Arbitrary Lagrangian Eulerian Framework Research Articles

Numerical Investigation of Two-Phase Flows in Corrugated Channel with Single and Multiples Drops

This work aims at investigating numerically the effects of channel corrugation in two-phase flows with single and multiples drops subject to buoyancy-driven motion. A state-of-the-art model is employed to accurately compute the dynamics of the drop’s interface deformation using a modern moving frame/moving mesh technique within the arbitrary Lagrangian–Eulerian framework, which allows one to simulate very large domains. The results reveal a complex and interesting dynamics when more than one drop is present in the system, leading eventually in coalescence due to the amplitude of the corrugated sinusoidal channel and distance between drops.

Simulation of vowel-vowel utterances using a 3D biomechanical-acoustic model.

A link is established between biomechanical and acoustic 3D models for the numerical simulation of vowel-vowel utterances. The former rely on the activation and contraction of relevant muscles for voice production, which displace and distort speech organs. However, biomechanical models do not provide a closed computational domain of the 3D vocal tract airway where to simulate sound wave propagation. An algorithm is thus proposed to extract the vocal tract boundary from the surrounding anatomical structures at each time step of the transition between vowels. The resulting 3D geometries are fed into a 3D finite element acoustic model that solves the mixed wave equation for the acoustic pressure and particle velocity. An arbitrary Lagrangian-Eulerian framework is considered to account for the evolving vocal tract. Examples include six static vowels and three dynamic vowel-vowel utterances. Plausible muscle activation patterns are first determined for the static vowel sounds following an inverse method. Dynamic utterances are then generated by linearly interpolating the muscle activation of the static vowels. Results exhibit nonlinear trajectory of the vocal tract geometry, similar to that observed in electromagnetic midsagittal articulography. Clear differences are appreciated when comparing the generated sound with that obtained from direct linear interpolation of the vocal tract geometry. That is, interpolation between the starting and ending vocal tract geometries of an utterance, without resorting to any biomechanical model.

Numerical treatment of frictional contact in ALE formulation for disc brake assemblies

The Arbitrary-Lagrangian-Eulerian (ALE) formulation is a popular approach for finite element (FE) models governed by large rigid body movement. It facilitates the decomposition of guiding motions and relative deformation inside the model by considering the main motion of the system in the form of a material flow inside the FE mesh. This way, similar to fluid dynamics, an observer focused on spatial positions rather than individual material particles is introduced. Among others, this approach serves as an established tool in the simulation of tire rolling contact and its dynamics. However, while allowing for specific mesh refinement and simplifying issues such as finite displacements due to rigid body motions, it complicates contact computations. The history of contacting particles is not inherently tracked, demanding a special treatment of their relative displacement for evaluating their frictional contact state. The handling of stationary problems in this context is well-studied for the examples of elastic-rigid rolling contact, including inelastic material properties and thermo-mechanical coupling. This work attends to the remaining area of frictional contact in transient ALE problems and extends the methods towards elastic-elastic contact coupling in a three dimensional framework. To this end, classic Zero-Thickness contact elements, whose application in these surroundings is enabled by the relative kinematic ALE framework in the first place, are extended towards ALE compatibility. The proposed approach involves a slip-based contact formulation including inelastic slip variables and is applied in the field of disc brake assemblies. In here, several stationary and transient studies of emerging contact patterns and limit cycles are accomplished, outlining the capabilities of the proposed implementation.

3D Cell-centered hydrodynamics with subscale closure model and multi-material remap

We extend a higher-order finite volume cell-centered hydrodynamic (CCH) formulation to include an interface-aware subscale closure model and a multi-material remap for simulating 3D compressible hydrodynamic problems within an arbitrary Lagrangian-Eulerian (ALE) framework. This CCH formulation involves a multidirectional approximate Riemann solution using quadratic polynomial reconstructions of the stress tensor and the velocity. At the subscale level, we determine pair-wise material interactions by solving a distinct approximate Riemann problem at the common interface, using the volume of fluids (VOF) method to find the interface. Material interactions are constrained to ensure smooth pressure equilibration among materials. The accuracy and robustness of the ALE method is demonstrated by simulating a suite of 3D Cartesian multi-material problems covering both gas and solid dynamics, where each test case has two or more materials.

