Lagrangian Eulerian Research Articles

Penetrator Penetrating Lunar Soil: Accurate Modeling and Parameter Analysis

This paper focuses on how to establish an accurate numerical model of penetrator penetration into the lunar soil and study the process of penetration. The lunar soil simulant was prepared according to published literature, and its plastic parameters were obtained by triaxial compression test. To determine the material model and parameters of the lunar soil simulant in numerical simulation, the corresponding numerical simulation was carried out for the triaxial test. Under the comprehensive consideration of accuracy and computational efficiency, the applicability of four numerical algorithms, including finite element method (FEM), arbitrary Lagrangian–Eulerian, smoothed particle hydrodynamics (SPH), and SPH–FEM, in the penetration simulation is evaluated. Finally, SPH–FEM is the optimal algorithm for the numerical simulation of the penetrator penetrating lunar soil. The convergence of the penetration model is tested. Finally, an accurate numerical model of the penetrator penetrating lunar soil is obtained. Then, the effects of incident velocity, incident angle, and angle of attack on penetration performance are studied, respectively, which provide reference for the design and process control of the penetrator project.

Gas–solid two-phase erosion of multi-elbow relief line under supercritical condition

The Eulerian–Lagrangian method was used to establish a numerical model of a multi-elbow pipeline erosion and examine the erosion of the relief line elbow under supercritical conditions. The model was used to examine the effects of the discharge volume, sand content, and shape factor of sand on multi-elbow erosion and those of the connection length on double-elbow erosion. The erosion mechanism and law of multi-elbow lines under gas–solid two-phase flow conditions were revealed. The results indicated that the most severe erosion of the elbow occurred on the impact surface. However, the erosion area of the multi-elbow was relatively long. In a multi-elbow pipeline, the maximum erosion rate of the outlet elbow was 84 times that of the inlet elbow. For the multi-elbow pipeline, the erosion rate was related to the discharge volume, sand content, particle shape, and length of the connection between elbows. The erosion rate of the elbow at the outlet decreased gradually and stabilized with an increase in the connection length.

Second-Order Time-Accurate ALE Schemes for Flow Computations with Moving and Topologically Changing Grids

In computations of unsteady flow problems by the arbitrary Lagrangian–Eulerian (ALE) method, the introduction of the grid velocity in the transport terms of the governing equations is not a sufficient condition for conservativeness if topology changes in the dynamic mesh are present and the number of mesh cells changes. We discuss an extension to second-order time differencing schemes (Implicit Euler and Crank–Nicolson) in the finite volume framework, to achieve second-order time-accuracy of the solution. Numerical experiments are given to illustrate the effectiveness of the presented method.

Structure-preserving discretizations of two-phase Navier–Stokes flow using fitted and unfitted approaches

We consider the numerical approximation of a sharp-interface model for two-phase flow, which is given by the incompressible Navier–Stokes equations in the bulk domain together with the classical interface conditions on the interface. We propose structure-preserving finite element methods for the model, meaning in particular that volume preservation and energy decay are satisfied on the discrete level. For the evolving fluid interface, we employ parametric finite element approximations that introduce an implicit tangential velocity to improve the quality of the interface mesh. For the two-phase Navier–Stokes equations, we consider two different approaches: an unfitted and a fitted finite element method, respectively. In the unfitted approach, the constructed method is based on an Eulerian weak formulation, while in the fitted approach a novel arbitrary Lagrangian–Eulerian (ALE) weak formulation is introduced. Using suitable discretizations of these two formulations, we introduce two finite element methods and prove their structure-preserving properties. Numerical results are presented to show the accuracy and efficiency of the introduced methods.

