Arbitrary Lagrangian-Eulerian Frame Research Articles

A finite element framework for fluid–structure interaction of turbulent cavitating flows with flexible structures

We present a finite element framework for the numerical prediction of cavitating turbulent flows interacting with flexible structures. The vapor–fluid phases are captured through a homogeneous mixture model, with a scalar transport equation governing the spatio-temporal evolution of cavitation dynamics. High-density gradients in the two-phase cavitating flow motivate the use of a positivity-preserving Petrov–Galerkin stabilization method in the variational framework. A mass transfer source term introduces local compressibility effects arising as a consequence of phase change. The turbulent fluid flow is modeled through a dynamic subgrid-scale method for large eddy simulations. The flexible structure is represented by a set of eigenmodes, obtained through a modal decomposition of the linear elasticity equations. While a partitioned iterative approach is adopted to couple the structural dynamics and cavitating fluid flow, the deforming flow domain is described by an arbitrary Lagrangian–Eulerian frame of reference. We establish the fidelity of the proposed framework by comparing it against experimental and numerical studies for both rigid and flexible hydrofoils in cavitating flows. Under unstable partial cavitating conditions, we identify specific vortical structures leading to cloud cavity collapse. We further explore features of cavitating flow past a rigid body such as re-entrant jet and turbulence-cavity interactions during cloud cavity collapse. Based on the validation study conducted over a flexible NACA66 rectangular hydrofoil, we elucidate the role of cavity and vortex shedding in governing the structural dynamics. Subsequently, we identify a broad spectrum frequency band whose central peak does not correlate to the frequency content of the cavitation dynamics or the natural frequencies of the structure, indicating the induction of unsteady flow patterns around the hydrofoil. Finally, we discuss the coupled fluid–structure dynamics during a cavitation cycle and the underlying mechanism associated with the promotion and mitigation of cavitation.

Internal resonance induced nonlinear vibration and acoustic radiation of a hyperelastic cantilever structure immersed in fluid

Hyperelastic structures can exhibit peculiar nonlinear vibration and acoustic behavior due to the existence of geometric and material nonlinearities. In order to comprehend the nonlinear vibro-acoustic responses of a hyperelastic cantilever structure configured with internal resonances, a finite element model that considers both the geometric and material nonlinearities in the structure and the finite-amplitude acoustic waves has been developed. The vibro-acoustic problem is described in an arbitrary Lagrangian-Eulerian frame with perfectly matched layers. Based on the finite element model, a loosely coupled serially staggered procedure is applied to solve the nonlinear governing equations. It is found that with large harmonic axial excitation applied to the hyperelastic cantilever structure, the structural vibration undergoes an axial-to-bending transition due to the 2:1 internal resonance, resulting in the variation of acoustic directivity and frequency characteristics. This transition of vibration form occurs quickly when the excitation amplitude is larger or the excitation frequency is closer to the frequency of the axial mode. A threshold of dissipation energy is proposed to determine the transition of the structural and acoustic responses due to nonlinear internal resonance. Saturation of the structural vibration and acoustic responses is observed in the frequency components associated with the first axial mode of the hyperelastic structure, which only exists in the case of the occurrence of the axial-to-bending transition.

Exploiting high‐contrast Stokes preconditioners to efficiently solve incompressible fluid–structure interaction problems

AbstractIn this work, we develop a new algorithm to solve large‐scale incompressible time‐dependent fluid–structure interaction problems using a matrix‐free finite element method in arbitrary Lagrangian–Eulerian frame of reference. We derive a semi‐implicit time integration scheme which improves the geometry‐convective explicit scheme for problems involving the interaction between incompressible hyperelastic solids and incompressible fluids. The proposed algorithm relies on the reformulation of the time‐discrete problem as a generalized Stokes problem with strongly variable coefficients, for which optimal preconditioners have recently been developed. The resulting algorithm is scalable, optimal, and robust: we test our implementation on model problems that mimic classical Turek–Hron benchmarks in two and three dimensions, and investigate timing and scalability results.

