Arbitrary Lagrangian Eulerian Simulations Research Articles

Conservative high-order data transfer method on generalized polygonal meshes

A conservative data transfer (remap) between two meshes is an important step of arbitrary Lagrangian-Eulerian (ALE) hydrodynamics simulations. High-order numerical methods for ALE simulations require both high-order (curvilinear) meshes and high-order remap algorithms. We develop a conservative and bounds-preserving method for accurate remapping of discrete fields on generalized polygonal meshes with curvilinear edges. The properties of the proposed method are studied theoretically and numerically for various (smooth and non-smooth) mesh deformations and discrete fields that represent smooth and discontinuous functions.

Multi-material swept face remapping on polyhedral meshes

Remapping is a conservative interpolation of a discretized intensive quantity between two meshes. In this article, we propose a novel multi-material flux remapping method that avoids the geometric computation of mesh-mesh intersections needed for an accurate intersection based remap. The flux remap is applicable to scalar quantities such as material density describing the multi-material flow between meshes with the same connectivity but small mesh displacements. The method is described for two- and three-dimensional polygonal/polyhedral meshes as it is implemented in Portage.1 Another open source library, Tangram,2 is used to calculate material interfaces in cells containing more than one material. Performance and accuracy of the flux remap are discussed with respect to Arbitrary Lagrangian-Eulerian simulations and compared to an accurate intersection based remap. In particular, cyclic remapping shows that the accuracy of the flux remap is limited to first order on material boundaries while maintaining second order accuracy in pure material regions.

Development, verification, and validation of comprehensive acoustic fluid‐structure interaction capabilities in an open‐source computational platform

AbstractThe acoustic fluid‐structure interaction (FSI) formulation is a practical numerical approach for the seismic analysis of fluid‐filled tanks. However, there are no verification and validation studies reported in the literature that demonstrate the ability of an acoustic FSI numerical model to predict responses important to structural and mechanical design for intense translational and rotational earthquake inputs. Herein, an acoustic FSI formulation is implemented in the open‐source Multiphysics Object‐Oriented Simulation Environment (MOOSE), and is formally verified and validated using analytical solutions and code‐to‐code verification, and experimental data, respectively. The analytical solutions are for small amplitude, unidirectional seismic inputs. The code‐to‐code verification utilizes a previously verified and validated Arbitrary Lagrangian‐Eulerian (ALE) numerical model in the commercial finite element code LS‐DYNA. The validation studies utilize a comprehensive data set assembled from results of 3D earthquake‐simulator tests of a fluid‐filled vessel. The acoustic numerical model in MOOSE is verified and validated for hydrodynamic pressures and support reactions except for cases that involve significant convective response. For small amplitude inputs, numerically predicted wave heights match those of the analytical solutions. The numerical model is not verified and validated for wave height calculations under intense 3D seismic inputs. The run times for the acoustic FSI simulations in MOOSE are an order of magnitude, or more, shorter than for the corresponding ALE simulations in LS‐DYNA. The utility of the MOOSE acoustic FSI implementation is demonstrated by seismic analysis of a building equipped with a fluid‐filled, advanced nuclear reactor.

The staged Z-pinch as a potential fusion energy source

A staged Z-pinch is a fusion concept where a high atomic number liner compresses a fusion fuel (deuterium–deuterium or deuterium–tritium) target. Here, we report new results from Lagrangian, Eulerian, and Arbitrary Lagrangian–Eulerian simulations, which confirm that, with proper treatment of the vacuum region outside from the pinch, fusion energy production >1 MJ can be expected without alpha heating and significantly higher if alpha heating is included. It is shown that shock waves play an important role in preheating the target plasma and in piling up liner mass at the liner/target interface. This results in higher ram pressure just before the pinch stagnation time and ultimately in higher energy density target plasma.

MPM and ALE Simulations of Large Deformations Geotechnics Instability Problems

Problems involving large deformations are the focus of numerical modeling researches in recent decades due to the challenge of finding a kinematic appropriate description of the continuum. In recent years, different formulations have been used to describe such problems as the Arbitrary Lagrangian Eulerian (ALE) method and the Material Point Method (MPM). These two methods allow to perform dynamic analyzes involving large deformations. In this way, this work aims to present a comparison of problems applied to Geotechnics involving large deformations and large displacements, using MPM and FEM associated with the ALE method. For this purpose, three problems are simulated: sliding of blocks on an inclined plane, runout process of sand and instability of a slope using the MPM and the FEM associated with the ALE method. In all cases a comparison of the results is presented, and the advantages and disadvantages of each method are discussed.

