Finite elements in CFD: What lies ahead

The current state of the art of Finite Element Methods in Computational Fluid Dynamics is reviewed. The aim of this review is to point out what appear currently as the main shortcomings of Finite Element Methods, so as to concentrate the efforts to remove them. The analysis of flows using Finite Elements will only be successful if all steps involved in it are optimized. Therefore, I deem it necessary to describe explicit and implicit flow solvers, unstructured multigrid methods, adaptive refinement schemes, grid generation and graphics in 3-D, the effective use of supercomputer hardware and memory, as well as the combination of structured and unstructured grids.     

Segregated Runge–Kutta methods for the incompressible Navier–Stokes equations

In this work, we propose Runge–Kutta time integration schemes for the incompressible Navier–Stokes equations with two salient properties. First, velocity and pressure computations are segregated at the time integration level, without the need to perform additional fractional step techniques that spoil high orders of accuracy. Second, the proposed methods keep the same order of accuracy for both velocities and pressures. The segregated Runge–Kutta methods are motivated as an implicit–explicit Runge–Kutta time integration of the projected Navier–Stokes system onto the discrete divergence‐free space, and its re‐statement in a velocity–pressure setting using a discrete pressure Poisson equation. We have analysed the preservation of the discrete divergence constraint for segregated Runge–Kutta methods and their relation (in their fully explicit version) with existing half‐explicit methods. We have performed a detailed numerical experimentation for a wide set of schemes (from first to third order), including implicit and IMEX integration of viscous and convective terms, for incompressible laminar and turbulent flows. Further, segregated Runge–Kutta schemes with adaptive time stepping are proposed. Copyright © 2015 John Wiley & Sons, Ltd.     

Effects of time integration schemes on discontinuous deformation simulations using the numerical manifold method

Numerical stability by using certain time integration scheme is a critical issue for accurate simulation of discontinuous deformations of solids. To investigate the effects of the time integration schemes on the numerical stability of the numerical manifold method, the implicit time integration schemes, ie, the Newmark, the HHT‐α, and the WBZ‐α methods, and the explicit time integration algorithms, ie, the central difference, the Zhai's, and Chung‐Lee methods, are implemented. Their performance is examined by conducting transient response analysis of an elastic strip subjected to constant loading, impact analysis of an elastic rod with an initial velocity, and excavation analysis of jointed rock masses, respectively. Parametric studies using different time steps are conducted for different time integration algorithms, and the convergence efficiency of the open‐close iterations for the contact problems is also investigated. It is proved that the Hilber‐Hughes‐Taylor‐α (HHT‐α), Wood‐Bossak‐Zienkiewicz‐α (WBZ‐α), Zhai's, and Chung‐Lee methods are more attractive in solving discontinuous deformation problems involving nonlinear contacts. It is also found that the examined explicit algorithms showed higher computational efficiency compared to those implicit algorithms within acceptable computational accuracy.     

Some computational and algorithmic developments in computational mechanics of discontinua

Discontinua simulations are becoming an important part of computational mechanics to the extent that computational mechanics of discontinua is becoming a separate sub-discipline of computational mechanics. Among the most widely used methods of computational mechanics of discontinua are discrete-element methods, combined finite-discrete-element methods and discontinuum deformation analysis methods. A range of key algorithmic procedures is common to most of these methods. These include contact detection, explicit solvers, fracture and fragmentation models, handling of complex geometric considerations when processing interaction in three dimensions (contact kinematics) and fluid coupling. In recent years, there have been major breakthroughs in almost all of these key algorithmic aspects. These include linear contact-detection procedures (NBS, C-grid), discretized contact solutions, fracture and fragmentation solutions, together with fluid pressure driven fracture process and three-dimensional explicit solvers incorporating finite rotations. Many of these breakthroughs have not yet been applied across the full range of relevant discontinuum problems. The major reason for this is that discrete-element method, discontinuum deformation analysis and combined finite-discrete-element method publications are spread over a wide range of specialist journals and conferences. Thus in this paper, the main features of a selection of algorithmic breakthroughs are reviewed for the first time, enabling researchers in different fields to apply these compatible developments to their specific applications.     

