Arbitrary Lagrangian Eulerian Formulation Research Articles

Continuous roll forming including in-line welding and post-cut within an ALE formalism

Cold roll forming is a rather old process for which there is a renewed interest due to its capacity to form ultra high-strength steels. For the first time ever in the literature, the manufacturing chain involving both the continuous cold roll-forming process, the in-line welding operation for closed sections and the post-cut operation is numerically modelled. The first phase of this process sequence consists in computing the hopefully steady state configuration of the strip for the total length of the roll-forming mill, including the in-line welding phase. For addressing this problem, the Arbitrary Lagrangian Eulerian (ALE) formalism is used. Once the ALE steady state is reached, the computation is pursued with a second Lagrangian phase which is aimed at simulating the post-cut operation that releases the formed section from the rolling tools of the mill, enabling to determine the final geometry of the product. In this paper, the computational modelling framework employed within the in-house finite element code METAFOR is described. In particular, the proposed techniques — which are definitely original within an ALE formalism — for modelling the in-line welding operation and the post-cut operation are extensively detailed. The welding is considered with three different methods: (1) symmetry boundary conditions coupled with a well-suited node relocation procedure, (2) a closed mesh of the closed section coupled with a well-suited node relocation procedure, and (3) sticking elements based on a spring constitutive formulation. A set of simple numerical examples demonstrates the confidence in all the proposed modelling methods. Finally, these methods are successfully applied in the cases of two complex roll-forming mills of closed tubular sections.

An Arbitrary Lagrangian–Eulerian Formulation for Modelling Cavitation in the Elastohydrodynamic Lubrication of Line Contacts

In this article an arbitrary Lagrangian–Eulerian (ALE) formulation for modelling cavitation in elastohydrodynamic lubrication (EHL) is derived and applied to line contact geometry. The method is developed in order to locate the position of cavitation onset along the length of the contacting region which gives the transition from liquid to vapour in the fluid. The ALE is implemented by introducing a spatial frame of reference in which the solution is required and a material frame of reference in which the governing equations are solved. The spatial frame is moved from the material frame according to the error in the Neumann pressure gradient constraint required at the cavitation location when Dirichlet constraints are imposed for pressure in the liquid phase. Results are calculated under both steady-state and transient operating conditions using a multigrid solver. The solutions obtained are compared to established literature and conventional approaches to modelling cavitation which show that the ALE formulation is an alternative, straightforward and accurate means of implementing such conditions in EHL. This is achieved without the penalties associated with the numerical modelling of Heaviside functions or free boundaries.

An ALE formulation for compressible flows based on multi-moment finite volume method

ABSTRACTA direct Arbitrary Lagrangian Eulerian (ALE) formulation using multi-moment finite volume method for viscous compressible flows on unstructured moving grids has been developed and presented in this paper. High-order polynomials are reconstructed over a compact mesh stencil by making use of both volume integrated average value (VIA) and point values (PV) at the vertices of mesh cells which change in time. By formulating the governing equation in ALE integral form and differential form, the computational variables, VIA and PV, are updated, respectively, by a finite volume scheme and a point-wise discretization using multi-moments (VIA and PV). A simple and efficient formulation is derived for moving mesh which satisfies the geometrical conservation law. In the computations involving moving-body, the PVs of velocity at the vertices of cells adjacent to the body surface are coincided with the motion of the body, which eliminates the numerical approximation to find the cell boundary values on the body surface in conventional finite volume method. Numerical tests are presented to verify the present method as a high-order ALE formulation for compressible flows on both stationary and moving grids.

Finite Element Simulation and Experimental Validation of Laser Assisted Machining of Inconel 718

Laser assisted machining (LAM) is a promising technology for machining of difficult to cut materials with reduced cutting forces, improved surface finish and tool life. This paper is mainly focused on study of cutting forces during laser assisted machining of Inconel 718 superalloy using finite element (FE) approach. An iterative procedure was adopted in this work to determine the coefficient of friction at tool chip interface to obtain rational values of cutting force for the identified cutting range. Arbitrary Lagrangian Eulerian (ALE) formulation was included to simulate the chip formation process. Johnson-cook material model for flow stress behaviour of workpiece material was incorporated during simulation. To validate the FE model, LAM experimentations were done under orthogonal cutting condition. The experiments were conducted according to Taguchi’s design of experiments. The cutting force results obtained through experimentation and simulation were compared and presented here. The FE prediction of cutting forces agrees with the experimental conditions for most of the cases within 15% deviation.

