Lagrangian-Eulerian Finite Element Method Research Articles

Numerical analysis of a new Eulerian–Lagrangian finite element method applied to steady‐state hot rolling processes

AbstractA finite element code for steady‐state hot rolling processes of rigid–visco‐plastic materials under plane–strain conditions was developed in a mixed Eulerian–Lagrangian framework. This special set up allows for a direct calculation of the local deformations occurring at the free surfaces outside the contact region between the strip and the work roll. It further simplifies the implementation of displacement boundary conditions, such as the impenetrability condition. When applied to different practical hot rolling situations, ranging from thick slab to ultra‐thin strip rolling, the velocity–displacement based model (briefly denoted as vu‐model) in this mixed Eulerian–Lagrangian reference system proves to be a robust and efficient method.The vu‐model is validated against a solely velocity‐based model (vv‐model) and against elementary methods based on the Kármán–Siebel and Orowan differential equations. The latter methods, when calibrated, are known to be in line with experimental results for homogeneous deformation cases. For a massive deformation it is further validated against the commercial finite‐element software package Abaqus/Explicit. It is shown that the results obtained with the vu‐model are in excellent agreement with the predictions of the vv‐model and that the vu‐model is even more robust than its vv‐counterpart. Throughout the study we assumed a rigid cylindrical work roll; only for the homogeneous test case, we also investigated the effect of an elastically deformable work roll within the frame of the Jortner Green's function method.The new modelling approach combines the advantages of conventional Eulerian and Lagrangian modelling concepts and can be extended to three dimensions in a straightforward manner. Copyright © 2004 John Wiley & Sons, Ltd.

Multiphysics Simulation of Left Ventricular Filling Dynamics Using Fluid-Structure Interaction Finite Element Method

To relate the subcellular molecular events to organ level physiology in heart, we have developed a three-dimensional finite-element-based simulation program incorporating the cellular mechanisms of excitation-contraction coupling and its propagation, and simulated the fluid-structure interaction involved in the contraction and relaxation of the human left ventricle. The FitzHugh-Nagumo model and four-state model representing the cross-bridge kinetics were adopted for cellular model. Both ventricular wall and blood in the cavity were modeled by finite element mesh. An arbitrary Lagrangian Eulerian finite element method with automatic mesh updating has been formulated for large domain changes, and a strong coupling strategy has been taken. Using electrical analog of pulmonary circulation and left atrium as a preload and the windkessel model as an afterload, dynamics of ventricular filling as well as ejection was simulated. We successfully reproduced the biphasic filling flow consisting of early rapid filling and atrial contraction similar to that reported in clinical observation. Furthermore, fluid-structure analysis enabled us to analyze the wave propagation velocity of filling flow. This simulator can be a powerful tool for establishing a link between molecular abnormality and the clinical disorder at the macroscopic level.

Finite element simulation of lubricated contact in rolling using the arbitrary Lagrangian–Eulerian formulation

In this paper, the lubrication problem in numerical simulation of forming processes is considered. Two efficient convection algorithms are presented in order to update the value stored at the Gauss points during the Eulerian step of an arbitrary Lagrangian Eulerian computation in solid mechanics. They are based on the finite volume method and on the streamline upwind Petrov Galerkin method. After enumerating the difficulties encountered when trying to solve the lubrication problem with the finite element method, a generalization of the formulation introduced by Liu [Simulation of Ring Rolling Process by Arbitrary Lagrangian Eulerian Finite Element Method, vol. 20, p. 225; ASME Winter Meeting, Dallas, Tx, 1990, Int. J. Numer. Methods Engrg. 36 (1993) 855; Numerical Methods in Industrial Forming Processes, Valbonne, France, 1992, p. 731] for the thin film hydrodynamic lubrication regime is presented. This method is then applied to a strip rolling simulation, using the arbitrary Lagrangian Eulerian formalism.

