Second-order sign-preserving conservative interpolation (remapping) on general grids

An accurate conservative interpolation (remapping) algorithm is an essential component of most arbitrary Lagrangian–Eulerian (ALE) methods. In this paper we describe a local remapping algorithm for a positive scalar function. This algorithm is second-order accurate, conservative, and sign preserving. The algorithm is based on estimating the mass exchanged between cells at their common interface, and so is equally applicable to structured and unstructured grids. We construct the algorithm in a series of steps, clearly delineating the assumptions and errors made at each step. We validate our theory with a suite of numerical examples, analyzing the results from the viewpoint of accuracy and order of convergence.     

ReALE: A reconnection-based arbitrary-Lagrangian–Eulerian method

We present a new reconnection-based arbitrary-Lagrangian–Eulerian (ALE) method. The main elements in a standard ALE simulation are an explicit Lagrangian phase in which the solution and grid are updated, a rezoning phase in which a new grid is defined, and a remapping phase in which the Lagrangian solution is transferred (conservatively interpolated) onto the new grid. In standard ALE methods the new mesh from the rezone phase is obtained by moving grid nodes without changing connectivity of the mesh. Such rezone strategy has its limitation due to the fixed topology of the mesh. In our new method we allow connectivity of the mesh to change in rezone phase, which leads to general polygonal mesh and allows to follow Lagrangian features of the mesh much better than for standard ALE methods. Rezone strategy with reconnection is based on using Voronoi tessellation. We demonstrate performance of our new method on series of numerical examples and show it superiority in comparison with standard ALE methods without reconnection.     

ALE计算中基于声速分布的网格优化技术

基于声速分布,提出一种拉氏流体力学计算中大变形网格优化的数值技术.该方法不但可以优化网格的几何形状且可以提高拉氏流体计算的时间步长.介绍基于声速分布的网格松弛泛函、修正梯度流方程的推导、离散和求解方法,启动/终止网格优化过程的条件,及基于这种网格优化方法的ALE算法.给出Rayleigh-Taylor不稳定性问题等数值算例,用以证明该方法的有效性.     

一种基于MOF界面重构的二维中心型MMALE方法

发展了一种基于MOF（Moment of Fluid）界面重构的二维中心型MMALE（Multi-Material Arbitrary Lagrangian-Eulerian）方法.其中,流体力学方程组采用中心型拉氏方法进行离散求解.混合网格的热力学封闭采用Tipton压力松弛模型.混合网格内的界面重构采用MOF方法,并对MOF方法作了简化和改进.重映步采用一种基于多边形剪裁算法的精确积分守恒重映方法.计算了若干数值例子,包括二维漩涡发展问题、Sedov问题、激波与氦气泡相互作用问题、水中强激波与空气泡相互作用问题、二维RT不稳定性问题等.数值算例表明,该方法具有二阶精度,能够计算界面两侧密度比和压力比很大的问题,并且其健壮性优于交错型MMALE方法,适合计算多介质复杂流体动力学问题.     

Two-step hybrid conservative remapping for multimaterial arbitrary Lagrangian–Eulerian methods

We present a new hybrid conservative remapping algorithm for multimaterial Arbitrary Lagrangian–Eulerian (ALE) methods. The hybrid remapping is performed in two steps. In the first step, only nodes of the grid that lie inside subdomains occupied by single materials are moved. At this stage, computationally cheap swept-region remapping is used. In the second step, nodes that are vertices of mixed cells (cells containing several materials) and vertices of some cells in a buffer zone around mixed cells are moved. At this stage, intersection-based remapping is used. The hybrid algorithm results in computational expense that lies between swept-region and intersection-based remapping We demonstrate the performance of our new method for both structured and unstructured polygonal grids in two dimensions, as well as for cell-centered and staggered discretizations.     

