Arbitrary Lagrangian Eulerian Formulation Research Articles

Numerical study of liquid sloshing in three-dimensional flexible cylindrical tank under multi-directional seismic excitation

The present study aims to evaluate the effect of the flexibility of three-dimensional (3D) cylindrical tank on liquid sloshing inside the tank as well as the influence of the hydrodynamic forces on the structure's deformation. A ground-supported cylindrical flexible tank filled with water and subjected to seismic excitation is investigated using a numerical coupling methodology to take into account the fluid–structure interactions. The Navier–Stokes equations in the fluid domain are solved using the finite volume method, while the finite element method is used to solve the linear-elastic equations of the structure. The arbitrary Lagrangian–Eulerian formulation is applied for the moving mesh in two-phase flow system. The simulations are conducted using free open-source software: OpenFOAM for fluid dynamics and FEniCS for solid mechanics, both utilize the preCICE library for data exchanging and fluid–structure coupling. The numerical methodology is validated by the experimental and numerical results given in literature. A multi-directional earthquake ground motion is then considered as external loading, and the elasticity of the tank wall is varied to investigate the effect on the fluid sloshing. It has been shown that the more flexible the tank walls are, the greater will be both the sloshing height and the structural deformations during an earthquake event. Several flow field information and structural responses have been provided, such as the sloshing height, hydrodynamic pressure, structure deformations, and structural stress distribution. The potential elephant's foot buckling phenomenon at the lower part of tank can also be predicted.

Treatment of inelastic material models within a dynamic ALE formulation for structures subjected to moving loads

AbstractThis article showcases the development of a dynamic Arbitrary Lagrangian Eulerian (ALE) formulation to account for inelastic material models within a finite element framework. Such a formulation is commonly utilized in research domains like fluid mechanics, fluid‐structure interaction, quasi static remeshing techniques, and quasi static load movement. The work at hand describes the application of the ALE formulation to efficiently analyse structures subjected to moving loads in the field of transient inelastic solid mechanics. In particular, structures such as pavements, gantry crane girders etc., which are subjected to moving loads, can be numerically simulated, and their transient response in the relevant region around the load can be obtained without relying on moving loads. The focus of this article is to facilitate the treatment of history variables stemming from inelastic material models. Of particular interest is the advection procedure required to transport the history variables through the mesh, as the material appears to flow through it. The mathematical framework necessary to treat this advection process is described in detail, considering a nonlinear viscoelastic material model on a neo‐Hookean base at finite deformations. Then, four methods for numerically achieving the advection are implemented within a transient finite element ALE formulation. These methods are compared against each other, and additionally with the conventional Lagrangian method for validation. The results demonstrate satisfactory agreement with conventional simulation methods, while offering a significant improvement in terms of computation speed. With the work at hand, the dynamic response of inelastic materials subjected to moving loads can be numerically simulated in a computationally efficient manner.

An efficient moving-mesh strategy for predicting crack propagation in unidirectional composites: Application to materials reinforced with aligned CNTs

This paper presents an efficient numerical approach for reproducing the process of crack propagation inside unidirectional composites subjected to general loading conditions, with special reference to epoxy materials enhanced with embedded aligned CNTs. This approach involves a traditional FE framework improved by the Moving Mesh (MM) technique based on the Arbitrary Lagrangian-Eulerian (ALE) formulation and the Interaction Integral Method (M−integral). The MM serves as a powerful numerical tool to simulate the discrete crack advance with minimal remeshing, thus reducing the computational complexities. Instead, the M−Integral method, formulated for generally anisotropic materials, has been employed to extract the mixed-mode Stress Intensity Factors (SIFs), which are necessary to define the crack onset condition and propagation direction on the basis of the modified Maximum Hoop Stress Criterion. The proposed strategy includes the extended rule of mixtures to evaluate the homogenized elastic properties of nano-reinforced composites. The validity of the proposed methodology has been assessed through comparisons with experimental data and numerical results available in the literature.