Investigation of the peak punch-through capacity of spudcan on sand overlying spatially variable clay

Accurately estimating peak penetration resistance in punch-through sensitive deposits is crucial for managing risks during jack-up rig installations. While offshore clays exhibit significant spatial variability, necessitating consideration of spatial variability in predicting peak penetration resistance. This study investigates the effects of spatial variability of underlying clay on the failure mechanism and magnitude of peak penetration resistance during punch-through incidents. The Modified Linear Estimation Method was used to generate three-dimensional random fields, incorporated into coupled Eulerian Lagrangian finite element analyses to simulate spudcan penetration from the ground surface. Results show that while spatial variability has an insignificant effect on the failure mechanism in the upper sand layer, it significantly alters soil flow in the underlying clay, leading to variations in peak penetration resistance. The peak penetration resistance follows a log-normal distribution, and a relationship was established between the probability of failure and the factor of safety. This study highlights the importance of considering soil spatial variability and demonstrates the significant impact of underlying clay on the failure mechanism and magnitude of peak penetration resistance during punch-through incidents, which might be helpful for the safety management of industry.

Effect of Laser Beam Profile on Thermal Transfer, Fluid Flow and Solidification Parameters during Laser-Based Directed Energy Deposition of Inconel 718.

The profile of the laser beam plays a significant role in determining the heat input on the deposition surface, further affecting the molten pool dynamics during laser-based directed energy deposition. The evolution of molten pool under two types of laser beam, super-Gaussian beam (SGB) and Gaussian beam (GB), was simulated using a three-dimensional numerical model. Two basic physical processes, the laser-powder interaction and the molten pool dynamics, were considered in the model. The deposition surface of the molten pool was calculated using the Arbitrary Lagrangian Eulerian moving mesh approach. Several dimensionless numbers were used to explain the underlying physical phenomena under different laser beams. Moreover, the solidification parameters were calculated using the thermal history at the solidification front. It is found that the peak temperature and liquid velocity in the molten pool under the SGB case were lower compared with those for the GB case. Dimensionless numbers analysis indicated that the fluid flow played a more pronounced role in heat transfer compared to conduction, especially in the GB case. The cooling rate was higher for the SGB case, indicating that the grain size could be finer compared with that for the GB case. Finally, the reliability of the numerical simulation was verified by comparing the computed and experimental clad geometry. The work provides a theoretical basis for understanding the thermal behavior and solidification characteristics under different laser input profile during directed energy deposition.

Blast-resistant design approach for RC bridge piers

Piers are the critical components to support the superstructure of bridge, but susceptible to the intentional and accidental explosions due to its easy accessibility. The blast-induced damage to the pier may cause the entire bridge to collapse, resulting in huge casualties and economy loss, and thus it is necessary to perform the blast-resistant design for the bridge pier. At present, the analytical model and blast-resistant design approach for blast-loaded reinforced concrete (RC) pier considering both the direct shear and flexural responses were studied. Firstly, six shots of field ground explosion test on 1/5-scale circular and square RC columns were carried out, in which the TNT charges with the scaled distances of 0.86–1.22 m/kg1/3 were adopted. Then, the corresponding numerical simulations incorporating the newly developed Structured Arbitrary Lagrangian Eulerian (SALE) solver were conducted, and the validity of finite element (FE) analysis approach was verified by comparing the experimental incident and reflected overpressure-time histories acting on the periphery of pier with deviations for most measuring points less than 20%. Afterwards, total 176 explosion scenarios were numerically simulated, and a blast loading distribution model on the pier for ground explosions was proposed, which considers the pier cross-section shapes (circular and square) and pier diameter/edge length (0.2–1.6 m), the scaled distances of explosive charge (0.3–2.3 m/kg1/3), as well as the blast wave reflection and diffraction by the pier. Furthermore, a Timoshenko beam-based multi-degree of freedom (MDOF) analytical model was established, and validated by both the existing explosion tests on RC columns and refined numerical simulations of blast-loaded bridge piers with corresponding deviations no more than 5%. Finally, a blast-resistant design procedure incorporating the above-proposed blast loading distribution and MDOF models was given, providing a reliable and efficient blast-resistant evaluation and design tool for bridge piers.

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.

