Lagrangian Eulerian Research Articles

Preform geometry determination for a connecting rod forging by CEL model in Abaqus™

AbstractForging is a widely used manufacturing process, and its design and modeling are important to reducing production costs, increasing die lifespan, and improving the mechanical properties of the final product. In this study, the forging process of a connecting rod was modeled using 3D coupled Eulerian Lagrangian (CEL) analysis by FEM. The methodology adopted achieved to determine a preform geometry that reduces final flash and forging load, while ensuring complete filling of the stamp. Starting from the final geometry, the final die was designed. After the first result for an approximately 27% of flash, the material distribution was adjusted decreasing it at the regions where the flash was too large. After an iterative method was applied to determine better preform, a proposal was found that reduced forging force by approximately 42% and the percentage of flash volume by 64% in comparison with the first one. A final flash of about 10% is considered a good objective to reach. Lower values may cause many iterations, not a significant difference in forging loads, the risk of an unfilled die, and complex preform geometries.

Strain Rate Sensitivity and Constitutive Law of Closed-Cell PVC Foams under Shock.

The mechanisms of the strain rate dependence of closed-cell PVC foams under shock were numerically studied based on a cell-based model combined with the Coupled Eulerian-Lagrangian (CEL) method in this paper. The strain rate effect of the base material and the entrapped gas effect were focused. The results show that the strain rate effect of the base material has a significant influence on the stress magnitude in the regions before and after the shock front, and the entrapped gas mainly affects the velocity field. Both the strain rate effect of the base material and the entrapped gas have a notable influence on the strain distribution. Taking PVC foam with a relative density of 0.07 as an example, the strain rate effect of the base material will increase the impact stress by 45% and reduce the impact strain by 0.04. The entrapped gas will reduce the impact strain by 0.18, and its effect on the impact stress can be ignored. Finally, two constitutive laws considering the strain rate effect and entrapped gas effect were proposed and compared for the PVC foam under shock with one based on the Hugoniot relationship and the other based on the D-RPH model.

Hybrid Eulerian–Lagrangian framework for structural full-field vibration quantification and modal shape visualization

Vibration measurement and modal analysis under a high spatial and temporal resolution have high significance for revealing inner structural characteristics. This work develops a hybrid Eulerian–Lagrangian framework for structural full-field dynamic vibration quantification and modal shape visualization and validates its performance through vision-based vibration experiments. By assuming a linear vibration system, modal responses that are estimated from the vibration signals on a structural surface provide a reliable basis for manipulating temporal and spatial vibrations in Eulerian motion processing. Then, dynamic spatial motion can be quantified by using the improved non-rigid image registration algorithm. Meanwhile, from the Lagrangian perspective, high-quality visualization of modal shapes is implemented by compensating spatial motion according to magnified dense displacement fields. The experiment results showed that the proposed hybrid framework has superiority in controlled modal tests and has potential for more practical applications.

Modeling the thermophoretic impact on nanoparticle production in an enclosed FSP reactor

For the accurate modeling and simulation of the flame spray pyrolysis (FSP) process, all the essential phenomena need to be taken into account. A validated model could then be used for process optimization and design with reasonably reduced costs. In that sense, this work uses a combination of computational fluid dynamics (CFD) and a population balance model (PBM) to investigate the influence of thermophoresis and other parameters (related to sintering and turbulent particle diffusion) in the final produced particle size. This work simulates the flame spray using CFD under an Eulerian–Lagrangian framework. Furthermore, a monodisperse solver for a bivariate PBM is implemented into ANSYS Fluent to simulate the formation and evolution of nanoparticles inside the reactor. We investigate the production of zirconium dioxide (ZrO2) nanoparticles and use experimental data from the literature for model validation. The model produces accurate results for some of the investigated cases.

Tension Characteristics of Cable-Driven System Based on ALE Formulation

This paper mainly studies the tension characteristics of cables in Cable-Driven Parallel Mechanisms (CDPMs). The exact catenary model and cable model based on the Arbitrary Lagrangian Eulerian description (ALE) are used. First, the ALE cable model and the exact catenary model are used to analyze and compare the tension at each point of a single cable, which proves the calculation accuracy of the ALE cable model. Then, the cable model described by ALE is used to model the Cable-Driven Parallel Mechanisms, the end effector is enabled to do a specific helical motion, the tension characteristics of the cable are extracted and analyzed, and the mesh independence of the ALE elements and calculation convergence are verified.