Mesh refinement algorithms in an unstructured solver for multiphase flow simulation using discrete particles

This study developed spray-adaptive mesh refinement algorithms with directional sensitivity in an unstructured solver to improve spray simulation for internal combustion engine application. Inadequate spatial resolution is often found to cause inaccuracies in spray simulation using the Lagrangian–Eulerian approach due to the over-estimated diffusion and inappropriate liquid–gas phase coupling. Dynamic mesh refinement algorithms adaptive to fuel sprays and vapor gradients were developed in order to increase the grid resolution in the spray region to improve simulation accuracy. The local refinement introduced the coarse-fine face interface that requires advanced numerical schemes for flux calculation and grid rezoning with moving boundaries. To resolve the issue in flux calculation, this work implemented the refinement/coarsening algorithms into a collocated solver to avoid tedious interpolations in solving the momentum equations. A pressure correction method was applied to address unphysical pressure oscillations due to the collocation of pressure and velocity. An edge-based algorithm was used to evaluate the edge-centered quantities in order to account for the contributions from all the cells around an edge at the coarse-fine interface. A quasi-second-order upwind scheme with strong monotonicity was also modified to accommodate the coarse-fine interface for convective fluxes. To resolve the issue related to grid rezoning, rezoning was applied to the initial baseline mesh only and the new locations of the refined grids were obtained by interpolating the updated baseline mesh. The time step constraints were also re-evaluated to account for the change resulting from mesh refinement. The present refinement algorithm was used in simulating fuel sprays in an engine combustion chamber. It was found that the present approach could produce the same level of results as those using the uniformly fine mesh with substantially reduced computer time. Results also showed that this approach could alleviate the artifacts related to the Lagrangian discrete modeling of spray drops due to insufficient spatial resolution.     

One-step hybrid remapping algorithm for multi-material arbitrary Lagrangian–Eulerian methods

In this paper, a new flux-based one-step hybrid remapping method for multi-material arbitrary Lagrangian–Eulerian (ALE) approach is introduced. In the vicinity of material interfaces, the swept region is intersected with pure material polygons in the Lagrangian mesh to construct the material fluxes. Far from interfaces, the fluxes are constructed in a standard swept-region manner without intersections. This method is conservative, second-order accurate and linearity-preserving (in case of straight material interfaces), and faster than method based on intersections, as shown on selected numerical examples.     

An immersed boundary energy-based method for incompressible viscoelasticity

Incompressible viscoelastic materials are prevalent in biological applications. In this paper we present a method for incompressible viscoelasticity in which the elasticity of the material is described in Lagrangian form (i.e. in material coordinates), and Eulerian (spatial) coordinates are used for the equations of motion and to enforce the incompressibility condition. The elastic forces are computed directly from an energy functional without the use of stress tensors, and the immersed boundary method is used to communicate between Lagrangian and Eulerian variables. The method is first applied to a warm-up problem, in which a viscoelastic incompressible material fills a two-dimensional periodic domain. For this problem, we study convergence of the velocity field, the deformation map, and the Eulerian force density. The numerical results indicate that the velocity field and deformation map converge strongly at second order and the Eulerian force density converges weakly at second order. Incompressibility is well maintained, as indicated by area conservation in this 2D problem. Finally, the method is applied to a three-dimensional fluid–structure interaction problem with two different materials: an isotropic neo-Hookean model and an anisotropic fiber-reinforced model.     

Metric-based mesh adaptation for 2D Lagrangian compressible flows

In this paper, we present a method to compute compressible flows in 2D. It uses two steps: a Lagrangian step and a metric-based triangular mesh adaptation step. Computational mesh is locally adapted according to some metric field that depends on physical or geometrical data. This mesh adaptation step embeds a conservative remapping procedure to satisfy consistency with Euler equations. The whole method is no more Lagrangian.After describing mesh adaptation patterns, we recall the metric formalism. Then, we detail an appropriate remapping procedure which is first-order and relies on exact intersections.We give some hints about the parallel implementation. Finally, we present various numerical experiments which demonstrate the good properties of the algorithm.     

On high velocity impact of micro-sized metallic particles in cold spraying

In this study, a systematic examination of particle deformation behaviour in cold spraying was conducted for Cu particle using both the Lagrangian and Arbitrary Lagrangian Eulerian (ALE) methods. It is found that the meshing size in modelling by Largrangian method influences significantly the localized shear instability at interface areas. With refining the meshing size the onset velocity for interface shear instability decreases. The extrapolation of these results yields a reasonable critical velocity comparable to the actual one in cold spray practice. The results indicate that both the flattening ratio and compression ratio of the deformed particles increase with the increase in particle velocity, which are in good agreement with the experiment results. The ALE method provides a suitable way to examine the particle deformation in cold spraying. Moreover, the numerical results also show that there exists the similarity for the deformation of particles of different diameters.     