Prediction of safe zone for osteonecrosis in the cutting process of bovine cortical femur bone using Arbitrary Lagrangian-Eulerian method and multi-objective optimization

In this study, the Arbitrary Lagrangian-Eulerian (ALE) method was used to investigate the osteonecrosis in orthopedic surgery of cortical bone. As a result of mesh motion control in the ALE method, element distortions were prevented and more accurate results predicted. To consider the temperature and reaction force variation in the orthogonal cutting of bovine cortical femur bone, the coupled thermo-mechanical simulations were used. Effects of cutting depth, cutting speed, and rake angle that may cause osteonecrosis on the reaction forces and the bone temperature variations were studied. The Johnson-Cook material model was employed to observe the strain rate effects on the cutting force and temperature distribution. The results of ALE simulations represented less than 10% error in comparison with the experimental data which could be used as the alternative method for experimental predictions. Furthermore, the response surface of design of experiments (DOE) method was used for performing a multi-objective optimization. Based on the obtained results, decreasing the rake angle less than 5° is the most significant term in the reduction of osteonecrosis.

Hydro-plastic response of beams and stiffened panels subjected to extreme water slamming at small impact angles, part II: Numerical verification and analysis

An analytical model has been proposed for the response of beams and stiffened panels subjected to extreme flat or nearly flat water impacts in Part I of the two-part companion paper. The model aims to capture the significant hydro-plastic coupling between large plastic structural deformations and the hydrodynamic pressure. Governing non-dimensional parameters for the hydro-plastic slamming phenomenon were identified and discussed.This Part II paper verifies the analytical model proposed in Part I by comparison with the hydro-plastic slamming response of beams and stiffened panels using multi-material Arbitrary Lagrangian Eulerian (ALE) methods in LS-DYNA. Numerical modelling and settings with the ALE simulations are firstly validated by comparison against drop-test experiments of a rigid wedge and of an elastic plate. Then, water entry simulations of flat plates and stiffened panels are carried out, where structural deformations go into the plastic regime. The simulated scenarios cover different plate thicknesses/cross sectional dimensions of stiffened panels, and various initial water-entry velocities. The analytical model is discussed with respect to the fluid flow, structural deflections, the pressure history and the impulse. Validity of assumptions of the analytical model is also discussed. Potential applications and limitations are indicated. The proposed design curves are well suited to be utilized in rules and standards for designing against extreme water slamming.

A coupled ALE–AMR method for shock hydrodynamics

We present a numerical method combining adaptive mesh refinement (AMR) with arbitrary Lagrangian–Eulerian (ALE) mesh motion for the simulation of shock hydrodynamics on unstructured grids. The primary goal of the coupled method is to use AMR to reduce numerical error in ALE simulations at reduced computational expense relative to uniform fine mesh calculations, in the same manner that AMR has been used in Eulerian simulations. We also identify deficiencies with ALE methods that AMR is able to mitigate, and discuss the unique coupling challenges. The coupled method is demonstrated using three-dimensional unstructured meshes of up to O(107) tetrahedral cells. Convergence of ALE–AMR solutions towards both uniform fine mesh ALE results and analytic solutions is demonstrated. Speed-ups of 5–10 × for a given level of error are observed relative to uniform fine mesh calculations.

Arbitrary Lagrangian–Eulerian simulations of highly electrically charged micro-droplet Coulomb explosion deformation pathways

Numerical simulation results of the Coulomb explosion pathways of cooled water and heated glycol droplets electrically charged to the critical Rayleigh limit are presented, calculated using an axi-symmetric finite element scheme previously used for the same problem [Radcliffe A. J., Non-conforming finite elements for axisymmetric charged droplet deformation dynamics and Coulomb explosions, Int. J. Num. Meth. Fluids 71:249–268, (2013), doi:10.1002/fld.3667.] which has been adapted to use arbitrary Lagrangian–Eulerian (ALE) methods and a novel tip reconstruction technique in order to greatly improve its accuracy in matching available experimental data.