High-order compact gas-kinetic scheme in arbitrary Lagrangian-Eulerian formulation

This study proposes an extension of the high-order compact gas-kinetic scheme (CGKS) to compressible flow simulation in an arbitrary Lagrangian-Eulerian (ALE) formulation in unstructured mesh. The ALE method is achieved by subdividing arbitrary mesh into tetrahedrons and integrating flux function in a local coordinate system at the cell interface to ensure geometric conservation law. The scheme incorporates a compact reconstruction with third-order accuracy for updating both cell-averaged conservative flow variables and their gradients. HWENO-type nonlinear reconstruction and gradient compression factors are adopted to improve the accuracy and robustness of the scheme. A multi-stage multi-derivative (MSMD) time-stepping method is also implemented to achieve high-order time accuracy with fewer middle stages. The scheme is used to study problems involving moving boundaries. The numerical experiments demonstrate the effectiveness of the scheme in capturing the accurate solutions of both low-speed smooth flow and highly compressible ones with strong shock waves.

Numerical modeling of systems based on SMA wires and pulleys: A finite element formulation within the Arbitrary Lagrangian–Eulerian framework

Thanks to their properties of superelasticity and shape memory effects, SMA wires are often considered for the design of damping devices or actuators coupled to pulleys. In order to dimension such devices for commercialization, it is necessary to develop reliable, robust and fast numerical tools. To this end, simplifying the management of the wire/pulley contact using an Arbitrary Lagrangian–Eulerian (ALE) formulation is an original and effective solution. In this work, the wires are modeled by three-dimensional truss thermo-mechanical finite elements associated with a superelastic law. For each node, the curvilinear abscissa is used as an additional degree of freedom in order to deal with material flow from one element to another. After presenting the ALE formalism and the necessary precautions (advection of variables), as well as the material behavior and thermo-mechanical contact laws at the pulleys, elementary validation tests are studied. Finally, two devices taken from the literature are modeled using the ALE formalism. The results obtained demonstrate the relevance and effectiveness of the approach adopted, as well as the importance of not neglecting friction at the pulleys and thermal effects on the material mechanical response.

Prediction of rolling resistance and wheel force for a passenger car tire: A comparative study on the use of different material models and numerical approaches

In this research, the characteristics of tire-road interaction of a 185/65R14 88H passenger car tire are investigated using the Finite Element Method in Abaqus commercial software. Moreover, the effect of various material models on tire performance is studied by implementing Visco-Hyperelastic, Parallel Rheological Framework, and Mullins effect. The novelty of this research is devoted to the development of the complex material models particularly considering the Mullins effect of the rubber compounds in the tire structure for the load-displacement criteria. For this purpose, a tire finite element model was generated using Abaqus/Standard command line in two different methods including an Arbitrary Lagrangian-Eulerian formulation for steady state rolling and implementing a pure Lagrangian approach for the transient dynamic analysis carried out implicit and explicit process respectively. Rolling resistance force was computed according to ISO 28580 with 210 kPa inflation pressure and 4155 N vertical load. The footprint test results were extracted in both static and transient dynamic analyses. Additionally, the wheel reaction force was predicted using an indirect method by extracting the tire-terrain contact patch reaction force in Abaqus/Explicit to observe the effect of the material convection along with stress softening phenomena of the rubber compounds of tire structure. In the post-processing analysis, the wheel reaction was filtered by implementing SAE60 filter to reduce the numerical noise in the final response.

