Single-Step Arbitrary Lagrangian-Eulerian Discontinuous Galerkin Method for 1-D Euler Equations

We propose an explicit,single-step discontinuous Galerkin method on moving grids using the arbitrary Lagrangian-Eulerian approach for one-dimensional Euler equations.The grid is moved with the local fluid velocity modified by some smoothing,which is found to con-siderably reduce the numerical dissipation introduced by Riemann solvers.The scheme preserves constant states for any mesh motion and we also study its positivity preservation property.Local grid refinement and coarsening are performed to maintain the mesh qual-ity and avoid the appearance of very small or large cells.Second,higher order methods are developed and several test cases are provided to demonstrate the accuracy of the proposed scheme.

Negative Norm Estimates for Arbitrary Lagrangian-Eulerian Discontinuous Galerkin Method for Nonlinear Hyperbolic Equations

In this paper,we present the negative norm estimates for the arbitrary Lagrangian-Eulerian discontinuous Galerkin(ALE-DG)method solving nonlinear hyperbolic equations with smooth solutions.The smoothness-increasing accuracy-conserving(SIAC)filter is a post-processing technique to enhance the accuracy of the discontinuous Galerkin(DG)solutions.This work is the essential step to extend the SIAC filter to the moving mesh for nonlinear problems.By the post-processing theory,the negative norm estimates are vital to get the superconvergence error estimates of the solutions after post-processing in the L2 norm.Although the SIAC filter has been extended to nonuniform mesh,the analysis of fil-tered solutions on the nonuniform mesh is complicated.We prove superconvergence error estimates in the negative norm for the ALE-DG method on moving meshes.The main dif-ficulties of the analysis are the terms in the ALE-DG scheme brought by the grid velocity field,and the time-dependent function space.The mapping from time-dependent cells to reference cells is very crucial in the proof.The numerical results also confirm the theoreti-cal proof.

ALE方法评估黏土场地的自升式平台桩靴承载力

为研究自升式平台桩靴在黏土场地中的极限承载力,建立Abaqus桩靴-土模型,采用任意拉格朗日-欧拉(Arbitrary Lagrangian-Eulerian,ALE)方法对桩靴的贯入过程进行模拟,提出大值法、中值法、小值法等3种数据处理方法,并拟合桩靴承载力估算公式。将模拟计算结果与SNAME规范公式计算结果进行对比分析,对比结果证明ALE方法模拟桩靴贯入的可行性。为提高工程的安全性,提出对SNAME公式计算结果进行折减的具体公式,以使其与ALE小值法处理结果更吻合。

A hybrid subcell-remapping algorithm for staggered multi-material arbitrary Lagrangian-Eulerian methods

A new flux-based hybrid subcell-remapping algorithm for staggered multimaterial arbitrary Lagrangian-Eulerian (MMALE) methods is presented. This new method is an effective generalization of the original subcell-remapping method to the multi-material regime (LOUBERE, R. and SHASHKOV,M. A subcell remapping method on staggered polygonal grids for arbitrary-Lagrangian-Eulerian methods. Journal of Computational Physics, 209, 105–138 (2005)). A complete remapping procedure of all fluid quantities is described detailedly in this paper. In the pure material regions, remapping of mass and internal energy is performed by using the original subcell-remapping method. In the regions near the material interfaces, remapping of mass and internal energy is performed with the intersection-based fluxes where intersections are performed between the swept regions and pure material polygons in the Lagrangian mesh, and an approximate approach is then introduced for constructing the subcell mass fluxes. In remapping of the subcell momentum, the mass fluxes are used to construct the momentum fluxes by multiplying a reconstructed velocity in the swept region. The nodal velocity is then conservatively recovered. Some numerical examples simulated in the full MMALE regime and several purely cyclic remapping examples are presented to prove the properties of the remapping method.

Arbitrary Lagrangian‑Eulerian Discontinuous Galerkin Methods for KdV Type Equations