A strongly-coupled cell-based smoothed finite element solver for unsteady viscoelastic fluid–structure interaction

This paper describes a cell-based smoothed finite element method (CS-FEM) for computing unsteady viscoelastic fluid–structure interaction (VFSI) within the arbitrary Lagrangian–Eulerian framework. The incompressible Navier–Stokes equations incorporating the Oldroyd-B constitutive relation are decoupled via the smoothed characteristic-based split scheme that allows equal low-order interpolations for the triple primitive variables. The elastodynamics equation of a geometrically nonlinear solid is solved by the modified Newton–Raphson method in conjunction with the generalized-α method. Following an efficient moving mesh strategy, the block-Gauss–Seidel procedure is utilized to tightly couple three interacting fields. In particular, CS-FEM is adopted to spatially discretize the global VFSI system where all gradient related terms are readily smoothed. The cell-based smoothing concept is also introduced to evaluate viscoelastic fluid forces acting on the deformable body. Two transient VFSI examples are analyzed to demonstrate the enlarged applicability of CS-FEM. The main characteristics of viscoelastic flow-induced oscillations are successfully captured.

A numerical simulation method for bionic fish self-propelled swimming under control based on deep reinforcement learning

In order to simulate the under control self-propelled swimming of bionic fishes, a coupling method of hydrodynamics/kinematics/motion-control is presented in this paper. The Navier-Stokes equations in the arbitrary Lagrangian-Eulerian framework are solved in parallel based on the computational domain decomposition to simulate the unsteady flow field efficiently. The flow dynamics is coupled with the fish dynamics in an implicit way by a dual-time stepping approach. In order to discretize the computational domain during a wide range maneuver, an overset grid approach with a parallel implicit hole-cutting technique is adopted and coupled with morphing hybrid grids around the undulation body. The motion control of the fish swimming is realized by a deep reinforcement learning algorithm, which makes the fish model choose proper undulation manner according to a specific purpose. By adding random disturbances in the training process of fish swimming along a straight line, a simplified two-dimensional fish model obtains the ability to swim along a specific trajectory. Then in subsequent tests, the two-dimensional fish model is able to swim along more complex curves with obstacles. Finally, the starting process of a three-dimensional tuna-like model is simulated preliminarily to validate the ability of the coupling method for three-dimensional complex configurations. The numerical results demonstrate that this study could be used to explore the swimming mechanism of fishes in complex environments and to guide how robotic fishes can be controlled to accomplish their tasks.

Arbitrary Lagrangian-Eulerian unstructured finite-volume lattice-Boltzmann method for computing two-dimensional compressible inviscid flows over moving bodies.

The objective of this study is to develop and apply an arbitrary Lagrangian-Eulerian unstructured finite-volume lattice-Boltzmann method (ALE-FVLBM) for solving two-dimensional compressible inviscid flows around moving bodies. The two-dimensional compressible form of the LB equation is considered and the resulting LB equation is formulated in the ALE framework on an unstructured body-fitted mesh to correctly model the body shape and properly incorporate the mesh movement due to the body motion. The spatial discretization of the resulting system of equations is performed by a second-order cell-centered finite-volume method on arbitrary quadrilateral meshes and an implicit dual-time stepping method is utilized for the time integration. To stabilize the numerical solution, appropriate numerical dissipation terms are added to the formulation. At first, the shock tube problem is computed to examine the accuracy of the solution obtained by applying the proposed FVLBM for this unsteady test case which includes shock, expansion wave, and contact discontinuity in the flow domain. Then, the stationary isentropic vortex is simulated on both the stationary and moving meshes to assess the implementation of the geometric conservation law in enhancing the solution accuracy of the ALE-FVLBM. The compressible inviscid flow in the transonic regime is then computed around the stationary NACA0012 airfoil in order to further study the sensitivity of the solution method to the user defined parameters. Now, the transonic inviscid flow is simulated over the pitching or plunging NACA0012 airfoil to investigate the accuracy and capability of the proposed solution method (ALE-FVLBM) for the computation of the compressible flows over moving bodies. Finally, the pitching or plunging NACA0012 airfoil near the ground in the transonic inviscid flow is simulated as a practical and challenging problem to study the ground effect on the aerodynamic characteristics of the airfoil. It is indicated that the solution methodology proposed based on the finite-volume LBM formulated in the arbitrary Lagrangian-Eulerian framework (ALE-FVLBM) is capable of accurately computing the compressible inviscid flows around the moving bodies with and without the ground effect.