Lagrangian and Arbitrary Lagrangian Eulerian Simulations of Cold Roll Forming Processes

Lagrangian and Arbitrary Lagrangian Eulerian Simulations of Cold Roll Forming Processes

Depiction of interfacial characteristic changes during impact welding using computational methods: Comparison between Arbitrary Lagrangian - Eulerian and Eulerian simulations

This paper investigates the development and evolution of various interfacial features during high speed impact welding processes using numerical simulations. Arbitrary Lagrangian - Eulerian (ALE) and Eulerian computations are performed and compared with experimental features of welded joints. ALE method reveals poor mesh quality while the interface experiences severe shear deformation. Therefore, ALE simulations cannot accurately capture the morphological transition toward a wavy shape and cannot compute the successive jet formation from the interface. But, the Eulerian method becomes a good alternative to overcome those limitations of the ALE method. Computed interfacial features using Eulerian simulations are agreed well with the experimental observations of various interfacial phenomena. The actual kinematics along the whole length of the weld including wavy pattern, localized plastic deformation and shear strain orientation are clearly reproduced. Eulerian simulation also provides understanding for the potential cause of defects that corresponds to the experimental observations, in terms of morphology and site occurrence.

Lagrangian and arbitrary Lagrangian Eulerian simulations of complex roll-forming processes

The Arbitrary Lagrangian Eulerian (ALE) formalism is a breakthrough technique in the numerical simulation of the continuous-type roll-forming process. In contrast to the classical Lagrangian approach, the ALE formalism can compute the hopefully stationary state for the entire mill length with definitely effortless set-up tasks thanks to a nearly-stationary mesh. In this paper, advantages of ALE and Lagrangian formalisms are extensively discussed for simulating such continuous-type processes. Through a highly complex industrial application, the ease of use of ALE modelling is illustrated with the in-house code METAFOR. ALE and Lagrangian results are in good agreement with each other.

Dielectrophoretic behavior of a single cell when manipulated by optoelectronic tweezers: A study based on COMSOL ALE simulations

Abstract Optoelectronic tweezers uses optically induced dielectrophoretic (DEP) force for manipulating cells in aqueous solution, which has shown potential applications in biology and tissue engineering among other possibilities. To effectively design the optoelectronic tweezers (OETs) chip, detailed knowledge about the behavior of cells in response to DEP force in an aqueous layer is needed. In this paper, the trajectories of an SMMC-77721 cell, simulated as a rigid dielectric sphere, in the induced electric field of optical trapping are studied using both an Arbitrary Lagrangian-Eulerian (ALE) method and a particle-tracing method (PTM) available within the COMSOL multiphysics software platform. Because the ALE method involves solving the distorted electric field around the cell and taking a full account of the Maxwell stress tensor (MST), it is expected to provide more accurate predictions about the spherical cell velocity than PTM that involves dipole moment approximation. Our ALE results show noticeably greater cell velocity than that predicted by the classical DEP expression based on dipole moment approximation. The ALE model can help gain new insights for analyzing cell motions in aqueous solution under sophisticated optical spot patterns.

Side Crash Pressure Sensor Prediction for Unitized Vehicles: An ALE Approach

With a goal to help develop pressure sensor calibration and deployment algorithms using computer simulations, an Arbitrary Lagrangian Eulerian (ALE) approach was adopted in this research to predict the responses of side crash pressure sensors for unitized vehicles. For occupant protection, acceleration-based crash sensors have been used in the automotive industry to deploy restraint devices when vehicle crashes occur. With improvements in the crash sensor technology, pressure sensors that detect pressure changes in door cavities have been developed recently for vehicle crash safety applications. Instead of using acceleration (or deceleration) in the acceleration-based crash sensors, the pressure sensors utilize pressure change in a door structure to determine the deployment of restraint devices. The crash pulses recorded by the acceleration-based crash sensors usually exhibit high frequency and noisy responses. Different from those of the acceleration-based crash sensors, the data obtained from the pressure sensors exhibit lower frequency and less noisy responses. Due to its ability to discriminate crash severities and allow the restraint devices to deploy earlier, the pressure sensor technology has gained its popularity for side crash applications. The lower frequency and less noisy characteristics are also more suitable for non-linear finite element codes to predict.Fifteen different benchmark problems were designed and tested in the first stage of this research to investigate the responses of pressure sensors in different impact conditions and the capabilities of the ALE method in the predictions of different pressure sensor responses. The fifteen benchmark problems were divided into three groups to examine the capabilities of the ALE method in detail. Different structures, gases, hole locations, sensor locations, hole sizes, impact speeds, and impactors, were chosen in the fifteen benchmark problems so that the sensitivity of the pressure responses to different factors could be obtained and understood. Computer simulations conducted by employing the ALE method for all fifteen benchmark problems were compared to their corresponding theoretical solutions or test data. The correlations between the tests and the computer simulations were found to be reasonable as reported in a paper published previously.The research was advanced into its final stage, full vehicle tests, after the positive results obtained from the benchmark study. The full vehicle study included two major vehicle architectures, body-on-frame and unitized, that are commonly used to design vehicles in the automotive industry. This paper focuses on the unitized vehicles. A total of thirteen tests, including different body styles, powertrains, drivetrains, test modes, and impact speeds, were investigated.A simulation methodology was developed in this study to correlate the structural responses and to predict the pressure sensor responses for unitized vehicles. The results obtained from the developed methodology using the ALE simulations are compared to those obtained from the corresponding tests. In the full vehicle study, low speed impact conditions were found to be more challenging to predict compared to those of the high speed impact conditions. This is because the pressure responses for the low speed impacts are usually much weaker than those of the high speed impacts. The numerical errors obtained from the simulations become more significant when the magnitudes of the pressure responses are low. The numerical errors induced by the coupling of Lagrangian and Eulerian calculation need to be distinguished and ignored (or filtered) when processing the pressure information. The slopes, peak values, and overall shapes of the predicted pressure responses correlate reasonably with most of the full vehicle tests selected. The correlations of the pre-peak responses are better than those of the post-peak responses which involve air leakage. The oblique pole, IIHS MDB, and FMVSS 214 MDB test modes create distinct door deformations and pressure responses which can be predicted by the computer simulations reasonably. Sensor engineers analyzed the results obtained from a FMVSS 214 simulation and confirmed that replacing the test data with the predicted results would result in the same deployment algorithm. Language: en

