Arbitrary Lagrangian-Eulerian Finite Element Research Articles

Spatial energy evolution of focused waves generated in numerical wave tank

This paper numerically investigates the energy evolution of the focused wave and its correlation with input parameters for the wave generation. The focused wave is numerically generated based on the fully nonlinear potential wave theory which is solved by the Quasi Arbitrary Lagrangian Eulerian-Finite Element Method. The investigation is performed by dividing the generated wave energy into three categories according to their frequency intervals, i.e., the initially assigned frequency interval [f1fN], the frequencies lower (f < f1) and higher (f > fN) than the initial interval. The amount of the generated energy falling into three energy categories and the energy distribution in the initial frequency interval are analysed against amplitude parameters and frequency bands under three amplitude spectra. Four indicators are proposed to indicate the actual energy variations comparing to the initial design. It is found that for all the three frequency intervals the energies increase when the amplitude parameters increases or the given frequency shifts towards the lower frequency domain. The choice of wave amplitude spectrum plays a significant role to maintain the distribution of the energy in the design frequency range and to minimise the energies out of the design frequency range. The results from the second-order wave group theory are used to assist analyses of the mechanism of the energy evolution alongside the focused wave generation. It is found that the lower-frequency energy in the vicinity of the focusing point is dominated by the second-order difference waves, whereas the higher-frequency energy is mainly produced by the wave flap. The waves which are higher than the second-order and the complex interactions not considered by the second-order wave theory are found significantly disturbing the energy distribution in the originally assigned wave-frequency interval.

Arbitrary Lagrangian-Eulerian finite element approximations for axisymmetric two-phase flow

We analyze numerical approximations for axisymmetric two-phase flow in the arbitrary Lagrangian-Eulerian (ALE) framework. We consider a parametric formulation for the evolving fluid interface in terms of a one-dimensional generating curve. For the two-phase Navier-Stokes equations, we introduce both conservative and nonconservative ALE weak formulations in the 2d meridian half-plane. Piecewise linear parametric elements are employed for discretizing the moving interface, which is then coupled to a moving finite element approximation of the bulk equations. This leads to a variety of ALE methods, which enjoy either an equidistribution property or unconditional stability. Furthermore, we adapt these introduced methods with the help of suitable time-weighted discrete normals, so that the volume of the two phases is exactly preserved on the discrete level. Numerical results for rising bubbles and oscillating droplets are presented to show the efficiency and accuracy of these introduced methods.

Numerical simulations of the dynamics of a deformable particle in a viscoelastic liquid subjected to Poiseuille flow in a cylindrical microcapillary at non-negligible inertia

The manipulation and control of particles in microfluidic devices through non-intrusive methods is pivotal in many application fields, e.g., cell focusing and sorting. Inertial microfluidics is rapidly gaining attention in the scientific community because of the considerable advantages in terms of throughput. In addition to inertia, other factors can trigger the cross-stream migration of particles in liquids undergoing pressure-driven channel flows, such as the deformability of the particles themselves and/or the viscoelasticity of the carrier fluid. For this reason, the dynamics of an initially spherical elastic particle suspended in a viscoelastic liquid subjected to pressure-driven flow in a cylindrical channel at non-negligible inertia is studied through three-dimensional arbitrary Lagrangian–Eulerian finite-element numerical simulations. The mechanical behavior of the particle is described through the neo-Hookean hyper-elastic constitutive equation, whereas the rheological behavior of the carrier liquid is described through the Giesekus model. The Reynolds number Re, measuring the relative importance of inertial and viscous forces in the tube, the elastic capillary number Cae, measuring the relative importance of liquid viscous stress and solid elastic stress, and the Deborah number De, measuring the ratio of the liquid relaxation time and the flow characteristic time, are varied. The particle migrates transversally to the flow direction until reaching a radial equilibrium position depending on Re, Cae, and De. Different dynamics are observed depending on the interplay among inertia and elasticity of both the liquid and the solid phase: one, two, or even three stable equilibrium positions can be found along the tube radius.

