On Taylor weak statement finite element methods for computational fluid dynamics

A Taylor series augmentation of a weak statement (a ‘Taylor weak statement’ or ‘Taylor-Galerkin’ method) is used to systematically reduce the dispersion error in a finite element approximation of the one-dimensional transient advection equation. A frequency analysis is applied to determine the phase velocity of semi-implicit linear, quadratic and cubic basis one-dimensional finite element methods and of several comparative finite difference/finite volume algorithms. The finite element methods analysed include both Galerkin and Taylor weak statements. The frequency analysis is used to obtain an improved linear basis Taylor weak statement finite element algorithm. Solutions are reported for verification problems in one and two dimensions and are compared with finite volume solutions. The improved finite element algorithms have sufficient phase accuracy to achieve highly accurate linear transient solutions with little or no artificial diffusion.     

A conservative flux-corrected continuous finite element method for fluid interface dynamics

This paper presents a continuous finite element solution for fluid flows with interfaces. The method is founded on the sign-preserving flux correction transport methodology and extends nonoscillatory finite element algorithm capabilities to predict interface motion efficiently. The procedure is composed of three main stages, along the lines of the conservative level set method: transport of phase function, reconstruction of phase function, and solution of equations of motion of two incompressible fluids. The flux correction technique takes action on the three steps. Limiting process incorporates a straightforward refinement to remove global mass residuals present in the earliest version of the algorithm. This is of particular importance in the transport step. Moreover, new method retains the efficacy of the original. To reconstruct the phase function after transport, a novel nonlinear (and conservative) streamlined diffusion equation is proposed, with an anisotropic diffusivity comprising artificial compression and diffusive fluxes along interface displacements direction. A substantial reduction of unphysical overshoots along the interface is reached by an improved bound estimation that includes interface information. Complete operation of the correction algorithm for two incompressible fluids flows requires two pressure solutions. We explore a reduced form to circumvent this extra burden. Numerical experiments verify the formulation by reproducing stringent benchmarks both for transport/reinitialization and for two-fluid interface propagation.     

Coarse-graining atomistic dynamics of brittle fracture by finite element method

In this work we present the finite element (FE) implementation of an atomistic formulation of balance equations and its application to coarse-grained (CG) simulation of dynamic fracture. First, we simulate a notched specimen that contains about 1.8 million atoms by the CG-FE method, and we compare the CG-FE results with that by all-atom molecular dynamics (MD) simulations. We find that CG-FE simulations with about 5% degrees of freedom of the MD simulation can capture the essential dynamic features, not in exact correspondence, but qualitatively and quantitatively similar to that obtained by MD simulations. We then proceed to simulate a series of micron-sized specimens by the CF-FE method. We find that it is the interaction of the forward propagating crack with the stress waves being reflected back by the boundaries of the specimen that triggers the dynamic instability and hence the branching of cracks in micron-sized specimens. The potential application of the method and future work for improvements are discussed.     

Multiresolution reproducing kernel particle method for computational fluid dynamics

Multiresolution analysis based on the reproducing kernel particle method (RKPM) is developed for computational fluid dynamics. An algorithm incorporating multiple-scale adaptive refinement is introduced. The concept of using a wavelet solution as an error indicator is also presented. A few representative numerical examples are solved to illustrate the performance of this new meshless method. Results show that the RKPM is a good candidate for tackling the widespread large-scale problems in fluid dynamics. © 1997 John Wiley & Sons, Ltd.     

A finite amplitude analysis of tunnel deformation by the finite element method with highly viscous fluid

A finite element method for highly viscous fluid is used to calculate the velocity and stress fields in the surrounding soft rock of a tunnel. In order to fit the calculated values with the measured displacement of tunnel wall, we inverted the boundary forces and the mechanical parameters of the surrounding rocks.     

A finite element method for compressible viscous fluid flows

This work presents a mixed three‐dimensional finite element formulation for analyzing compressible viscous flows. The formulation is based on the primitive variables velocity, density, temperature and pressure. The goal of this work is to present a ‘stable’ numerical formulation, and, thus, the interpolation functions for the field variables are chosen so as to satisfy the inf–sup conditions. An exact tangent stiffness matrix is derived for the formulation, which ensures a quadratic rate of convergence. The good performance of the proposed strategy is shown in a number of steady‐state and transient problems where compressibility effects are important such as high Mach number flows, natural convection, Riemann problems, etc., and also on problems where the fluid can be treated as almost incompressible. Copyright © 2010 John Wiley & Sons, Ltd.     

A Hermite finite element method for incompressible fluid flow

We describe some Hermite stream function and velocity finite elements and a divergence‐free finite element method for the computation of incompressible flow. Divergence‐free velocity bases defined on (but not limited to) rectangles are presented, which produce pointwise divergence‐free flow fields (∇· u _h≡0). The discrete velocity satisfies a flow equation that does not involve pressure. The pressure can be recovered as a function of the velocity if needed. The method is formulated in primitive variables and applied to the stationary lid‐driven cavity and backward‐facing step test problems. Copyright © 2009 John Wiley & Sons, Ltd.     

