Arbitrary Lagrangian Eulerian Formulation Research Articles

An Arbitrary Lagrangian-Eulerian, coupled neutronics-shock physics model for the analysis of shockwave implosion of solid fissile materials

The aim of this work is to present a coupled neutronics-shock physics model for the study of shockwave compression of solid fissile materials. The shock-physics solver implements multi-material continuum mechanics balance equations, a hydrodynamic material response model and a dynamic mesh to describe shock-induced deformations in solid bodies. In addition, an Arbitrary Lagrangian-Eulerian (ALE) formulation of the governing equations is adopted, in order to avoid mesh distortion and tangling problems in case of large deformations. As for neutronics, a multigroup SP3 neutron transport model is selected for the estimation of the neutron flux.Several case studies are presented to validate the developed models, demonstrate the coupling between the two physics and highlight the advantages of the ALE approach.The proposed model can be a useful tool for the simulation of shock implosion of fissile materials, such as in subcritical plutonium experiments or in reactivity accidents initiated by strong energetic events.

A Parallel Newton Multigrid Framework for Monolithic Fluid-Structure Interactions

We present a monolithic parallel Newton-multigrid solver for nonlinear nonstationary three dimensional fluid-structure interactions in arbitrary Lagrangian Eulerian (ALE) formulation. We start with a finite element discretization of the coupled problem, based on a remapping of the Navier–Stokes equations onto a fixed reference framework. The strongly coupled fluid-structure interaction problem is discretized with finite elements in space and finite differences in time. The resulting nonlinear and linear systems of equations are large and show a very high condition number. We present a novel Newton approach that is based on two essential ideas: First, a condensation of the solid deformation by exploiting the discretized velocity-deformation relation d_t mathbf {u}= mathbf {v}, second, the Jacobian of the fluid-structure interaction system is simplified by neglecting all derivatives with respect to the ALE deformation, an approximation that has shown to have little impact. The resulting system of equations decouples into a joint momentum equation and into two separate equations for the deformation fields in solid and fluid. Besides a reduction of the problem sizes, the approximation has a positive effect on the conditioning of the systems such that multigrid solvers with simple smoothers like a parallel Vanka-iteration can be applied. We demonstrate the efficiency of the resulting solver infrastructure on a well-studied 2d test-case and we also introduce a challenging 3d problem.

Arbitrary Lagrangian–Eulerian finite element method for curved and deforming surfaces: I. General theory and application to fluid interfaces

An arbitrary Lagrangian–Eulerian (ALE) finite element method for arbitrarily curved and deforming two-dimensional materials and interfaces is presented here. An ALE theory is developed by endowing the surface with a mesh whose in-plane velocity need not depend on the in-plane material velocity, and can be specified arbitrarily. A finite element implementation of the theory is formulated and applied to curved and deforming surfaces with in-plane incompressible flows. Numerical inf–sup instabilities associated with in-plane incompressibility are removed by locally projecting the surface tension onto a discontinuous space of piecewise linear functions. The general isoparametric finite element method, based on an arbitrary surface parametrization with curvilinear coordinates, is tested and validated against several numerical benchmarks. A new physical insight is obtained by applying the ALE developments to cylindrical fluid films, which are computationally and analytically found to be stable to non-axisymmetric perturbations, and unstable with respect to long-wavelength axisymmetric perturbations when their length exceeds their circumference. A Lagrangian scheme is attained as a special case of the ALE formulation. Though unable to model fluid films with sustained shear flows, the Lagrangian scheme is validated by reproducing the cylindrical instability. However, relative to the ALE results, the Lagrangian simulations are found to have spatially unresolved regions with few nodes, and thus larger errors.

Finite element analysis of an arbitrary Lagrangian–Eulerian method for Stokes/parabolic moving interface problem with jump coefficients

In this paper, a type of arbitrary Lagrangian–Eulerian (ALE) finite element method in the monolithic frame is developed for a linearized fluid–structure interaction (FSI) problem — an unsteady Stokes/parabolic interface problem with jump coefficients and moving interface, where, the corresponding mixed finite element approximation in both semi- and fully discrete scheme are developed and analyzed based upon one type of ALE formulation and a novel H1-projection technique associated with a moving interface problem, and the stability and optimal convergence properties in the energy norm are obtained for both discretizations to approximate the solution of a transient Stokes/parabolic interface problem that is equipped with a low regularity. Numerical experiments further validate all theoretical results. The developed analytical approaches and numerical implementations can be similarly extended to a realistic FSI problem in the future.