Application of the Constrained Formulation to the Nonlinear Sloshing Problem Based on the Arbitrary Lagrangian–Eulerian Method

Abstract This study deals with an application of constrained formulation to a nonlinear sloshing problem based on the arbitrary Lagrangian–Eulerian finite element method (ALE). The ALE method incorporates a discretized form of equations of motion with mesh updating algorithms in order to prevent a problem of mesh distortion. This paper focuses on an analytical aspect of such treatments as constrained systems in the formulation of the ALE method. Since the mesh updating algorithms give algebraic relations for nodal coordinates, this study treats these relations as constraints. Then, we introduce formulation for constrained systems based on the method of Lagrange multipliers. As a result of this formulation, equations of motion are given by differential algebraic equations (DAEs) consisting of differential equations for time evolution of physical quantities and algebraic equations (constraints). The present method can be classified into a kind of augmented formulation. Moreover, we present a size-reduction technique used in the Newton–Raphson method in order to remove a part of the redundant degrees-of-freedom in the iterative procedures, because the resulting set of DAEs involves a larger number of unknowns than the minimal number of degrees-of-freedom due to the introduction of the constrained formulation. The derived reduced system still holds the physically essential part of equations of motion for the sloshing problem. Even though it is not reduced to the minimal degrees-of-freedom, it does not involve algebraically complicated and inefficient procedures in computation. In addition, this study presents a method to introduce damping effects defined in the modal space into the FEM models. The proposed approach is validated by comparisons with experimental data in the time domain analysis.

Finite Deformation of Scleral Tissue under Electrical Stimulation: An Arbitrary Lagrangian-Eulerian Finite Element Method.

The sclera is considered as the principal load-bearing tissue within the eye. The sclera is negatively charged; thus, it exhibits mechanical response to electrical stimulation. We recently demonstrated the electroactive behavior of sclera by performing experimental measurements that captured the deformation of the tip of scleral strips subjected to electric voltage. We also numerically analyzed the electromechanical response of the tissue using a chemo-electro-mechanical model. In the pre-sent study, we extended our previous work by experimentally characterizing the deformation profile of scleral strips along their length under electrical stimulation. In addition, we improved our previous mathematical model such that it could numerically capture the large deformation of samples. For this purpose, we considered the transient variability of the fixed charge density and the coupling between mechanical and chemo-electrical phenomena. These improvements in-creased the accuracy of the computational model, resulting in a better numerical representation of experimentally measured bending angles.

Estimation of spudcan penetration in variable sand deposits with the Arbitrary Lagrangian Eulerian Finite Element Method

Offshore jack-up rigs are most commonly founded on large-diameter conical “spudcan” foundations, which are frequently designed using traditional analytical methods for shallow footings. This paper presents the design of a spudcan installed off the coast of Tunisia. The maximum penetration depth of the footing under the available preload is predicted by a combination of analytical techniques, 2-dimensional axisymmetric modelling and 3-dimensional Finite Element Methods (FEM) using large strain arbitrary Lagrangian-Eulerian (ALE) techniques. Spudcan penetration based on FEM simulation of CPT soil profiles forms the basis of a comparison with results from the Society of Naval Architects and Marine Engineers (SNAME) guidelines. Particular attention is given to model calibration using the limited site investigation data available. Results are presented for the effect of penetrating footings on the behaviour of neighbouring footings, showing good agreement with conventional prediction methods.

Comparative modeling of the mitral valve in normal and prolapse conditions.

Computational modeling is one of the best non-invasive approaches to predicting the functional behavior of the mitral valve (MV) in health and disease. Mitral valve prolapse (MVP) due to partial or complete chordae tendineae rapture is the most common valvular disease and results in mitral regurgitation (MR). In this study, Image-based fluid-structure interaction (FSI) models of the human MV are developed in the normal physiological and posterior leaflet prolapse conditions. Detailed geometry of the healthy human MV is derived from Computed Tomography imaging data. To provide prolapse condition, some chords attached to the posterior leaflet are removed from the healthy valve. Both normal and prolapsed valves are embedded separately in a straight tubular blood volume and simulated under physiological systolic pressure loads. The Arbitrary Lagrangian-Eulerian finite element method is used to accommodate the deforming intersection boundaries of the blood and MV. The stress values in the mitral components, and also flow patterns including the regurgitant flow rates are obtained and compared in both conditions through the simulation. These simulations have the potential to improve the treatment of patients with MVP, and also help surgeons to have more realistic insight into the dynamics of the MV in health and prolapse. In the prolapse model, computational results show incomplete leaflet coaptation, higher MR severity, and also a significant increment of posterior leaflet stress compared to the normal valve. Moreover, it is found more deviation of the regurgitant jet towards the left atrium wall due to the posterior leaflet prolapse.

Study on the Formation Mechanism of Cutting Dead Metal Zone for Turning AISI4340 with Different Chamfering Tools.