WITHDRAWN: Effect of geo-material on dynamic response of tunnel subjected to surface explosion

Prime materials involved in a problem such as underground structures are concrete, reinforcement steel and geo-material surrounding the tunnel. Among these three materials, concrete and steel are manufactured materials and their properties can be controlled up to a certain extent. However, geomaterial is a naturally occurring material whose constitutive properties vary from region to region making it highly unpredictable. Findings from one study cannot be applied to other geotechnical problems directly, especially, in the case of tunnels subjected to surface explosion. The blast wave generated has to travel through the geo-material before it interacts with the tunnel. As the shock wave propagates radially its characteristics are most likely to get altered by the geo-material. Limited study has been carried out considering this problem. In the present study, the effect of various types of geo-material on the blast response of tunnels subjected to surface explosion is investigated. Finite element analysis has been carried out using LS-DYNA® wherein the problem has been modelled using multi-material arbitrary Lagrangian Eulerian (MM-ALE) method. Materials with fluid behavior such as air, explosive and soil are modelled using ALE formulation. Other materials including tunnel lining, reinforcing steel and rock are modelled using Lagrangian formulation. Blast loading is simulated using Jones-Wilkins-Lee (JWL) equation of state. Geo-materials considered for the comparative study are sandy loam, sandstone, and granite. Vertical displacement measured at the crown of the tunnel is used to determine the response of tunnel. Sandy loam soil being a highly compressible soil, exhibits non-linear and fluid-like behavior under high strain loading such as explosion. Tunnel undergoes extreme deformation in the case of sandy loam soil as compared rock cases. Additionally, the effect of weathering of rock on the tunnel response is investigated in the case of sandstone and granite. It was observed that weathering rock led to more displacement of tunnel crown as compared to intact rock.

A 2D frequency domain finite element formulation for solving the wave equation in the presence of rotating obstacles

A numerical method is presented to solve the propagation of sound waves in a two-dimensional domain in the presence of rotating obstacles without flow. It relies on a domain decomposition whereby rotating components are all embedded in a circular domain and the Arbitrary Lagrangian Eulerian framework which consists in writing the wave equation in the rotating reference frame. The transmission conditions at the interface between both domains is accomplished via the Frequency Scattering Boundary Conditions which, after classical discretization with the Finite Element Method (FEM), 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 of increasing complexity.

Experimental and Numerical Study on the Residual Axial Capacity of RC Bridge Piers after Contact Explosion

Bridge piers damaged by accidental or deliberate explosions lose a portion of their axial carrying capacity, which can induce partial and/or complete collapse of the entire bridge structure. With the aim of evaluating blast-damage and establishing blast-resistant design approaches for bridge piers, the dynamic behaviors and residual axial-load-bearing capacities (ALC) of RC piers subjected to contact explosions were studied experimentally and numerically. First, five circular cross-sectional 1/2-scale RC piers, with a height of 3 m and diameter of 450 mm, were fabricated. A field contact-explosion test on three pier specimens was conducted using TNT charge weights of 0.5, 1.0, and 2.0 kg. Then, these three postblast plus another two intact RC piers were transported to the laboratory to experimentally examine the axial bearing capacities using a hydraulic testing machine. The test data obtained included the incident overpressure–time histories and quantitative postblast damage profiles of the piers, and full curves for the axial force-displacement. Second, corresponding high-fidelity finite-element (FE) models of both the contact explosion and the succeeding axial compression tests were established. The refined numerical simulations were performed by adopting the fluid–structure interaction, multimaterial arbitrary–Lagrangian–Eulerian, and erosion algorithms implemented in the FE program LS-DYNA. The material model and the FE analysis approach were comprehensively verified by comparing the numerical simulated results with the test data. A series of numerical simulations were also conducted on seismically designed prototype bridge piers in order to examine the parametric influences on the damage mode and residual ALC, and the corresponding damage index of the piers. Finally, based on the parametric study, several blast-resistant design suggestions are proposed for prototype RC bridge piers against contact explosions.

