Eulerian Finite Element Method 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.

An Eulerian finite element method for the linearized Navier–Stokes problem in an evolving domain

Abstract The paper addresses an error analysis of an Eulerian finite element method used for solving a linearized Navier–Stokes problem in a time-dependent domain. In this study, the domain’s evolution is assumed to be known and independent of the solution to the problem at hand. The numerical method employed in the study combines a standard backward differentiation formula-type time-stepping procedure with a geometrically unfitted finite element discretization technique. Additionally, Nitsche’s method is utilized to enforce the boundary conditions. The paper presents a convergence estimate for several velocity–pressure elements that are inf-sup stable. The estimate demonstrates optimal order convergence in the energy norm for the velocity component and a scaled $L^{2}(H^{1})$-type norm for the pressure component.

Numerical investigation of the underwater explosion of a cylindrical explosive with the Eulerian finite-element method

The shock wave and bubble dynamics of an underwater explosion are significant in various fields. When the charge is non-spherical, the detonation process will remarkably affect the shock wave formation and the subsequent bubble motion. In this work, the underwater explosion of a cylindrical explosive is investigated numerically with the Eulerian finite-element method combined with the programed burn model treating the detonation process. The present model is validated by comparing the simulated results with the experimental ones. Then, several cases with different slenderness of the explosive charge in various buoyancy environments are simulated and analyzed. The results demonstrate a notable variation of the shock wave in different directions. The shock wave will reach the highest pressure peak and shortest pulse width at a certain angle determined by the ratio between the speeds of the detonation wave and the shock wave. Furthermore, the non-spherical initial expansion of the bubble casts a significant influence on the subsequent bubble evolution. Three typical jet morphologies are identified with different combinations of buoyancy parameter and oblateness ratios of the bubble, featured by a slightly oblique upward jet penetrating the bubble, a laminar jet that failed to penetrate the bubble continuously, and a pair of opposite horizontal jets penetrating the bubble. Meanwhile, the horizontal jets that happen under a weak buoyancy environment will reduce the upward migration.

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.

An accurate and robust Eulerian finite element method for partial differential equations on evolving surfaces

An accurate and robust Eulerian finite element method for partial differential equations on evolving surfaces

Numerical investigation on the interaction of an oscillating bubble with the interface of a non-Newtonian fluid

The dynamics of an oscillating bubble near a liquid–liquid interface is a complex multiphase flow problem due to the highly nonlinear interaction, such as interface fragmentation and bubble tearing. When one of the liquid mediums is non-Newtonian, its constitution would significantly influence both the bubble motion and the interface evolution. In this study, a numerical model is established based on the Eulerian finite element method with the non-Newtonian fluid described by the Herschel–Bulkley model. The numerical model is validated by comparing with experimental results for a non-spherical pulsating bubble at a water–oil interface and the analytical solution for the laminar flow of non-Newtonian fluids in a circular tube. According to the simulation and analysis with different case parameters, our findings suggest that the non-Newtonian fluid forms a crater when squeezed by the bubble, and the downward jet can penetrate the bubble and further deepen the crater. As the distance parameter increases, the crater gradually disappears or even bulges. Furthermore, the yield shear stress can give non-Newtonian fluid plastic properties similar to a solid, effectively reducing the bubble's pulsation and jet load. Additionally, the flow behavior index n comes from the power-law model for non-Newtonian fluids and significantly impacts the jet's impact process. When n≤1, the crater is likelier to become funnel-shaped, whereas when n &amp;gt; 1, it is likelier to become bullet-head-shaped. In addition to enhancing the bubble's nonsphericity, the reduction in Reynolds number also weakens the crimp deformation of the interface. When the distance parameter is zero, the larger the buoyancy parameter and the less deformable the non-Newtonian fluid, the easier the bubble to split by the annular jet.