Tools with chamfered edges are often used in high speed machining of hard materials because they provide compelling cutting toughness and reduced tool wear. Chamfered tools are also responsible for the dead metal zone (DMZ). Through numerical simulation of orthogonal cutting with AISI 4340 steel, this paper examines the mechanism of the DMZ, the cutting speed, the impacts of the chamfer angle, and the coefficient of friction on the generation of the DMZ. The analysis is based upon the Arbitrary Lagrangian-Eulerian (ALE) finite element method (FEM) for the continuous process of chip formation. The different chamfered angles, cutting speeds, and friction coefficient conditions are utilized in the simulation. The research demonstrates that a zone of trapped material called DMZ has been formed beneath the chamfer and serves as an effective cutting edge of the tool. Additionally, the dead metal zone DMZ becomes smaller while the cutting speed increases or the friction coefficient decreases. The machining forces rise with increasing chamfer angles, rise with increasing friction coefficients, and fall with increasing cutting speed in both the cutting and thrust directions. In this paper, the effect of different chamfering tools on AISI 4340 steel using carbide tools in the simulation environment is studied. It has certain reference significance for studying the formation mechanism of the dead zone of difficult-to-machine materials such as AISI4340 and improving the processing efficiency and workpiece surface quality.

Numerical simulations of cell sorting through inertial microfluidics

The dynamics of a cell suspended in a Newtonian liquid subjected to a pressure-driven flow at non-negligible inertia in cylindrical and square cross section microfluidic channels is studied through three-dimensional arbitrary Lagrangian–Eulerian finite-element numerical simulations. The cell is modeled through the neo-Hookean hyper-elastic constitutive equation, which can describe biological particles undergoing moderate deformations. The cell-to-channel relative dimension is fixed to 0.2, whereas the Reynolds number Re, measuring the relative importance of liquid inertial and viscous forces, and the elastic capillary number Cae, measuring the relative importance of liquid viscous stress and solid elastic stress, are varied by several orders of magnitude. In a cylindrical tube, the cell migrates transversally to the flow direction until reaching a radial equilibrium position depending on Re and Cae. Given Re, the softer the cell (i.e., the larger Cae) the closer its equilibrium position to the tube axis, thus allowing for the separation of healthy and diseased cells which have similar dimensions but different mechanical properties. In a channel with a square cross section, a much more complex dynamics is found. Depending on Re and Cae, the cell can either migrate to the channel centerline, to the closest median of the channel cross section (thus, four equilibrium positions can be identified due to symmetry), to the closest diagonal (again, four equilibrium positions), or to an intermediate position in between the median and the diagonal (eight equilibrium positions).

Numerical simulation of turn and zigzag Maneuvres of trimaran in calm water and waves by a hybrid method

As one of the widely used high-performance ships, the hydrodynamic characteristics of the trimaran ships have been widely investigated in recent years. But, the study on the maneuverability and the skill of the maneuvering simulation are still limited. In this paper, a hybrid method coupling the FNPT-based (fully nonlinear potential flow theory) QALE-FEM (quasi arbitrary Lagrangian-Eulerian finite element method) with the viscous flow method is applied to simulate the turn and zigzag maneuvers of the trimaran in both calm water and waves. The environment of calm water and incident waves is simulated by the numerical tank of QALE-FEM. The maneuvering of the trimaran is carried out in the scope of the numerical tank by the viscous flow method. The grid convergence test is carried out first, and the computed results are compared with the experimental results. Then, the turn and zigzag maneuvers of a trimaran model in both calm water and waves are simulated to study the trimaran's maneuvering characteristics.

An energy diminishing arbitrary Lagrangian–Eulerian finite element method for two-phase Navier–Stokes flow

A linearized fully discrete arbitrary Lagrangian–Eulerian finite element method is proposed for solving the two-phase Navier–Stokes flow system and to preserve the energy-diminishing structure of the system at the discrete level, by taking account of the kinetic, potential and surface energy. Two benchmark problems of rising bubbles in fluids in both two and three dimensions are presented to illustrate the convergence and performance of the proposed method.

Energy stable arbitrary Lagrangian Eulerian finite element scheme for simulating flow dynamics of droplets on non–homogeneous surfaces

An energy stable finite element scheme within arbitrary Lagrangian Eulerian (ALE) framework is derived for simulating the dynamics of millimetric droplets in contact with solid surfaces. Supporting surfaces considered may exhibit non–homogeneous properties which are incorporated into the governing system through generalized Navier boundary conditions (GNBC). Numerical scheme is constructed such that the counterpart of (continuous) energy balance holds on the discrete level. This ensures that no spurious energy is introduced into the discrete system, i.e. the discrete formulation is stable in the energy norm. The newly proposed scheme is numerically validated to confirm the theoretical predictions. Of a particular interest is the case of droplet on a non–homogeneous inclined surface. This case shows the capabilities of the scheme to capture the complex droplet dynamics (sliding and rolling) while maintaining stability during the long time simulation.