Aeroacoustic simulation of human phonation based on the flow-induced vocal fold vibrations including their contact

This paper deals with the numerical simulation of the flow induced sound as generated in the human phonation process including vocal folds (VF) contact. The hybrid approach is used allowing to separate the acoustic and the biomechanical description of the considered problem. The two-dimensional fluid–structure interaction (FSI) problem is modelled by a simplified linear elastic model coupled to the incompressible Navier–Stokes equations in the arbitrary Lagrangian–Eulerian form and special attention is paid to the treatment of the contact problem in the FSI context. Following approach of [Sváček and Horáček, 2021] the incompressible flow model is modified by an addition of fictitious porosity term and the penalization inlet boundary condition is used. Further, the Lighthill acoustic analogy is applied for calculation of sound sources and their propagation through a vocal tract model. All partial differential equations, describing the elastic body vibration, the fluid flow and the acoustics, are approximated by finite element method. Numerical results of one-sided vocal fold paralysis are presented.

Sedimentation of a spheroidal particle in an elastoviscoplastic fluid

The sedimentation dynamics of a prolate spheroidal particle in an unbounded elastoviscoplastic (EVP) fluid is studied by direct finite element simulations under inertialess flow conditions. The Saramito–Giesekus constitutive equation is employed to model the suspending liquid. The arbitrary Lagrangian–Eulerian formulation is used to handle the particle motion. The sedimentation, lift, and angular velocities of spheroids with aspect ratio between 1 and 8 are computed as the initial orientation, Bingham, and Weissenberg numbers are varied. Similar to the purely viscoelastic case, a spheroid in an EVP fluid rotates up to align its major axis with the applied force. As the Bingham number increases, the settling rate monotonically reduces, while the angular velocity first increases and then decreases. The initial orientation has a relevant effect on the particle stoppage because of the different drag experienced by the spheroid as its orientation is varied. The yielded and unyielded regions around the spheroid reveal that, for particle oriented transversely to the force, the yielded envelope shrinks near the tips due to the fast spatial decay of the stresses, and unyielded regions appear along the surface of the particle, similar to the solid caps observed at the front and back of a sphere. Fluid plasticity enhances the negative wake phenomenon that is observed at Weissenberg numbers significantly lower than the purely viscoelastic case. The results of the drag correction coefficient for particles aligned with longest axis along the force are presented.

Fluid‐structure interaction study of an oscillating heat source effect on the natural convection flow

AbstractThe objective of this paper is to investigate fluid‐structure interaction (FSI) within conjugate natural convection. An oscillating fin, featuring a heat source placed at different locations, is examined using the Arbitrary Lagrangian–Eulerian formulation. The Galerkin finite element method is utilized to solve nonlinear dimensionless equations. Verification of grid independence is conducted and the model undergoes validation. Simulation outcomes for three fin positions (left, center, and right) and three heat source locations (bottom, middle, and top) are presented, illustrating streamlines, isotherms, and the average Nusselt number. The governing equations and boundary conditions are addressed using the finite element method. Temperature profiles in four scenarios are analyzed, along with horizontal velocities at different levels (0, D/2, D from the bottom wall). Dimensionless time (10−5 ≤ t ≤ 3), Ra = 106, Kr = 10, E = 1011 are used as parameters. The impact of the heat source position on vibratory motion is evaluated through Nusselt number variation, affecting heat exchange in different cases. The results show that heat transfer is minimal for a source location at the center of the fin (c). These findings also offer insights into FSI applications in economics and industry, guiding practical design considerations. Additionally, coupling the vibratory motion of the heat source with the flexible oscillating fin at the same frequency enhances understanding of the system.

Numerical investigation of liquid sloshing in 2D flexible tanks subjected to complex external loading