3D Numerical Simulation of Seamless Pipe Piercing Process by Fluid-Structure Interaction Method

Seamless pipes are produced using piercing rolling process in which round bars are fed between two rolls and pierced by stationary plug. During this process, the material undergoes severe deformation which renders it impractical to perform the numerical simulations with conventional finite element methods. In this paper, three dimensional numerical simulations of the piercing process are performed with Fluid-Structure Interaction (FSI) Method using Arbitrary Lagrangian-Eulerian (ALE) Formulation with LS DYNA software. The results of numerical simulations agree with experimental data of Plasticine workpiece and the validity of the analysis method is confirmed.

Numerical simulation of fluid flow through simplified blade cascade with prescribed harmonic motion using discontinuous Galerkin method

This paper deals with a numerical simulation of compressible viscous fluid flow around three flat plates with prescribed harmonic motion. This arrangement presents a simplified blade cascade with forward wave motion. The aim of this simulation is to determine the aerodynamic forces acting on the flat plates. The mathematical model describing this problem is formed by Favre-averaged system of Navier-Stokes equations in arbitrary Lagrangian-Eulerian (ALE) formulation completed by one-equation Spalart-Allmaras turbulence model. The simulation was performed using the developed in-house CFD software based on discontinuous Galerkin method, which offers high order of accuracy.

Numerical simulation of fluid flow through simplified blade cascade with prescribed harmonic motion using discontinuous Galerkin method

This paper deals with a numerical simulation of compressible viscous fluid flow around three flat plates with prescribed harmonic motion. This arrangement presents a simplified blade cascade with forward wave motion. The aim of this simulation is to determine the aerodynamic forces acting on the flat plates. The mathematical model describing this problem is formed by Favre-averaged system of Navier-Stokes equations in arbitrary Lagrangian-Eulerian (ALE) formulation completed by one-equation Spalart-Allmaras turbulence model. The simulation was performed using the developed in-house CFD software based on discontinuous Galerkin method, which offers high order of accuracy.

On the solution of the problem of a drop falling against a plane by using a level set – moving mesh – immersed boundary method

A coupled Conservative Level Set – Moving Mesh – Immersed Boundary method is formulated and validated against the three-dimensional gravity-driven falling drop problem. First, by employing Conservative Level-Set (CLS) method, the multiphase domain can be successfully handled, while the mass conservation is controlled. Then, by using an Arbitrary Lagrangian-Eulerian formulation (i.e. a moving mesh), the simulation domain can be optimized by reducing the domain size and by allowing an improved mesh, resulting in a computational resources saving. Finally, the use of an Immersed Boundary (IB) method allows to deal with intricate geometries. All these functionalities result in a versatile and robustness method to simulate bubbles/drops problems in complex geometries. The mentioned method was successfully used to thoroughly study the falling of a drop against a plane surface, providing detailed results including velocity evolution, mesh independence study, evolution of the vertical position of the drop, streamlines and vorticity fields, and profiles evolution.

A consistent viscoelastic formulation for the numerical analysis of steady state rolling tires

The analysis of rolling rubber wheels under stationary conditions is commonly done using an Arbitrary Lagrangian-Eulerian formulation, especially suitable for finite element analysis under consideration of elastic or hyperelastic materials. While the reference frame is neither attached to the material, nor fixed in space, all equations can be expressed independent of time. However, this leads to the challenging task of taking internal variables into account for inelastic materials. In this contribution, a coupled approach for the analysis of steady state rolling bodies with inelastic properties is presented. A subsequent viscoelastic formulation takes into account the inelastic contribution of all material points along a streamline. Furthermore, coupling terms are obtained, which have to be considered in the system of equations. The reliability and application of this formulation to tire analysis is discussed and important remarks close this presentation.