In this paper,several arbitrary Lagrangian-Eulerian discontinuous Galerkin(ALE-DG)methods are presented for Korteweg-de Vries(KdV)type equations on moving meshes.Based on the L^(2) conservation law of KdV equations,we adopt the conservative and dissipative numerical fuxes for the nonlinear convection and linear dispersive terms,respectively.Thus,one conservative and three dissipative ALE-DG schemes are proposed for the equations.The invariant preserving property for the conservative scheme and the corresponding dissipative properties for the other three dissipative schemes are all presented and proved in this paper.In addition,the L^(2)-norm error estimates are also proved for two schemes,whose numerical fuxes for the linear dispersive term are both dissipative type.More precisely,when choosing the approximation space with the piecewise kth degree polynomials,the error estimate provides the kth order of convergence rate in L^(2)-norm for the scheme with the conservative numerical fuxes applied for the nonlinear convection term.Furthermore,the(k+1∕2)th order of accuracy can be proved for the ALE-DG scheme with dissipative numerical fuxes applied for the convection term.Moreover,a Hamiltonian conservative ALE-DG scheme is also presented based on the conservation of the Hamiltonian for KdV equations.Numerical examples are shown to demonstrate the accuracy and capability of the moving mesh ALE-DG methods and compare with stationary DG methods.

Second-Order Invariant Domain Preserving ALE Approximation of Euler Equations

An invariant domain preserving arbitrary Lagrangian-Eulerian method for solving non-linear hyperbolic systems is developed.The numerical scheme is explicit in time and the approximation in space is done with continuous finite elements.The method is made invar-iant domain preserving for the Euler equations using convex limiting and is tested on vari-ous benchmarks.

ALE框架下几种不同Godunov型格式的数值比较

给出一种有限体积Godunov型的ALE方法,用于求解多介质大变形的可压缩流动问题.由于方法具有任意的网格移动速度,可在拉氏、欧氏和ALE之间切换,具有较强的适应性.通过数值算例对这3种框架下的数值特性进行了比较研究.同时还研究了几种不同Godunov型格式的数值行为特性,分析比较了它们对激波和接触间断的分辨效果.

A Vertex-Centered Arbitrary Lagrangian-Eulerian Finite Volume Method with Sub-Cells for Two-Dimensional Compressible Flow

In this paper,we present a new vertex-centered arbitrary LagrangianEulerian(ALE)finite volume scheme for two-dimensional compressible flow.In our scheme,the momentum equation is discretized on the vertex control volume,while the mass equation and the energy equation are discretized on the sub-cells which are included in the vertex control volume.We attain the average of the fluid velocity on the vertex control volume directly by solving the conservation equations.Then we can obtain the fluid velocity at vertex with the reconstructed polynomial of the velocity.This fluid velocity is chosen as the mesh velocity,which makes the mesh move in a Lagrangian manner.Two WENO(Weighted Essentially Non-Oscillatory)reconstructions for the density(the total energy)and the velocity are used to make our scheme achieve the anticipated accuracy.Compared with the general vertexcentered schemes,our scheme with the new approach for the space discretization can simulate some multi-material flows which do not involve large deformations.In addition,our scheme has good robustness,and some numerical examples are presented to demonstrate the anticipated accuracy and the good properties of our scheme.

阵列起爆爆轰波阵面形状变化及其特征研究

为提升药包的爆炸冲击效能,针对爆轰冲击波传播过程,建立了水下爆炸药包内爆轰波与水中形成的冲击波流固耦合数值模型。引入了阵列角研究考虑攻角效应的单源药包内边缘阵列起爆,并基于爆轰波叠加原理模拟了阵列起爆后爆轰波耦合增强及跨介质传递初始冲击波过程。此外,通过改变阵列角探究了药包内边缘阵列起爆对爆轰波阵面形状、爆轰波压和爆轰波速及水中形成的初始冲击波特征参数的影响。研究表明:两侧对称爆轰波相互作用会导致其传播方向转变为垂直向上,耦合后波压升高,波速小幅度增加;阵列角为20°~45°时,阵列角越小,则爆轰波交汇耦合角度越大,交汇后的耦合爆轰波越早形成类平面波的波阵面。此外,考虑攻角效应的单源药包内边缘阵列起爆对冲击波载荷峰值的增益效果比对波阵面传播速度的增益效果更为显著。本文研究可为水下阵列爆炸的冲击效能优化提供参考。

