ALE Formulation Research Articles

Remapping Method for Transferring Data between Two Meshes Using a Modified Iterative SPR Approach for Parallel Resolution

In metal forming simulations, the transfer of data from one mesh to another can be very often required such as, in ALE formulation at each time step or in Lagrangian formulation at each remeshing step. Its accuracy is one of the main concerns for researchers, since the accumulation of generated diffusion can lead to larger numerical error and create convergence problems. Since 1992, the Super convergent Patch Recovery method (SPR) introduced by Zienkiewicz & Zhu [1], was a major breakthrough for the methods to estimate errors in FE solution. Later, it has also been used to recover nodal fields from integration points, in order to transfer data between two meshes [10]. The original method raises some difficulties to treat the domain boundaries. They have not quite properly been handled, in particular in the frame of parallel implementation, despite the fact that, surface phenomena, such as contact and friction, play such an important role in metal forming applications. In the present paper, it is presented a modified iterative SPR method which deals with boundary points with the same order of accuracy as the interior points. It does not require increasing the patch size and it is easier to implement in parallel environment. Also, when interpolating the field on new mesh, a new and consistent technique (P0+ transport) of enriched field has been used. It involves building a P1+ field by mixing the recovered SPR nodal field with the known field at integration points. Results are presented in form of convergence rate and L2 error norm, for several analytical functions for a SPR based transfer operator against a simple volumetric based transfer operator, before being applied to an actual metal forming problem. All computations presented here have been done by using four processors in parallel environment.

High-order numerical simulation of fluid-structure interaction in the human larynx

We present a 2D model of the human larynx able to reproduce the self–sustained oscillating interaction between the airflow and the vocal folds. We describe the interaction of the compressible flow and the elastic structure in an ALE formulation. Our solver uses globally fourth–order accurate Summation By Parts (SBP) finite difference operators in space and a fourth–order explicit method in time. We derive Simultaneous Approximation Term (SAT) expressions that impose the velocity and traction boundary conditions weakly. We present results from simulations with realistic parameters for human phonation.

Arbitrary Lagrangian Eulerian formulation for soil structure interaction problems

Simulation of soil structure interaction problems becomes more and more the focus of computational engineering in the last decades in the civil engineering community. In many applications in civil engineering, the Lagrangian formulation is the main approach for soil structure interaction problems. For the last decades, various approaches have been investigated in fluid and structure mechanics, to solve fluid structure interaction problems, including Arbitrary Eulerian Lagrangian (ALE) formulation and meshless methods as Smoothing Particle Hydrodynamics (SPH). These two approaches are more common in fluid mechanics than structure and civil engineering, where the classical Lagrangian formulation is still dominant for numerical simulations. The Lagrangian formulation, where the mesh moves with material is mainly used to solve problems in solid mechanics, and civil engineering for soil structure interaction problems. For small deformation, Lagrangian formulation can solve soil structure interface and material boundary accurately; the main limitation of the formulation is high mesh distortion for large deformation and moving structure. One of the commonly used approaches to solve these problems is the ALE formulation which has been used with success in the simulation of fluid structure interaction with large structure motion such as sloshing fuel tank in automotive industry and bird impact in aeronautic industry. In this paper, devoted to ALE formulation for civil engineering problems, the mathematical and numerical implementation of the ALE formulation and the contact algorithm are described. In order to validate the ALE algorithms, and prove its ability for civil engineering applications, we consider two different applications. First we perform the simulation of a shock wave propagation generated by a detonation of an explosive. In the second application, we consider a soil structure coupling problem, using an ALE formulation for the soil which undergoes large mesh deformation, and a Lagrangian formulation for the structure. A contact algorithm is performed at the soil structure interface.