Formula omitted]adaptation for Shallow Water flows: conservation, well balancedness, efficiency

We investigate the potential of the so-called “relocation” mesh adaptation in terms of resolution and efficiency for the simulation of free surface flows in the near shore region. Our work is developed in three main steps. First, we consider several Arbitrary Lagrangian Eulerian (ALE) formulations of the shallow water equations on moving grids, and provide discrete analogue in the Finite Volume and Residual Distribution framework. The compliance to all the physical constraints, often in competition, is taken into account. We consider different formulations allowing to combine volume conservation (DGCL) and equilibrium (Well-Balancedness), and we clarify the relations between the so-called pre-balanced form of the equations [7]), and the classical upwinding of the bathymetry term gradients [1]. Moreover, we propose a simple remap of the bathymetry based on high accurate quadrature on the moving mesh which, while preserving an accurate representation of the initial data, also allows to retain mass conservation within an arbitrary accuracy. Second, the coupling of the resulting schemes with a mesh partial differential equation is studied. Since the flow solver is based on genuinely explicit time stepping, we investigate the efficiency of three coupling strategies in terms of cost overhead w.r.t. the flow solver. We analyze the role of the solution remap necessary to evaluate the error monitor controlling the adaptation, and propose simplified formulations allowing a reduction in computational cost. The resulting ALE algorithm is compared with the rezoning Eulerian approach with interpolation proposed e.g. in Tang and Tang [17]. An alternative cost effective Eulerian approach, still allowing a full decoupling between adaptation and flow evolution steps is also proposed. Finally, a thorough numerical evaluation of the methods discussed is performed. Numerical results on propagation, and inundation problems shows that the best compromise between accuracy and CPU time is provided by a full ALE formulation. If a loose coupling with the mesh adaptation is sought, then the cheaper Eulerian approach proposed is shown to provide results quite close to the ALE.

High-order non-inertial computation method for helicopter rotor flows in forward flight

Purpose This paper aims to improve the computational efficiency and to achieve high-order accuracy for the computation of helicopter rotor unsteady flows in forward flight during the industrial preliminary design stage. Design/methodology/approach The integral arbitrary Lagrangian–Eulerian form of unsteady compressible Navier–Stokes equations with low Mach number preconditioned pseudo time terms based on non-inertial frame of reference undergoing rotating and translating was derived and discretized in the framework of multi-block structured finite volume grid using three types of spatial reconstruction schemes, i.e. the third-order accurate monotonic upwind scheme for conservation laws, the fifth-order accurate weighted essentially non-oscillatory and the fifth-order accurate weighted compact nonlinear schemes. Findings The results show that the present non-inertial computational method can obtain comparable results with other methods, such as the dynamic overset method, and make sure that the higher-order spatial schemes can significantly improve the tip vortex resolution. Originality/value The computational grid used by the present method remained static during the whole unsteady computation process, with only local deformations induced by blade cyclic pitch and other operating motions, which greatly reduced the complexity of grid motion and enhanced the efficiency and robustness.

A comprehensive analysis of solidity for cambered darrieus VAWTs

Wind energy technology has seen steady growth in the energy market as a clean, renewable source of energy. This has brought attention to areas with moderate wind energy potentials. Darrieus type Vertical Axis Wind Turbines (VAWTs) allow capturing of this potential for energy production at a cost-effective scale. To improve the performance of these turbines their design needs to be optimized. With better manufacturing methods currently available cambered blades are being investigated to improve the performance of these turbines. In particular, turbine solidity is investigated in this paper due to different airfoil chord lengths and different number of blades. The analysis is conducted following high fidelity CFD modeling in unsteady, turbulent regimes in Arbitrary Lagrangian Eulerian formulation accounting for the sliding/rotating rotor's domain configuration that emphasizes the role of rotor blades interaction. The study showed that low solidity turbines with cambered blades operate at low coefficients of performance (CP) over a large range of tip speed ratios (TSRs). High solidity turbines, with solidity close to unity, have much higher CP, but at smaller TSRs and a short range of TSRs. Medium solidity turbines, as well as turbines with solidity greater than unity, are faced with considerable interaction which compromises their operation at TSRs as low as one. The larger chord length helps improve the torque production of the turbine. The change in number of blades leads to a large dispersed wake behind the rotor which is unfavorable for deployment of an array of turbines.