A variational formulation and a double-grid method for meso-scale modeling of stressed grain growth in polycrystalline materials

A deterministic approach for meso-scale modeling of grain growth in stressed polycrystalline materials based on the principle of virtual power is presented. The variational equation is formulated based on the power balance of the system associated with grain boundary surface tension and curvature, rate of strain energy stored in each grain, strain energy density jump across the grain boundaries, and external work rate. The numerical solution of stressed grain growth variational equation requires discretization of grain interiors and grain boundaries. This cannot be effectively modeled by Lagrangian, Eulerian, or Arbitrary Lagrangian Eulerian finite element method if grain boundary migration (moving interfaces) and topological changes of grain boundary geometry are considered. This paper presents a double-grid method to resolve the above mentioned difficulty. In this approach, the material grid points carry material kinematic variables, whereas the grain boundary grid points carry grain boundary kinematic variables. The material domain is discretized by a moving least squares reproducing kernel approximation with strain discontinuity enrichment across the grain boundaries. The grain boundaries, on the other hand, are discretized by the standard finite elements. An interface enrichment function to accurately capture strain jump conditions across the grain boundaries is introduced. A reproducing kernel approximation that includes the periodicity of the unit cell in the construction of reproducing kernel shape function for material velocity is also presented. This proposed double-grid method allows modeling of arbitrary evolution of grain boundaries without remeshing.

Simulation of Fluid-Structure Interaction Using Voxel Method

A Voxel grid method combined with cut cells has been developed for simulating two-dimensional unsteady, viscous, and incompressible flows with arbitrary boundaries. The volume of fluid (VOF) approach based on finite volume method (FVM) is employed to accurately discretize the governing equation in Voxel cells that are cut by irregular boundaries. The precision of solution in the cut cells is ensured by special discretizations near the embedded boundaries. The accuracy and robustness of the proposed method are validated by simulating viscous flow around a stationary cylinder. Moreover, in order to verify the ability of this method to simulate the flow fields with arbitrarily moving boundaries, the fluid-structure interaction of bluff body in forced oscillation and vortex-induced vibration are also investigated by employing Voxel method developed in this paper, Calculation results show that lock-in phenomenon in wind engineering, in which the vortex-shedding is synchronized with the vibration of cylinder, is consistent with experimental data, the cylinder resonance state caused by vortex-induced vibration is also in reasonable agreement with results calculated by Arbitrary Lagrangian Eulerian (ALE) finite element method (FEM)

An arbitrary Lagrangian–Eulerian finite element method for finite strain plasticity

AbstractThis paper presents a new arbitrary Lagrangian–Eulerian (ALE) finite element formulation for finite strain plasticity in non‐linear solid mechanics. We consider the models of finite strain plasticity defined by the multiplicative decomposition of the deformation gradient in an elastic and a plastic part (F = FeFp), with the stresses given by a hyperelastic relation. In contrast with more classical ALE approaches based on plastic models of the hypoelastic type, the ALE formulation presented herein considers the direct interpolation of the motion of the material with respect to the reference mesh together with the motion of the spatial mesh with respect to this same reference mesh. This aspect is shown to be crucial for a simple treatment of the advection of the plastic internal variables and dynamic variables. In fact, this advection is carried out exactly through a particle tracking in the reference mesh, a calculation that can be accomplished very efficiently with the use of the connectivity graph of the fixed reference mesh. A staggered scheme defined by three steps (the smoothing, the advection and the Lagrangian steps) leads to an efficient method for the solution of the resulting equations. We present several representative numerical simulations that illustrate the performance of the newly proposed methods. Both quasi‐static and dynamic conditions are considered in these model examples. Copyright © 2003 John Wiley & Sons, Ltd.

An arbitrary Lagrangian–Eulerian finite element method for interactions of airflow and a moving AGV in a cleanroom

The motions of the airflow induced by the movement of an automatic guided vehicle (AGV) in a cleanroom are numerically studied. This subject is an important issue of microcontamination control for the semiconductor process. The characteristics of the airflow induced by the movement of the AGV are dynamic and are classified as a type of moving boundary problems. An arbitrary Lagrangian–Eulerian finite element method with moving meshes is adopted to analyze this problem. Two different moving velocities of the AGV and two different positions of the wafer cassette under Reynolds number Re=500 are considered in details. The results show that the recirculation zones, which inhibit removing microcontaminants, are observed around the AGV and wafer cassette, and the formations of recirculation zones are remarkably dependent upon the moving velocity of the AGV and the position of the wafer cassette. These phenomena are remarkably different from those of a moving AGV regarded as a stationary one.