Numerical analysis of nonlinear interaction between a gas bubble and free surface in a viscous compressible liquid

Liquid viscosity has a potential effect on bubble dynamics. This paper is concerned with bubble dynamics in a compressible viscous liquid near the free surface. The liquid–gas flow is modeled using the Eulerian finite element method coupled with the volume of fluid method. The numerical results have been shown to be in excellent agreement with those from the spherical bubble theory and experiment. Parametric studies are carried out regarding the Reynolds number Re and the stand-off parameter γd. It clearly demonstrated that the liquid viscosity inhibits bubble pulsation, jet flow, free surface jet, and bubble splitting. Quantitatively, as Reynolds number Re decreases, the maximum bubble volume, jet tip velocity, free surface spike, and crown height decrease, and the toroidal bubble splitting weakens. As the stand-off parameter γd increases, the maximum bubble volume, jet velocity, and bubble average pressure peak increase while the height of the free surface spike decreases. Close observation reveals that the free surface crown tends to disappear at small Re or large γd, further indicating the complex mechanism behind the crown spike evolution.

Characteristics and load reduction method of the deep-sea implosion of the ellipsoidal and egg-shaped pressure hulls

Pressure hulls is an essential part of the deep-sea submersible and may implode to emit shock pressure wave. This paper establishes a deep-sea implosion model based on the Eulerian finite element method, and two implosion conditions, ellipsoidal and egg-shaped, are studied. It is found that the pressure load at the gauging point in the long semi-axis direction is always greater than that in the short semi-axis direction when the two models implosion. Introducing the radius ratio β, it is found that the pressure load after the implosion of the two models increases first and then decreases with the increase of β. Then the dimensionless number Ma is introduced to simulate the change of pressure load under implosion in different water depths. It is found that pressure peak Pm∗ decreases by a factor of Ma−0.5 as Ma increases, and the time Tm∗ to reach the pressure peak increases linearly with the rise of Ma. Therefore, the pressure load caused by implosion can be reduced by reasonably selecting the radius ratio of the pressure hull. In addition, the implosion load when a plurality of pressure hulls are arranged together can be reduced by organizing them along the short semi-axis direction.

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.

Numerical investigation on jet penetration capacity of hypervelocity shaped charge in underwater explosion

Shaped charges are widely employed in both military and civilian applications owing to their strong penetration capacities, such as weapon production, oil exploitation, mining, and tunneling. To augment the armor-breaking capability of a shaped charge, it is of utmost significance to enhance the velocity of the metal jet. Therefore, the hypervelocity shaped charge (HSC) has been proposed as a potential solution. Compared with ordinary shaped charges, HSCs can cause severer damage to the target owing to the faster velocity of its jet head. The Eulerian Finite Element Method (EFEM) is developed to build the axisymmetric numerical model, and it can simulate shockwave propagation, jet penetration, and target destruction. The model is validated by the experiments, and the convergence analysis is also performed. And the fundamental characteristics of HSC in underwater explosions including the metal jet and shockwave are investigated. Subsequently, the HSCs with the different liner truncation heights HL and liner materials are simulated to obtain the optimal parameter (range) for penetration capacity. Analyzing the numerical results, when 8mm≤HL≤13mm or the liner is made of aluminum, the metal jet generated by HSC will have a greater penetration capacity to the steel plate through the comparison of the crevasse diameter and residual velocity.

Experimental and numerical investigations on the explosions nearby a free surface from both sides

The explosion near a free surface is a complex multiphase flow problem involving large deformation and even fragmentation of the fluid interface. Experiments and numerical simulations are presented to investigate the transient fluid dynamics of explosions approaching the free surface from the air and water sides. The numerical model is established based on the Eulerian finite element method and the Volume of Fluid method. The explosion experiments are conducted in a water tank. We classify the problem into cavities and bubbles according to whether the explosive is exposed to the air. Explosions above the free surface generate hemispherical cavities that evolve into W-shaped ones after an upward jet develops from the bottom. The hemispherical cavity will close if the explosion is close enough to the free surface. The simulations reveal that the distance parameter dominates the bubble’s bursting and that the explosive equivalent and distance parameter compete to determine the cavity’s closure. In addition, the impact pressure characteristics generated by the explosion near the free surface are also analyzed.