Effect of Wall Oscillation Period on Fluid Flow in Branched Channel with a Moving Indentation

The numerical simulation of two-dimensional fluid flow in T-shaped and Y-shaped channels having a single moving indented wall is performed by the finite element method in the Arbitrary Lagrangian-Eulerian frame. The motion of the indented wall is defined as a hyperbolic function, and it is located at a small segment at the bottom wall in the parent channel. The smallest value of the wall oscillation period causes the waviest core flow in the main channel, resulting in bigger vortices and greater flow separation region in the branches, especially during the outward indentation motion. This flow disturbance pattern is found more severe in T-channel as compared to that of in Y-channel.

Development of an In house Computer Code for the Simulation of Detonation Shock Dynamics in Underwater Explosion Scenario

Detonation Shock Dynamics (DSD), involved in an underwater explosion scenario is numerically simulated by an in-house computer code, ‘DSSDYN’. The simulation is based on Chapman-Jouguet (CJ) theory, formulated in Arbitrary Lagrangian Eulerian frame work. Specifically, the propagation of detonation front is simulated with‘Burn Fraction Model’. The classical burn fraction model is improved for achieving better computational efficiency. The simulation capability of DSS-DYN is demonstrated through a case study on explosion of PETN charge under the deep-water medium.Through this study, the salient features of DSD with better insight have been brought out. Besides, the physical parameters, such as work potential of PETN, are predicted efficiently.The apportionment of energy distributions indicates that about 70 % of chemical energy of explosive is transmitted to the surrounding water that is the major contribution of damage potential of the explosive. The predictions of peak velocity and peak pressure values by DSS-DYN and LS-DYNA show satisfactory comparison.DSS-DYN consumes lesser computational time (~1h), compared to LS-DYNA (~3h).

An arbitrary Lagrangian-Eulerian method for nonlinear structural-acoustic interaction of hyperelastic solid and compressible viscous fluid

This paper develops a numerical method for predicting the nonlinear structural-acoustic interactions of a hyperelastic solid immersed in an unbounded compressible viscous fluid. The nonlinear acoustic wave model with perfectly matched layers adopted is proposed in an arbitrary Lagrangian-Eulerian frame to absorb acoustic waves at truncated fluid boundaries, whereas the geometric and material nonlinearities of the hyperelastic solid are characterized in a total Lagrangian form. Both the fluid and the solid are discretized by the finite element method, and both models are coupled together by the momentum equilibrium and kinematic compatibility on the common interface. The convergence and accuracy of the proposed method are assessed by comparing the computed results with existing solutions. A hyperelastic ring in a viscous fluid is analyzed to provide physical insights into the nonlinear dynamics behaviors of the coupled hyperelastic solid and acoustic fluid system. The hyperelastic ring in acoustic fluid exhibits deformation instability under a uniform distributed excitation load, producing a series of subharmonics, ultrasubharmonics, and superharmonics in the structural and acoustic responses. Nonlinear internal resonance and bifurcation phenomena in the vibration of the ring and the acoustic waves of the fluid are also discussed.

Complex-valued derivative propagation method with adaptive moving grids for electronic nonadiabatic dynamics

The coupled complex quantum Hamilton–Jacobi equations in the diabatic representation are solved on adaptive moving grids for electronic nonadiabatic dynamics. Substitution of the wave functions in exponential form into the coupled time-dependent Schrödinger equations yields the coupled complex quantum Hamilton–Jacobi equations. Analytic equations of motion are derived for the spatial derivatives of the complex action functions for both surfaces. The equations of motion for the complex actions and their spatial derivatives are converted into the arbitrary Lagrangian–Eulerian frame, allowing us to have many ways to choose the velocities for the moving points. The grid points can be specified to move along with the probability fluid. In contrast, choosing the boundary grid points to follow Lagrangian trajectories and forcing the internal grid points to be equally spaced at each time step, we obtain a moving structured grid of uniform spacing during the wave packet evolution. Several examples are presented to illustrate main features of the adaptive grid method, including a single avoided crossing system, a dual avoided crossing system, an ultrashort laser pulse excitation system, and a three-dimensional nonadiabatic system. Accurate computational results demonstrate that the adaptive grid method provides an alternative way to study electronic nonadiabatic dynamics.