An implicit meshless method for application in computational fluid dynamics

An implicit meshless scheme is developed for solving the Euler equations, as well as the laminar and Reynolds‐averaged Navier–Stokes equations. Spatial derivatives are approximated using a least squares method on clouds of points. The system of equations is linearised, and solved implicitly using approximate, analytical Jacobian matrices and a preconditioned Krylov subspace iterative method. The details of the spatial discretisation, linear solver and construction of the Jacobian matrix are discussed; and results that demonstrate the performance of the scheme are presented for steady and unsteady two dimensional fluid flows. Copyright © 2012 John Wiley & Sons, Ltd.     

Expectations for computational fluid dynamics

This article presents a sampling of the author's expectations for the field of computational fluid dynamics (CFD) in the areas of research, development and application. The primary focus of the discussion herein is related to the non-linear transonic flow regime, and more specifically, for calculations about commercial transport aircraft. However, many of these topics are pertinent to all flow field regimes and aircraft designs. The underlying goal is to enable the automation of multi-disciplinary design processes, which utilize state-of-the-art numerical simulation methods. These include issues pertaining to accuracy, robustness, efficiency, ease-of-use, uncertainty requirements and other challenges.     

模糊随机有限元在转子动力学分析中的应用

综合考虑转子系统的模糊和随机不确定性,根据信息熵理论,在保证模糊熵不变的前提下将模糊量转化为随机变量.然后,采用基于Neumann展开式的随机有限元法分析转子系统动力学问题.将Neumann展开式与Newmark-β法结合起来,从而使基于Neumann展开式的随机有限元法可以应用于分析非线性转子系统.以考虑模糊和随机因素的转子系统的临界转速和非线性响应为例,验证了模糊随机有限元方法在转子系统动力学分析中的适用性.     

Direct modeling for computational fluid dynamics

All fluid dynamic equations are valid under their modeling scales, such as the particle mean free path and mean collision time scale of the Boltzmann equation and the hydrodynamic scale of the Navier–Stokes(NS) equations.The current computational fluid dynamics(CFD) focuses on the numerical solution of partial differential equations(PDEs), and its aim is to get the accurate solution of these governing equations. Under such a CFD practice, it is hard to develop a unified scheme that covers flow physics from kinetic to hydrodynamic scales continuously because there is no such governing equation which could make a smooth transition from the Boltzmann to the NS modeling. The study of fluid dynamics needs to go beyond the traditional numerical partial differential equations. The emerging engineering applications, such as air-vehicle design for near-space flight and flow and heat transfer in micro-devices, do require further expansion of the concept of gas dynamics to a larger domain of physical reality, rather than the traditional distinguishable governing equations. At the current stage, the non-equilibrium flow physics has not yet been well explored or clearly understood due to the lack of appropriate tools.Unfortunately, under the current numerical PDE approach, it is hard to develop such a meaningful tool due to the absence of valid PDEs. In order to construct multiscale and multiphysics simulation methods similar to the modeling process of constructing the Boltzmann or the NS governing equations, the development of a numerical algorithm should be based on the first principle of physical modeling. In this paper, instead of following the traditional numerical PDE path, we introduce direct modeling as a principle for CFD algorithm development. Since all computations are conducted in a discretized space with limited cell resolution, the flow physics to be modeled has to be done in the mesh size and time step scales.Here, the CFD is more or less a direct construction of discrete numerical evolution equations, where the mesh size and time step will play dynamic roles in the modeling process.With the variation of the ratio between mesh size and local particle mean free path, the scheme will capture flow physics from the kinetic particle transport and collision to the hydrodynamic wave propagation. Based on the direct modeling, a continuous dynamics of flow motion will be captured in the unified gas-kinetic scheme. This scheme can be faithfully used to study the unexplored non-equilibrium flow physics in the transition regime.     

A least squares finite element method for viscoelastic fluid flow problems

Viscoelastic flows remain a demanding class of problems for approximate analysis, particularly at increasing Weissenberg numbers. Part of the difficulty stems from the convective behavior and in the treatment of the stress field as a primary unknown. This latter aspect has led to the use of higher-order piecewise approximations for the stress approximation spaces in recent finite element research. The computational complexity of the discretized problem is increased significantly by this approach but at present it appears the most viable technique for solving these problems. Motivated by recent success in treating mixed systems and convective problems, we formulate here a least squares finite element method for the viscoelastic flow problem. Numerical experiments are conducted to test the method and examine its strengths and limitations. Some difficulties and open issues are identified through the numerical experiments. We consider the use of high degree elements (p refinement) to improve performance and accuracy.     

A new finite element formulation for both compressible and nearly incompressible fluid dynamics

A new finite element formulation designed for both compressible and nearly incompressible viscous flows is presented. The formulation combines conservative and non‐conservative dependent variables, namely, the mass–velocity (density * velocity), internal energy and pressure. The central feature of the method is the derivation of a discretized equation for pressure, where pressure contributions arising from the mass, momentum and energy balances are taken implicitly in the time discretization. The method is applied to the analysis of laminar flows governed by the Navier–Stokes equations in both compressible and nearly incompressible regimes. Numerical examples, covering a wide range of Mach number, demonstrate the robustness and versatility of the new method. Copyright © 2000 John Wiley & Sons, Ltd.     