Fluid Structure Interaction for Hydraulic Problems

Fluid Structure interaction plays an important role in engineering applications. Physical phenomena such as flow induced vibration in nuclear industry, fuel sloshing tank in automotive industry or rotor stator interaction in turbo machinery, can lead to structure deformation and sometimes to failure.In order to solve fluid structure interaction problems, the majority of numerical tests consists in using two different codes to separately solve pressure of the fluid and structural displacements. In this paper, a unique code with an ALE formulation approach is used to implicitly calculate the pressure of an incompressible fluid applied to the structure. The development of the ALE method as well as the coupling in a computational structural dynamic code, allows to solve more large industrial problems related to fluid structure coupling.

Fluid-structure interaction with an application to a body immersed and anchored in a fluid flow

In this paper, an implicit coupling algorithm for fluid–structure interaction problems with under-time steps for the solid is presented. Its implementation on two configurations is achieved by using the CASTEM finite-elements code. First, the free oscillations of a cylinder in an annular fluid domain where its movement is determined by the coupled fluid–solid action is considered in the case of viscous fluid. It should be noted that the implicit coupling algorithm gives the best prediction of the structure oscillations. The under-time steps for the solid are introduced in order to obtain better results. Then, an application whose final objective is to model a floating barrage is studied. The main goal of this application is to predict the displacements of a ring completely immersed and anchored by a cable to the lower boundary of the fluid domain. The finite-element discretization of the Navier–Stokes equations in the ALE formulation is used

Modelling the structural response of GLARE panels to blast load

This paper deals with the structural response of fully-clamped quadrangular GLARE panels subjected to an intense air-blast load using the commercial finite element software, LS-DYNA. A cohesive tie-break algorithm is implemented to model interfacial debonding between adjacent plies. The blast loads was simulated using a ConWep blast algorithm and a multi-material ALE formulation with fluid–structure interaction to determine the performance of each method. Numerical model validation have been performed considering case studies of GLARE panels subjected to spherical explosive charges of C-4, for which experimental data on the back face-displacement and post-damage observations were available. Excellent agreement of mid-point deflections and evidence of severe yield line deformation were presented and discussed against the performed blast tests.

An ALE mesh movement scheme for long‐term in‐flight ice accretion

SUMMARYThe rather irregular shapes that glaze ice may grow into while accreting over the surface of an aircraft represent a major difficulty in the numerical simulation of long periods of in‐flight icing. There is a constant need for remeshing: a wasteful procedure. In the framework of ALE formulations, a mesh movement scheme is presented, in which frame and elasticity analogies are loosely coupled. The resulting deformed mesh preserves the quality of elements, especially in the near‐wall region, where accurate prediction of heat flux and shear stresses is required. The proposed scheme handles mesh deformation in a computationally efficient manner by localizing the mesh deformation. The 2D problem of ice accretion over single and multi‐element airfoils is considered here as a numerical experiment. Experimentally measured glaze ice shapes were used to evaluate the performance of the present approach. Copyright © 2011 John Wiley & Sons, Ltd.

Numerical modelling of partially filled aircraft fuel tanks submitted to Hydrodynamic Ram

Hydrodynamic Ram (HRAM) is a phenomenon that occurs when a high-kinetic energy object penetrates a fluid-filled container. The projectile transfers its momentum and kinetic energy through the fluid to the surrounding structure, increasing the risk of catastrophic failure and excessive structural damage. This is of particular concern in the design of wing fuel tanks for aircraft since it has been identified as one of the important factors in aircraft vulnerability. Usually the HRAM phenomenon is analyzed considering completely filled tanks, but its effect on partially filled containers should also be taken into account due to the fact that tanks use to be impacted under these conditions. In the present paper, the commercial finite element code LS-DYNA has been used to simulate an HRAM event created by a steel spherical projectile impacting a partially water-filled aluminium square tube. The ALE formulation is employed to reproduce the event. Experimental tests which indicate the pressure at different points of the fluid, displacement of the walls and cavity evolution for different impact velocities, are compared with the numerical results in order to assess the validity and accuracy of the ALE technique in reproducing such a complex phenomenon.