An ALE-FE method for two-phase flows with dynamic boundaries

The present work aims at developing a new flexible computational framework to simulate macro and microscale two-phase flows with dynamic boundaries. Such a technique is extremely useful for periodic and very large domains which requires exhaustive computational resources, consequently reducing the required numerical domain. In this article an interface tracking Finite Element (FE) method is used to solve the equations governing the motion of two immiscible incompressible fluids in the Arbitrary Lagrangian–Eulerian framework (ALE). The equations are written in axisymmetric coordinates, however the proposed moving boundary technique can be easily extended to 3-dimensional flows and other methods using the ALE framework such as the finite volume method. The two-phase interface separating the fluids is a subset of the domain mesh, therefore a layer of zero thickness is achieved assuring sharp transition of properties among phases. At the scale of interest, surface tension plays an important role and is thus considered in the flow equations. Several validations and results are presented for gravity dominated problem, including the sessile drop test and rising of spherical and Taylor bubbles, as well as the divergent and sinusoidal channels, showing accuracy for modeling two-phase flows in large and periodic domains.

Arbitrary Lagrangian-Eulerian-type discrete unified gas kinetic scheme for low-speed continuum and rarefied flow simulations with moving boundaries.

In this paper, the original discrete unified gas kinetic scheme (DUGKS) is extended to the arbitrary Lagrangian-Eulerian (ALE) framework to enable simulation of low-speed continuum and rarefied flows with moving boundaries. For the proposed ALE-type DUGKS, the mesh motion velocity is introduced in the Boltzmann-BGK equation and a remapping-free scheme is used to discretize the governing equation. Under this coupling framework, the complex rezoning and remapping phases implemented in the traditional ALE method are avoided. In some application areas, large discretization errors are introduced in the simulation if the geometric conservation law (GCL) is not guaranteed. Therefore, three GCL-compliant approaches are discussed, and a uniform flow test case is conducted to validate these schemes. Further, to illustrate the performance of the proposed method, four test cases are simulated, including the continuum flow around an oscillating circular cylinder, the continuum flow around a pitching NACA0012 airfoil, a moving piston driven by a rarefied gas, and the rarefied flow caused by a plate oscillating in the normal direction. Finally, an extended test case considering the rarefied flow over an oscillating circular cylinder is also studied, as this condition is not sufficiently researched. Consistent and good results obtained from the above test cases demonstrate the capability of the proposed ALE-type DUGKS to simulate moving boundary problems in different flow regimes.

A 3D conservative sharp interface method for simulation of compressible two-phase flows