Vulnerability analysis of HSR bridge under near-field blast based on response surface method

In this paper, the vulnerability analysis of a high-speed railway (HSR) bridge system attacked by the near-field blasts is conducted using the response surface method. An “explosive-air-ground” coupling analysis model is established by the ANSYS/LS-DYNA software. Through a comparison of the numerical simulated and the experimental results, the effectiveness of the Arbitrary Lagrangian Eulerian (ALE) algorithm and the accuracy of the analysis model are verified. Then, the dynamic responses of an HSR simply-supported bridge subjected to four types of car-bomb blasts are calculated. The damage levels of the bridge components are evaluated using the residual bearing capacities of the pier and beam, and the shear displacement of the bearings. By adopting the central composite design method, the response surface models for the evaluation indices of the bridge components are fitted. Furthermore, the failure probabilities of each component at various damage levels are calculated, and the vulnerability of the bridge system is analyzed based on the Boundary Estimation Method. The proposed method could predict the damage probability of the bridge system under various car-bomb blasts, and provide a reference for exploring a probability-based evaluation method for the post-blast safety of the HSR bridges.

Analysis of the Effect of Wear on Tire Cornering Characteristics Based on Grounding Characteristics

Electric vehicles can lead to accelerated tire wear, an inevitable phenomenon during tire usage that can affect the cornering characteristics determining handling stability. In order to simulate tire wear, a finite element model for tire wear was established using the UMESHMOTION subroutine and Arbitrary Lagrangian–Eulerian (ALE) adaptive meshing in ABAQUS, which is based on the Archard theory. The tire’s cornering characteristics were analyzed based on the obtained worn tire. The research results demonstrate that as the wear amount increases, the cornering stiffness and aligning stiffness of the tire also increase. When there are differences in wear on both tire shoulders with the same global wear, the change in cornering stiffness is not significant, while the aligning stiffness exhibits noticeable differences. To explain the above phenomenon, grounding characteristics were incorporated as mediator variables. The analysis results indicate that wear has an impact on the grounding characteristics. Additionally, statistically significant correlations exist between grounding parameters and cornering characteristics. In conclusion, wear affects the tire’s cornering characteristics by changing the grounding characteristics.

A computational framework for crack propagation along contact interfaces and surfaces under load

We present the first implicit computational framework for simulating crack propagation along contact interfaces and surfaces under load in three-dimensional bodies, which is distinct from modelling the contact interaction associated with crack closure. We restrict ourselves to brittle fracture and frictionless contact and focus on numerical challenges associated with the coupling of unilateral constraints emerging from the Griffith’s criterion and the contact conditions. The formulation is based on the configurational mechanics framework and is solved using the finite element method. The approach utilises a monolithic Arbitrary Lagrangian–Eulerian formulation permitting simultaneous resolution of crack propagation and unilateral contact constraints. Contact is embedded in the model using the well-known mortar contact formulation. Evolving cracks are explicitly modelled as displacement discontinuities within the mesh. Heterogeneous approximation of arbitrary order is used to discretise spatial displacements, enabling hp-adaptive refinement around the crack front and the contact interfaces traversed by the crack. The result is a holistic approach which handles issues associated with thermodynamic consistency, numerical accuracy and robustness of the computational scheme. Several numerical examples are presented to verify the model formulation and implementation; they also highlight how contact pressure and load applied on surfaces traversed by cracks influence their propagation. The robustness of the approach is validated by comparison of our simulations with existing numerical results and an industrial experiment involving cracks of complex morphologies propagating along contact interfaces between multiple deformable bodies.

An arbitrary Lagrangian–Eulerian method for analyzing finite-amplitude viscous acoustic waves radiated from vibrational solid boundaries: An implicit method

This paper is concerned with the development of an arbitrary Lagrangian–Eulerian (ALE) method for predicting nonlinear acoustic waves radiated from large-amplitude vibrational solid boundaries immersed in viscous fluids of infinite extent. The method is formulated based on a nonlinear acoustic model characterized by velocity potential and acoustic pressure. Unsplit implementations of three perfectly matched layer (PML) techniques are presented, namely conventional PML, convolution PML, and complex frequency-shifted PML (CFS-PML), to enable bounded-domain modeling of nonlinear acoustic waves propagating in an unbounded fluid medium. The long-time stability characteristics of the three distinct PML techniques are examined through theoretical analysis and numerical verification of their ability to absorb evanescent waves. Our results reveal that the CFS-PML technique exhibits superior long-time stability performance compared to the other two techniques. The CFS-PML is also compared to two different absorbing layer techniques: the Kosloff Absorbing Layer (KAL) technique and the Acoustic Black Hole (ABH) technique. The finite element method is adopted for spatial discretization of the finite-amplitude acoustic wave model, and an implicit predictor–corrector algorithm is proposed for time advancement. Several benchmark problems, including acoustic waves produced by a vibrational piston, and transversely oscillating and pulsating cylinders, are analyzed by the proposed method. It is found that the proposed method attains a second-order rate of convergence for spatial discretization, and the convergence rate in terms of the time step size is about first-order. The application of the method to acoustic levitation problems is also demonstrated, and the nonlinear phenomenon of acoustic waves in a single-axis acoustic levitator is discussed.