Hyperviscosity for unstructured ALE meshes

An artificial viscosity, originally designed for Eulerian schemes, is adapted for use in arbitrary Lagrangian–Eulerian simulations. Changes to the Eulerian model (dubbed ‘hyperviscosity’) are discussed, which enable it to work within a Lagrangian framework. New features include a velocity-weighted grid scale and a generalised filtering procedure, applicable to either structured or unstructured grids. The model employs an artificial shear viscosity for treating small-scale vorticity and an artificial bulk viscosity for shock capturing. The model is based on the Navier–Stokes form of the viscous stress tensor, including the diagonal rate-of-expansion tensor. A second-order version of the model is presented, in which Laplacian operators act on the velocity divergence and the grid-weighted strain-rate magnitude to ensure that the velocity field remains smooth at the grid scale. Unlike sound-speed-based artificial viscosities, the hyperviscosity model is compatible with the low Mach number limit. The new model outperforms a commonly used Lagrangian artificial viscosity on a variety of test problems.

Properties of a diffuse interface model based on a porous medium theory for solid–liquid dissolution problems

In this paper, a local non-equilibrium diffuse interface model is introduced for describing solid–liquid dissolution problems. The model is developed based on the analysis of Golfier et al. (J Fluid Mech 457:213–254, 2002) upon the dissolution of a porous domain, with the additional requirement that density variations with the mass fraction are taken into account. The control equations are generated by the upscaling of the balance equations for a solid–liquid dissolution using a volume averaging theory. This results into a diffuse interface model (DIM) that does not require an explicit treatment of the dissolving interface, e.g., the use of arbitrary Lagrangian–Eulerian (ALE) methods, for instance. Test cases were performed to study the features and influences of the effective coefficients inside the DIM. In particular, an optimum expression for the solid–liquid exchange coefficient is obtained from a comparison with the referenced solution by ALE simulations. Finally, a Ra–Pe diagram illustrates the interaction of natural convection and forced convection in the dissolution problem.

Numerical and experimental studies on extrudate swell of branched polyethylene through axisymmetric and planar dies

Abstract Extrudate swell is simulated using an Arbitrary Lagrangian Eulerian (ALE) technique based finite element formulation and the same has been validated by comparing the results with reported numerical and experimental studies. In the present work we compare our ALE simulations with our own experimental data on the extrudate swell of commercial grade low density polyethylene (LDPE) resin. The resins were characterized for their rheological behavior in both shear and uniaxial extension. The polymers were extruded from a capillary under isothermal conditions and the extrudates were observed on-line using a video camera. ALE simulations were performed using molecular constitutive model like eXtended Pom–Pom (XPP) for branched (LDPE). The simulated extrudate swell was a good match with the experimental data. It was found that the swell values of LPDE through planar die are higher than the axisymmetric die.