On the potential of computational methods and numerical simulation in ice mechanics

This paper deals with the challenge of developing better methods and tools for analysing interaction between sea ice and structures and, in particular, to be able to calculate ice loads on these structures. Ice loads have traditionally been estimated using empirical data and "engineering judgment". However, it is believed that computational mechanics and advanced computer simulations of ice-structure interaction can play an important role in developing safer and more efficient structures, especially for irregular structural configurations. The paper explains the complexity of ice as a material in computational mechanics terms. Some key words here are large displacements and deformations, multi-body contact mechanics, instabilities, multi-phase materials, inelasticity, time dependency and creep, thermal effects, fracture and crushing, and multi-scale effects. The paper points towards the use of advanced methods like ALE formulations, mesh-less methods, particle methods, XFEM, and multi-domain formulations in order to deal with these challenges. Some examples involving numerical simulation of interaction and loads between level sea ice and offshore structures are presented. It is concluded that computational mechanics may prove to become a very useful tool for analysing structures in ice; however, much research is still needed to achieve satisfactory reliability and versatility of these methods.

Viscoelasticity-induced migration of a rigid sphere in confined shear flow

Suspensions of solid particles in liquids are often made to flow in devices with characteristic dimensions comparable to that of the suspended particles, the so-called confined situation, as in the case of several microfluidic applications. Combination of confinement with viscoelasticity of the suspending liquid can lead to peculiar effects. In this paper we present the first 3D simulation of the dynamics of a particle suspended in a viscoelastic liquid under imposed confined shear flow. The full system of equations is solved through the finite element method. A DEVSS/SUPG formulation with a log-representation of the conformation tensor is implemented, assuring stable and convergent results up to high flow rates. Particle motion is handled through an ALE formulation. To optimize the computational effort and to reduce the remeshing and projection steps required when the mesh becomes too distorted, a rigid motion of the grid in the flow direction is performed, so that, in fact, the particle moves along the cross-streamline direction only. Confinement and viscoelasticity are found to induce particle migration, i.e., transverse motion across the main flow direction, towards the closest wall. Under continuous shearing, three different dynamical regimes are recognized, related to the particle-wall distance. A simple heuristic argument is given to link the cross-flow migration to normal stresses in the suspending liquid. The analysis is then extended to a time-dependent shear flow imposed by periodically inverting the direction of wall motion. A slower migration is found for higher forcing frequency. A peculiar effect arises if the inversion period is chosen close to the fluid relaxation time: the migration velocity oscillates around zero, and the overall migration is suppressed. Such novel prediction of a dynamic instability scenario, with the particle escaping the center plane of the channel, and many features of the computed results, are in nice agreement with recent experiments reported in the literature [14].

Numerical Simulation of Tower Rotor Interaction for Downwind Wind Turbine

Downwind wind turbines have lower upwind rotor misalignment, and thus lower turning moment and self‐steered advantage over the upwind configuration. In this paper, numerical simulation to the downwind turbine is conducted to investigate the interaction between the tower and the blade during the intrinsic passage of the rotor in the wake of the tower. The moving rotor has been accounted for via ALE formulation of the incompressible, unsteady, turbulent Navier‐Stokes equations. The localized CP, CL, and CD are computed and compared to undisturbed flow evaluated by Panel method. The time history of the CP, aerodynamic forces (CL and CD), as well as moments were evaluated for three cross‐sectional tower; asymmetrical airfoil (NACA0012) having four times the rotor′s chord length, and two circular cross‐sections having four and two chords lengths of the rotor′s chord. 5%, 17%, and 57% reductions of the aerodynamic lift forces during the blade passage in the wake of the symmetrical airfoil tower, small circular cross‐section tower and large circular cross‐section tower were observed, respectively. The pronounced reduction, however, is confined to a short time/distance of three rotor chords. A net forward impulsive force is also observed on the tower due to the high speed rotor motion.