激光靶耦合问题的二阶精度重映算法研究

 任意拉格朗日欧拉方法被认为是解决2维流体力学大变形数值模拟的有效途径之一。以激光黑腔靶耦合问题为背景，结合质量流的物理意义，提出了一种二阶守恒物理量重映的算法，并利用1维激光靶耦合程序对方法及其可行性进行了考察，结合物理问题调整网格，用较少的网格得到了与较多网格拉氏计算同样的结果。为算法在2维LARED-H程序中的应用奠定了基础。     

一类基于ENO插值的守恒重映算法

在大变形流体力学问题的数值模拟中,常常会涉及到计算网格的重分.基于不同网格的物理量传递便是所谓的重映技术.基于ENO插值的思想,发展了一类适用于任意网格的ENO守恒重映算法,并给出了数值结果.     

Eulerian–Lagrangian multiscale methods for solving scalar equations – Application to incompressible two-phase flows

The present article proposes a new hybrid Eulerian–Lagrangian numerical method, based on a volume particle meshing of the Eulerian grid, for solving transport equations. The approach, called Volume Of Fluid Sub-Mesh method (VOF-SM), has the advantage of being able to deal with interface tracking as well as advection–diffusion transport equations of scalar quantities. The Eulerian evolutions of a scalar field could be obtained on any orthogonal curvilinear grid thanks to the Lagrangian advection and a redistribution of particles on the Eulerian grid. The Eulerian concentrations result from the projection of the volume and scalar informations handled by the particles. The particle velocities are interpolated from the Eulerian velocity field. The VOF-SM method is validated on several scalar interface tracking and transport problems and is compared to existing schemes within the literature. It is finally coupled to a Navier–Stokes solver and applied to the simulation of two free-surface flows, i.e. the two-dimensional buckling of a viscous jet during the filling of a square mold and the three-dimensional dam-break flow in a tank.     

A conservative level-set based method for compressible solid/fluid problems on fixed grids

A three-dimensional Eulerian method is presented for simulating dynamic systems comprising multiple compressible solid and fluid components where internal boundaries are tracked using level-set functions. Aside from the interface interaction calculation within mixed cells, each material is treated independently and the governing constitutive laws solved using a conservative finite volume discretisation based upon the solution of Riemann problems to determine the numerical fluxes. The required reconstruction of mixed cell volume fractions and cut cell geometries is presented in detail using the level-set fields. High-order accuracy is achieved by incorporating the weighted-essentially non-oscillatory (WENO) method and Runge–Kutta time integration. A model for elastoplastic solid dynamics is employed formulated using the tensor of elastic deformation gradients permitting the equations to be written in divergence form. The scheme is demonstrated using selected one-dimensional initial value problems for which exact solutions are derived, a two-dimensional void collapse, and a three-dimensional simulation of a confined explosion.     

Modeling material responses by arbitrary Lagrangian Eulerian formulation and adaptive mesh refinement method

In this paper we report an efficient numerical method combining a staggered arbitrary Lagrangian Eulerian (ALE) formulation with the adaptive mesh refinement (AMR) method for materials modeling including elastic–plastic flows, material failure, and fragmentation predictions. Unlike traditional AMR applied on fixed domains, our investigation focuses on the application to moving and deforming meshes resulting from Lagrangian motion. We give details of this numerical method with a capability to simulate elastic–plastic flows and predict material failure and fragmentation, and our main focus of this paper is to create an efficient method which combines ALE and AMR methods to simulate the dynamics of material responses with deformation and failure mechanisms. The interlevel operators and boundary conditions for these problems in AMR meshes have been investigated, and error indicators to locate material deformation and failure regions are studied. The method has been applied on several test problems, and the solutions of the problems obtained with the ALE–AMR method are reported. Parallel performance and software design for the ALE–AMR method are also discussed.     