Fluid Dynamics Calculation in SF6 Circuit Breaker during Breaking as a Prerequisite for the Digital Twin Creation

The requirements to switching the capacities of SF6 circuit breakers submitted by Russian Grid companies are difficult to satisfy. The first limitation is related to material and financial costs in order to create a new requirement-satisfying switching device. The second limitation is dictated by the necessity of calculating complex physical processes in a circuit braker interrupter during fault–current making or breaking before creating a prototype. The latter task is reduced to the problem of simulating the processes of interaction between the switching arc and the SF6 gas flow. This paper deals with the solution of the problem both analytically by a special method and numerically by a numerical software package through the creation of a mathematical model of the interaction process. The switching arc is taken into account as a form of a temperature source, based on experimental data on measuring the temperature of the arc column. The key feature of the research is to use the finite element method based on a moving mesh—the Arbitrary Lagrangian Eulerian (ALE) method. Such a problem statement allows us to take the contact separation curve of the circuit breaker into account as the input data of the model. The calculations were carried out during fault-current breaking by a 110 kV SF6 dead-tank circuit breaker. The calculations of pressure and mass flow in the under-piston volume change, gas flow speed, and temperature depending on the contact separation are given. The proposed model of the switching arc was used to simulate the process of 25 kA symmetrical fault–current breaking and was compared with an experiment.

Conservative semi-Lagrangian method for continuity equation with various time steps in different parts of computational domain

The system of Navier Stokes differential equations describes gas flow near rocket nozzle. To find solution of this system in general case, scientists and engineers use numerical methods. Modern numerical methods do not allow engineers to model all features of gas flow. It happened because of sophisticated physical processes and limitations of computational hardware. So, at least where are two ways to improve it: to enlarge hardware or reduce computational complicacy. The global aim of many science investigations is to develop numerical approach with reduced computational complicacy and at the same time without loss of computational accuracy. In 1959, Aksel C. Wiin-Nielsen proposed a new and effective trajectories method for problem of numerical forecasting. In 1966, K. M. Magomedov developed similar approach (method of characteristics) for numerical modelling of space (three dimensional on space) gas flow. In 1982, O. Pironneau showed a new and effective approach for two dimensional approximation of the Navier Stokes problem. It was based on method of characteristics also. Nowadays, these methods are called semi-Lagrangian or Eulerian Lagrangian methods. They use Lagrangian nature of the transport process. To apply this advantage, scientists decompose each equation of Navier Stokes system into three parts: convective part (hyperbolic type), elliptic part and part of right-hand side. Scientists use semi-Lagrangian approach to approximate the convective parts of equations. To develop and test modern algorithms from family of semi-Lagrangian methods, we use continuity equation from the system of Navier Stokes equations. Conservative versions of semi-Lagrangian approach are based on Gauss Ostrogradsky (divergence) theorem. It allows scientists to get conservation low (balance equation) for numerical solution in norm of L1 space. Aim of our investigation is to use different time steps in different parts of computation domain. It enables us to obtain at the same time three advantages: convergence of numerical solution to exact solution, reduction of computational complicacy, implementation of conservation low (balance equation) without weight coefficients. For this purpose we decompose computational domain into two parts (subdomains) and use different time steps in them. Main complication is design algorithm in the boundaries of computational subdomains. We use one dimensional (on space) problem to demonstrate ability of developing numerical method with described advantages. Generalization of considered approach for two- or even three-dimensional cases allows engineers to model gas flow more accurately and without artificial viscosity. It is essentially important in the parts of computational domain with high level of solution gradient.