Investigation of hydrodynamics of water impact and tail slamming of high-speed water entry with a novel immersed boundary method

High-speed water entry is a transient hydrodynamic process that is accompanied by strongly compressible flow, free surface splash, cavity evolution and other nonlinear hydrodynamic phenomena. To address these problems, a novel fluid–structure interaction (FSI) scheme based on the immersed boundary method is proposed which is suitable for strongly compressible multiphase flows. In this scheme, considering the multiphase interfaces at the immersed boundary, an improved immersed boundary method for effectively suppressing the non-physical force oscillation is proposed. Additionally, a quaternion-based six degrees of freedom motion system is used to describe rigid body motion, and the multiphase flow Eulerian finite element method is applied as the fluid solver. Using analytical solutions, experimental data and literature data, the accuracy and robustness of the FSI scheme are validated. Finally, the high-speed water entry of the slender body with different noses is investigated, and the hydrodynamic loads including the axial and normal drag forces and the bending moment are extensively discussed. The hydrodynamic load and motion trajectory are determined by the nose configuration. The tail slamming phenomenon is the primary focus, and it is revealed that its formation is primarily related to the pitch moment formed at the stage of crossing the free surface. Tail slamming also causes violent impact loads, especially bending moments, which may cause slender projectiles to break off. Finally, to combine the features of the flat and hemispherical noses, the water entry of the projectile with a truncated hemispherical nose is simulated and discussed.

Analysis and mitigation of spatial integration errors for the material point method

SummaryThe material point method (MPM) is a robust numerical simulation approach for continuum mechanics problems involving large material deformations coupled to changing surface topographies. These types of problems are present in many different engineering contexts, from understanding the failure processes of earthen slopes to predicting the strengths and failure mechanisms of body armor to modeling the impact forces of waves in fluid tanks. By using a set of persistent material point tracers to follow the motion and deformation of the continuum material while solving the equations of motion on a static simulation grid, the MPM avoids several shortcomings of more traditional numerical approaches including blurring of material surfaces — as in Eulerian finite element or finite volume methods (FEMs or FVMs) — and mesh tangling — as in Lagrangian FEMs. Despite its robustness, MPM is known to develop significant numerical errors: namely, (i) the particle ringing instability and (ii) solution dependent discretization and integration errors. In this work, we present an analysis of local‐in‐time, spatial integration errors in the MPM and several techniques designed to mitigate these errors. Error mitigation approaches previously described in the literature are compared to a new method we propose for problems involving very large material deformations. The proposed method is shown to offer substantial improvement over standard MPM for simulations of fluid‐like materials without requiring significant augmentation of existing MPM frameworks.

Coupling between a bubble and a liquid-liquid interface in viscous flow

The bubble dynamics are of great significance in the scientific community and have wide applications. Due to the complex coupling effect, high nonlinearities arise when the bubble is initiated at a liquid-liquid interface. Based on the Eulerian finite element method, the coupling model of a bubble and the liquid-liquid interface is established considering the viscosity and is verified by comparing with the experiments. It is found that the bubble only generates a downward jet at a large Reynolds number. In contrast, a pinch-off effect occurs due to an annular jet impact, which generates two opposite jets at a small Reynolds number. And the bubble migrates upward with the increase of buoyancy parameters. Moreover, through dynamic analysis, we find three reasons for the pressure peaks above the bubble: the annular jet impact, the downward jet impact, and the upward jet passing the measuring point. It is found that the pressure peaks are nonlinearly related to viscosity and buoyancy, and they can enhance each other to generate higher pressure peaks.

Jetting and migration of a laser-induced cavitation bubble in a rectangular channel

The jetting behaviour and migratory characteristics of a laser-induced cavitation bubble in a rectangular channel are investigated both experimentally and numerically, for various combinations of the geometric and physical parameters of the system. High-speed photography is used to visualize the temporal development of the bubble shape, the formation of liquid jets during bubble collapse, and the bubble displacement in contact with the sidewalls of the channel during two oscillation cycles of the bubble. The bubble profiles, pressure contours and velocity vectors ambient to the bubble are obtained through numerical simulation results by using an Eulerian finite element method with a compressible liquid impact model. The jetting behaviour of the bubble varies between single jet formation and the formation of three liquid jets directed towards each wall of the channel. The numerical calculations indicate that the liquid jets directed towards the sidewalls of the channel reach maximum velocities of 100 m s−1while the peak velocity of the liquid jet directed towards the channel endwall is about 55 m s−1. A small bubble generated close to a sidewall of the channel develops only a single inclined jet during collapse. Such jets can reach velocities of up to 110 m s−1. A bubble displacement in contact with the sidewalls of the channels of 350 μm was observed during the first two oscillation cycles for a bubble with a maximum diameter slightly smaller than the height of the channel. The results of our investigations are compared to previous results obtained in similar configurations.