Validation of underwater explosion response analysis for airbag inflator using a fluid-structure interaction algorithm

Air gun shock systems are commonly used as alternative explosion energy sources for underwater explosion (UNDEX) shock tests owing to their low cost and environmental impact. The airbag inflator of automotive airbag systems is also very useful to generate extremely rapid underwater gas release in lab-scale tests. To overcome the restrictions on the very small computational time step owing to the very fine fluid mesh around the nozzle hole in the explicit integration algorithm, and also the absence of a commercial solver and software for gas UNDEX of airbag inflator, an idealized airbag inflator and fluid mesh modeling technique was developed using nozzle holes of relatively large size and several small TNT charges instead of gas inside the airbag inflator. The objective of this study is to validate the results of an UNDEX response analysis of one and two idealized airbag inflators by comparison with the results of shock tests in a small water tank. This comparison was performed using the multi-material Arbitrary Lagrangian-Eulerian formulation and fluid-structure interaction algorithm. The number, size, vertical distance from the nozzle outlet, detonation velocity, and lighting times of small TNT charges were determined. Through mesh size convergence tests, the UNDEX response analysis and idealized airbag inflator modeling were validated.

Equivalence of Lagrange’s equations for non-material volume and the principle of virtual work in ALE formulation

The Arbitrary Lagrangian–Eulerian (ALE) formulation provides a high-efficiency approach to simulate the moving contact behaviors in mechanical systems using the relative movement between material points and mesh nodes. Two approaches have been widely adopted to obtain the governing equations of ALE elements, namely (i) Lagrange’s equations for non-material volume and (ii) the principle of virtual work. Indeed, consistent numerical results should be obtained through these two approaches; however, their mathematical expressions are quite different at first glance. In this paper, the equivalence of the above-mentioned two approaches demonstrated analytically for a general case and a specific example of three-dimensional two-node ALE cable elements. Additionally, the existence and disappearance conditions for additional terms $${\mathbf {L}}_{1}$$ and $${\mathbf {L}}_{2}$$ of the generalized inertial forces, which distinguish the ALE formulation from the Lagrangian formulation, are explicitly presented. They are physically caused by the material flow through the boundary and mathematically caused by the dependence of the mass matrix $${\mathbf {M}}$$ on the material coordinates $$p_{1}$$ and $$p_{2}$$. When no mass is transported through the boundary surface or the kinetic energy of the material points flowing into or out of the control volume is zero, $${\mathbf {L}}_{1}$$ and $${\mathbf {L}}_{2}$$ automatically disappear, and the governing equations degenerate to the classical form of Lagrange’s equations. Finally, an example of preloaded wire rope with variable length is used to verify the above conclusions.

Blast-induced damage and evaluation method of concrete gravity dam subjected to near-field underwater explosion

Bomb attacks from terrorists have become potential risks to important infrastructures, most of which were built without considering their vulnerability to such events. This paper focuses on the blast-induced damage to concrete gravity dams subjected to near-field underwater explosions. Different from direct damage mode analysis, a novel vibration-based damage evaluation method is proposed to illustrate the damage state of a dam after an explosion. To this end, a three-dimensional fluid-solid coupling numerical model in LS-DYNA is proposed for simulating the shock wave propagation and its interaction with dam structures, in which the explosive, air, and water are meshed by an arbitrary Lagrangian-Eulerian (ALE) formulation, while the dam and its foundation are meshed by the Lagrange formulation. Then, the structural responses and damage characteristics of the dam are investigated under various explosion scenarios considering changes in explosive charge, standoff distance and detonation depth. On this basis, the optimized vibration characteristics, including peak velocity summation (PVS) and mean frequency (MF), are adopted to evaluate the vulnerability of concrete gravity dams subjected to underwater explosions, which is better than the traditional damage evaluation method based solely on the peak particle velocity (PPV). The PVS-MF spectrum criterion proposed in this study is feasible to evaluate the damage state of concrete gravity dams after underwater explosions, and the results can be used in the blast-resistance design of similar hydraulic engineering.

Local projection stabilization with discontinuous Galerkin method in time applied to convection dominated problems in time-dependent domains

This paper presents the numerical analysis of a stabilized finite element scheme with discontinuous Galerkin (dG) discretization in time for the solution of a transient convection–diffusion–reaction equation in time-dependent domains. In particular, the local projection stabilization and the higher order dG time stepping scheme are used for convection dominated problems. Further, an arbitrary Lagrangian–Eulerian formulation is used to handle the time-dependent domain. The stability and error estimates are given for the proposed numerical scheme. The validation of the proposed local projection stabilization scheme with higher order dG time discretization is demonstrated with appropriate numerical examples.