Numerical Study of Dry Reforming of Methane in Packed and Fluidized Beds: Effects of Key Operating Parameters

Replacing the conventionally used steam reforming of methane (SRM) with a process that has a smaller carbon footprint, such as dry reforming of methane (DRM), has been found to greatly improve the industry’s utilization of greenhouse gases (GHGs). In this study, we numerically modeled a DRM process in lab-scale packed and fluidized beds using the Eulerian–Lagrangian approach. The simulation results agree well with the available experimental data. Based on these validated models, we investigated the effects of temperature, inlet composition, and contact spatial time on DRM in packed beds. The impacts of the side effects on the DRM process were also examined, particularly the role the methane decomposition reaction plays in coke formation at high temperatures. It was found that the coking amount reached thermodynamic equilibrium after 900 K. Additionally, the conversion rate in the fluidized bed was found to be slightly greater than that in the packed bed under the initial fluidization regime, and less coking was observed in the fluidized bed. The simulation results show that the adopted CFD approach was reliable for modeling complex flow and reaction phenomena at different scales and regimes.

Uncertainty quantification analysis of Reynolds-averaged Navier–Stokes simulation of spray swirling jets undergoing vortex breakdown

The computational fluid dynamics-based design of next-generation aeronautical combustion chambers is challenging due to many geometrical and operational parameters to be optimized and several sources of uncertainty that arise from numerical modeling. The present work highlights the potential benefits of exploiting Bayesian uncertainty quantification at the preliminary design stage. A prototypical configuration of an acetone/air spray swirling jet is investigated through an Eulerian–Lagrangian method under non-reactive conditions. Two direct numerical simulations (DNSs) provide reference data, coping with different vortex breakdown states. Consequently, a set of Reynolds-averaged Navier–Stokes simulations is conducted. Polynomial chaos expansion (PCE) is adopted to propagate the uncertainty associated with the spray dispersion model and the turbulent Schmidt number, delivering confidence intervals and the sensitivity of the output variance to each uncertain input. Consequently, the most significant sources of modeling uncertainty may be identified and eventually removed via a calibration procedure, thus making it possible to carry out a combustion chamber optimization process that is no longer affected by numerical biases. The uncertainty quantification analysis in the current study demonstrates that the spray dispersion model slightly affects the fuel vapor spatial distribution under vortex breakdown flow conditions, compared with the output variance induced by the selection of the turbulent Schmidt number. As a result, additional high-fidelity experimental and numerical campaigns should exclusively address the development of an ad hoc model characterizing the spatial distribution of the latter in the presence of vortex breakdown phenomenology, discarding any effort to improve the spray dispersion formulation.

Torsional installation and vertical tensile capacity of helical piles in clay

Helical piles are being increasingly considered for offshore applications as they avoid the acoustic emissions associated with pile driving, and they provide additional capacity relative to a driven pile. In this paper, the installation and tensile capacity of helical piles is considered through a combination of centrifuge experiments and large-deformation finite-element analyses within a coupled Eulerian–Lagrangian framework. The experiments provide a basis for validating the numerical simulations, but also quantify the expected installation torque and undrained tensile capacity, including the variation with time after installation. The numerical simulations extend the parameter space investigated experimentally, considering the number of helices, their spacing and pitch, in addition to the ratio of pile shaft to helix diameter and the profile of undrained shear strength. Mechanisms revealed through the numerical simulations are reflected in a new analytical model for calculating undrained tensile capacity, which is seen to agree reasonably well with the numerically and experimentally determined capacities.

An arbitrarily Lagrangian–Eulerian SPH scheme with implicit iterative particle shifting procedure

This work presents an Arbitrarily Lagrangian–Eulerian Smoothed Particle Hydrodynamics (ALE-SPH) scheme for Navier–Stokes equations where the particle distribution is optimized by an Implicit Iterative Particle Shifting (IIPS) correction capable of improving the accuracy of the SPH spatial operators. Here two different possible approaches, based respectively on Moving Least Square (MLS) and on convective fluxes have been adopted to embed the IIPS formulation in an ALE-SPH scheme. The theoretical second-order convergence rate has been achieved for the Taylor–Green flow at Reynolds number Re=100. Results obtained with the proposed numerical scheme for the Moving box and the Impinging jet test cases confirm that the accuracy is significantly improved, with respect to existing quasi-Lagrangian ALE-SPH schemes.