Establishing stable time‐steps for DEM simulations of non‐collinear planar collisions with linear contact laws

The discrete element method, developed by Cundall and Strack, typically uses some variations of the central difference numerical integration scheme. However, like all explicit schemes, the scheme is only conditionally stable, with the stability determined by the size of the time‐step. The current methods for estimating appropriate discrete element method time‐steps are based on many assumptions; therefore, large factors of safety are usually applied to the time‐step to ensure stability, which substantially increases the computational cost of a simulation. This work introduces a general framework for estimating critical time‐steps for any planar rigid body subject to linear damping and forcing. A numerical investigation of how system damping, coupled with non‐collinear impact, affects the critical time‐step is also presented. It is shown that the critical time‐step is proportional to if a linear contact model is adopted, where m and k represent mass and stiffness, respectively. The term which multiplies this factor is a function of known physical parameters of the system. The stability of a system is independent of the initial conditions. © 2016 The Authors. International Journal for Numerical Methods in Engineering Published by John Wiley & Sons Ltd.     

Fourth‐order energy‐preserving locally implicit time discretization for linear wave equations

A family of fourth‐order coupled implicit–explicit time schemes is presented as a special case of fourth‐order coupled implicit schemes for linear wave equations. The domain of interest is decomposed into several regions where different fourth‐order time discretizations are used, chosen among a family of implicit or explicit fourth‐order schemes. The coupling is based on a Lagrangian formulation on the boundaries between the several non‐conforming meshes of the regions. A global discrete energy is shown to be preserved and leads to global fourth‐order consistency in time. Numerical results in 1D and 2D for the acoustic and elastodynamics equations illustrate the good behavior of the schemes and their potential for the simulation of realistic highly heterogeneous media or strongly refined geometries, for which using everywhere an explicit scheme can be extremely penalizing. Accuracy up to fourth order reduces the numerical dispersion inherent to implicit methods used with a large time step and makes this family of schemes attractive compared with second‐order accurate methods. Copyright © 2015 John Wiley & Sons, Ltd.     

Predicting dust emissions – Experimental study compared to coupled DEM/CFD simulations using a reference test bulk material

The awareness of dust emissions is crucial regarding safe industrial processes, environmental protection and health care. For this purpose, closely linked experimental and numerical investigations are performed. This work presents the results of an experimental study which is used for the calibration of a modelling framework based on the Discrete Element Method (DEM) coupled with Computational Fluid Dynamics (CFD) and applied for the calculation of dust emissions for predictive purposes. The key objective of the approach is to come up with a dust source term which enables to describe and to quantify the release of particle emissions. For the presented experimental study, a wind tunnel and a rotating drum setup, which cover various handling types of bulk materials, are used in order to gain data about parameters having an impact on the dust release. The special feature of the investigations is the use of a reference test bulk material which represents a bulk material in its generally main fractions, the fine and the coarse material, keeping the discrepancy between experiments and simulations low. With the help of the experimental results the calibration of the simulation model was carried out and followed by a comparison.     

Gradient and dilatational stabilizations for stress‐point integration in the element‐free Galerkin method

Stabilized stress‐point integration schemes based on gradient stabilization and dilatational stabilization methods are presented for linear elastostaticity problems in the framework of element‐free Galerkin (EFG) method. The instability in stress fields associated with the stress‐point integration is treated by the addition to the Galerkin weak form of stabilization terms which contain product of the gradient of the residual or the trace of the gradient of the residual; the latter is called dilatational stabilization. Numerical results show that the oscillations in the stress fields are successfully removed by the presented stabilization methods, and that the convergence and stability properties of direct stress‐point integration are greatly improved. These stabilization methods are particularly suitable for the solution of non‐linear continua with explicit time integration methods. Copyright © 2008 John Wiley & Sons, Ltd.     

Time integration for extended discontinuous Galerkin methods with moving domains

We study time integration schemes for discontinuous Galerkin discretizations of problems with moving immersed interfaces. Two approaches have been discussed in literature so far: splitting schemes and space‐time methods. Splitting schemes are cheap and easy to implement, but are non‐conservative, inherently limited to low orders of accuracy, and require extremely small time steps. Space‐time methods, on the other hand, are conservative, allow for large time steps, and generalize to arbitrary orders of accuracy. However, these advantages come at the expense of a severe growth the systems to be solved. Within this work, we present a generic strategy that combines the advantages of both concepts by evaluating numerical fluxes in a moving reference frame and by making use of a conservative cell‐agglomeration strategy. We study the performance of this strategy in combination with backward‐difference‐formulas and explicit Runge‐Kutta schemes in the context of the scalar transport equation, the Burgers equation, and the heat equation. Our results indicate that higher order spatial and temporal convergence rates can be achieved without increasing the size of the systems to be solved.     

