF coordinates(HTF1–HTF4)with similar slopeVL but different vertical transition methods of thickVL are designed,and the relationship between thickVL and numerical errors in each coordinate is compared in the classic idealized thermal convection[two-dimensional(2D)rising bubble]experiment over steep *** errors of potential temperatureθand vertical velocity w are reduced most,by approximately 70%and 40%,respectively,in the HTF1 coordinate,with a monotonic increase in thickVL according to the increasing height;however,the errors ofθincreased in all the other HTF coordinates,with nonmonotonic ***,analyses of the errors of vertical pressure gradient force(VPGF)show that due to the interpolation errors of thickVL,the inflection points in the vertical transition of thickVL induce the initial VPGF errors;therefore,the HTF1 coordinate with a monotonic increase in thickVL has the smallest errors among all the *** importantly,the temporal evolution of VPGF errors manifests top-type VPGF errors that propagate upward gradually during the time *** the HTF1 and HTF4 coordinates with a monotonic increase in thickVL near the top of the terrain can suppress this *** optimized HTF coordinate(i.e.,HTF1)can be a reference for designing a vertical thickVL in a numerical model.

关键词： hybrid terrain-following coordinate thickness of vertical layer numerical error pressure gradient force two-dimensional(2D)rising bubble experiment steep terrain

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numerical errors at walls: on the sensitivity of RANS models to near-wall cell size

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INTERNATIONAL JOURNAL OF COMPUTATIONAL FLUID DYNAMICS 2020年第3期34卷 204-225页

作者： Lloyd, C. J. Peakall, J. Burns, A. D. Keevil, G. M. Dorrell, R. M. Univ Leeds Ctr Doctoral Training Fluid Dynam Leeds W Yorkshire England Univ Leeds Sch Earth & Environm Leeds W Yorkshire England Univ Leeds Sch Chem & Proc Engn Leeds W Yorkshire England Univ Hull Energy & Environm Inst Kingston Upon Hull N Humberside England

The sensitivity of a large range of turbulence closures to the near-wall grid resolution is assessed through analytical and numerical techniques. We determine model boundedness, limiting behaviour, and solution dependence of channel flow simulations to the near-wall cell size . The sensitivity of present simulations to is strongly dependent on the choice of scale determining variable;and based models can lead to friction velocity errors of 1.5% and 4.74% at , the latter due to the singular behaviour of close to the wall. Here this is mitigated by adopting mathematically equivalent , or models, which reduce errors by an order of magnitude to at . The model is insensitive to motivating a new closure with updated coefficients and a cross diffusion term, with the aim to ensure its applicability to a wider range of flows.

关键词： RANS numerical error boundary layer near-wall grid resolution k - omega alternatives

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Full operator preconditioning and the accuracy of solving linear systems

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IMA JOURNAL OF numerical ANALYSIS 2024年第6期44卷 3259-3279页

作者： Mohr, Stephan Nakatsukasa, Yuji Urzua-Torres, Carolina Tech Univ Munich Dept Phys James Franck Str 1 D-85748 Garching Germany Univ Oxford Math Inst Woodstock Rd Oxford OX2 6GG England Delft Univ Technol Delft Inst Appl Math Mekelweg 4 NL-2628 CD Delft Netherlands

Unless special conditions apply, the attempt to solve ill-conditioned systems of linear equations with standard numerical methods leads to uncontrollably high numerical error and often slow convergence of an iterative solver. In many cases, such systems arise from the discretization of operator equations with a large number of discrete variables and the ill-conditioning is tackled by means of preconditioning. A key observation in this paper is the sometimes overlooked fact that while traditional preconditioning effectively accelerates convergence of iterative methods, it generally does not improve the accuracy of the solution. Nonetheless, it is sometimes possible to overcome this barrier: accuracy can be improved significantly if the equation is transformed before discretization, a process we refer to as full operator preconditioning (FOP). We highlight that this principle is already used in various areas, including second kind integral equations and Olver-Townsend's spectral method. We formulate a sufficient condition under which high accuracy can be obtained by FOP. We illustrate this for a fourth order differential equation which is discretized using finite elements.

