Arbitrary Lagrangian Eulerian Formulation Research Articles

Numerical simulations on the dynamics of a spheroid in a viscoelastic liquid in a wide-slit microchannel

The translational and orientational dynamics of a spheroid suspended in a viscoelastic liquid flowing in a wide-slit microchannel is studied by 3D finite element simulations. An Arbitrary Lagrangian-Eulerian formulation is adopted to handle particle motion. The particle dynamics, with specific interest on the migration phenomenon and on the long-time orientation, is studied by computing the trajectories of the spheroid and the orbits described by its orientation vector. The effects of the initial distance from the channel centerplane and of the initial orientation on the particle motion is investigated, for low and moderate Weissenberg numbers. A ‘migration state diagram’ giving the long-time equilibrium positions of the particle for all possible initial configurations is derived: in most cases, the particle goes to the centerplane, but there are circumstances where also the wall is an attractor.

Vortex-induced vibrations of an elliptic cylinder with both transverse and rotational degrees of freedom

Two-degree-of-freedom (2-DOF) vortex-induced vibration (VIV) of an elliptic cylinder is numerically investigated at a Reynolds number Re=150. The incompressible Navier–Stokes equations in arbitrary Lagrangian–Eulerian formulation are solved by a four-step fractional finite element method. The elliptic cylinder, with a variety of aspect ratios (AR=1.0–2.5), is free to vibrate in both transverse and azimuthal directions. At each AR, the computations are conducted within a wide range of reduced velocities (2≤Ur≤15). For comparison, the one-degree-of-freedom (1-DOF) responses of the cylinder vibrating only transversely are also provided. The results show that significant rotations occur under a condition of frequency synchronization similar to the lock-in condition of VIV in the transverse direction. Under the rotation effect, the transverse peak amplitude at AR=1.0 reaches a higher value about 20% larger than its counterpart of 1-DOF VIV, whereas it fells by more than half at AR=2.0 and 2.5, and the corresponding wake patterns present distinct characteristics. The typical phenomenon of phase switch can be captured successfully between the torque and rotation response, which occurs at the same reduced velocity to the phasing between the transverse force and motion. Our simulation results also elucidate that the free rotation may significantly affect the statistics of the drag, lift and moment coefficients. The most striking observation is that all of the force responses at AR=2.0 and 2.5 collapse onto a very low level compared to those of the transverse-only VIV. This is consistent with the dramatic drop in the transverse vibration amplitude at the same AR.

Numerical study of heat transfer from a synthetic impinging jet with a detailed model of the actuator membrane

Synthetic jets are produced by the oscillatory movement of a membrane inside a cavity, causing the fluid to enter and leave through a small orifice. The present study is focused on investigating the cooling capabilities of a synthetic jet enclosed between two parallel isothermal plates with an imposed temperature difference. The unsteady three-dimensional Navier-Stokes equations have been solved for a range of Reynolds numbers from 50 to 1000 using time-accurate numerical simulations. A detailed model based on an Arbitrary Lagrangian-Eulerian (ALE) formulation is used to account for the movement of the actuator membrane. All the resulting flows are inherently three-dimensional and dominated by two major vortices, which find their counterparts inside the actuator cavity. A new structure, which is not found in open cavities, appears as an interaction of the synthetic jet flow with the bottom wall and results in a change on the jet's heat transfer mechanisms. Analysis of the outlet temperature has shown that assuming a uniform profile is reasonable if the Reynolds number is high enough, however, the outlet jet temperature is significantly higher than the cold plate temperature. Finally, this study proposes correlations for the heat transfer at the hot wall and the outlet temperature with the Reynolds number, which can be used to account for the cavity effects without the computationally expensive ALE model.

Fluid–structure interactions modeling the venous valve

In this study, a fully coupled fluid–structure interaction computational model for a pulsatile flow in the venous valve was developed. An arbitrary Lagrangian–Eulerian formulation is used in the coupled model, considering two domains, specifically the blood flow and the valve leaflets. The effect of the gap width between the valve leaflets (venous valve incompetence) on the valve reverse flow (reflux) has been investigated.