Alternative time-marching schemes for elastodynamic analysis with the domain boundary element method formulation

This work presents alternative time-marching schemes for performing elastodynamic analysis by the Boundary Element Method. The use of the static fundamental solution and the maintenance of the domain integral associated to the accelerations characterize the formulation employed in this work. It is called D-BEM, D meaning domain. Time response is obtained by employing step-by-step time-marching procedures similar to those adopted in the Finite Element Method. Among all integration procedures, Houbolt scheme became the most popular used to march in time with D-BEM formulation, in spite of the presence of a high numerical damping. In order to improve the integration, this work presents alternative schemes that can be used to perform elastodynamic analysis by the BEM with a better damping control. In order to verify the accuracy of the proposed scheme, three examples are presented and discussed at the end of this work.     

On the numerical implementation of 3D rate‐dependent single crystal plasticity formulations

In this paper, several important numerical issues are addressed for three‐dimensional (3D) rate‐dependent single crystal plasticity. After a thorough comparison of different constitutive algorithms, we classify the integration methods into three approaches, namely, the implicit elastic/plastic deformation gradient approach, the implicit slip‐rate approach, and the explicit slip‐rate approach. As part of this algorithmic study, we focus on five different schemes to enforce the plastic incompressibility, four ways to update the texture, and one convergence criteria. The numerical performance of these different methods is illustrated. The contribution of this study is three‐fold: a stable scheme for the incompressibility enforcement, an improved implicit algorithm, and a fully explicit algorithm. Copyright © 2005 John Wiley & Sons, Ltd.     

Critical time step for DEM simulations of dynamic systems using a Hertzian contact model

The discrete element method (DEM) typically uses an explicit numerical integration scheme to solve the equations of motion. However, like all explicit schemes, the scheme is only conditionally stable, with the stability determined by the size of the time step. Currently, there are no comprehensive techniques for estimating appropriate DEM time steps when a nonlinear contact interaction is used. It is common practice to apply a large factor of safety to these estimates to ensure stability, which unnecessarily increases the computational cost of these simulations. This work introduces an alternative framework for selecting a stable time step for nonlinear contact laws, specifically for the Hertz-Mindlin contact law. This approach uses the fact that the discretised equations of motion take the form of a nonlinear map and can be analysed as such. Using this framework, we analyse the effects of both system damping and the initial relative velocity of collision on the critical time step for a Hertz-Mindlin contact event between spherical particles.     

NOPD的椭球状散体元建模

提出了能模拟不同颗粒形状的微颗粒组合体的椭球状散体元模型,从理论上解决了椭球单元的接触判断及运动学、动力学问题。采用该模型对NOPD复合阻尼结构进行了动力学计算,计算结果与实验结果具有很好的一致性,为进一步研究NOPD的阻尼机理提供了一种理论分析方法。     

A semi‐implicit integration scheme for a combined viscoelastic‐damage model of plastic bonded explosives

This paper presents a new implementation of a constitutive model commonly used to represent plastic bonded explosives in finite element simulations of thermomechanical response. The constitutive model, viscoSCRAM, combines linear viscoelasticity with isotropic damage evolution. The original implementation was focused on short duration transient events; thus, an explicit update scheme was used. For longer duration simulations that employ significantly larger time step sizes, the explicit update scheme is inadequate. This work presents a new semi‐implicit update scheme suitable for simulations using relatively large time steps. The algorithm solves a nonlinear system of equations to ensure that the stress, damaged state, and internal stresses are in agreement with implicit update equations at the end of each increment. The crack growth is advanced in time using a sub‐incremental explicit scheme; thus, the entire implementation is semi‐implicit. The theory is briefly discussed along with previous explicit integration schemes. The new integration algorithm and its implementation into the finite element code, Abaqus, are detailed. Finally, the new and old algorithms are compared via simulations of uniaxial compression and beam bending. The semi‐implicit scheme has been demonstrated to provide higher accuracy for a given allocated computational time for the quasistatic cases considered here. Published 2014. This article is a US Government work and is in the public domain in the USA.     