A design method of hopper shape optimization with improved mass flow pattern and reduced particle segregation

A shape optimization method is presented in this paper to re-design hopper shapes for improving the flow patterns in silos. This method combines a continuum model of granular matter based on the Eulerian Finite Element Method, the optimization algorithms of genetic algorithm and the gradient descent method. Starting from a classic conical shape, the optimization method searches for the optimal shape of the hopper under given geometrical constraints. An optimized curve design can increase the mass flow zone in the hopper by more than 90%. The sensitivity of this method suggests that the shape optimization is of particular significance for initially funnel flow hoppers and granular materials with high internal friction. The optimized design was further characterized by using the Discrete Element Method. The DEM results demonstrate that the optimized hopper can also dramatically reduce particle segregation by around 70% during hopper discharge.

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.

Optimal convergence of arbitrary Lagrangian–Eulerian iso-parametric finite element methods for parabolic equations in an evolving domain

Abstract An optimal-order error estimate is presented for the arbitrary Lagrangian–Eulerian (ALE) finite element method for a parabolic equation in an evolving domain, using high-order iso-parametric finite elements with flat simplices in the interior of the domain. The mesh velocity can be a linear approximation of a given bulk velocity field or a numerical solution of the Laplace equation with specified boundary value matching the velocity of the boundary. The optimal order of convergence is obtained by comparing the numerical solution with the ALE-Ritz projection of the exact solution, and by establishing an optimal-order estimate for the material derivative of the ALE-Ritz projection error.

Innovative meshless approach for shaped charges applications

We focus here on modelling shaped charges. Combining large deformations, numerous interfaceproductions, and strong damage mechanisms, those events are particularly challenging from a numerical point of view. Eulerian finite element methods are classically used for such modeling.However, they induce very long computation times, accuracy losses (projection algorithms), anddifficulties with opening criteria related to jet fragmentation. Among the Lagrangian approaches, the meshless method called Smoothed Particle Hydrodynamics (SPH) appears as a relevant alternative to prevent such shortcomings. Based on a set of moving interpolation points, it disregards any connectivity between its elements which makes it naturally well suited to handle material failure. Nevertheless, SPH schemes suffer from well-known instabilities questioning their accuracy and activating nonphysical processes,such as numerical fragmentation. Many stabilizing tools are available in the literature however, they either raise conservation and consistency issues or drastically increase the computation times. We propose then to use an alternative scheme called γ-SPH-ALE. Based on the ALE framework, it achieves robust and consistent stabilization through an arbitrary description of motion. Thanks to CFL-like conditions obtained through a nonlinear stability analysis, the scheme stability is ensured. By preventing spurious oscillations in elastic waves and correcting the so-called tensile instability, both stability and accuracy are increased regarding classical approaches. Also, taking advantage of GPU computing, such results are achieved in reduced computation times contrary to classical CPUimplementations. Its implementation on a “Viper” shaped charge shows that the scheme handles the jet generation process as well as its resulting interaction with a target.

Research on characteristics of deep-sea implosion based on Eulerian finite element method

Marine structures operating in the deep-sea, such as submersibles and photomultiplier tubes, bear enormous hydrostatic pressure of deep-sea, which may cause structural damage and implosion. And a pressure wave will emit from the collapse of the cavity and pose a threat to nearby structures. On the basis of the Eulerian finite element method, the numerical model of multiphase flow for deep-sea implosion is established in this paper. The numerical method is verified by comparing the results with those from Keller equation and literature. The physical process under the spherical simplification is firstly analyzed in our work. The basic phenomena are studied, and the pressure changes under different distances and Mach numbers are given. Subsequently, the basic physical process and pressure distribution under cylindrical simplification are introduced. Then, the bubble evolutions under different Mach numbers and initial aspect ratios are given. In the early stage of bubble movement, a rarefaction wave will radiate to the surrounding. The multi-peak phenomenon will occur in the cylindrical model because of the bubble sides breakage and the bubble pulsation. Finally, asymmetric three-dimensional cylindrical bubble implosion near the solid wall with different standoff distances are present in this paper, where the typical oblique jet is observed and analyzed.