Multiscale and monolithic arbitrary Lagrangian–Eulerian finite element method for a hemodynamic fluid-structure interaction problem involving aneurysms

In this paper, a multiscale and monolithic arbitrary Lagrangian–Eulerian finite element method (ALE-FEM) is developed for a multiscale hemodynamic fluid-structure interaction (FSI) problem involving an aortic aneurysm growth to quantitatively predict the long-term aneurysm risk in the cardiovascular environment, where the blood fluid profile, the hyperelastic arterial wall, and the aneurysm pathophysiology are integrated into one hemodynamic FSI model, together with no-slip interface conditions between the blood fluid and the arterial wall. Additionally, two different time scales are involved: a fast time scale for the blood fluid-arterial wall interaction process in terms of seconds, and a slow time scale for the biological (abdominal aortic aneurysms (AAA) progression) process in terms of years. Two types of multiscale methods, the heterogeneous multiscale method (HMM) and the seamless multiscale method (SMM), are employed to tackle different time scales while the arbitrary Lagrangian–Eulerian (ALE) method is adopted to generate the moving blood fluid meshes that adapt to the deformation of the hyperelastic arterial wall all the time, based on which the variable time-stepping/mixed finite element method (FEM) is defined in the ALE frame to discretize the developed hemodynamic FSI model involving aneurysms. A two-dimensional schematic blood fluid-artery-aneurysm interaction example and a three-dimensional realistic cardiovascular FSI problem with an aortic aneurysm growth based upon the patients' CT scan data are simulated to validate the accuracy and the efficiency of our developed HMM(SMM)/ALE-FEM, and a medically reasonable long-term prediction is obtained for the aneurysm growth as well.

A novel arbitrary Lagrangian–Eulerian finite element method for a parabolic/mixed parabolic moving interface problem

In this paper, a monolithic arbitrary Lagrangian–Eulerian (ALE)-finite element method (FEM) is developed based upon a novel ALE mapping for a type of parabolic/mixed parabolic moving interface problem with jump coefficients. A stable Stokes-pair mixed FEM within a specific stabilization technique and a novel ALE time-difference scheme are developed to discretize this interface problem in both semi- and fully discrete fashion, for which the stability and error estimate analyses are conducted in the ALE frame. Numerical experiments are carried out to validate all theoretical results in different cases. The developed novel ALE-FEM can be extended to a moving interface problem that involves the pore fluid (Darcy) equation or Biot’s model in the future.

Effect of vortex-induced vibration of finned cylinders on heat transfer enhancement

Two-degree-of-freedom vortex-induced vibration (VIV) of a finned cylinder with heat transfer is studied numerically at the Reynolds number Re = 150. The governing equations in the Arbitrary Lagrangian-Eulerian frame are solved by the finite volume method. The dynamics of the oscillating cylinder (with or without fins) in the fluid flow was approximated as a mass-spring system. The effects of the number and arrangement of the fins (14 different cases) on the vortex shedding pattern, vibration amplitude, and frequency and heat transfer of the cylinder are investigated and discussed. The results indicate that in comparison with the stationary state, the effects of the number and arrangement of the fins on the wake pattern and the heat transfer enhancement in the VIV state are significant. Different vortex shedding pattern like 2S, P, 2P, S + P and combination of them with stable or unstable interactions between vortices and cylinders are observed in an oscillating cylinder. In the vibration state of finned cylinders, the heat transfer enhances up to 50.4% with respect to the stationary state and increases up to 64% with respect to the stationary smooth cylinder.