A three-dimensional (3D) conservative sharp interface (CSI) method, which combines 3D cut-cell methods and finite volume methods in the arbitrary Lagrangian-Eulerian framework, is proposed to simulate compressible inviscid two-phase flows. The 3D cut-cell method allows to reconstruct interfaces on a Cartesian mesh and update unstructured interface cells in the vicinity of the interface, making the reconstructed interface always coinciding with the cell faces of the interface cells. The complexity in generating 3D interface cells is significantly reduced by exploiting the symmetry and rotational symmetry of fluid positions in a cut cell. Compared to our previous two-dimensional work (Lin et al. (2017) [17]), the occurrence of large interface cells is also avoided by using the new strategy of cell assembly. With the help of the 3D cut-cell method, a second-order finite volume method in the arbitrary Lagrangian-Eulerian framework can be used nearly in the whole domain, except for the interface cells with under-resolved interfaces or topology changes, at which a first-order finite volume method is used instead. The convergence of the 3D CSI method is examined by studying the Rayleigh collapse of a gas bubble in water, and second-order accuracy has been obtained for the interface evolution. Simulation of shock-induced bubble collapse demonstrates that the axisymmetricity of the interface is well preserved in the 3D computation. Moreover, the 3D CSI methods are also given experimental validation, including the interaction of a shock with 3D curved gas-gas interface and droplet deformation after shock impact. In all cases, a good agreement has been achieved with the experimental data, with respect to the interface evolution and flow features.

Numerical Solution of Viscoelastic Fluid‐Structure‐Diffusion Systems with Applications in Ophthalmology

AbstractThe research of fluid‐structure interaction problems is a continuously growing field, especially regarding applications in medicine and biology. We present the coupling of a potentially viscoelastic fluid with multiple hyperelastic structures incorporating chemical processes in the arbitrary Lagrangian Eulerian framework. This monolithic formulation allows a robust numerical solution with Newton's method. The discretization is based on the backward Euler scheme for temporal discretization and the Galerkin finite element method for spatial discretization. This fluid‐structure interaction problem is applied to ophthalmology in order to improve the medical treatment of retinal diseases. The physiological processes include the elastic response of various structures like the sclera, lens and iris coupled to the fluid‐like vitreous which is modeled by a viscoelastic Burgers type model for the healthy case and by the Newtonian Navier‐Stokes equations for the pathological case. Since most medical treatments are based on the injection of medicine we furthermore study the drug distribution, which is modeled by convection‐diffusion‐reaction equations, in the whole eye for healthy and non‐healthy pathologies.

Mechanics of plastic flow past a narrow-angle wedge indenter

ABSTRACTThe indentation of a metal specimen by a narrow-angle wedge produces extreme plastic deformation, with an effect akin to cutting into the metal. Simulation of such processes is challenging, and complicated by the need to model material separation along the indentation symmetry axis. Here we use an Arbitrary Lagrangian Eulerian (ALE) framework to enforce the symmetry boundary conditions (bcs) in their original, `strong’ form, as well as conventional Lagrangian FE to impose the bcs in a complementary, `weak’ form. Taken together these two cases, representing perfectly strong and perfectly weak interfaces, produce accurate bounds on the mechanical response for indentation by wedges with semi-apical angles as small as 15 degrees, and encompass intermediate cases that would require complicated models of ductile failure. The method accurately predicts the transition from the cutting pattern to the non-cutting (radially compressive) pattern as the apical angle is increased. In combination with Lagrangian particle tracking, the simulations reveal the deformation pattern as well as strain, strain-rate, and velocity fields in narrow angle indentation at high resolution. Interestingly, the strong form predicts a thin (tens of microns), near-wall layer of intense plastic strain, which has been observed recently in indentation experiments. With the exception of this feature, the strong and weak bc solutions are quite similar. The present approach reveals insights about plastic flow past narrow obstacles in a range of related problems including cone penetration and machining, and suggests using narrow-angle indentation as a way to probe material failure.

On the Differentiability of Fluid–Structure Interaction Problems with Respect to the Problem Data

A coupled system of stationary fluid–structure equations in an arbitrary Lagrangian–Eulerian framework is considered in this work. Existence results presented in the literature are extended to show differentiability of the solutions to a stationary fluid–structure interaction problem with respect to the given data, volume forces and boundary values, provided a small data assumption holds. Numerical experiments are used to substantiate the theoretical findings.