Dynamic modeling of levitation of a superconducting bulk by coupled H-magnetic field and arbitrary Lagrangian–Eulerian formulations

Intrinsically stable magnetic levitation between superconductors and permanent magnets can be exploited in a variety of applications of great technical interest in the field of transportation (rail transportation), energy (flywheels) and industry. In this contribution, we present a new model for the calculation of levitation forces between superconducting bulks and permanent magnets, based on the H-formulation of Maxwell’s equations coupled with an arbitrary Lagrangian–Eulerian formulation. The model uses a moving mesh that adapts at each time step based on the time-change of the distance between a superconductor bulk and a permanent magnet. The model is validated against a fixed mesh model (recently in turn validated against experiments) that uses an analytical approach for calculating the magnetic field generated by the moving permanent magnet. It is then used to analyze the magnetic field dynamics both in field-cooled and zero-field-cooled conditions and successively used to test different configurations of permanent magnets and to compare them in terms of levitation forces. The ease of implementation of this model and its flexibility in handling different geometries, material properties, and application scenarios makes it an attractive tool for the analysis and optimization of magnetic levitation-based applications.

Evaluation of Peak Wall Stress in an Ascending Thoracic Aortic Aneurysm Using FSI Simulations: Effects of Aortic Stiffness and Peripheral Resistance.

It has been reported clinically that rupture or dissections in thoracic aortic aneurysms (TAA) often occur due to hypertension which may be modelled with sudden increase of peripheral resistance, inducing acute changes of blood volumes in the aorta. There is clinical evidence that more compliant aneurysms are less prone to rupture as they can sustain such changes of volume. The aim of the current paper is to verify this paradigm by evaluating computationally the role played by the variation of peripheral resistance and the impact of aortic stiffness onto peak wall stress in ascending TAA. Fluid-structure interaction (FSI) analyses were performed using patient-specific geometries and boundary conditions derived from 4D MRI datasets acquired on a patient. Blood was assumed incompressible and was treated as a non-Newtonian fluid using the Carreau model while the wall mechanical properties were obtained from the bulge inflation tests carried out in vitro after surgical repair. The Navier-Stokes equations were solved in ANSYS Fluent. The Arbitrary Lagrangian-Eulerian formulation was used to account for the wall deformations. At the interface between the solid domain and the fluid domain, the fluid pressure was transferred to the wall and the displacement of the wall was transferred to the fluid. The two systems were connected by the System Coupling component which controls the solver execution of fluid and solid simulations in ANSYS. Fluid and solid domains were solved sequentially starting from the fluid simulations. Distributions of blood flow, wall shear stress and wall stress were evaluated in the ascending thoracic aorta using the FSI analyses. We always observed a significant flow eccentricity in the simulations, in very good agreement with velocity profiles measured using 4D MRI. The results also showed significant increase of peak wall stress due to the increase of peripheral resistance and aortic stiffness. In the worst case scenario, the largest peripheral resistance (1010kgsm-4) and stiffness (10MPa) resulted in a maximal principal stress equal to 702kPa, whereas it was only 77kPa in normal conditions. This is the first time that the risk of rupture of an aTAA is quantified in case of the combined effects of hypertension and aortic stiffness increase. Our findings suggest that a stiffer TAA may have the most altered distribution of wall stress and an acute change of peripheral vascular resistance could significantly increase the risk of rupture for a stiffer aneurysm.

Fluid-structure interaction of heart valve dynamics in comparison to finite-element analysis

Abstract An established therapy for aortic valve stenosis and insufficiency is the transcatheter aortic valve replacement. By means of numerical simulation the valve dynamics can be investigated to improve the valve prostheses performance. This study examines the influence of the hemodynamic properties on the valve dynamics utilizing fluidstructure interaction (FSI) compared with results of finiteelement analysis (FEA). FEA and FSI were conducted using a previously published aortic valve model combined with a new developed model of the aortic root. Boundary conditions for a physiological pressurization were based on measurements of ventricular and aortic pressure from in vitro hydrodynamic studies of a commercially available heart valve prosthesis using a pulse duplicator system. A linear elastic behavior was assumed for leaflet material properties and blood was specified as a homogeneous, Newtonian incompressible fluid. The type of fluid domain discretization can be described with an arbitrary Lagrangian-Eulerian formulation. Comparison of significant points of time and the leaflet opening area were used to investigate the valve opening behavior of both analyses. Numerical results show that total valve opening modelled by FEA is faster compared to FSI by a factor of 5. In conclusion the inertia of the fluid, which surrounds the valve leaflets, has an important influence on leaflet deformation. Therefore, fluid dynamics should not be neglected in numerical analysis of heart valve prostheses.