Effect of soluble surfactant on the interface dynamics of a rising droplet

In this study, we present numerical simulations examining the impact of soluble surfactant on the interface dynamics of a rising droplet. To achieve this, the droplet interface is tracked using an arbitrary Lagrangian–Eulerian approach, and the bulk and interfacial surfactant concentration evolution equations fully coupled with the incompressible Navier–Stokes equations are solved. We systematically evaluate the boundary of interfacial dynamics evolution by varying certain dimensionless parameters. Specifically, we study the effects of changes in parameters such as the Langmuir number, the Biot number, the Damkohler number, the bulk Peclet number, and the elastic number on interfacial tangential velocity, interfacial concentration and its gradient, interfacial viscous shear stress, and droplet rising velocity. Our findings confirm the validity of the stagnant-cap model for describing the interfacial fluidity of a surfactant-laden rising droplet. Increasing the Langmuir number and decreasing the Damkohler number can inhibit interface fluidity, but there is a threshold for the Damkohler number. Additionally, the overall increase in interface tension may mask the hindering effect of the locally increased concentration gradient on the interfacial fluidity. The Biot number has no impact on the steady state of the interface, but a slow adsorption rate may result in a bimodal retardation before the interface reaches a steady state. A clear threshold exists for the Peclet number to hinder the interface velocity, and a too high Peclet number leads to strong nonlinearity in the interface physical quantities. Variations in the elastic number significantly affect the evolution of the interface, causing the interface velocity to pass through several states, ranging from almost no retardation, uniform retardation, stagnant-cap retardation to complete retardation.

Migration and heat transfer modeling of a neutrally buoyant melting particle in Poiseuille flow

A computational model is developed to simulate the hydrodynamic and heat transfer behavior of a melting cylindrical solid particle in plane Poiseuille flow between horizontal parallel plates. The two-dimensional transient conservation equations for mass, momentum, and energy are solved using a finite-volume scheme implemented on a deforming mesh, accounting for the rotation and non-uniform melting of the particle. An arbitrary Lagrangian–Eulerian (ALE) method is employed to directly track the moving and deforming solid–liquid interface during the particle migration and phase change. The developed model was validated by comparison with the previously reported numerical results for migration of a non-melting neutrally buoyant cylindrical particle in plane Poiseuille flow with heat transfer. The effects of flow Reynolds (Re), Grashof (Gr), and Stefan (Ste) numbers, as well as the initial position of the melting particle across the channel on the particle trajectory, melting rate, and average Nusselt (Nu) number were investigated. It was found that the melting rate increased by increasing Gr and Ste and decreased by increasing Re. The Nu did not change more than 20% with increasing Re number from 100 to 1000, but increased significantly by increasing Gr. It was also observed that by increasing Re, the particle migrates toward the channel center because of the stronger Magnus effect. Increasing Gr, on the other hand, pushes the particle to the bottom wall due to the stronger downward flow adjacent to the particle induced by buoyancy-driven convection (natural convection).

Temperature Distribution of Friction Stir Welded Al 7075 alloy using Finite Element Simulation along with Experimental Validation

Finite element simulation (FE simulation) and experimental validation of Friction Stir Welding (FSW) has been performed on 3 mm thick plates of AA7075 at rotational speed 1000 rpm and transverse speed 50 mm/min. It is difficult to determine the temperature distribution of the workpiece and tool interface experimentally, wherein temperature distribution of workpiece is responsible for frictional heat generation. The frictional heat generation is one of the important factors affecting the material flow and joint quality of FSW. In present work, finite element simulation using Coupled Eulerian Lagrangian approach has been performed to estimate the temperature of the workpiece and the tool. The simulation result shows the temperature distribution of the workpiece and the tool. The maximum temperature was attained by the tool. The FE simulation predicts 308 C temperature adjacent to the shoulder which corresponds to experimentally measured temperature ∼ 290 C. The simulation predicts 531 C as the maximum temperature at the tool shoulder whereas the experimentally measured maximum temperature at the tool shoulder is found to be > 400 C, showing good agreement between simulation and experiment. The tensile strength 62.74 % of the base alloy and elongation of 11.5 % have been achieved in the experimentally produced FSW sample at 1000 rpm rotational speed and 50 mm/min transverse speed. The nugget zone of FSW has higher hardness due to the recrystallized fine grain and HAZ has comparatively lower hardness due to grain coarsening.