On the mesh motion for ALE modeling of metal forming processes

Arbitrary Lagrangian–Eulerian finite element method is being increasingly used for simulation of metal-forming operations, due to its capability to combine the features of both Lagrangian and Eulerian approaches in a single analysis. An important consideration in applying this approach is an algorithm which decides upon the motion of all degrees of freedom of the finite element mesh in every step of the analysis, such that the resulted mesh retains an optimal shape and condition. Such an algorithm has to be effective, general, and computationally efficient. In this paper, an overview of the mesh motion problem is provided in which guidelines are given for designing mesh motion strategies for particular problems, and numerical schemes are proposed for moving the mesh on external boundaries and in the interior of the body consistent with the ALE principles. Some numerical examples are provided which demonstrate that with suitable mesh motion schemes, ALE formulation can be used effectively to solve problems which are otherwise difficult to solve in Lagrangian or Eulerian frameworks.

Numerical Analysis on Molten Metal in Cold Crucible Using A-φ Method

It has become increasingly important to calculate magneto-hydrodynamics (MHD) flows under alternating-current (AC) magnetic fields, in order to control the molten metal processing step in the furnace by electromagnetic force. In this paper, three dimensional problems for an arbitrary-shape model are solved by using the hybrid finite element method-boundary element method (FEM-BEM) with the A-φ (magnetic vector potential-electric scalar potential) method in the electromagnetic field and using the arbitrary Lagrangian Eulerian-finite element method (ALE-FEM) in the velocity field with a free surface. The strong point of the ALE method is that the element itself moves along the velocity field of the fluid, and consequently, the boundary between the fluid and air is distinct. Using these methods, we can observe the complicatedly tangled physical parameters of the electromagnetic field in the cold crucible and confirm the unique shape and vibration of the surface of the molten metal.

HBGC123D: a high-performance computer model of coupled hydrogeological and biogeochemical processes

Groundwater flow and transport models have been used to assist management of subsurface water resources and water quality. The needs of more efficient use of technical and financial resources have recently motivated the development of more effective remediation techniques and complex models of coupled hydrogeological and biogeochemical processes. We present a high-performance computer model of the coupled processes, HBGC123D. The model uses a hybrid Eulerian–Lagrangian finite element method to solve the solute transport equation and a Newton's method to solve the system of nonlinear, mixed kinetics and equilibrium reaction equations. Application of the model to a laboratory soil column with multispecies tracer injection suggests that one may use the model to derive important parameters of subsurface solute fate and transport. These parameters may be used for predictive purpose in similar field problems. To this end, we present a three-dimensional, hypothetical bioremediation simulation on an aquifer contaminated by CoNTA. The simulation suggests that, using oxygen alone to stimulate the biodegradation of the contaminant, one may reduce the waste to 40% in 10 years. Using a refined mesh of this three-dimensional model, we also conduct a performance study of HBGC123D on an array of SGI Origin 2000 distributed shared-memory processors. Both the computational kernels and the entire model show very good performance up to 32 processors. The CPU time is essentially reduced by 20-fold using 64 processors. This result suggests that HBGC123D may be a useful tool in assisting environmental restoration efforts such as waste site characterization and remediation.

Overall numerical simulation of extrusion blow molding process

AbstractThis paper focuses on the overall numerical simulation of the parison formation and inflation process of extrusion blow molding. The competing effects due to swell and drawdown in the parison formation process were analyzed by a Lagrangian Eulerian (LE) finite element method (FEM) using an automatic remeshing technique. The parison extruded through an annular die was modeled as an axisymmetric unsteady nonisothermal flow with free surfaces and its viscoelastic properties were described by a K‐BKZ integral constitutive equation. An unsteady die‐swell simulation was performed to predict the time course of the extrudate parison shape under the influence of gravity and the parison controller. In addition, an unsteady large deformation analysis of the parison inflation process was also carried out using a three‐dimensional membrane FEM for viscoelastic material. The inflation sequence for the parison molded into a complex‐shaped mold cavity was analyzed. The numerical results were verified using experimental data from each of the sub‐processes. The greatest advantage of the overall simulation is that the variation in the parison dimension caused by the swell and drawdown effect can be incorporated into the inflation analysis, and consequently, the accuracy of the numerical prediction can be enhanced. The overall simulation technique provides a rational means to assist the mold design and the determination of the optimal process conditions.