Simulating macroscopic behavior of self-compacting mixtures with DEM

Development of proper rheological models and suitable numerical methods are necessary for a thorough understanding of the basic flow properties of fresh mortar or concrete. Main challenge for models is to find a quantitative correlation between the model parameters and the properties and proportions of the mix ingredients. This paper presents a modeling approach for the rheological behavior of fresh self-compacting mixtures using a Discrete Element Method (DEM). The employed method is based on a conceptual idea where the grain-paste-grain interactions are explicitly described as an interactive two-phase paste-bridge system. Each mixture is considered to be an assembly of mutually interacting “grain-paste-grain” systems which can be characterized according to the mix composition with help of the “excess paste theory”. Macroscopic slump flow predictions are evaluated by laboratory tests. Simulations and experimental test results show good agreement.     

散状物料转载系统设计DEM仿真方法的研究

颗粒仿真技术可以对散状物料的运动进行观察、机理分析、受力分析、磨损(寿命)估计和系统优化.比较了散状物料转载计算方法,给出了采用DEM方法的建模与模型检验的基本步骤.针对DEM计算方法存在所需计算时间过长的问题,实验比较了降低剪切模量和增大颗粒粒度等方法对DEM计算时间和实验结果的影响,结果表明,在不影响计算结果的前提下可通过降低剪切模量和增大颗粒粒度来缩短DEM仿真计算时间.利用EDEM软件实现了散料转载过程的可视化,比较分析了直线型溜槽、折线型溜槽和曲线型溜槽中物料转载效果.仿真表明：采用变曲率半径效果更好;U型溜槽截面能够减少物料不对中情况的发生.     

Robust and provably second‐order explicit–explicit and implicit–explicit staggered time‐integrators for highly non‐linear compressible fluid–structure interaction problems

An explicit–explicit staggered time‐integration algorithm and an implicit–explicit counterpart are presented for the solution of non‐linear transient fluid–structure interaction problems in the Arbitrary Lagrangian–Eulerian (ALE) setting. In the explicit–explicit case where the usually desirable simultaneous updating of the fluid and structural states is both natural and trivial, staggering is shown to improve numerical stability. Using rigorous ALE extensions of the two‐stage explicit Runge–Kutta and three‐point backward difference methods for the fluid, and in both cases the explicit central difference scheme for the structure, second‐order time‐accuracy is achieved for the coupled explicit–explicit and implicit–explicit fluid–structure time‐integration methods, respectively, via suitable predictors and careful stagings of the computational steps. The robustness of both methods and their proven second‐order time‐accuracy are verified for sample application problems. Their potential for the solution of highly non‐linear fluid–structure interaction problems is demonstrated and validated with the simulation of the dynamic collapse of a cylindrical shell submerged in water. The obtained numerical results demonstrate that, even for fluid–structure applications with strong added mass effects, a carefully designed staggered and subiteration‐free time‐integrator can achieve numerical stability and robustness with respect to the slenderness of the structure, as long as the fluid is justifiably modeled as a compressible medium. Copyright © 2010 John Wiley & Sons, Ltd.     

An adaptive time integration strategy based on displacement history curvature

This work introduces a time‐adaptive strategy that uses a refinement estimator on the basis of the first Frenet curvature. In dynamics, a time‐adaptive strategy is a mechanism that interactively proposes changes to the time step used in iterative methods of solution. These changes aim to improve the relation between quality of response and computational cost. The method here proposed is suitable for a variety of numerical time integration problems, for example, in the study of bodies subjected to dynamical loads. The motion equation in its space‐discrete form is used as reference to derive the formulation presented in this paper. Our method is contrasted with other ones based on local error estimator and apparent frequencies. We check the performance of our proposal when employed with the central difference, the explicit generalized‐ α and the Chung‐Lee integration methods. The proposed refinement estimator demands low computational resources, being easily applied to several direct integration methods. Copyright © 2012 John Wiley & Sons, Ltd.     

Multi‐model Arlequin method for transient structural dynamics with explicit time integration

This paper addresses the explicit time integration for solving multi‐model structural dynamics by the Arlequin method. Our study focuses on the stability of the central difference scheme in the Arlequin framework. Although the Arlequin coupling matrices can introduce a weak instability, the time integrator remains stable as long as the initial kinematic conditions of both models agree on the coupling zone. After showing that the Arlequin weights have an adverse impact on the critical time step, we present two approaches to circumvent this issue. Computational tests confirm that the two approaches effectively preserve a feasible critical time step and show the efficiency of the Arlequin method for structural explicit dynamic simulations. Copyright © 2017 John Wiley & Sons, Ltd.