Extrudate swell of linear and branched polyethylenes: ALE simulations and comparison with experiments

Extrudate swell is a common phenomenon observed in the polymer extrusion industry. Accurate prediction of the dimensions of an extrudate is important for appropriate design of dies for profile extrusion applications. Prediction of extrudate swell has been challenging due to (i) difficulties associated with accurate representation of the constitutive behavior of polymer melts, and (ii) difficulties associated with the simulation of free surfaces, which requires special techniques in the traditionally used Eulerian framework. In a previous work we had argued that an Arbitrary Lagrangian Eulerian (ALE) based finite element formulation may have advantages in simulating free surface deformations such as in extrudate swell. In the present work we reinforce this argument by comparing our ALE simulations with experimental data on the extrudate swell of commercial grades of linear polyethylene (LLDPE) and branched polyethylene (LDPE). Rheological behavior of the polymers was characterized in shear and uniaxial extensional deformations, and the data was modeled using either the Phan–Thien Tanner (PTT) model or the eXtended Pom–Pom (XPP) model. Additionally, flow birefringence and pressure drop measurements were done using a 10:1 contraction–expansion (CE) slit geometry in a MultiPass Rheometer. Simulated pressure drop and contours of the principal stress difference were compared with experimental data and were found to match well. This provided an independent test for the accuracy of the ALE code and the constitutive equations for simulating a processing-like flow. The polymers were extruded from long ( L/ D = 30) and short ( L/ D = 10) capillaries dies at 190 °C. ALE simulations were performed for the same extrusion conditions and the simulated extrudate swell showed good agreement with the experimental data.

The impulsive motion of a small cylinder at an interface

We study the unsteady motion caused by an impulse acting at time t=0 on a small cylinder floating horizontally at a liquid–gas interface. This is a model for the impact of a cylinder onto a liquid surface after the initial splash. Following the impulse, the motion of the cylinder is determined by its weight per unit length (pulling it into the bulk liquid) and resistance from the liquid, which acts to keep the cylinder at the interface. The range of cylinder radii r and impact speeds U considered is such that the resistance from the liquid comes from both the interfacial tension and hydrodynamic pressures. We use two theoretical approaches to investigate this problem. In the first, we apply the arbitrary Lagrangian Eulerian (ALE) method developed by Li et al. [“An arbitrary Lagrangian Eulerian method for moving-boundary problems and its application to jumping over water,” J. Comput. Phys. 208, 289 (2005)] to compute the fluid flow caused by the impulse and the (coupled) motion of the cylinder. We show that at early times the interfacial deformation is given by a family of shapes parametrized by r/t2/3. We also find that for a given density and radius there is a critical impulse speed below which the cylinder is captured by the interface and floats but above which it pierces the interface and sinks. Our second theoretical approach is a simplified one in which we assume that the interface is in equilibrium and derive an ordinary differential equation for the motion of the cylinder. Solving this we again find the existence of a critical impulse speed for sinking giving us some quantitative understanding of the results from the ALE simulations. Finally, we compare our theoretical predictions with the results of experiments for cylinder impacts by Vella and Metcalfe [“Surface tension dominated impact,” Phys. Fluids 19, 072108 (2007)]. This comparison suggests that the influence of contact line effects, neglected here, may be important in the transition from floating to sinking.

Perspective on the geometric conservation law and finite element methods for ALE simulations of incompressible flow

This paper takes a fresh look at the geometric conservation law (GCL) from the perspective of the finite element method (FEM) for incompressible flows. The GCL arises naturally in the context of Arbitrary Lagrangian Eulerian (ALE) formulations for solving problems on deforming domains. GCL compliance is traditionally interpreted as a consistency criterion for applying an unsteady flow solution algorithm to simulate exactly a uniform flow on a deforming domain. We introduce an additional requirement: the time integrator must maintain its fixed mesh accuracy when applied to deforming meshes. A review of the literature shows that while many authors use an ALE FEM, few of them discuss the GCL issues. We show how a fixed mesh unsteady FEM using high order time integrator (up to fifth order in time) can be transposed to solve problems on deforming meshes and preserve its fixed mesh high order temporal accuracy. An appropriate construction of the divergence of the mesh velocity guarantees GCL compliance while a separate construction of the mesh velocity itself allows the time-integrator to deliver its fixed mesh high order temporal accuracy on deforming domains. Analytical error analysis of problems with closed form solutions provides insight on the behavior of the time integrators. It also explains why high order temporal accuracy is achieved with a conservative formulation of the incompressible Navier–Stokes equations, while only first order time accuracy is observed with the non-conservative formulation and all time-integrators investigated here. We present thorough time-step and grid refinement studies for simple problems with closed form solutions and for a manufactured solution with a non-trivial flow on a deforming mesh. In all cases studied, the proposed reconstructions of the mesh velocity and its divergence for the conservative formulation lead to optimal time accuracy on deforming grids.