Remapping-free ALE-type kinetic method for flow computations

Based on the integral form of the fluid dynamic equations, a finite volume kinetic scheme with arbitrary control volume and mesh velocity is developed. Different from the earlier unified moving mesh gas-kinetic method [C.Q. Jin, K. Xu, An unified moving grid gas-kinetic method in Eulerian space for viscous flow computation, J. Comput. Phys. 222 (2007) 155–175], the coupling of the fluid equations and geometrical conservation laws has been removed in order to make the scheme applicable for any quadrilateral or unstructured mesh rather than parallelogram in 2D case. Since a purely Lagrangian method is always associated with mesh entangling, in order to avoid computational collapsing in multidimensional flow simulation, the mesh velocity is constructed by considering both fluid velocity (Lagrangian methodology) and diffusive velocity (Regenerating Eulerian mesh function). Therefore, we obtain a generalized Arbitrary-Lagrangian–Eulerian (ALE) method by properly designing a mesh velocity instead of re-generating a new mesh after distortion. As a result, the remapping step to interpolate flow variables from old mesh to new mesh is avoided. The current method provides a general framework, which can be considered as a remapping-free ALE-type method. Since there is great freedom in choosing mesh velocity, in order to improve the accuracy and robustness of the method, the adaptive moving mesh method [H.Z. Tang, T. Tang, Adaptive mesh methods for one-and two-dimensional hyperbolic conservation laws, SIAM J. Numer. Anal. 41 (2003) 487–515] can be also used to construct a mesh velocity to concentrate mesh to regions with high flow gradients.     

Optimization-based synchronized flux-corrected conservative interpolation (remapping) of mass and momentum for arbitrary Lagrangian–Eulerian methods

A new optimization-based synchronized flux-corrected conservative interpolation (remapping) of mass and momentum for arbitrary Lagrangian–Eulerian hydro methods is described. Fluxes of conserved variables – mass and momentum – are limited in a synchronous way to preserve local bounds of primitive variables – density and velocity.     

Three-dimensional multi-relaxation-time lattice Boltzmann front-tracking method for two-phase flow

We developed a three-dimensional multi-relaxation-time lattice Boltzmann method for incompressible and immiscible two-phase flow by coupling with a front-tracking technique. The flow field was simulated by using an Eulerian grid, an adaptive unstructured triangular Lagrangian grid was applied to track explicitly the motion of the two-fluid interface, and an indicator function was introduced to update accurately the fluid properties. The surface tension was computed directly on a triangular Lagrangian grid, and then the surface tension was distributed to the background Eulerian grid. Three benchmarks of two-phase flow, including the Laplace law for a stationary drop, the oscillation of a three-dimensional ellipsoidal drop,and the drop deformation in a shear flow, were simulated to validate the present model.     

An improved penalty immersed boundary method for fluid–flexible body interaction

An improved penalty immersed boundary (pIB) method has been proposed for simulation of fluid–flexible body interaction problems. In the proposed method, the fluid motion is defined on the Eulerian domain, while the solid motion is described by the Lagrangian variables. To account for the interaction, the flexible body is assumed to be composed of two parts: massive material points and massless material points, which are assumed to be linked closely by a stiff spring with damping. The massive material points are subjected to the elastic force of solid deformation but do not interact with the fluid directly, while the massless material points interact with the fluid by moving with the local fluid velocity. The flow solver and the solid solver are coupled in this framework and are developed separately by different methods. The fractional step method is adopted to solve the incompressible fluid motion on a staggered Cartesian grid, while the finite element method is developed to simulate the solid motion using an unstructured triangular mesh. The interaction force is just the restoring force of the stiff spring with damping, and is spread from the Lagrangian coordinates to the Eulerian grids by a smoothed approximation of the Dirac delta function. In the numerical simulations, we first validate the solid solver by using a vibrating circular ring in vacuum, and a second-order spatial accuracy is observed. Then both two- and three-dimensional simulations of fluid–flexible body interaction are carried out, including a circular disk in a linear shear flow, an elastic circular disk moving through a constricted channel, a spherical capsule in a linear shear flow, and a windsock in a uniform flow. The spatial accuracy is shown to be between first-order and second-order for both the fluid velocities and the solid positions. Comparisons between the numerical results and the theoretical solutions are also presented.     

Lattice-Boltzmann simulation of particle-laden flow over a backward-facing step

This paper deals with the numerical simulation of gas－solid two-phase flows in an Eulerian－Lagrangian scheme. The particle tracks are calculated using a recently developed exponential Lagrangian scheme, and the approach presently used for the computation of fluid phase is based on a modified Lattice-BGK model. Different from earlier publications, the present study employs a two-way coupling mechanism to handle the interactions between carrier phase and dispersed phase in the model. This new model is applicable to simulating gas－solid two-phase flows. For example, based on the scheme, we have recaptured some phenomena of planar laminar particle-laden flow over a backward-facing step in this research, and found a new interesting phenomenon.