Simulation of the orthogonal metal cutting process using an arbitrary Lagrangian–Eulerian finite-element method

Two different finite-element formulations, the Lagrangian and the Eulerian, have been used extensively in the modeling of the orthogonal metal cutting process. Each of these formulations has some disadvantages that make it inefficient for modeling the cutting process. In this paper, it is shown that a more general formulation, the arbitrary Lagrangian–Eulerian (ALE) method may be used to combine the advantages and avoid the shortcomings of both of the previous methods. It is also shown that due to the characteristics of the cutting process, this formulation offers the most efficient modeling approach. Some preliminary results of this approach are presented to demonstrate its capabilities and potential in simulating the cutting process.

Numerical Analysis on Molten Metal in Cold Crucible using GSMAC-ALE Method.

Nowadays, it becomes more and more important to calculate magneto-hydrodynamics (MHD) flows under AC (Alternating current) magnetic fields, in order to control the molten metal processing in the furnace by the electromagnetic force. In this paper, three dimensional problems for the arbitrary shape model are solved by using the hybrid FEM-BEM (hybrid finite element method-boundary element method) with the A-〓 (magnetic vector potential-electric scalar potential) method in the electromagnetic field and using the ALE-FEM (arbitrary Lagrangian Eulerian-finite element method) in the velocity field with free surface. The strong point of the ALE method is that the element itself is moved along the velocity of the fluid, and consequently, the boundary between fluid and the air is shown precisely. Using these distinct methods in each field, we can observe the complicatedly tangled physical parameters of the electromagnetic field in the cold crucible and confirm the unique shape and vibration of the surface.

NUMERICAL SIMULATION OF BUBBLE GROWTH IN NUCLEATE BOILING - EFFECTS OF SYSTEM PARAMETER

A numerical simulation scheme was developed in this paper to analyze the bubble growth process in nucleate boiling. A single bubble was assumed to grow axisymmetrically on an isothermal wall with heat being supplied through the microlayer beneath the growing bubble and from the superheated liquid outside the bubble cap. In the simulation no assumption was made on the bubble shape during its growth. Stephan and Hammer model was used for the evaporation in the micro-region beneath the bubble. In the macro-region around the bubble cap, transient Navier-Stokes and energy equations were solved using a two-dimensional arbitrary Lagrangian-Eulerian finite element method (ALE-FEM) based on quadrilateral elements. Numerical calculation was carried out in nucleate boiling under saturated, subcooled, terrestrial- and reduced-gravity conditions. The present simulation scheme offers a useful means to consistently investigate the growth process of an isolated vapor bubble in various conditions with a minimum number of assumptions and simplifications. It enables to describe quantitatively the flow and temperature fields around the growing bubble from the inception through departure to subsequent rising in stagnant fluid.

Solving 3D Subsurface Flow and Transport with Adaptive Multigrid

This paper presents an adaptive multigrid approach, combining multigrid methods and adaptive local grid refinement, to solve 3D density-dependent flow and transport problems in the subsurface. This approach is incorporated with the Galerkin finite-element method to solve the modified Richards' equation in the flow module and the Lagrangian-Eulerian finite-element method to solve the advection-dispersion equation in the transport module, which are coupled through density effects. With multigrid methods, the linear/linearized matrix equations can be solved with O(n) manipulations to save computer time. With adaptive local grid refinement, computational accuracy is improved by refining only rough regions of the problem domain and the computation is efficiently achieved because computer efforts are focused on those regions. In this work, rough regions are determined by examining the mesh Peclet number during each nonlinear iteration of the flow module and by checking the smoothness of the Lagrangian concentra...