Discretization Approaches to Model Metal Cutting with Lagrangian, Arbitrary Lagrangian Eulerian and Smooth Particle Hydrodynamics formulations

AbstractWith significant technological growth and computing power it is possible to simulate metal cutting processes with different discretization techniques. Classically the Lagrangian or Eulerian finite element formulations are used to model metal cutting process. Lagrangian approach is accurate with it's representation of the domain boundary, but requires a re‐meshing procedure to avoid element distortions. Eulerian approach provides a steady state solution of the chip‐workpiece separation, however its limitation lies in the treatment of convective terms during motion. The Arbitrary Lagrangian‐Eulerian method can be used to combine the advantages of both methods and avoid the disadvantages. In the Lagrangian framework, use of a meshless technique– Smooth Particle Hydrodynamics (SPH) has its advantage in large strain deformation problems without the need for re‐meshing algorithms. This work compares the LAG, ALE and SPH approaches by modelling a turning process.

Numerical Simulation of the Ice Resistance in Pack Ice Conditions

In this study, the ice resistance of ship under the pack ice condition has been estimated using Multi-Material Arbitrary Lagrangian–Eulerian formulation and penalty-based fluid–structure interaction technique. Especially, the hydrodynamic loads occurring during the ship–ice interaction were combined in the numerical model, which could be important for simulating pack ice conditions. The feasibility of the hydrostatic and hydrodynamic performance in the fluid model was verified, respectively. By using an ice field generation algorithm to generate randomly distributed pack ices, the ice resistance of an oil tanker has been calculated and validated by comparing with the empirical formula. Moreover, the necessity to consider water effect in the interaction between ship and ice was also discussed.

An efficient multibody dynamic model of three-dimensional meshing contacts in helical gear-shaft system and its solution

The dynamics of helical gear-shaft systems are characterized by three-dimensional (3D) meshing contacts that have significant variations in the location and size of the contact area, resulting in noise that is unavoidably transmitted to the gearbox through the shaft. Accurate and efficient predictions of the dynamic behaviors of helical gear and shaft are indispensable in reliable and cost-effective gearbox design. Available analytical methods, though computationally feasible, cannot consider multi-point contacts and uneven tooth-load distribution. In contrast, the finite element (FE) method provides a high-fidelity approach to compute the dynamic behaviors of a general gear-shaft system at high expenses of computation. This paper aims to establish a high-efficiency multibody dynamic model for 3D contacts in helical gear-shaft systems, in which the helical gear is pertinently represented under the framework of Arbitrary Lagrangian Eulerian formulation and the shaft is discretized by 3D Timoshenko beam elements. The computational efficiency is greatly improved through the following four steps. First, the low-frequency approximation technique is adopted to reduce the degrees of freedom (DOFs) resulting from the fixed boundary normal modes. Second, under the framework of ALE formulation, only the FE nodes of three meshing tooth-faces are defined as boundary nodes. Then, the dynamic equations and Jacobian matrix are simplified by ignoring the inertial forces associated with deformation. Finally, a two-step algorithm is adopted to accelerate the contact detection process. The accuracy and efficiency of the proposed method are demonstrated through five numerical tests with correlation to commercial nonlinear finite element software.

Model order reduction applied to ALE‐fluid dynamics

AbstractThe Arbitrary Lagrangian Eulerian formulation allows the description of fluid dynamics involving moving and deforming fluid domains as they are found in many fluid‐structure interaction problems. This paper discusses a reduced order approach for ALE incompressible Navier‐Stokes flow formulated on a fixed reference configuration. Based on the proper orthogonal decomposition (POD) for generation of the reduced basis, the supermizer technique is adapted to the formulation involving a non‐affine term and used to ensure the fulfilment of the inf‐sup condition among the velocity and pressure basis. The approach is studied for an extended version of the classical driven‐cavity setup involving a non‐affine parameterization of the prescribed mesh deformation and a time‐dependent non‐homogeneous lid‐flow boundary condition.