A numerical study on the transient impulsive pressure of a water jet impacting nonplanar solid surfaces

This paper presents an in-depth investigation into the transient impulsive pressure of an arc-curved water jet impacting a solid surface. The emphasis of this study is on the variations of the surface shape, which are classified into four types: The flat surface, the concave surface, the convex surface and the inclined surface. The numerical tool of arbitrary Lagrangian-Eulerian formulations is used to model the arc-curved jet impacting these different solid surface types. Elaborately designed experiments were conducted to test the impulsive pressure profile; the experimental results are found to be in approximate agreement with the numerical results. The impulsive pressure profiles of water jet impacting the flat and inclined solid surface are observed to exhibit two quintessential stages, in line with the traditional pressure profile; however, a double/multiple-peaked pressure structure is observed for the cases of the water jet impacting the concave and convex solid surfaces. Additionally, the value of the peak pressure is found to be a quadratic representation with the jet velocity, and the duration of the peak pressure is found to be an exponential representation with the jet velocity. The compression degrees of the liquid jet impacting the different surfaces are validated to be the root cause for the discrepancy of the impulsive pressure.

A Predictive Model of Thrombus Growth in Stenosed Vessels with Dynamic Geometries

This paper introduces a dynamic model of platelet-rich thrombus growth in stenosed vessels using computational fluid dynamics methods. Platelet adhesion, aggregation and activation kinetics are modeled by solving mass transport equations for blood components involved in thrombosis. Arbitrary Lagrangian–Eulerian formulation is used to model the growing thrombi with variable geometry. The wall boundaries are discretely moved based on the amount of platelet deposition that occurs on vessel wall. To emulate the dynamic behavior of platelet adhesion kinetics during thrombus growth, a validated model for platelet adhesion, which calculates platelet-surface adhesion rates as a function of stenosis severity and Reynolds number, is applied to the model. Results of the present model for vessel occlusion times and platelet deposition in stenosed region are compared to ex vivo and in vitro experimental data. The model successfully predicts the nonlinear growth of thrombi in the stenosed area. These simulations provide a useful guideline to understand the effect of growing thrombus on thrombus growth rate, platelet activation kinetics and recurrence of embolism in highly stenosed arteries.

Developing a parallel density-based implicit solver with mesh deformation in OpenFOAM

Numerical simulations like the aeroelastic computation and aerodynamic shape optimization usually involve moving boundaries, and are always carried out with supersonic compressible flows. In this paper, based on the OpenFOAM platform, we present a parallel density-based implicit solver with mesh deformation to address this kind of problems, and fill the gap for the deficiencies of implicit solvers. The core implementation details of the solver based on OpenFOAM are introduced. The Godunov type schemes in ALE (Arbitrary Lagrangian Eulerian) formulation and the dual-time stepping LU-SGS (Lower-Upper Symmetric Gauss-Seidel) algorithm are applied to solve the Navier-Stokes equations with turbulence model. For dynamic meshes, the novel parallel mesh deformation approach based on the Support Vector Machine is implemented. Four typical cases are tested to demonstrate the validity and parallel efficiency of the proposed solver. The result shows that the scalability of the solver reaches its limit at around 128 MPI (Message Passing Interface) tasks, and the cost of MPI communication is the bottleneck for large-scale simulations. Overall, our solver is applicable for various dynamic mesh compressible problems and all types of grids.

A numerical study on hemodynamics in the left coronary bifurcation with normal and hypertension conditions.