Computational and experimental investigation of flow over a transient pitching hydrofoil

The present study is developed within the framework of marine structure design operating in transient regimes. It deals with an experimental and numerical investigation of the time–space distribution of the wall-pressure field on a NACA66 hydrofoil undergoing a transient up-and-down pitching motion from 0° to 15° at four pitching velocities and a Reynolds number Re = 0.75 × 10 6. The experimental investigation is performed using an array of wall-pressure transducers located on the suction side and by means of time–frequency analysis and Empirical Modal Decomposition method. The numerical study is conducted for the same flow conditions. It is based on a 2D RANS code including mesh reconstruction and an ALE formulation in order to take into account the foil rotation and the tunnel walls. Due to the moderate Reynolds number, a laminar to turbulent transition model was also activated. For the operating flow conditions of the study, experimental and numerical flow analysis revealed that the flow experiences complex boundary layer events as leading-edge laminar separation bubble, laminar to turbulent transition, trailing-edge separation and flow detachment at stall. Although the flow is relatively complex, the calculated wall pressure shows a quite good agreement with the experiment provided that the mesh resolution and the temporal discretization are carefully selected depending on the pitching velocity. It is particularly shown that the general trend of the wall pressure (low frequency) is rather well predicted for the four pitching velocities with for instance a net inflection of the wall pressure when transition occurs. The inflection zone is reduced as the pitching velocity increases and tends to disappear for the highest pitching velocity. Conversely, high frequency wall-pressure fluctuations observed experimentally are not captured by the RANS model. Based on the good agreement with experiment, the model is then used to investigate the effects of the pitching velocity on boundary layer events and on hydrodynamic loadings. It is shown that increasing the pitching velocity tends to delay the laminar-to-turbulence transition and even to suppress it for the highest pitching velocity during the pitch-up motion. It induces also an increase of the stall angle (compared to quasi-static one) and an increase of the hysteresis effect during pitch-down motion resulting to a significant increase of the hydrodynamic loading.

Prediction of mechanistic cutting force coefficients using ALE formulation

This paper demonstrated the use of an efficient and accurate numerical tool (i.e., FEA) in simulating the cutting process and determining both the average and instantaneous cutting force coefficients. The main advantage of this approach compared to other available methods is that it eliminates the need for experimental calibrations. In this approach, an Arbitrary Lagrangian Formulation was employed in the finite element method simulations. This formulation has been gaining more recognition in structural analysis for its combined advantages of both Lagrangian and Eulerian formulations in a single model. Based on the work of Kline et al. (ASME J Eng Ind 104:272–278, 10), the tool is discretised along the axis into segments and the cutting forces acting on the cutting edge segment are presented in terms of cutting force coefficients. Cutting force coefficients are obtained using the least squares method and cutting force predictions using evaluated coefficients are shown to match experimental results with satisfactory accuracy.

Numerical simulation of a film coating flow at low capillary numbers

The drag-out problem in film coating has long been the subject of academic study and the question is of great interest for industrial processes. Landau’s theoretical analysis [Landau L, Levich B. Dragging of a liquid by moving a plate. Acta Physicochim URSS 1942;17:42–54] provides a formula which permits the prediction of the final film thickness but only for a steady flow at small capillary numbers Ca . Unfortunately, certain more recent experimental results [Kizito J, Kamotani Y, Ostrach S. Experimental free coating flows at high capillary and Reynolds number. Exp Fluids 1999;27:235–43] have shown the formula cannot be used to predict the mean value of the thickness for several regimes at low capillary numbers. The aim of this paper is to develop a reliable numerical code to correctly predict the hydrodynamics field for this configuration. This numerical simulation requires an unsteady Navier–Stokes code using the ALE formulation and a semi-implicit front tracking method for the moving free surface. The code, developed here, uses the finite element environment Freefem++ and provides quantitative results which fit the experimental data. It also shows that the problem requires the description of two dimensionless parameters (capillary number Ca and Morton number m ). Moreover, the steady state is found only for a range of parameters, Ca and m , and wavy states can produce fluctuations which can reach 10% of the mean value of the film thickness.