A continuum framework for modeling liquid-vapor interfaces out of local thermal equilibrium

Continuum numerical methods modeling liquid-vapor phase change processes typically assume local thermal equilibrium at the liquid-vapor interface – continuity of temperature at phase interfaces, and a relation between interfacial saturation pressure and temperature based on phase co-existence curves. Several standard macroscopic problems have been solved accurately by adhering to this assumption. However, in micro-scale and certain non-standard macro-scale applications, significant jumps in temperature are observed at liquid-vapor interfaces during phase change, and vapor pressure can be far from equilibrium values. In this paper, we present a locally discontinuous arbitrary Lagrangian Eulerian finite element formulation with temperature discontinuities at interfaces. A penalty-based approach is used to weakly enforce the temperature jump condition. Furthermore, we use kinetic-theory-based Schrage relationship to evaluate the rate of phase change. We apply our methodology to solve the problem of flowing vapor in a planar heat pipe. The flow rates predicted by our continuum simulations are in excellent agreement with recently published molecular dynamics simulation results of the same problem. Interestingly, accounting for temperature discontinuities leads to only a slight improvement in the prediction of mass fluxes as compared to the case when temperature continuity is assumed. This is in contrast to the large improvement in the prediction of temperature profiles and is a consequence of a weak dependence of the evaporation/condensation rates on the vapor temperature.

Entropy stable spectral collocation schemes for the 3-D Navier-Stokes equations on dynamic unstructured grids

New entropy stable spectral collocation schemes of arbitrary order of accuracy are developed for the unsteady 3-D Euler and Navier-Stokes equations on dynamic unstructured grids. To take into account the grid motion and deformation, we use an arbitrary Lagrangian-Eulerian formulation. As a result, moving and deforming hexahedral grid elements are individually mapped onto a cube in the fixed reference system of coordinates. The proposed scheme is constructed by using the skew-symmetric form of the Navier-Stokes equations, which are discretized by using summation-by-parts spectral collocation operators that preserve the conservation properties of the original governing equations. Furthermore, the metric coefficients are approximated such that the geometric conservation laws are satisfied exactly on both static and dynamic grids. To make the scheme entropy stable, a new entropy conservative flux is derived for the 3-D Euler and Navier-Stokes equations on dynamic unstructured grids. The new flux preserves the design order of accuracy of the original spectral collocation scheme and guarantees entropy conservation on moving and deforming grids. We present numerical results demonstrating design order of accuracy and freestream preservation properties of the new schemes for both the Euler and Navier-Stokes equations on dynamic unstructured grids.

Fully coupled thermomechanical simulation of friction stir welding of aluminum 6061-T6 alloy T-joint

The current study executes a fully coupled thermomechanical simulation of friction stir welding (FSW) process of aluminum 6061-T6 alloy T-joint type using finite element method. The analysis simulation accounts for the three steps of the FSW process which includes: plunging, dwelling, and moving stages. The temperature history, associated stresses and strains generated through the FSW phases, tool reaction force, and time-dependence of the energy dissipation were evaluated. To overcome the shortcomings of purely Lagrangian and Eulerian descriptions, Arbitrary Lagrangian Eulerian (ALE) formulation, adaptive meshing, and the mass scaling were used as techniques to improve sequence modeling of the friction stir welding process. Coulomb’s friction law with nonlinear friction coefficient was used to model contact between the tool and T-joint configuration. To verify the numerical results, an experimental setup was constructed to carry out the FSW of a T-joint. Meanwhile, embedded thermocouples were used in the advancing side to measure temperature close to the welding line. The results obtained throughout the simulation study showed that the temperature was symmetrically distributed across the T-joint width, and the temperature contour displayed a high gradient in the weld stirring zone with a V type shape after the plunge stage. Good correlation between numerical and experimental temperature results was obtained with a little shift in the peak value. The results of von-Mises value showed that the maximum stress has been moved from the skin part gradually into the stringer part with the advancement of the tool during plunging. The plastic strain value was higher on the advancing side in comparison with the retreating side. Moreover, the heat generation calculation showed that the frictional dissipation energy was responsible for generating most of the heat needed to obtain a successful FSW.

Role of fluid-structure interaction in mixed convection from a circular cylinder in a square enclosure with double flexible oscillating fins

The problem of unsteady mixed convection in a square enclosure containing a heated circular cylinder is studied in this paper. Two solid elastic thin fins are attached on the top cold wall. The governing equations for fluid flow and energy in the fluid and structures domains are written in Arbitrary Lagrangian-Eulerian (ALE) formulation. Finite element method was used to obtain the approximate solutions of the governing equation. The governing parameters considered are the oscillating fin direction, fin amplitude and length with various elasticity values and cylinder sizes. The results show that the instantaneous heat transfer characteristics of the contrary oscillation differs greatly from the identical oscillation. The influence of the elasticity value on the heat transfer rate is significant at a sufficiently small amplitude oscillation.