In this study, a three-dimensional analysis of the non-Newtonian blood flow was carried out in the left coronary bifurcation. The Casson model and hyperelastic and rigid models were used as the constitutive equation for blood flow and vessel wall model, respectively. Physiological conditions were considered first normal and then compliant with hypertension disease with the aim of evaluating hemodynamic parameters and a better understanding of the onset and progression of atherosclerosis plaques in the coronary artery bifurcation. Two-way fluid-structure interaction method applying a fully implicit second-order backward Euler differencing scheme has been used which is performed in the commercial code ANSYS and ANSYS CFX (version 15.0). When artery deformations and blood pressure are associated, arbitrary Lagrangian-Eulerian formulation is employed to calculate the artery domain response using the temporal blood response. As a result of bifurcation, noticeable velocity reduction and backflow formation decrease shear stress and made it oscillatory at the starting point of the LCx branch which caused the shear stress to be less than 1 and 2Pa in the LCx and the LAD branches, respectively. Oscillatory shear index (OSI) as a hemodynamic parameter represents the increase in residence time and oscillatory wall shear stress. Because of using the ideal 3D geometry and realistic physiological conditions, the values obtained for shear stress are more accurate than the previous studies. Comparing the results of this study with previous clinical investigations shows that the regions with low wall shear stress less than 1.20Pa and with high OSI value more than 0.3 are in more potential risk to the atherosclerosis plaque development, especially in the posterior after the bifurcation.

Tetrahedral mesh adaptation for Lagrangian shock hydrodynamics

Lagrangian shock hydrodynamics simulations will fail to proceed past a certain time if the mesh is approaching tangling. A common solution is an Arbitrary Lagrangian Eulerian (ALE) form, in which the mesh is improved (remeshing) and the solution is remapped onto the improved mesh. The simplest remeshing techniques involve moving only the nodes of the mesh. More advanced remeshing techniques involve altering the mesh connectivity in portions of the domain in order to prevent tangling. Work has been done using Voronoi-based polygonal mesh generators and 2D quad/triangle mesh adaptation. This paper presents the use of tetrahedral mesh adaptation methods as the remeshing step in an otherwise Lagrangian finite element shock hydrodynamics code called Alexa.

A monolithic fluid–structure interaction formulation for solid and liquid membranes including free-surface contact

A unified fluid–structure interaction (FSI) formulation is presented for solid, liquid and mixed membranes. Nonlinear finite elements (FE) and the generalized-α scheme are used for the spatial and temporal discretization. The membrane discretization is based on curvilinear surface elements that can describe large deformations and rotations, and also provide a straightforward description for contact. The fluid is described by the incompressible Navier–Stokes equations, and its discretization is based on stabilized Petrov–Galerkin FE. The coupling between fluid and structure uses a conforming sharp interface discretization, and the resulting non-linear FE equations are solved monolithically within the Newton–Raphson scheme. An arbitrary Lagrangian–Eulerian formulation is used for the fluid in order to account for the mesh motion around the structure. The formulation is very general and admits diverse applications that include contact at free surfaces. This is demonstrated by two analytical and three numerical examples exhibiting strong coupling between fluid and structure. The examples include balloon inflation, droplet rolling and flapping flags. They span a Reynolds-number range from 0.001 to 2000. One of the examples considers the extension to rotation-free shells using isogeometric FE.

A monolithic multiphase porous medium framework for (a-)vascular tumor growth

We present a dynamic vascular tumor model combining a multiphase porous medium framework for avascular tumor growth in a consistent Arbitrary Lagrangian Eulerian formulation and a novel approach to incorporate angiogenesis. The multiphase model is based on Thermodynamically Constrained Averaging Theory and comprises the extracellular matrix as a porous solid phase and three fluid phases: (living and necrotic) tumor cells, host cells and the interstitial fluid. Angiogenesis is modeled by treating the neovasculature as a proper additional phase with volume fraction or blood vessel density. This allows us to define consistent inter-phase exchange terms between the neovasculature and the interstitial fluid. As a consequence, transcapillary leakage and lymphatic drainage can be modeled. By including these important processes we are able to reproduce the increased interstitial pressure in tumors which is a crucial factor in drug delivery and, thus, therapeutic outcome. Different coupling schemes to solve the resulting five-phase problem are realized and compared with respect to robustness and computational efficiency. We find that a fully monolithic approach is superior to both the standard partitioned and a hybrid monolithic-partitioned scheme for a wide range of parameters. The flexible implementation of the novel model makes further extensions (e.g., inclusion of additional phases and species) straightforward.