耦合拉格朗日-欧拉方法及其在海洋工程中的应用

Combining the strengths of Lagrangian and Eulerian descriptions,the coupled Lagrangian–Eulerian methods play an increasingly important role in various subjects.This work reviews their development and application in ocean engineering.Initially,we briefly outline the advantages and disadvantages of the Lagrangian and Eulerian descriptions and the main characteristics of the coupled Lagrangian–Eulerian approach.Then,following the developmental trajectory of these methods,the fundamental formulations and the frameworks of various approaches,including the arbitrary Lagrangian–Eulerian finite element method,the particle-in-cell method,the material point method,and the recently developed Lagrangian–Eulerian stabilized collocation method,are detailedly reviewed.In addition,the article reviews the research progress of these methods with applications in ocean hydrodynamics,focusing on free surface flows,numerical wave generation,wave overturning and breaking,interactions between waves and coastal structures,fluid–rigid body interactions,fluid–elastic body interactions,multiphase flow problems and visualization of ocean flows,etc.Furthermore,the latest research advancements in the numerical stability,accuracy,efficiency,and consistency of the coupled Lagrangian–Eulerian particle methods are reviewed;these advancements enable efficient and highly accurate simulation of complicated multiphysics problems in ocean and coastal engineering.By building on these works,the current challenges and future directions of the hybrid Lagrangian–Eulerian particle methods are summarized.

ALE Fractional Step Finite Element Method for Fluid-Structure Nonlinear Interaction Problem

A computational procedure is developed to solve the problems of coupled motion of a structure and a viscous incompressible fluid. In order to incorporate the effect of the moving surface of the structure as well as the free surface motion, the arbitrary Lagrangian-Eulerian formulation is employed as the basis of the finite element spatial discretization. For numerical integration in time, the fraction,step method is used. This method is useful because one can use the same linear interpolation function for both velocity and pressure. The method is applied to the nonlinear interaction of a structure and a tuned liquid damper. All computations are performed with a personal computer.

Comparisons of some difference forms for compressible flow in cylindrical geometry on arbitrary Lagrangian and Eulerian framework

The study of cylindrically symmetric compressible fluid is interesting from both theoretical and numerical points of view. In this paper, the typical spherical sym- metry properties of the numerical schemes are discussed, and an area weighted scheme is extended from a Lagrangian method to an arbitrary Lagrangian and Eulerian （ALE） method. Numerical results are presented to compare three discrete configurations, i.e., the control volume scheme, the area weighted scheme, and the plane scheme with the addition of a geometrical source. The fact that the singularity arises from the geometri- cal source term in the plane scheme is illustrated. A suggestion for choosing the discrete formulation is given when the strong shock wave problems are simulated.

FINITE ELEMENT ANALYSIS FOR CHIP FORMATION IN HIGH SPEED TURNING OPERATIONS BY ARBITRARY LAGRANGIAN EULERIAN METHOD

A two-dimensional finite element （FE） model for the high speed turning operations when orthogonally machining AISI H13 tool steel at 49HRC using poly crystalline cubic boron nitride （PCBN） is described. An arbitrary Lagrangian Eulerian （ALE） method has been adopted which does not need any chip separation criteria as opposed to the traditional Lagrangian approach. Through FE simulations temperature and stresses distributions are presented that could be helpful in predicting tool life and improving process parameters. The results show that high temperatures are generated along the tool rake face as compared to the shear zone temperatures due to high thermal conductivity of PCBN tools.

Simulation of Sloshing Phenomena Using ALE Approach

Sloshing has a widespread application in many industries including automotive, aerospace, ship building and motorcycle manufacturing. The goals of sloshing simulation is to first study the sloshing pattern and then improve the tank design to reduce noise levels, stresses on the structure and optimize the baffle arrangements. In this project simulation of the fluid in tank is studied and the design modification with baffle plate is considered to minimize the sloshing phenomena using Arbitrary Langrangian Eulerian （ALE） method. Also it is explained that there is need to analyze the sloshing phenomena in detail. Arbitrary Langrangian Eulerian finite element methods gain interest for the capability to control mesh geometry independently from material geometry, the ALE methods are used to create a new undistorted mesh for the fluid domain. In this paper we use the ALE technique to solve fuel slosh problem. Fuel slosh is an important design consideration not only for the fuel tank, but also for the structure supporting the fuel tank. Fuel slosh can be generated by many ways： abrupt changes in acceleration （braking）, as well as abrupt changes in direction （highway exit-ramp）. Repetitive motion can also be involved if a sloshing resonance is generated. These sloshing events can in turn affect the overall performance 0fthe parent structure. A finite element analysis method has been developed to analyze this complex event. A new ALE formulation for the fluid mesh can be used to keep the fluid mesh integrity during the motion of the tank. This paper explains the analysis capabilities on a technical level.

Simulation of extrusion process of complicated aluminium profile and die trial

The arbitrary Lagrangian-Eulerian（ALE） adaptive remeshing technology and the HyperXtrude software of transient finite element simulations were used on analogue simulation of aluminium extrusion processing.The field distributions of strain rate,stress,temperature and velocity of metal flow were obtained.The results are basically consistent with the experiment,which indicates that this method may successfully predict the defects in the actual extrusion process.