Initiation and evolution of debonding phenomena in layered structures

A new methodology based on a moving mesh technique and a multilayer formulation is proposed with the purpose to predict crack onset, evolution and coalescence of the interlaminar damage mechanisms. In particular, a numerical approach based on Arbitrary Lagrangian Eulerian (ALE) formulation combined with shear-deformable beam elements is implemented in a numerical procedure based on the finite element method. The moving mesh technique combined with a multilayer formulation ensures a reduction of the computational costs, required to predict crack onset and subsequent evolution of the debonding phenomena. The analysis is performed with respect to several onset configurations, including the case in which multiple debonding mechanisms with coalescence affects the interfaces. The proposed approach is quite robust to be implemented also in dynamics, in which the process zone is known to advance at high speed ranges. In order to verify the proposed modeling, a sensitivity study in terms of convergence and stability of the solution is proposed. Moreover, comparisons with existing results available from the literature are developed with the purpose to verify the reliability and the consistency of the proposed formulation.

Dynamic analysis of continuous-flow microwave reactor with a screw propeller

To solve nonuniform heating when microwave energy is applied in a continuous-flow applicator, a microwave heating system with a screw propeller is designed. By introducing the implicit function and level set methods, the electromagnetic heat source input into the temperature field is updated with the rotation of the screw propeller. Combined with Arbitrary Lagrangian-Eulerian Formulation (ALE) machinery, the temperature distribution in the heating process is obtained. Water is taken as the sample in the experiment to verify the computational model. By using the proposed model, we also calculate the temperature distribution of two other commonly used liquids (ethylene glycol and glycerol). To analyze the effect of a screw propeller on heating uniformity, the temperature distribution of the cases with a screw propeller is quantitatively compared to that of the cases without a screw propeller.

10-MW Wind Turbine Performance Under Pitching and Yawing Motion

The possibility of a wind turbine entering vortex ring state (VRS) during pitching oscillations is explored in this paper. The work first validated the employed computational fluid dynamics (CFD) method, and continued with computations at fixed yaw of the NREL phase VI wind turbine. The aerodynamic performance of the rotor was computed using the helicopter multiblock (HMB) flow solver. This code solves the Navier–Stokes equations in integral form using the arbitrary Lagrangian–Eulerian formulation for time-dependent domains with moving boundaries. With confidence on the established method, yawing and pitching oscillations were performed suggesting partial vortex ring state during pitching motion. The results also show the strong effect of the frequency and amplitude of oscillations on the wind turbine performance.

A parallel monolithic approach for fluid-structure interaction in a cerebral aneurysm

Abstract A parallel fully-coupled numerical algorithm has been developed for the fluid-structure interaction problem in a cerebral artery with aneurysm. For the fluid part of the problem, an Arbitrary Lagrangian-Eulerian formulation based on the side-centered unstructured finite volume method is employed for the governing incompressible Navier–Stokes equations. The deformation of the solid domain is governed by the constitutive laws for the nonlinear Saint Venant-Kirchhoff material and the classical Galerkin finite element method is used to discretise the governing equations in a Lagrangian frame. The time integration method for the structure domain is based on the Newmark type generalized − α method while the second-order backward difference (BDF2) is used in the fluid domain. A special attention is given to construct an algorithm obeying the local/global discrete geometric conservation laws (DGCL) in order to conserve fluid volume at machine precision when the fluid domain is entirely enclosed by solid domain boundary. Therefore, a compatible kinematic boundary condition is applied at the interface between the solid and fluid domains. The parallel implementation of the present fully coupled unstructured fluid-structure solver is based on the PETSc library and a one-level restricted additive Schwarz preconditioner with a block-incomplete factorization within each partitioned sub-domains is utilized for the resulting fully coupled system. The proposed algorithm is initially validated for a pressure pulse propagating in a flexible tube and the mass conservation accuracy is tested for a thin elastic sphere filled with an incompressible fluid in a circular tube. Then the numerical method is applied to a complicated problem involving unsteady pulsatile blood flow in a cerebral artery with aneurysm as a realistic fluid-structure interaction problem encountered in biomechanics. Various hemodynamic quantities of interest like fluid velocities, blood pressure and wall shear stresses (WSS) are computed as well as the time dependent artery wall deformations.