Multi-moment finite volume method for incompressible flows on unstructured moving grids and its application to fluid-rigid body interactions

Multi-moment finite volume method for unsteady incompressible flows on unstructured moving grids is developed under the Arbitrary Lagrangian Eulerian (ALE) framework. Two moments, volume integrated average (VIA) and point values (PV), are treated as computational variables which allow to construct quadratic or higher-order polynomials within a compact stencil. The VIAs are computed by a finite volume method (FVM) that ensures the rigorous numerical conservativeness, while the PVs are defined at cell vertices and updated efficiently by a formulation based on the governing equations in a differential form. By employing PVs at cell vertices as additional variables updated at each time step, the present multi-moment finite volume method shows great advantage when applied to the ALE framework for interactions among multiple materials, where the solution points can always coincide with the interfaces between different materials. Thus, any extra numerical step to approximate the values onto the interfaces is not necessary. We have devised two coupling schemes, i.e. an explicit weak coupling scheme and a semi-implicit strong coupling scheme, to formulate the interactions between fluid and moving solid with a wide range of mass ratios. An accurate and robust fluid-solid interaction (FSI) solver has been built by integrating the above numerical components with a radial basis function (RBF) interpolation technique for mesh movement. We in this paper present various benchmark tests to extensively verify the proposed numerical solver, which demonstrate its appealing performance in solving a large class of FSI problems.

A stabilized ALE method for computational fluid–structure interaction analysis of passive morphing in turbomachinery

Computational fluid–structure interaction (FSI) and flow analysis now have a significant role in design and performance evaluation of turbomachinery systems, such as wind turbines, fans, and turbochargers. With increasing scope and fidelity, computational analysis can help improve the design and performance. For example, it can help add a passive morphing attachment (MA) to the blades of an axial fan for the purpose of controlling the blade load and section stall. We present a stabilized Arbitrary Lagrangian–Eulerian (ALE) method for computational FSI analysis of passive morphing in turbomachinery. The main components of the method are the Streamline-Upwind/Petrov–Galerkin (SUPG) and Pressure-Stabilizing/Petrov–Galerkin (PSPG) stabilizations in the ALE framework, mesh moving with Jacobian-based stiffening, and block-iterative FSI coupling. The turbulent-flow nature of the analysis is handled with a Reynolds-Averaged Navier–Stokes (RANS) model and SUPG/PSPG stabilization, supplemented with the “DRDJ” stabilization. As the structure moves, the fluid mechanics mesh moves with the Jacobian-based stiffening method, which reduces the deformation of the smaller elements placed near the solid surfaces. The FSI coupling between the blocks of the fully-discretized equation system representing the fluid mechanics, structural mechanics, and mesh moving equations is handled with the block-iterative coupling method. We present two-dimensional (2D) and three-dimensional (3D) computational FSI studies for an MA added to an axial-fan blade. The results from the 2D study are used in determining the spanwise length of the MA in the 3D study.

An ALE pairwise-relaxing meshless method for compressible flows

In this paper, we present a pairwise-relaxing meshless (PRM) method for solving the Euler equations of compressible flows within the Arbitrary Lagrangian Eulerian (ALE) framework. Derived from the moving particle semi-implicit (MPS) method and the finite volume particle (FVP) method, the PRM approximates the derivatives from the value defined at the midpoint of each interacting particle pairs through a kernel-based formulation. Pairwise-relaxing constants are introduced to the kernels to provide degree of freedom to enforce the Taylor-series consistency condition while mass, momentum and energy are conserved exactly. An upwind high-order reconstruction scheme via a corrective procedure and variable cut-off radius is also developed for this PRM method. The HLLC approximate Riemann solver is adopted to solve Riemann problem. One and two-dimensional numerical tests are presented to demonstrate the performance of the PRM method.