DYNAMIC SIMULATION OF FLUID-STRUCTURE INTERACTION PROBLEMS INVOLVING LARGE-AMPLITUDE SLOSHING

An effective computational method is developed for dynamic analysis offluid-structure interaction problems involving large-amplitude sloshing of the fluid andlarge-displacement motion of the structure. The structure is modeled as a rigid container supportedby a system consisting of springs and dashpots. The motion of the fluid is decomposed into twoparts: the large-displacement motion with the container and the large-amplitude sloshing relative tothe container. The former is conveniently dealt with by defining a container-fixed noninertiallocal frame, while the latter is easily handled by adopting an ALE kinematical description. Thisleads to an easy and accurate treatment of both the fluid-structure interface and the fluid freesurface without producing excessive distortion of the computational mesh. The coupling between thefluid and the structure is accomplished through the coupling matrices that can be easilyestablished. Two numerical examples, including a TLD-structure system and a simplified liquid-loadedvehicle system, are presented to demonstrate the effectiveness and reliability of the proposedmethod. The present work can also be applied to simulate fluid-structure problems incorporatingmultibody systems and several fluid domains.

A numerical study of passenger side airbag deployment based on arbitrary Lagrangian-eulerian method

The passenger side airbags(PAB)are usually larger than the driver airbags.Therefore,the inflator of PAB is more powerful with high mass rate.In this paper,an Arbitrary Lagrangian-Eulerian(ALE)method based computational method is developed to simulate the deployment of a PAB.The tank test is used to test the property of the inflator.Through comparison of numerical and experimental results,the ALE method is validated.Based on a failed airbag test,a smaller sub-airbag is placed inside PAB to disperse the gas flow to directions which are less damaging.By applying dynamic relaxation,the initial mesh corresponding to the experimental terms is obtained.The results indicate that the interior pressure and impact force coincide with the test data,and the method in this paper is capable of capturing airbag deploying process of the PAB module accurately.

Effects of die geometry on non-equal channel lateral extrusion(NECLE) of AZ80 magnesium alloy

Non-equal channel lateral extrusion（NECLE） is a new process that can be used to attain higher grain refinement in comparison with equal channel lateral extrusion（ECLE）. The die design for this process was numerically and experimentally studied. After finding a good correlation between the numerical and experimental results, more comprehensive FE analyses were carried out. Different die geometrical parameters were considered and their effects on the induced plastic strain, stress distribution, velocity field and forming load of the process were investigated. It was found that by this process with a suitable set of die geometrical parameters, higher induced effective strain and more homogeneous strain distribution could be achieved in comparison with ECLE operation.

Axially variable-length solid element of absolute nodal coordinate formulation

An axially variable-length solid element with eight nodes is proposed by integrating the arbitrary Lagrangian-Eulerian (ALE) formulation and the absolute nodal coordinate formulation (ANCF). In addition to the nodal positions and slopes of eight nodes, two material coordinates in the axial direction are used as the generalized coordinates. As a consequence, the nodes in the ALE-ANCF are not associated with any specific material points and the axial length of the solid element can be varied over time. These two material coordinates give rise to a variable mass matrix and an additional inertial force vector. Computationally efficient formulae of the additional inertial forces and elastic forces, as well as their Jacobians, are also derived. The dynamic equation of a flexible multibody system (FMBS) with variable-length bodies is presented. The maximum and minimum lengths of the boundary elements of an FMBS have to be appropriately defined to ensure accuracy and non-singularity when solving the dynamic equation. Three numerical examples of static and dynamic problems are given to validate the variable-length solid elements of ALE-ANCF and show their capability.

Dynamic Response of Falling Liquid Storage Container Under Transient Impact

The dynamic response performance of a large,cylindrical,fluid-filled steel container under high-speed impact is evaluated through fluid-structure interaction analysis using arbitrary Lagrange-Eulerian(ALE)method.The ALE method is adopted to accurately calculate the structural behavior induced by the internal liquid impact of the container.The stress and strain results obtained from the finite element analysis are in line with the experimental shell impact data.The influences of drop angle,drop height,and flow impact frequency are discussed.Calculation results indicate that the impact stress and damage of the container increase with drop height.However,the amplitude of the oscillation and the impact stress increase when the container and flow impact resonance occur at a certain drop height.The impact stress shows a nonlinear relationship with drop angle.