Due to the complexity of fluid–structure interactions (FSI), the majority of studies in the literature dealing with the sloshing problem are restricted to rigid tanks. This paper is devoted to a numerical investigation of the liquid sloshing behavior in a flexible tank subjected to external loading. A numerical methodology is proposed, taking into account the FSI problem by coupling two open-source codes: OpenFOAM for the fluid and FEniCS for the solid, using the preCICE library, a free library for fluid–structure interaction. The Arbitrary Lagrangian–Eulerian formulation is used for the two-phase flow system to solve the Navier–Stokes equations in the fluid domain using the finite volume method. Simultaneously, the linear-elastic equation of the structure is solved using the finite element method. An implicit coupling scheme is considered at the fluid–structure interface. The numerical methodology is validated by the results given in literature for harmonic excitation at different frequencies. Subsequently, an analysis of complex external loading, such as Gabor wavelets and earthquake ground motion, is conducted to highlight the significant impact of the wall flexibility on sloshing, as well as the influence of hydrodynamic forces on the structure’s deformation. The proposed coupling methodology is robust and effective, it can be applied to all types of liquids and materials. A dataset of one of the studied cases is given as a supplement to the paper (Kha et al., 2024).

Experimental and numerical investigation of badarpur sand subjected to air-blast loading

In the present study, the results of experimental and numerical investigation of Badarpur sand subjected to air-blast loading generated from a laboratory-scale blast simulator are presented. The air-blast loading on the sand specimen is simulated experimentally using a vertical shock tube test facility to investigate the stress wave attenuation and vibrational characteristics of the Badarpur sand. The peak stress wave pressure and peak particle acceleration are recorded using the synchronized pressure transducers and triaxial accelerometers embedded inside the sand specimen. A stress enhancement of about 1.25-2.69 times the peak over-pressure is observed at the top one-third layer of the sand due to its compaction by the stress wave generated upon the blast wave impact. The stress wave intensity first increases and then rapidly decreases with the depth of the sand specimen. The amplitude of the peak particle acceleration is also observed to attenuate with the depth of the sand specimen. Further, based on the experimental results, the empirical equations are developed to predict peak particle velocity against the scaled distance with a power law index of 3.9, 2.5, and 3.0 for dense-coarse, dense-fine, and medium-dense fine sand, respectively. The numerical model of the vertical shock tube and sand specimens are also developed based on the Arbitrary Lagrangian-Eulerian formulation, commercially available in finite element simulation package LS DYNA and the results are validated with the experimental data.

Load history estimation for ballistic impacts with bullet-splash

The paper validates two simplified approaches to simulate the interactions between impactor and target during the so-called bullet splash phenomenon, i.e., the full fragmentation of the impactor without any penetration of the target. The approaches are based on treating the interactions between impactor and target as the deflection of a fluid flow; in one case the load history is estimated by the mass distribution of the bullet and its impact velocity; in the second case the interaction is simulated by means of an arbitrary Lagrangian Eulerian formulation. The two methods are applied and experimentally validated to simulate a monolithic .308 rifle bullet impacting on a high strength steel plate. The results demonstrate the suitability of the load history method for large scale finite element explicit simulations for the assessment of ballistic protection systems.

A three-dimensional finite element formulation coupling electrochemistry and solid mechanics on resolved microstructures of all-solid-state lithium-ion batteries

In this paper we present an all-solid-state lithium-ion battery (ASSB) model that considers electrochemistry and solid mechanics in a fully coupled manner. The model spatially resolves the three-dimensional microstructure of an ASSB cell and is based on non-linear continuum mechanics. The coupling of electrochemistry and solid mechanics is incorporated via lithiation-dependent volumetric changes of the active materials and the consistent formulation of the electrochemical governing equations on deformed geometries resulting in changed percolation paths. As the volumetric changes due to (de-)lithiation are reported to be large for several active materials, we introduce this effect using a multiplicative split of the deformation gradient. To also allow for large deformations in the mass conservation equation, an Arbitrary Lagrangian–Eulerian formulation is employed. Furthermore, we show the central steps to deduce the finite element formulation of our model as well as the adopted monolithic solution procedure to solve this strongly coupled non-linear problem. The linear solver used for large problem setups is adjusted from the advanced, physics-oriented preconditioning technique published in our previous work. In the numerical examples, we investigate three different geometries with up to about 1.3 million degrees of freedom and thereby prove that our approach is applicable to geometrically large and complex scenarios. Moreover, we show that our approach consistently introduces the mass and charge conservation on deformed meshes. Furthermore, the numerical examples demonstrate the high relevance to incorporate current collectors into the ASSB cell simulations. Finally, we show that solid mechanics and the coupling effects have indeed a large impact on the ASSB cell performance and their incorporation is therefore vital for precise results and predictions.