Advances in the simulation of multi‐fluid flows with the particle finite element method. Application to bubble dynamics

In this work, we extend the Particle Finite Element Method (PFEM) to multi‐fluid flow problems with the aim of exploiting the fact that Lagrangian methods are specially well suited for tracking interfaces. We develop a numerical scheme able to deal with large jumps in the physical properties, included surface tension, and able to accurately represent all types of discontinuities in the flow variables. The scheme is based on decoupling the velocity and pressure variables through a pressure segregation method that takes into account the interface conditions. The interface is defined to be aligned with the moving mesh, so that it remains sharp along time, and pressure degrees of freedom are duplicated at the interface nodes to represent the discontinuity of this variable due to surface tension and variable viscosity. Furthermore, the mesh is refined in the vicinity of the interface to improve the accuracy and the efficiency of the computations. We apply the resulting scheme to the benchmark problem of a two‐dimensional bubble rising in a liquid column presented in Hysing et al. (International Journal for Numerical Methods in Fluids 2009; 60 : 1259–1288), and propose two breakup and coalescence problems to assess the ability of a multi‐fluid code to model topology changes. Copyright © 2010 John Wiley & Sons, Ltd.     

Memory access optimization for particle operations in computational fluid dynamics-discrete element method simulations

Computational Fluid Dynamics – Discrete Element Method is used to model gas-solid systems in several applications in energy, pharmaceutical and petrochemical industries. Computational performance bottlenecks often limit the problem sizes that can be simulated at industrial scale. The data structures used to store several millions of particles in such large-scale simulations have a large memory footprint that does not fit into the processor cache hierarchies on current high-performance-computing platforms, leading to reduced computational performance. This paper specifically addresses this aspect of memory access bottlenecks in industrial scale simulations. The use of space-filling curves to improve memory access patterns is described and their impact on computational performance is quantified in both shared and distributed memory parallelization paradigms. The Morton space filling curve applied to uniform grids and k-dimensional tree partitions are used to reorder the particle data-structure thus improving spatial and temporal locality in memory. The performance impact of these techniques when applied to two benchmark problems, namely the homogeneous-cooling-system and a fluidized-bed, are presented. These optimization techniques lead to approximately two-fold performance improvement in particle focused operations such as neighbor-list creation and data-exchange, with ∼ 1.5 times overall improvement in a fluidization simulation with 1.27 million particles.     

Exploring alternative approaches to computational fluid dynamics

This paper discusses advances in two areas which may have the potential to impact future simulation capabilities through advanced algorithms. This includes spectral multigrid (MG) solvers for high-order accurate spatial discretizations and efficient MG solvers for kinetic-based schemes. Preliminiary evidence is given illustrating the promise of these approaches for application to engineering simulations.     

Understanding coherent vortices through computational fluid dynamics

A new definition of coherent vortices in turbulence is proposed, where the vorticity equation reduces to a cyclostrophic balance. Afterward, we describe five fundamental vortex interactions, the sheet, the spiral, the pairing, the even longitudinal, and the odd longitudinal modes. Numerous examples of these interactions are provided from direct numerical or large-eddy simulations. The resulting vortices are responsible for the internal intermittent character of turbulence, with highly nongaussian tails for the probability density functions of vorticity, passive scalar, and low pressure. In a mixing layer, the combination of the odd longitudinal and the pairing modes (helical pairing) is inhibited by compressibility, above a convective Mach number of 0.7. When turbulence is submitted to a solid-body rotation, anticyclonic vortices of local Rossby number of the order of 1 transform into intense perpendicular Görtler-type alternate longitudinal vortices.The support of CCVR, CEA, CNRS, Dassault/CNES, DRET, LHF, and Région Rhône-Alpes is acknowledged. This paper is the text of an invited lecture given at the IUTAM Symposium on Eddy Structure Identification in Free Turbulent Shear Flows, Poitiers, 12–14 October 1992.     

Domain decomposition methods in computational fluid dynamics

The divide-and-conquer paradigm of iterative domain decomposition or substructuring has become a practical tool in computational fluid dynamics applications because of its flexibility in accommodating adaptive refinement through locally uniform (or quasi-uniform) grids, its ability to exploit multiple discretizations of the operator equations, and the modular pathway it provides towards parallelism. We illustrate these features on the classic model problem of flow over a backstep using Newton's method as the non-linear iteration. Multiple discretizations (second-order in the operator and first-order in the preconditioner) and locally uniform mesh refinement pay dividends separately and can be combined synergistically. We include sample performance results from an Intel iPSC/860 hypercube implementation.     

Simulation of metal-forming processes by the finite element method

An algorithm to model the contact, with friction, between a deformable body and rigid surfaces is presented. The method shown is to integrate the equation of motion in the actual reference frame, taking into account the proper constraint set induced by the contact conditions; moreover, the various terms of the 3D stiffness tangent matrix are calculated. The model has been implemented in an FEM code suitable for dealing with finite deformation problems, and the results obtained in the simulation of a bulge-test case and a deep-drawing test are presented and compared with experimental data.