Continuum shape sensitivity analysis for aeroelastic gust using an arbitrary lagrangian-eulerian reference frame

Continuum Sensitivity Analysis (CSA), a method to determine response derivatives with respect to design variables, is derived here for the first time in an arbitrary Lagrangian-Eulerian (ALE) reference frame. CSA differentiates nonlinear governing system of equations to arrive at a linear system of partial differential continuum sensitivity equations (CSEs), here, for fluid-structure interaction (FSI). The CSEs and associated sensitivity boundary conditions are derived here for the first time for FSI, using the boundary velocity formulation, carefully distinguishing design velocity from flow velocity and ALE mesh velocity. Whereas boundary conditions must be differentiated using the material (total) derivative, it is sometimes advantageous to derive the CSEs using local (partial) derivatives. The benefit is that geometric sensitivity, known as design velocity, may not be required in the domain. It is shown here that this advantage is realized when the ALE frame undergoes only the rigid body motion associated with the structure to which it is attached. It is further shown that the advantage is not realized when the ALE mesh deforms due to the flexible motion of the fluid-structure interface. The equations for the transient gust response of a two-dimensional airfoil in compressible flow, flexibly attached to a rigid body mass, are presented as a model problem to illustrate a detailed derivation.

Numerical modeling of inelastic structures at loading of steady state rolling

In order to gain a deeper knowledge of the interactions in the coupled tire-pavement-system, e.g. for the future design of durable pavement structures, the paper presents recent results of research in the field of theoretical-numerical asphalt pavement modeling at material and structural level, whereby the focus is on a realistic and numerically efficient computation of pavements under rolling tire load by using the finite element method based on an Arbitrary Lagrangian Eulerian (ALE) formulation. Inelastic material descriptions are included into the ALE frame efficiently by a recently developed unsplit history update procedure. New is also the implementation of a viscoelastic cohesive zone model into the ALE pavement formulation to describe the interaction of the single pavement layers. The viscoelastic cohesive zone model is further extended to account for the normal pressure dependent shear behavior of the bonding layer. Another novelty is that thermo-mechanical effects are taken into account by a coupling of the mechanical ALE pavement computation to a transient thermal computation of the pavement cross-section to obtain the varying temperature distributions of the pavement due to climatic impact. Then, each ALE pavement simulation considers the temperature dependent asphalt material model that includes elastic, viscous and plastic behavior at finite strains and the temperature dependent viscoelastic cohesive zone formulation. The temperature dependent material parameters of the asphalt layers and the interfacial layers are fitted to experimental data. Results of coupled tire-pavement computations are presented to demonstrate potential fields of application.

Residual-based turbulence models and arbitrary Lagrangian–Eulerian framework for free surface flows

We present a residual-based turbulence model for problems with free surfaces. The method is derived based on variational multiscale ideas that assume a decomposition of the solution fields into overlapping scales that are termed as coarse and fine scales. The fine scales are further split hierarchically into fine-scales level-I and fine-scales level-II. The hierarchical variational problems that govern the two fine-scale components are modeled employing bubble functions approach. The model for level-II scales is variationally embedded in the mixed field level-I problem to yield a stable level-I formulation. Subsequently, the model for level-I scales that in fact constitutes the fine-scale turbulence model is then variationally injected in the coarse-scale variational form. A significant feature of the method is that it does not contain any embedded tunable parameters. To accommodate the moving boundaries we cast the formulation in an arbitrary Lagrangian–Eulerian frame of reference. The free surface boundary condition is imposed weakly which results in a formulation that conserves the volume of the fluid. A variety of benchmark problems show the accuracy and range of applicability of the proposed formulation and results are compared with published data. A wavy bed problem is investigated to show the interaction of turbulence generated at the bottom surface with the free surface thereby leading to irregular free surface elevations.