On the computation of compressible multiphase flows with heat and mass transfer in elastic pipelines

The present work is motivated by the engineering need to simulate the multiphase flows in the course of the start-up of a long-distance crude oil pipeline. We propose a model for compressible multiphase flows with heat and mass transfer and diffusion processes in elastic pipelines. The model is derived with two approaches, i.e., (a) the spatial averaging procedure of a single phase model and (b) the Arbitrary Lagrangian Eulerian (ALE) formulation of the Baer-Nunziato model with quasi-1D approximation. The diffusion processes include the viscous dissipation, wall friction, heat conduction, and heat exchange with external environment. In particular, the wall friction consists of steady friction term calculated by Darcy-Weisbach formula, and the unsteady friction determined by the instantaneous-acceleration-based (IAB) model. The unsteady friction modifies the characteristic structure of the hyperbolic part of the model. For the solution of the hyperbolic part, we propose a three-wave approximate Riemann solver incorporating the unsteady friction term. Mass and heat transfer are realized via instantaneous relaxations of the chemical potential and temperature, respectively. Efficient iterative relaxation procedures for N-phase flows have been proposed. We have validated the proposed model and numerical methods against some benchmark multiphase problems and applied the model to calculate the start-up of a realistic liquid pipeline with intermediate pump station boundary condition.

A fast dynamic smooth adaptive meshing scheme with applications to compressible flow

We develop a fast-running smooth adaptive meshing (SAM) algorithm for dynamic curvilinear mesh generation, which is based on a fast solution strategy of the time-dependent Monge-Ampère (MA) equation, det⁡∇ψ(x,t)=G∘ψ(x,t). The novelty of our approach is a new so-called perturbation formulation of MA, which constructs the solution map ψ via composition of a sequence of near-identity deformations of a reference mesh. Then, we formulate a new version of the deformation method [21] that results in a simple, fast, and high-order accurate numerical scheme and a dynamic SAM algorithm that is of optimal complexity when applied to time-dependent mesh generation for solutions to hyperbolic systems such as the Euler equations of gas dynamics. We perform a series of challenging 2D and 3D mesh generation experiments for grids with large deformations, and demonstrate that SAM is able to produce smooth meshes comparable to state-of-the-art solvers [22,18], while running approximately 200 times faster. The SAM algorithm is then coupled to a simple Arbitrary Lagrangian Eulerian (ALE) scheme for 2D gas dynamics. Specifically, we implement the C-method [64,65] and develop a new ALE interface tracking algorithm for contact discontinuities. We perform numerical experiments for both the Noh implosion problem as well as a classical Rayleigh-Taylor instability problem. Results confirm that low-resolution simulations using our SAM-ALE algorithm compare favorably with high-resolution uniform mesh runs.

Euler–Lagrangian Approach to Stochastic Euler Equations in Sobolev Spaces

Euler–Lagrangian Approach to Stochastic Euler Equations in Sobolev Spaces

Performance Investigation and Optimization of the Primary Separation Part of the Oil-Gas Separator

Oil-gas separators are significant in the safe and reliable operation of the natural gas storage system. The separation efficiency of the conventional primary separation part of the oil-gas separator was not high enough, causing the insufficient performance of the whole separator. Three modification schemes were proposed to optimize the primary separation part of the oil-gas separator, including adding a curved baffle at the entrance, forming a centrifugal separation through the tangential inlet, and forming a cyclone structure by adding an inner cylinder. The separation performance was obtained through the Eulerian–Lagrangian approach numerically. For each scheme, a variety of parameter combinations were designed by the orthogonal array and the structure with the highest separation efficiency was reasonably selected. The separation efficiency of the centrifugal structure was higher than that of the collision structure on the whole. To evaluate the feasibility of the practical application, the three structural schemes were compared coupled with the analytic hierarchy process (AHP) based on the separation efficiency, the pressure loss, the uniformity of filter inlet velocity, and the modification cost. The results showed that the scheme of adding a curved baffle with reasonable structural parameters at the entrance had better comprehensive performance, which could be used as the optimal scheme in improving the oil-gas separator applied in natural gas storage.

Damage mechanisms of full-scale ship under near-field underwater explosion

The near-field underwater explosion produced by the torpedo and other weapons was one of the most critical threats to naval ships. However, most underwater explosion tests were scale models, which would bring dissimilarities between the gravity and strain rate. We carried out a near-field underwater explosion test on a full-scale ship and recorded the dynamic response processes to make up for this shortcoming. The ship damage was mainly embodied in the loss of overall strength and damage of local structures. The Coupled Eulerian–Lagrangian (CEL) method was used to establish the model of the near-field underwater explosion, and the effectiveness of the calculation method was validated. Based on the analysis results, the near-field underwater explosion could be divided into three stages: shock wave and after flow, bubble pulsation, and water jet. The overall strength of the ship was lost under the combined action of these three stages. The “negative pressure” caused by bubble pulsation was the key to the sagging deformation of the ship. This study would guide the production of underwater weapons and ship protection design.