Modeling of 2D Density-Dependent Flow and Transport in the Subsurface

A 2-dimensional finite-element model for density-dependent flow and transport through saturated-unsaturated porous media has been developed. The combined flow and transport model can handle a wide range of real-world problems, including the simulations of flow alone, contaminant transport alone, and combined flow and transport. The conventional finite-element methods and a hybrid Lagrangian-Eulerian finite-element method were incorporated in the transport module. Saltwater intrusion problems and instability caused by denser water on the top were investigated in this paper. Because the fundamental mechanism causing saltwater intrusion most likely is caused by density-induced convection and dispersion, the developed model was used to assess the interplay between density-driven flow and the subsurface media through which the saltwater intrusion occurs. The mathematical formulation of the model is comprised of fluid flow and solute transport equations, coupled by fluid density. In the specific case of saltwater intrusion and unstable brine transport problems, this set of governing equations is nonlinear and requires iterative methods to solve them simultaneously. Three case studies, which include a wide spectrum of physical conditions, show the verification and effectiveness of the model by comparing previously published solutions from other researchers with the simulation results of the present model. Two demonstrated problems examine the model capabilities to handle saltwater intrusion problems through unsaturated-saturated porous media and density-dependent flow and transport under unstable conditions.

An arbitrary Lagrangian-Eulerian finite element method for solving three-dimensional free surface flows

This paper discusses numerical solution of unsteady three-dimensional free surface flows. The governing equilibrium equations are written in the framework of the Arbitrary Lagrangian-Eulerian kinematic description. The corresponding variational formulation is established afterwards. Since the variational problems are nonlinear with respect to the moving coordinates, a second-order approximate variational problem is derived after a consistent linearization of the referential motion. Stability of the discrete formulations is ensured with the help of a new stabilization method. A robust preconditioned GMRES algorithm is then used to solve the resulting set of nonlinear equations. Finally, the computational algorithms are assessed through numerical studies of various problems: a large sloshing flow in a three-dimensional reservoir, a discharge flow from a reservoir, simulation of a liquid vortex produced inside a cylindrical container with a disk rotating at the bottom and a three-dimensional practical hydraulic problem.

Start-up of flow of a FENE-fluid through a 4:1:4 constriction in a tube

The flow of a FENE-fluid through a 4:1:4 constriction in a tube is computed by a split Lagrangian–Eulerian finite element method. In steady flow it is found that the upstream vortex grows with increasing Deborah number, while the down-stream vortex diminishes and disappears. The steady pressure drop decreases with Deborah number unless the finite extensibility L is quite small. Starting from rest at high Deborah number, the upstream vortex grows in two stages, each with their own time scales. A simple model of this growth is proposed.

Localized metal forming simulation by r-s-adapted arbitrary Lagrangian-Eulerian finite element method

In this paper, an adaptive arbitrary Lagrangian—Eulerian (ALE) large deformation finite element method (FEM) is developed for solving metal forming problems with strain localization. The ALE mesh movement is coupled with r-adaptation of automatic node relocation to minimize mesh distortion during the process of deformation. A strain localization phenomenon is incorporated through constitutive relations for porous ductile materials. Prediction of localized deformation is achieved through a multilevel mesh superimposition method, called s-adaptation. A few metal forming problems are simulated to test the effectiveness of this model.

3-D Sloshing Analysis by an Arbitrary Lagrangian-Eulerian Finite Element Method

This paper presents a new Arbitrary Lagrangiae-Eulerian (ALE) finite element method for the analysis of three-dimensional large-amplitude sloshing problems. The present method is composed of three phases, i.e., Phase 1: Purely Lagrangian calculation. Phase 2: Remeshing procedure and Phase 3: Rezoning procedure. By applying these procedures, the present method has succeeded in overcoming the main difficulty of computation in the large-amplitude sloshing analysis. In the Lagrengian calculation, a new fractional step method that satisfies the free-surface boundary condition has been applied, and treatment of the free surface on the multi-sloped wall boundary has been advanced. In the remeshing procedure, a method based on boundary-fitted coordinates is introduced. Numerical results are compared with experimental results, and it is shown that this method is sufficiently accurate to calculate the three-dimensional sloshing. Furthermore, a numerical example of large-amplitude sloshing in a tank with roofs and chamfers illustrates the validity of the method for the computation of large-amplitude sloshing waves whose free surface configuration is a double-valued function.