A decoupled Arbitrary Lagrangian-Eulerian method for large deformation analysis of saturated sand

In the finite element analysis, large deformation caused by saturated sand liquefaction may lead to numerical problems owing to mesh distortion. Based on the soil–water two-phase mixing theory, the governing equations of the two-phase mixture are derived in the study. Using the operator splitting technique and the decoupled Arbitrary Lagrangian-Eulerian (ALE) frame, a decoupled ALE Finite Element Method is developed to solve the numerical problems caused by the possible mesh distortion in the liquefaction large deformation analysis. The method is applied to one-dimensional consolidation analysis and dynamic liquefaction large deformation analysis of an embankment. The results show that the method can keep the deformed mesh in a healthy state and ensure the accuracy of the numerical solutions in large deformation analysis.

Nonlinear dynamics analysis of variable length flexible beams using the ALE-ANCF method

For the flexible slender structure with time-variation of its length, the tip could vibrate violently. In recent years, flexible multibody dynamics has been used to analyze this phenomenon. On the other hand, when analyzing this phenomenon by the finite element method, the analysis becomes complicated because the nodes across the boundaries between constrained and unbounded areas, so it is necessary to discriminate the area for each node. For this problem, using the ALE (Arbitrary Lagrangian-Eulerian) finite element method, some nodes can be fixed at boundaries and the others can be arranged arbitrarily in each area. This will make it easier to analyze the flexible body with time-varying length. This study presents an analytical method for flexible bodies with time-varying length by using the ALE-ANC method. The ALE-ANC method, which is a combination of the ALE finite element method and the ANCF (Absolute Nodal Coordinate Formulation) that is one of the formulations of flexible multibody dynamics can place nodes independently of the motion. In this method, the element coordinates of nodes are used as the generalized coordinates. It is possible to express the length variation of the flexible body by changing these coordinates with time. In addition, we developed a numerical method that introduces an energy-momentum conservation scheme. This avoids complications in the analysis due to the time dependence of the mass matrix caused by the length variation of the element. In order to verify the validity of the proposed method, this study simulates variable length flexible beams using the presented model and compares the result with previous research.

Numerical research on the dynamic rock-breaking process of impact drilling with multi-nozzle water jets

Based on the arbitrary Lagrangian-Eulerian finite element method (ALE-FEM), a coupling model for simulating the rock mass destruction process under the impact of a high-pressure water jet was established. Through the analysis of multi-nozzle jets’ impact on rock surface stress changes, the process of rock breaking and pore formation, and the rock breaking dynamic damage evolution, the mechanism of rock breaking by multi-nozzle jets was revealed by analyzing the stress distribution and dynamic damage of the rock mass during the impact process. The results show that nozzle jets with different inclination angles exhibit different rock breaking methods, which are mainly tensile failure and radial shear fracturing caused by the impact of the water jets. Rock breaking using multi-nozzle jets can be divided into four stages: the elastic deformation stage, the plastic deformation stage, the micro-fracture stage, and the complete failure stage. Based on the spatial-temporal evolution of the damage, the damage depth and scope of rock mass caused by multi-nozzle jets with different tilt angles are different. The results of this study provide a theoretical basis for the effect and optimization of the impact of multi-nozzle jets on rock breaking.

Uncertainty due to discretization on the ALE algorithm for predicting water slamming loads

Numerical uncertainty due to discretization on the Arbitrary Lagrangian-Eulerian (ALE) Finite Element method is investigated in the study. The paper quantifies uncertainty using two ITTC recommended methods, and also applies a constant Courant-Friedrichs-Lewy (CFL) number based discretization approach, instead of performing the independent grid and time-based discretization recommended by ITTC. As a case study, water entry of a flat bottom rigid and flexible plate is simulated considering various entry velocities. The total slamming loads and structural responses on both the rigid and elastic bottom plates are predicted and validated against available experimental data. Results indicate that numerical errors due to discretization differ in the various parameters and from case to case. They do affect the analysis of slamming loads and associated structural responses, and the hydroelasticity analysis as well. The hydroelasticity effects on the slamming force generally increase as the entry velocity increases, however, the quantitative results differ much for models with different grids. For example, when the hydroelasticity effect is estimated using the finer model, the deviation of the total slamming force on the elastic plate relative to the one on the rigid body are 56%, 57%, and 63% respectively for the three constant entry velocities, whereas the estimations are −27%, −4% and 3% with the coarser model. The study concludes that the uncertainty due to discretization in ALE is not just case-specific, but also parameter specific. The uncertainty quantification procedures with a constant CFL number based refinement are recommended to investigate the uncertainty comparing to the individual grid and time step study, in particular for the ALE solution where the time step is adjusted automatically as the grid changes. Thus, consideration should be given to updating the ITTC guidelines to incorporate the constant CFL based discretization approach.