Lagrangian and ALE Formulations For Soil Structure Coupling with Explosive Detonation

Simulation of Soil-Structure Interaction becomes more and more the focus of computational engineering in civil and mechanical engineering, where FEM (Finite element Methods) for structural and soil mechanics and Finite Volume for CFD are dominant. New formulations have been developed for FSI applications using ALE (Arbitrary Lagrangian Eulerian) and mesh free methods as SPH method, (Smooth Particle Hydrodynamic). In defence industry, engineers have been developing protection systems for many years to reduce the vulnerability of light armoured vehicles (LAV) against mine blast using classical Lagrangian FEM methods. To improve simulations and assist in the development of these protections, experimental tests, and new numerical techniques are performed. To carry out these numerical calculations, initial conditions such as the loading prescribed by a mine on a structure need to be simulated adequately. The effects of blast on structures depend often on how these initial conditions are estimated and applied. In this report, two methods were used to simulate a mine blast: the classical Lagrangian and the ALE formulations. The comparative study was done for a simple and a more complex target. Particle methods as SPH method can also be used for soil structure interaction.

Finite element approximation of flow induced vibrations of human vocal folds model: Effects of inflow boundary conditions and the length of subglottal and supraglottal channel on phonation onset

The paper presents the numerical analysis of interaction of the vibrating simplified human vocal folds model with the incompressible viscous airflow in a channel modeling simplified subglottal and supraglottal spaces. The flow in the considered 2D computational fluid domain is governed by the Navier–Stokes equations written in the Arbitrary Lagrangian–Eulerian form. The stabilized finite element method is applied for numerical approximation and the choice of boundary conditions and their implementation is discussed. For the considered model problem the prescribed inlet velocity and prescribed pressure difference formulations were numerically analyzed. The prescribed inlet velocity formulation was successful in predicting of the flutter velocity value, whereas the prescribed pressure difference gave nonphysical results. Finally a modified inlet boundary condition motivated by the penalization approach is suggested. It is shown that this approach gives possibilities to optimize the inlet boundary condition related to a physical reality by changing smoothly the penalty parameter in the interval between the two extremes and to treat the complete closures of the channel.

Numerical study of Taylor bubbles rising in a stagnant liquid using a level-set/moving-mesh method

An Arbitrary Lagrangian-Eulerian formulation has been posed to solve the challenging problem of the three-dimensional Taylor bubble, within a Conservative Level Set (CLS) framework. By employing a domain optimization method (i.e. the moving mesh method), smaller domains can be used to simulate rising bubbles, thus saving computational resources. As the method requires the use of open boundaries, a careful treatment of both inflow and outflow boundary conditions has been carried out. The coupled CLS - moving mesh method has been verified by means of extensive numerical tests. The challenging problem of the full three-dimensional Taylor bubble has then been thoroughly addressed, providing a detailed description of its features. The study also includes a sensitivity analyses with respect to the initial shape of the bubble, the initial volume of the bubble, the flow regime and the inclination of the channel.

Investigation of Blood Flow Modeling in Artery Using ALE Formulation

The aim of the paper is to use Arbitrary Lagrangian Eulerian (ALE) formulation for fluid–structure interaction for modeling blood flow in artery. Predicting blood flow and its effects on arteries requires simulation of fluid–structure coupling with deformable mesh. For fluid simulation velocity–pressure formulation is used, we present the algorithm which allows to compute fluid velocity and pressure using explicit time integration. This method has been applied successfully for several applications including sloshing tank analysis. For the structure shell type elements with five points integration through the thickness to accurately represent bending effects, are modeled. Since the structure is deformable, to prevent high mesh distortion an elasticity material model for the mesh is used for mesh deformation. For fluid–structure coupling, explicit contact algorithm based on penalty method. Such a model can be used to study the profile of the flow and pressure waves as they propagate along the arteries. In the paper, the onset of a pressure pulse was simulated at the entrance of a three dimension straight artery blood vessel and the resulting dynamic response in the form of a propagating pulse wave through the wall was analyzed and compared. Good agreement was found between the numerical results and the theoretical description of an idealized artery. Work has also been done on implementing the material constitutive models specific for vascular applications.