An arbitrary Lagrangian–Eulerian formulation for the Reynolds equation considering the JFO boundary condition

AbstractIn the realm of lubrication theory, the hydrodynamics of the fluid lubricant are commonly elucidated through the Reynolds equation. Its establishment stems from using the Eulerian description method, restricting the fluid domain to a fixed spatial configuration. However, certain scenarios necessitate coupling the Reynolds equation with solid deformation, wherein the fluid boundary undergoes alterations concurrent with the displacement or deformation of the solid. In such cases, the employment of a movable fluid domain becomes imperative. To address this requirement, this contribution introduces the arbitrary Lagrangian–Eulerian method, which effectively transforms the reference system of the Reynolds equation into a fluid domain amenable to arbitrary movement. Subsequently, the weak formulation associated with the movable fluid domain is derived and synergistically combined with a preexisting cavitation formulation. This integration facilitates an efficient solution of the Reynolds equation utilizing the finite element method while duly accounting for the mass‐conserving cavitation effects. Ultimately, a transient hydrodynamic lubrication example is presented to demonstrate the feasibility of the newly developed formulations.

Addressing geometric and material nonlinearities in fluid-structure interaction with the ALE-SSM framework

A novel methodology was introduced for integrating skeleton-based structural models (SSMs) with an Arbitrary Lagrangian-Eulerian (ALE) formulation to perform fluid–structure interaction (FSI) simulations (i.e., ALE-SSM). In ALE-SSM, the structural domain consists of force-based line elements, which are favored in modeling solid structures because of their computational efficiency. This paper enhances ALE-SSM to account for geometrically nonlinear and inelastic material responses of the structural domain. Geometric nonlinearity is introduced in two stages: (1) at the basic coordinate system of the force-based elements, where second-order effects stemming from the transverse element deformations are accounted for in the element flexibility matrix; and (2) at the global coordinate system, where a corotational formulation is used to perform the geometric transformation from the basic to the global coordinate system. The proposed approach for modeling geometric nonlinearity in ALE-SSM is evaluated with benchmark problems involving large deformations of beam structures positioned parallel and perpendicular to one-phase flows. Next, ALE-SSM is employed to define a new benchmark study that evaluates the effects of geometric and material nonlinearities in FSI simulations with force-based elements.

A 3D frequency domain finite element formulation for solving the wave equation in the presence of rotating obstacles

A frequency domain numerical method aimed at solving the propagation of sound waves in a three-dimensional domain enclosing rotating obstacles is presented. It relies on a domain decomposition whereby rotating parts are all embedded in a cylindrically-shaped domain. Following the classical Arbitrary Lagrangian Eulerian formalism, the governing equation, which is written in the local rotating frame, takes the form of a convected wave equation. The transmission conditions at the interface between the fixed and rotating domains is accomplished via the Frequency Scattering Boundary Conditions which, after classical FEM discretization, give rise to a series of coupled problems associated with a discrete set of frequencies. The performances of the method are demonstrated through several test cases involving rotating sources and/or obstacles inserted in circular and rectangular ducts.

Finite Element Modeling and Verification of the Plunge Stage in Friction Stir Welding

In the present work, a finite element model of the first stage, plunge, of friction stir welding is considered. The welded parts are flat aluminum 1050 plates. The arbitrary Lagrangian-Eulerian formulation implemented in ABAQUS/Explicit is used. The material model of the welded parts is the Johnson-Cook law, and Coulomb’s law describes the friction between the tool and the weldment. Temperature field during the process is obtained, and the influence of the parameters concerning the algorithmic implementation of the finite element method is established. The simulation results are compared with experimental results obtained for this purpose.