Numerical Examination of Wave-induced Coupling Roll Motion and Fluid Resonance between twin Floating Barges in Proximity

Coupling effects of roll motion and fluid resonance between twin rectangular barges subjected to incident waves normal to the spanwise of the narrow gap are investigated employing a two-dimensional numerical wave flume based on a CFD package OpenFOAM. The governing equations for the fluid flow are the Navier-Stokes equations of incompressible Newtonian fluid. The interface between water and air phases is captured by a VOF (volume of fluid) method. The fluid-structure interaction (FSI) is treated with moving meshes in the Arbitrary Lagrangian-Eulerian frame. The present numerical model is validated against wave generation and propagation over a flat bed and the decay of a free rolling rectangular barge in still water. The comparisons with available numerical results show that the present numerical wave flume works well. It is then used to investigate the roll response of the twin floating barges in proximity. The numerical results of this work show that the resonant frequency of the fluid oscillation in the narrow gap between twin rolling barges is smaller than that between twin fixed barges. However, the resonant wave height in the gap between twin rolling barges is observed to be in the same magnitude as that between twin fixed barges. Moreover, the present numerical results indicate that the maximum rolling amplitudes of the twin barges are close to each other, and they appear at the same resonant frequency of the fluid oscillation in the narrow gap. At the low frequency band, the amplitude of the rear barge is larger than that of the leading one. Otherwise, in the frequency band higher than the resonant conditions, the leading barge has a larger response. The present numerical solutions also suggest that the twin barges roll in anti-phase.

Numerical simulation of pavement structures with inelastic material behaviour under rolling tyres based on an arbitrary Lagrangian Eulerian (ALE) formulation

In this research, a numerically efficient computational procedure for the simulation of pavement structures under the load of rolling tyres is presented. In order to avoid a time-consuming transient finite element computation, an arbitrary Lagrangian Eulerian (ALE) formulation is introduced for the pavement structure. In the ALE frame, the pavement is described in a moved reference configuration, so that the loading of the pavement with a steady state rolling tyre can be described as a time-independent process. For the consideration of the inelastic material behaviour of asphalt in the ALE frame, two different procedures are presented and compared. The developed novel inelastic unsplit ALE procedure is found to be adequate, reliable and efficient. As an example, a pavement structure under the load of a rolling truck tyre is simulated. Stresses, strains and displacements of the pavement structure as well as the effect of different driving velocities of the tyre are evaluated.

A DYNAMICS CODE FOR A DUSTY INTERSTELLAR CLOUD EMBEDDED IN AN INTER-CLOUD MEDIUM

To study the internal dynamics of dusty interstellar clouds bound by an inter-cloud medium, we have written a one-dimensional numerical code for dusty gas dynamics in two-phase media. The governing equations for both cloud and inter-cloud media are solved using a Godunov-type scheme. The interface between the cloud and inter-cloud media is tracked by a mesh in the arbitrary Lagrangian-Eulerian frame. The dust component is treated as either a pressureless continuum or super-particles. The dust-gas drag is described by either the Epstein law or the Baines relation. To minimize errors from source terms, we utilize (in time integration) an explicit fourth-order Runge-Kutta scheme. This code is specially designed and tuned for simulating the complicated dynamics of dusty interstellar clouds. We have run three sets of test problems: the first set comprises shock-tube tests. The resulting shocks are resolved with two or three grid points. The second set is to examine the degree of error convergence. The results at various spatial resolutions demonstrate the second-order spatial accuracy of our code. The third set involves the Bonnor-Ebert sphere. From a number of simulations, we could numerically determine the Bonnor-Ebert criterion with errors less than 6%. As an example of utilizing our code, we simulate the collapse of dusty clouds embedded in a radiation field. Inside the cloud, the dust grains drift inward by radiation field and the dust-to-gas ratio at the intermediate radius increases by a factor of 20.