Numerical simulations for studying the influence of friction in forging

ABSTRACT The modelling and simulation of forging has reached a matured stage, but the friction and friction models got only scant attention; this work intends to fill up that gap. Here, arbitrary Lagrangian-Eulerian (ALE) finite element method (FEM)-based numerical simulations are carried out for the forging of aluminium workpiece by considering two friction models – Coulomb-Siebel and Tresca. A relation between Coulomb’s coefficient of friction and friction factor has been established based on the slab method. The forging load is obtained by calculating the reaction force on the die as well as from energy method. Energy method provides more accurate and consistent results. Forging load and bulge increase with the friction, but at high friction, the effect is minimal. Bulge is found to be more sensitive to friction than the forging load. The distribution of normal stress is quite different on the die-workpiece interface compared to that on a parallel plane in the interior. The phenomenon of fold-over observed at high friction and high reduction observed in a Lagrangian formulation is absent in ALE formulation. The fold-over phenomenon occurs due to mesh distortion. It is not observed in a workpiece with round corners, even while using Lagrangian formulation.

Numerical simulation of the progressive development of soil arching in column-supported embankments

The soil arching mechanism is partly responsible for the transfer of stresses away from the soft subsoil and towards the relatively stiff column heads in column-supported embankments. Experimental studies have shown that the load transfer resulting from soil arching evolves progressively with increasing subsoil settlement. Past numerical studies exploring soil arching in column-supported embankments have typically not been able to capture this progressive development of load transfer. A series of improvements on past numerical studies are outlined that allow for improved simulation of the soil arching mechanism in column-supported embankments. These improvements include implementation of a strain-softening constitutive model, non-local integral type regularization and the application of the arbitrary Lagrangian–Eulerian finite element method. The benefits of these improvements are observed through comparison of simulation results to recent experimental studies of column-supported embankments. The comparison indicates that these techniques allow key aspects of soil arching kinematics and mechanics to be captured.

Extension of ALE method in large deformation analysis of saturated soil under earthquake loading

This article develops an Arbitrary Lagrangian Eulerian (ALE) finite element program for liquefaction dynamic analysis of saturated sand based on the updated Lagrange finite element program in U-P form. Based on the operator splitting technique, this method adopts the displacements of boundary points and Radial Basis Function (RBF) method to optimize the mesh, and employs super-convergence elements patch and Radial Basis Function (RBF) method to reconstruct variable fields and project variables between old and new configurations. The analytical solutions and numerical solutions of one-dimension consolidation are compared to verify the accuracy of the ALE method. Three typical dynamic liquefaction problems of saturated sand are implemented to measure the applicability of UL method and ALE method. The results show that both methods can provide similar results in the case of small earthquakes. Under strong earthquakes, the UL method fails for large-scale or local mesh distortion, while the proposed ALE can still maintain the health of the mesh and provide satisfactory results.

Cross-Flow-Induced Vibration of an Elastic Plate

The cross-flow over a surface-mounted elastic plate and its vibratory response are studied as a fundamental two-dimensional configuration to gain physical insight into the interaction of viscous flow with flexible structures. The governing equations are numerically solved on a deforming mesh using an arbitrary Lagrangian-Eulerian finite-element method. The turbulent flow is resolved using the unsteady Reynolds-averaged Navier–Stokes equations at a Reynolds number of 2.5×104 based on the plate height. The material properties of the plate are selected so that the structural frequency is close to the frequency of vortex shedding from the free edge of a rigid plate, which is studied initially as the reference case. The results show that the plate tip oscillates back and forth in response to unsteady fluid loading at twice the frequency of vortex shedding, which is attributable to the sequential formation of a primary vortex from the free edge and a secondary vortex near the base of the plate. The effects of the plate elasticity and density on the structural response are considered, and results are compiled in terms of the reduced velocity U* and the density ratio ρ*. The standard deviation of tip displacement increases with reduced velocity in the range 7.1⩽U*⩽18.4, irrespective of whether the elasticity or the density of the plate is varied. However, the average deflection of the plate in the streamwise direction displays different scaling with U* and ρ*, but scales almost linearly with the Cauchy number ∼U*2/ρ*. Interestingly, the synchronization between plate motion and vortex shedding ceases at U*=18.4, and the excitation mechanism in the latter case resembles flutter instability, rather than vortex-induced vibration found at lower U*.