关键词： preconditioning operator preconditioning condition number numerical error discretization error

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error in Ulps of the Multiplication or Division by a Correctly-Rounded Function or Constant in Binary Floating-Point Arithmetic

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IEEE TRANSACTIONS ON EMERGING TOPICS IN COMPUTING 2024年第2期12卷 656-666页

作者： Brisebarre, Nicolas Muller, Jean-Michel Picot, Joris CNRS Lab LIP ENS Lyon F-69342 Lyon France ENS Lyon Lab LIP F-69342 Lyon France

Assume we use a binary floating-point arith-metic and that RN is the round-to-nearest function. Also assume that c is a constant or a real function of one or more variables, and that we have at our disposal a correctly rounded implementation of c, sayc=RN(c). For evaluat-ing xc (resp.x/c or c/x), the natural way is to replace it by RN (xc) (***(x/c) or RN(c/x)), that is, to call functioncand to perform a floating-point multiplica-tion or division. This can be generalized to the approxima-tion of n/dbyRN(n/d) and the approximation of ndbyRN(nd), wheren=RN(n) andd=RN(d),and n and d are functions for which we have at our disposal acorrectly rounded implementation. We discuss tight error bounds in ulps of such approximations. From our results, one immediately obtains tight error bounds for calculations such as x & lowast;pi,ln(2)/x,x/(y+z),(x+y)& lowast;z,x/sqrt(y),sqrt(x)/y,(x+y)(z+t),(x+y)/(z+t),(x+y)/(zt),etc. in floating-point arithmetic

关键词： Floating-point arithmetic ulp numerical error correct rounding multiplication by a constant

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The meshless radial point interpolation method with ρ∞-Bathe implicit time discretization algorithm for transient elastodynamic analysis

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ENGINEERING ANALYSIS WITH BOUNDARY ELEMENTS 2024年 162卷 184-202页

作者： Zhang, Xiaoyan Xue, Hongjun Cheng, Jiaao Northwestern Polytech Univ Sch Marine Sci & Technol Xian 710072 Peoples R China Northwestern Polytech Univ Sch Aeronaut Xian 710072 Peoples R China

In this work the novel rho infinity-Bathe direct time integration algorithm is combined with the classical meshless radial point interpolation method to solve transient elastodynamic problems. The present combined numerical approach makes the best use of high computation accuracy of the RPIM in spatial discretization and high numerical stability of the rho infinity-Bathe method in temporal discretization. In addition, we theoretically show the total dispersion error of the obtained numerical solutions in transient elastodynamic analysis can be split into the spatial error and temporal error, which respectively show the monotonic convergence property with respect to the decreasing element sizes and time integration steps. The relationship among the spatial discretization error, temporal discretization error and total numerical error is also characterized in a simple and elegant form. The present study can explain theoretically why the present combined numerical approach possesses the important monotonic convergence property in solving transient elastodynamic problems. Due to this attractive property, the quality of the numerical solutions can be monotonically improved by directly using the decreasing time steps as long as a relatively reasonable spatial discretization is employed. Three typical numerical experiments are conducted to demonstrate that the important monotonic convergence property for transient elastodynamic analysis indeed can be achieved by the present combined numerical approach, hence the proposed combined numerical approach has a wide prospect for tackling very complex transient elastodynamic problems, such as the wave propagation in anisotropic media and laminated composite structures.

关键词： Meshfree techniques Finite element method Time integration Transient wave propagation numerical error

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The edge-based smoothed FEM with p.-Bathe implicit temporal discretization scheme for the analyses of underwater wave propagation problems

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OCEAN ENGINEERING 2023年第Part1期285卷

作者： Chai, Yingbin Li, Wei Zhang, Qifan Wang, Yingwei Wuhan Univ Technol Key Lab High Performance Ship Technol Minist Educ Wuhan 430063 Peoples R China Wuhan Univ Technol Sch Naval Architecture Ocean & Energy Power Engn Wuhan 430063 Peoples R China Huazhong Univ Sci & Technol Sch Naval Architecture & Ocean Engn Wuhan 430074 Peoples R China Wuhan Second Ship Design & Res Inst Wuhan 430205 Peoples R China

The underwater acoustic wave propagation problem is an important issue in ocean engineering field. The classical finite element approach with edge-based smoothed gradient smoothing technique is incorporated into the novel p.-Bathe implicit direct time integration scheme in this paper for solving underwater transient acoustic wave propagation problems. This edge-based smoothed FEM (ES-FEM) and p.-Bathe method are employed to perform the space and time domain discretization, respectively. Due to the appropriate softening effects from the ES-FEM, the space discretization errors can be markedly reduced. Meanwhile, the possible numerical error related to the temporal discretization also can be significantly suppressed by the novel p.-Bathe method which possesses the controllable numerical damping effects. Through detailed analysis of the total dispersion error, we find that the numerical results from the proposed scheme can converge monotonically to the exact solutions when employing the decreasing temporal discretization intervals and the possible numerical anisotropy effects are also effectively suppressed. These excellent numerical performance clearly distinguish the present method from the conventional approaches and make it to be particularly suitable for complicated acoustic wave propagation analysis. Finally, various typical numerical experiments are performed to illustrate the capacities of the proposed scheme in tackling underwater transient acoustic wave propagation problems.

关键词： Direct time integration Underwater acoustic wave propagation Edge-based smoothed FEM numerical error Temporal discretization

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