From the material behaviour to the thermo-mechanical long-term response of asphalt pavements and the alteration of surface drainage due to rutting: a sensitivity study

ABSTRACT In this contribution, a simulation chain for the thermo-mechanical and hydromechanical analysis (surface drainage) of a pavement structure during its whole service life is applied to a typical asphalt pavement (Bk100) used for motorways in Germany. Transient load signals for each tyre position of a truck-trailer combination (vehicle) are generated from a multibody analysis of the vehicle driving on a motorway with rough pavement surface. Material model parameters are derived from experimental identification tests of asphalt materials. The model parameters are used together with a continuum-mechanical description of the asphalt material representing the thermo-mechanical material behaviour at the material scale. The material response of the pavement layers is integrated on the structural scale by using the finite element method (FEM) in combination with an arbitrary Lagrangian-Eulerian (ALE) formulation and equivalent tyre loads of a representative, finite element (FE) discretised truck tyre rolling on an FE discretised pavement section. Different scenarios of rut formation are computed by varying different influence factors (climate temperature, vertical tyre force, type of asphalt material of the surface layers etc.). During the subsequent hydromechanical analysis of the pavement, the simulated deformed geometry of the pavement surface is used to numerically compute surface drainage characteristics.

A 2D elastic wall model to analyze the hyper‐viscous effects on blood flow across abdominal aortic aneurysm in COVID patients by fluid–structure interaction technique with ALE formulation

The worldwide dissemination of the coronavirus, as well as its dependencies, is being aided by detrimental urbanization. The impact of the viral infection upon an infected person is also thought to be influenced by any pre‐existing medical issues. Recent investigations have demonstrated that COVID‐19 patients have a higher degree of blood viscosity as a result of alterations in morphology in blood cells. Examining the importance of hyper‐viscosity in patients with COVID is crucial in determining the diseases of the arteries since viscosity is a key flow characteristic that affects the flow across a stenosis as well as an aneurysm. In the current study, a patient‐specific instance with an aneurysm across the abdominal aortal walls was taken into consideration. Given that abdominal aneurysms occur frequently in people, the high viscosity values are taken into account while examining the hyper‐viscous impact of fluid on its passage through the affected aorta. The current study is the initial endeavor to use the Arbitrary Lagrangian–Eulerian (ALE) Technique to study the impact of fluid–structure interaction (FSI) on the movement of fluid across an aneurysmal aortal section in conjunction with hyper‐viscous anomalous blood characteristics. The investigation backs up the different medical results that described the negative consequences of blood hyper‐viscosity. The computational findings from employing a finite element method (FEM) solver to resolve the fluid mechanical computations complemented alongside the solid mechanical principles indicate a possibility that the increased stress imposed by the hyper‐viscous propagates on the interior walls of the aneurysmal aorta may prompt the aneurysmal pouch to grow or breakdown. Also, the hyper‐viscous blood in COVID patients destroys the smooth muscle cells of the media layer endangering its normal structure and functions and leading to the development of new aneurysms.

A dynamic ALE formulation for structures under moving loads

This work describes the implementation of a novel dynamic Arbitrary LagrangianEulerian (ALE) formulation for the simulation of pavement structures loaded by rolling tires in a finite element framework. The proposed formulation enables the simulation of dynamic effects like acceleration, deceleration and variation of the wheel load on the pavement. The ALE scheme is described for a hyperelastic St. Venant-Kirchhoff material capable of finite deformations. With the adoption of this dynamic ALE formulation, a significant improvement in terms of speed and efficiency of the simulation is achieved in comparison to a classical transient Lagrangian formulation. This is primarily because only the relevant portion of the mesh around the applied load needs to be discretized and simulated. Another benefit is that a cumbersome moving load formulation does not need to be implemented. The results show satisfactory agreement with a conventional Lagrangian simulation with a moving load.