Arbitrary Lagrangian Eulerian-Type Finite Element Methods Formulation for PDEs on Time-Dependent Domains with Vanishing Discrete Space Conservation Law

The aim of this paper is to introduce a finite element formulation within an arbitrary Lagrangian Eulerian (ALE) framework with a vanishing discrete space conservation law (SCL) for differential equations on time-dependent domains. The novelty of the formulation is the method for temporal integration which results in preserving the SCL property and retaining the higher order accuracy at the same time. Once the time derivative is discretized (based on an integration or differentiation formula), the common approach for terms in differential equation which do not involve temporal derivative is classified to be a kind of “time averaging” between time steps. In the spirit of classical approaches, this involves evaluating these terms at several points in time between the current and the previous time step ($[t_n,t_{n+1}]$), and then averaging them in order to provide the satisfaction of discrete SCL. Here, we fully use the polynomial in time form of mapping through which the evolution of the domain is realized---the so-called ALE map---in order to avoid the problems arising due to the moving grids. We give a general recipe on temporal schemes that have to be employed once the discretization for the temporal derivative is chosen. Numerical investigations on stability, accuracy, and convergence are performed and the simulated results are compared with benchmark problems set up by other authors.

Smoothed particle hydrodynamics simulation of a laser pulse impact onto a liquid metal droplet.

The impact of a laser pulse onto a liquid metal droplet is numerically investigated by utilising a weakly compressible single phase model; the thermodynamic closure is achieved by the Tait equation of state (EoS) for the liquid metal. The smoothed particle hydrodynamics (SPH) method, which has been employed in the arbitrary Lagrangian Eulerian (ALE) framework, offers numerical efficiency, compared to grid related discretization methods. The latter would require modelling not only of the liquid metal phase, but also of the vacuum, which would necessitate special numerical schemes, suitable for high density ratios. In addition, SPH-ALE allows for the easy deformation handling of the droplet, compared to interface tracking methods where strong mesh deformation and most likely degenerate cells occur. Then, the laser-induced deformation of the droplet is simulated and cavitation formation is predicted. The ablation pattern due to the emitted shock wave and the two low pressure lobes created in the middle of the droplet because of the rarefaction waves are demonstrated. The liquid metal droplet is subject to material rupture, when the shock wave, the rarefaction wave and the free surface interact. Similar patterns regarding the wave dynamics and the hollow structure have been also noticed in prior experimental studies.

Magneto-fluid-structure interaction issues for vibrating rigid bodies in conducting fluids: The numerical and the analytical approaches

To study the magneto-fluid-structure interaction (MFSI) problems for rigid bodies and conducting fluids, a numerical method and an analytical approach have been carried out. The numerical scheme is based on a partitioned arbitrary Lagrangian-Eulerian framework, and is suitable for viscous, incompressible magneto-fluid-structure interaction simulation. A displacement prediction-pressure stabilization scheme has been established to enhance the stability and efficiency. Meanwhile, a consistent and conservative scheme for deforming configurations has been developed. This method can numerically ensure the divergence-free condition of the current density, and can conserve the momentum from the Lorentz forces after grids update. The analytical approach has considered a vibrating cylinder surrounded by confined fluids in a magnetic field. By assuming a small amplitude and a low magnetic Reynolds number, the analytical solution can describe the temporal and spatial distribution of the fluid fields, the electromagnetic fields, and the solid motion. These solutions are also suitable for general fluid-structure interaction (FSI) problems. Comparative results suggest good agreement between the two methods developed in this paper. Nonlinear effects of the magnetic fields were presented and discussed based on the numerical results. These cases are based on careful validations, and can hopefully be used for future verification and validation work.

An oil sloshing study: adaptive fixed-mesh ALE analysis and comparison with experiments

We report in this work a numerical analysis of the sloshing of a squared tank partially filled with a domestic vegetable oil. The tank is subject to controlled motions with a shake table. The free-surface evolution is captured using ultrasonic sensors and an image capturing method. Only confirmed data within the error range is reported. Filling depth, imposed amplitude and frequency effects on the sloshing wave pattern are specifically evaluated. The experiments also reveal the nonlinear wave behavior. The numerical model is based on a stabilized finite element method of the variational multi-scale type. The free-surface is captured using a level set technique developed to be used with adaptive meshes in Arbitrary Lagrangian–Eulerian framework. The numerical results are compared with the experiments for different sloshing conditions near the first sloshing mode. The simulations satisfactorily match the experiments, providing a reliable tool for the analysis of this kind of problems.