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.

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.

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.

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.

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.

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.

Analysis of Sheet Metal Extrusion Process Using Finite Element Method

Sheet bulk metal forming processes have been widely developed to the facilitate manufacture of complicated 3D parts. However, there is still not enough know-how available. In this paper, as one of the typical sheet bulk metal forming processes, the sheet metal extrusion process was studied. A reasonable finite element method （FEM） model of sheet metal extrusion process taking the influence of flow-stress curve with wide range of plastic strain and ductile damage into consideration was established and simulated by an arbitrary Lagrangian-Eulerian （ALE） FEM implemented in MSC.Marc. Validated by comparing the results with experiment, some phenomenological characteristics, such as metal flow behavior, shrinkage cavity, and the influence of different combinations of diameter of punch, diameter of extrusion outlet, and diameter of pre-punched hole were analyzed and concluded, which can be used as theoretical fundamental for the design of the sheet metal extrusion process.

SIMULATION OF FLUID-SOLID INTERACTION ON WATER DITCHING OF AN AIRPLANE BY ALE METHOD

Ditching is considered as one of the important aspects of safety performances of airplanes. It is related primarily with the fluid-solid interaction, whose studies mainly depend on experiments at the present time. Numerical and analytical methods for fluid-solid interaction by using 3-D full scale airplane＇s model will reduce the dependence on the expensive model tests. Numerical studies can be used to estimate the safety of ditching and provide a reference for the crashworthiness design. This article proposes a 3-D dynamical structural model after the real shape of an airplane and an Arbitrary Lagrange-Euler （ALE） fluid-field model, to simulate the fluid-solid interactions caused by low speed ditching. The simulation is based on interaction computational methods, within LS-DYNA nonlinear finite-element code. The results of pressure distributions and accelerating time histories of the airplane＇s subfloor are discussed in the context of the safety of ditching, and the simulation results and the analytical methods are verified.

AN ALE METHOD AND DDM WITH HIGH ACCURATE COMPACT SCHEMES FOR VORTEX-INDUCED VIBRATIONS OF AN ELASTIC CIRCULAR CYLINDER

A numerical study was conducted for the vortex-induced vibrations of anelastic circular cylinder at low Reynolds numbers. An Arbitrary Lagrangian-Eulerian (ALE) method wasemployed to deal with the fluid-structure interaction with an H-O type of non-staggered gridsincorporating the domain decomposition method (DDM), which could save the computational CPU time dueto re-meshing. The computational domain was divided into nine sub-domains including one ALEsub-domain and eight Eulerian sub-domains. The convection term and dissipation term in the N-Sequations were discretized using the third-order upwind compact scheme and the fourth-order centralcompact scheme, respectively. The motion of the cylinder was modeled by a spring-damper-mass systemand solved using the Runge-Kutta method. By simulating the non-linear fluid-structure interaction,the ''lock-in'', ''beating'' and ''phase switch'' phenomena were successfully captured, and the resultsagree with experimental data Furthermore, the vortex structure, the unsteady lift and drag on thecylinder, and the cylinder displacement at various natural frequency of the cylinder for Re = 200were discussed in detail, by which a jump transition of the wake structure was captured.

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.

Prediction of shear-related defect locations in semi-solid casting using numerical flow models

Contaminated surfaces of the feedstock materials in aluminum alloy casting processes often produce various types of defects which can affect the tensile properties of the final products as well as their fatigue reliabilities.Semi-solid processing takes advantage of a much higher apparent viscosity of the die cast materials by limiting the risk of oxides formed at the free surfaces to become incorporated into the casting when the material is injected into the die.Most of existing semi-solid processes that use billets as feedstock material are however tied up with a different type of contaminated surface.During the injection phase,the external-skin on the periphery of the billet,which has been in contact with air and lubricant during the transfer in the shot sleeve,can be incorporated into the casting.When subjected to a heat treatment,the lubricant is decomposed and produces lens shape porosities.This might be a cause of reject for most structural parts.To avoid this kind of defects,the paths along which the billet skin evolves must be controlled during filling.In order to investigate the possibility of skin inclusion into cast parts during injection of the billet,a two-phase finite element mixture model is employed to model the metal flow.The formation of a skin on the periphery of the billet is modeled by setting an initial solid phase concentration profile in the radial direction.Microscopic observations of the real castings show that the approach is able to model the shear layers and to predict the paths along which the"lens porosity"defects could be formed.An Arbitrary Eulerian-Lagangian(ALE) method is also investigated and appears to be very promising to follow the skin movement in the casting.

A NEW FREE SURFACE MESH TRACING METHOD

Free surface flow problems involving large free motions are analysed using finite element techniques. In solving these problems an Arbitrary Lagrangian-Eulerian(ALE)kinematical description of the fluid domain is adopted, in which the nodal points can be displaced independently of the fluid motion. A new mesh tracing method is proposed in this paper. To confirm the effectiveness of the new method, solitary wave propagation is analysed and the numerical results are compared with the analytical results. The behaviour of the viscous fluid flow with a free surface is expressed by the unsteady Navier-Stokes equation. For numerical integration in time the velocity correction fractional step method is used.

Numerical Simulation of Macrosegregation Caused by Thermal–Solutal Convection and Solidification Shrinkage Using ALE Model

Solidifi cation shrinkage has been recognized as an important factor for macrosegregation formation. An arbitrary Lagrangian–Eulerian(ALE) model is constructed to predict the macrosegregation caused by thermal–solutal convection and solidi-fi cation shrinkage. A novel mesh update algorithm is developed to account for the domain change induced by solidifi cation shrinkage. The velocity–pressure coupling between the non-homogenous mass conservation equation and momentum equation is addressed by a modifi ed pressure correction method. The governing equations are solved by the streamline-upwind/Petrov–Galerkin-stabilized fi nite element algorithm. The application of the model to the Pb-19.2 wt%Sn alloy solidifi cation problem is considered. The inverse segregation is successfully predicted, and reasonable agreement with the literature results is obtained. Thus, the ALE model is established to be a highly effective tool for predicting the macrosegregation caused by solidifi cation shrinkage and thermal–solutal convection. Finally, the effect of solidifi cation shrinkage is analyzed. The results demonstrate that solidifi cation shrinkage delays the advance of the solidifi cation front and intensifi es the segregation.

Numerical Simulation of Glottal Flow in Interaction with Self Oscillating Vocal Folds:Comparison of Finite Element Approximationwith a Simplified Model

In this paper the numerical method for solution of an aeroelastic model describing the interactions of air flow with vocal folds is described.The flow is modelled by the incompressible Navier-Stokes equations spatially discretizedwith the aid of the stabilized finite element method.The motion of the computational domain is treated with the aid of the Arbitrary Lagrangian Eulerian method.The structure dynamics is replaced by a mechanically equivalent system with the two degrees of freedom governed by a system of ordinary differential equations and discretized in time with the aid of an implicit multistep method and strongly coupled with the flow model.The influence of inlet/outlet boundary conditions is studied and the numerical analysis is performed and compared to the related results from literature.