Large eddy simulation of combustion instability in a subcritical hydrogen peroxide/kerosene liquid rocket engine: Intermittency route to period-2 thermoacoustic instability

This paper presents the first numerical evidence of an intermittency route to period-2 thermoacoustic instability in a subcritical single-element liquid rocket engine burning hydrogen peroxide/kerosene as we decrease the equivalence ratio (ϕ) from fuel-rich to fuel-lean. To achieve this, three-dimensional compressible large eddy simulation algorithms combined with the Euler–Lagrangian framework are used. A one-equation eddy sub-grid turbulence model with a partially stirred reactor sub-grid combustion model is employed to simulate the spray turbulent combustion process in a high-pressure liquid-fueled combustor based on open-source platform OpenFOAM. This paper focuses on examining the transition process of the dynamical states in the thermoacoustic system and the synchronization between multiple subsystems. The results indicate that, as the equivalence ratio reduces continuously (1.5 ≤ ϕ ≤ 0.5), the system dynamics shift from period-1 oscillations (ϕ = 1.5) to period-2 oscillations (ϕ = 0.5) via intermittency (1.3 ≤ ϕ ≤ 0.9). Under the equivalence ratio of 0.7 (ϕ = 0.7), a transient mode switching between period-1 and period-2 was also observed. The synchronization processes between the pressure and combustion subsystems in terms of phase-locking and frequency-locking are responsible for the emergence of complex dynamical states. The cycle snapshots analysis also provides more details on the synchronization processes between the pressure and the multiple subsystems, such as vortex dynamics, mixture fraction, and combustion heat release. In summary, this paper sheds light on the complex non-linear thermoacoustic oscillations and the underlying physical mechanisms related to the two-phase flow of spray combustion in liquid rocket engines using three-dimensional large eddy simulations, paving the way for developing passive or active control methods.

Configurational forces and ALE formulation for geometrically exact, sliding shells in non-material domains

The paper presents a novel Arbitrary Lagrangian–Eulerian formulation for dynamic problems of geometrically exact, sliding shells. In contrast to previous researches where the sliding motions are prescribed at sliding boundaries and the existence of configurational forces is not identified, the configurational momentum equation that governs the evolution of sliding motions is derived in a consistent variational framework in the present paper. To this end, the time-varying material domain due to the presence of sliding motion is mapped onto a time-invariant mesh domain, variations at fixed material and at fixed mesh coordinates are introduced, and their relationship with variation of material coordinates at fixed mesh coordinates are derived. Hamilton’s principle of variation of action is employed to derive the strong form of mechanical and configurational momentum equations together with natural boundary conditions at the sliding boundaries. In the finite element formulation, transfinite interpolation is employed to relate material coordinates of nodes inside the domain to the values at the sliding boundaries. The discrete form of Hamilton’s variational principle leads to discrete governing equations of the proposed ALE formulation. The generalized-α scheme is adjusted to integration the resulting mechanical and configurational momentum equations. Numerical examples are presented to validate correctness and efficiency of the proposed formulation.