Complex trajectories sans isochrones: Quantum barrier scattering with rectilinear constant velocity trajectories

One of the major obstacles in employing complex-valued trajectory methods for quantum barrier scattering calculations is the search for isochrones. In this study, complex-valued derivative propagation method trajectories in the arbitrary Lagrangian-Eulerian frame are employed to solve the complex Hamilton-Jacobi equation for quantum barrier scattering problems employing constant velocity trajectories moving along rectilinear paths whose initial points can be in the complex plane or even along the real axis. It is shown that this effectively removes the need for isochrones for barrier transmission problems. Model problems tested include the Eckart, Gaussian, and metastable quadratic+cubic potentials over a variety of wave packet energies. For comparison, the "exact" solution is computed from the time-dependent Schrodinger equation via pseudospectral methods.

A 3D adaptive mesh moving scheme

AbstractThis paper presents an adaptive mesh moving technique for three‐dimensional (3D) fluid flow problems that involve moving fluid boundaries and fluid–solid interfaces. Such mesh moving techniques are an essential ingredient of fluid–structure interaction methods that typically employ arbitrary Lagrangian–Eulerian (ALE) frameworks. In the ALE frame, the velocity field representing motion of the underlying continuum is integrated in the fluid flow equations. In the discretized setting, the velocity field of the underlying continuum gives rise to the mesh displacement field that needs to be solved for in addition to the flow equations and the structural equations. Emphasis in the present work is on the motion and deformation of 3D grids that are composed of linear tetrahedral and hexahedral elements in structured and unstructured configurations. The proposed method can easily be extended to higher‐order elements in 3D. A variety of moving mesh problems from different fields of engineering are presented that show the range of applicability of the proposed method and the class of problems that can be addressed with it. Copyright © 2007 John Wiley & Sons, Ltd.

Effects of Mesh Motion on the Stability and Convergence of ALE Based Formulations for Moving Boundary Flows

This paper investigates the effects of mesh motion on the stability of fluid-flow equations when written in an Arbitrary Lagrangian–Eulerian frame for solving moving boundary flow problems. Employing the advection-diffusion equation as a model problem we present a mathematical proof of the destabilizing effects induced by an arbitrary mesh motion on the stability and convergence of an otherwise stable scheme. We show that the satisfaction of the so-called geometric conservation laws is essential to the development of an identity that plays a crucial role in establishing stability. We explicitly show that the advection dominated case is susceptible to growth in error because of the motion of the computational grid. To retain the bound on the growth in error, the mesh motion techniques need to account for a domain based constraint that minimizes the relative mesh velocity. Analysis presented in this work can also be extended to the Navier–Stokes equations when written in an ALE frame for FSI problems.

Quantum hydrodynamics: Capturing a reactive scattering resonance

The hydrodynamic equations of motion associated with the de Broglie-Bohm formulation of quantum mechanics are solved using a meshless method based upon a moving least-squares approach. An arbitrary Lagrangian-Eulerian frame of reference and a regridding algorithm which adds and deletes computational points are used to maintain a uniform and nearly constant interparticle spacing. The methodology also uses averaged fields to maintain unitary time evolution. The numerical instabilities associated with the formation of nodes in the reflected portion of the wave packet are avoided by adding artificial viscosity to the equations of motion. A new and more robust artificial viscosity algorithm is presented which gives accurate scattering results and is capable of capturing quantum resonances. The methodology is applied to a one-dimensional model chemical reaction that is known to exhibit a quantum resonance. The correlation function approach is used to compute the reactive scattering matrix, reaction probability, and time delay as a function of energy. Excellent agreement is obtained between the scattering results based upon the quantum hydrodynamic approach and those based upon standard quantum mechanics. This is the first clear demonstration of the ability of moving grid approaches to accurately and robustly reproduce resonance structures in a scattering system.