Efficient and high-fidelity steering ability prediction of a slender drilling assembly

In drilling engineering, it is extremely challenging to drill a prescribed wellbore over several thousand meters. One of the main difficulties arises from accurately predicting and controlling the directional drilling performance, caused by the complex nonlinear dynamics of the slender drilling assembly and its interactions with the surrounding rocks. Nowadays, the simplified analytical geometry method, which has been adopted as the industry standard, can merely offer a rough estimation of the drilling direction, while the high-accuracy finite element method is computationally inefficient. This study is intended to provide a straightforward prediction of the drilling direction for a long drilling distance accurately and efficiently by proposing a dynamical simulation method based on the flexible multibody approach. Three techniques were adopted to achieve the critical objective of the paper. First, an Arbitrary Lagrangian–Eulerian formulation was used to provide a new approach to balance the efficiency and accuracy. Additionally, it can perfectly simulate the realistic drilling operation that drill pipes are continuously added to the drill string one by one through dynamically inserting new beam elements into the existing model. Second, the whole drill string and its interaction with the wellbore were all considered to carry out a high-fidelity simulation. Finally, the bit–rock interaction model was introduced to offer a straightforward way of evaluating the steerability of drilling assemblies. The presented method and model were validated by the consistency between the simulated wellbore trajectory and the in-field experimental data and are ready to be applied in drilling tools design, real-time drilling simulation, and drilling direction control.

Fluid-structure interaction analysis of transient convection heat transfer in a cavity containing inner solid cylinder and flexible right wall

PurposeThis paper aims to investigate the fluid structure interaction analysis of conjugate natural convection in a square containing internal solid cylinder and flexible right wall.Design/methodology/approachThe right wall of the cavity is flexible, which can be deformed due to the interaction with the natural convection flow in the cavity. The top and bottom walls of the cavity are insulated while the right wall is cold and the left wall is partially heated. The governing equations for heat, flow and elastic wall, as well as the grid deformation are written in Arbitrary Lagrangian–Eulerian formulation. The governing equations along with their boundary conditions are solved using the finite element method.FindingsThe results of the present study show that the presence of the solid cylinder strongly affects the transient solution at the initial times. The natural convection flow changes the shape of the flexible right wall of the cavity into S shape wall due to the interaction of the flow and the structure. It is found that the increase of the flexibility of the right wall increases the average Nusselt number of the hot wall up to 2 per cent.Originality/valueTo the best of the authors' knowledge, the unsteady natural convection in an enclosure having a flexible wall and inner solid cylinder has never been reported before.

Isogeometric Analysis for Tire Simulation at Steady-State Rolling

ABSTRACTThe use of isogeometric analysis (IGA) in industrial applications has increased in the past years. One of the main advantages is the combination of finite element analysis (FEA) with the capability of representing the exact geometry by means of non-uniform rational B-splines (NURBS). This framework has proven to be an efficient alternative to standard FEA in solid mechanics and fluid dynamics, in cases in which sensitivity to geometry is found. The numerical simulation of rolling tires requires a proper discretization for the curved boundaries and complex cross sections, which often leads to the use of higher-order or cylindrical elements. As remeshing operations are numerically costly in tire models, IGA stands as an attractive alternative for the modeling of rolling tires.In this contribution, an arbitrary Lagrangian Eulerian formulation is implemented into IGA to provide the basic tools for the numerical analysis of rolling bodies at steady-state conditions. The solid basis of the formulation allows the employment of standard material models, but tire constructive elements, such as reinforcing layers, require special attention. Streamlines are constructed based on the locations of the integration points, and therefore, linear and nonlinear viscoelastic models can be implemented. Numerical examples highlight the advantage of the new approach of requiring fewer degrees of freedom for an accurate description of the geometry.

Fluid-structure interaction analysis of entropy generation and mixed convection inside a cavity with flexible right wall and heated rotating cylinder

The current work concentrates on the transient entropy generation and mixed convection due to a rotating hot inner cylinder within a square cavity having a flexible side wall by using the finite element method and arbitrary Lagrangian-Eulerian formulation. Effects of various relevant parameters like Rayleigh number (104⩽Ra⩽107), angular rotational velocity (-1⩽Ω⩽1), dimensionless elasticity modulus (1012⩽E⩽1015) on the convective heat transfer characteristics and entropy generation rates are analyzed for dimensionless time 10-8⩽τ⩽3.5. It is observed that various complex shaped wall deformations are established depending on the non-dimensional elastic modulus of the flexible right wall and cylinder rotation direction. The local and average Nusselt numbers rise with Ra and secondary peaks in the local Nusselt number are established for lower values of Ra. The local heat transfer along the hot cylinder does not change for the case of clockwise rotation of the heated cylinder even if there is a wall deformation in the positive x-direction. The highest average heat transfer and global entropy generation rates are achieved for the case of counter-clockwise rotation of the circular cylinder and for lower values of the flexible wall deformation.