Interaction of two cylinders immersed in a viscous fluid. On the effect of moderate Keulegan–Carpenter numbers on the fluid forces

This work deals with the hydrodynamic interaction of two parallel circular cylinders, with identical radii, immersed in a viscous fluid initially at rest. One cylinder is stationary while the other one is imposed a harmonic motion with a moderate amplitude of vibration. The direction of motion is parallel to the line joining the centers of the two cylinders. The two dimensional fluid–structure problem is numerically solved by the Arbitrary Lagrangian–Eulerian method implemented in the open-source CFD code TrioCFD. First, we show that the fluid forces on the two cylinders are aligned with the direction of the imposed motion. Second, we show that the moderate oscillations of the moving cylinder create nonlinear effects in the fluid that strongly affect the characteristics (Fourier harmonics) of the hydrodynamic force acting on the stationary cylinder. The fluid force on the moving cylinder is shown to be poorly affected by the nonlinear effects, which makes it possible to extend the linear concept of self-added mass and damping coefficients. First, we show that the self-added coefficients decrease as Sk−1/2, with Sk the Stokes number (dimensionless number constructed from the imposed vibration frequency). Second, we show that the self-added mass (resp. damping) decreases (resp. increases) as −KC3 (resp. +KC3), with KC the Keulegan–Carpenter number (ratio between the imposed amplitude vibration and the separation distance between the cylinders). These variations are included in new power laws derived from nonlinear regressions of the numerical results. These new power laws for the self-added coefficients combine the effect of both Sk and KC, covering the viscous (Sk≥500) and weakly nonlinear (KC≤0.3) regimes.

Numerical Study on Asteroid Deflection by Penetrating Explosion Based on Single-Material ALE Method and FE-SPH Adaptive Method

An asteroid impact can potentially destroy life on this planet. Therefore, asteroids should be prevented from impacting the Earth to impede severe disasters. Nuclear explosions are currently the only option to prevent an incoming asteroid impact when the asteroid is large or the warning time is short. However, asteroids exist in an absolute vacuum, where the explosion energy propagation mechanism differs from that in an air environment. It is difficult to describe this process using standard numerical simulation methods. In this study, we used the single-material arbitrary Lagrangian–Eulerian (ALE) method and the finite element-smoothed particle hydrodynamics (FE-SPH) adaptive method to simulate the process of deflecting hazardous asteroids using penetrating explosions. The single-material ALE method can demonstrate the expansion process of explosion products and energy coupling in absolute vacuum. The FE-SPH adaptive method can transform failed elements into SPH particles during the simulation, avoiding system mass loss, energy loss, and element distortion. We analyzed the shock initiation and explosion damage process and obtained an effective simulation of the damage evolution, stress propagation, and fragment distribution of the asteroid. In addition, we decoupled the penetrating explosion into two processes: kinetic impact and static explosion at the impact crater. The corresponding asteroid damage modes, velocity changes, and fragmentation degrees were simulated and compared. Finally, the high efficiency of the nuclear explosion was confirmed by comparing the contribution rates of the kinetic impact and nuclear explosion in the penetrating explosion scheme.

Conditionally Eulerian–Lagrangian Method on the Example of the Problem on the Dynamics of an Intrathermocline Vortex Lens

Conditionally Eulerian–Lagrangian Method on the Example of the Problem on the Dynamics of an Intrathermocline Vortex Lens

Numerical investigations of the failure mechanism of spreading landslides

Spreading landslides are geohazards that often occur in sensitive clay areas. A unique characteristic of the spreads is that an initial small local slope failure may cause a catastrophic landslide above the horizontal failure surface, forming horsts and grabens. Although many hypotheses were proposed to explain the failure mechanism of spreads, some topography of the landslides observed from field investigations cannot be well explained by the existing hypotheses. This study revisits the 2010 Saint-Jude spreading landslide by using the coupled Eulerian–Lagrangian approach within ABAQUS to reveal the failure mechanism of spreading landslides. Two types of cross-sections (static and dynamic cross-sections) were selected to monitor the total horizontal force of the sliding mass during the process of migration. It was found that there are two spreading failure mechanisms to form horsts and grabens. The first one is the static spread failure mechanism that global failure occurs accompanied by spreading failure. The other one is the dynamic spreading failure mechanism that after the global failure, the sliding masses break into horsts and grabens during the forward movement under the pushing of the sliding mass at the back and the blocking action of the soil mass in the front. These two failure mechanisms of spreads can well explain various geomorphologic shapes found in the Saint-Jude landslide.