An arbitrary Lagrangian Eulerian approach to the three-dimensional simulation of electromagnetic forming

Electromagnetic metal forming is a contact-free high-speed forming process in which strain rates of more than 103s−1 are achieved. The deformation of the workpiece is driven by the Lorentz force, a material body force, that results from the interaction of a pulsed magnetic field with eddy currents induced in the workpiece by the magnetic field itself. In this work, a coupled 3D simulation of this process is presented. For the mechanical structure a thermoelastic, viscoplastic, electromagnetic material model is relevant, which is incorporated in a large-deformation dynamic formulation. The evolution of the electromagnetic fields is governed by Maxwell’s equations under quasi-static conditions. Their numerical solution in 3D requires particular arrangements due to problems connected with an adequate gauging of the fields. Hence, Nédélec elements are employed. Coupling between the thermomechanical and electromagnetic subsystems takes the form of the Lorentz force, the electromotive intensity, and the current geometry of the workpiece. A staggered scheme based on a Lagrangian mesh for the workpiece and an ALE formulation for the electromagnetic field is utilized to solve the coupled system, guaranteeing the efficiency and accuracy of the data transfer between the two meshes.

Numerical simulation of the flow around oscillating wind turbine airfoils Part 1: Forced oscillating airfoil

This is the first part of a two part paper on flow around vibrating wind turbine airfoils. In this part 1, the unsteady, incompressible, viscous and laminar flow over a forced oscillating airfoil is computed using a method based on a commercial Computational Fluid Dynamics (CFD) code. Beforehand, the Navier-Stokes equations are solved for the unsteady flow around a NACA 0012 airfoil at a fixed 20° incidence and the low Reynolds numbers of 103 and 104 to check the reliability of the CFD computations. Then the flow around a pitching airfoil is simulated for prescribed values of the reduced frequency. The Navier-Stokes equations are expressed in ALE formulation and solved with moving mesh. The effects of the discretization scheme and the moving mesh technique are investigated. The hysteresis loops of the dynamic stall phenomenon are highlighted.

Partitioned block-Gauss–Seidel coupling for dynamic fluid–structure interaction

This paper presents a fully coupled three-dimensional solver for the analysis of time-dependent fluid-structure interaction. A partitioned time marching algorithm is employed for the solution of the time-dependent coupled discretised problem, enabling the use of highly developed, robust and well-tested solvers for each field. Conservative transfer of information at the fluid-structure interface is combined with an effective block-Gauss-Seidel iterative scheme to enable implicit coupling of the interacting fields at each time increment. The three-dimensional unsteady incompressible fluid is solved using a powerful implicit dual time stepping technique with explicit multi-stage Runga-Kutta time stepping in pseudo-time and an ALE formulation for moving boundaries. A finite element dynamic analysis of the highly deformable structure is carried out with a numerical strategy combining the implicit Newmark time integration algorithm with a Newton-Raphson second-order optimisation method. Test cases are presented to benchmark the algorithm and to demonstrate the potential applications of this method.

A numerical method for a blood-artery interaction problem

We present a numerical method for the resolution of a bidimensional blood flow problem and more generally for a fluid flow surrounded by a time dependent domain. Our approach is based on an ALE formulation which is solved using a Galerkin method with an eigenvectors basis set on the initial fixed domain.

A preliminary comparison between finite element and meshless simulations of extrusion

In this paper the extrusion process of a cross-shaped profile was investigated. In particular, the study was focused on the distortion of extruding profiles when the workpiece and die axis are not aligned. The process was simulated using the finite element method (FEM) and the natural element method (NEM), both implemented in an updated-Lagrangian formulation, in order to avoid the burden associated with the description of free surfaces in ALE or Eulerian formulations. Furthermore, an experimental equipment was developed in order to obtain reliable data in terms of deformed entity, required process load and calculated pressure. At the end, a comparison between the numerical predictions and the experimentally measured data was carried out. The main results are presented in the paper.

Convergence of a Lagrange-Galerkin method for a fluid-rigid body system in ALE formulation

We propose a numerical scheme to compute the motion of a two-dimensional rigid body in a viscous fluid. Our method combines the method of characteristics with a finite element approximation to solve an ALE formulation of the problem. We derive error estimates implying the convergence of the scheme.