Modelling of guillotine cutting of multi-layered aluminum sheets

The paper presents a numerical modelling of the guillotine cutting process of sheet aluminum bundles. A finite element method with a smoothed particle hydrodynamics approach were coupled to simulate the cutting process. Moreover, arbitrary Lagrangian–Eulerian formulation was also proposed and applied to model the process. Experimental results of the cutting were presented for the validation purposes. The measured force characteristics in all numerical simulations and experiment were compared. The computational costs of the implemented methods were also analyzed. Additionally, a mesh (particles) sensitivity study was performed, and the influence of the mesh on the obtained results was assessed. Based on the outcomes the arbitrary Lagrangian–Eulerian model was selected as more suitable method of modelling the guillotine cutting process and it was validated in different cutting conditions and the influence of technological parameters such as knife velocity and cutting-edge angle was investigated.

Numerical approach to study bubbles and drops evolving through complex geometries by using a level set – Moving mesh – Immersed boundary method

The present work proposes a method to study problems of drops and bubbles evolving in complex geometries. First, a conservative level set (CLS) method is enforced to handle the multiphase domain while keeping the mass conservation under control. An Arbitrary Lagrangian-Eulerian (ALE) formulation is proposed to optimize the simulation domain. Thus, a moving mesh (MM) will follow the motion of the bubble, allowing the reduction of the computational domain size and the improvement of the mesh quality. This has a direct impact on the computational resources consumption which is notably reduced. Finally, the use of an Immersed Boundary (IB) method allows to deal with intricate geometries and to reproduce internal boundaries within an ALE framework. The resulting method is capable of dealing with full unstructured meshes. Different problems have been studied to assert the proposed formulation, both involving constricting and non-constricting geometries. In particular, the following problems have been addressed: a 2D gravity-driven bubble interacting with a highly-inclined plane, a 2D gravity-driven Taylor bubble turning into a curved channel, the 3D passage of a drop through a periodically constricted channel, and the impingement of a 3D drop on a flat plate. Good agreement was found for all these cases study, which proves the suitability of the proposed CLS + MM + IB method to study this type of problems.

A 3D common-refinement method for non-matching meshes in partitioned variational fluid–structure analysis

We present a three-dimensional (3D) common-refinement method for non-matching unstructured meshes between non-overlapping subdomains of incompressible turbulent fluid flow and nonlinear hyperelastic structure. The fluid flow is discretized using a stabilized Petrov–Galerkin method, and the large deformation structural formulation relies on a continuous Galerkin finite element method. An arbitrary Lagrangian–Eulerian formulation with a nonlinear iterative force correction (NIFC) coupling is achieved in a staggered partitioned manner by means of fully decoupled implicit procedures for the fluid and solid discretizations. To begin, we first investigate the accuracy of common-refinement method (CRM) to satisfy the traction equilibrium condition along the fluid-elastic interface with non-matching meshes. We systematically assess the accuracy of CRM against the matching grid solution by varying grid mismatch between the fluid and solid meshes over a tubular elastic body. We demonstrate the second-order accuracy of CRM through uniform refinements of fluid and solid meshes along the interface. We then extend the error analysis to transient data transfer across non-matching meshes between the fluid and solid solvers. We show that the common-refinement discretization across non-matching fluid–structure grids yields accurate transfer of the physical quantities across the fluid–solid interface. We next solve a 3D fluid–structure interaction (FSI) problem of a cantilevered hyperelastic plate behind a circular bluff body and verify the accuracy of coupled solutions with respect to the available solution in the literature. By varying the solid interface resolution, we generate various non-matching grid ratios and quantify the accuracy of CRM for the nonlinear structure interacting with a laminar flow. We illustrate that the CRM with the partitioned NIFC treatment is stable for low solid-to-fluid density ratio and non-matching meshes for the 3D FSI problem. Finally, we demonstrate the 3D parallel implementation of the common-refinement with the NIFC method for a realistic engineering problem of drilling riser undergoing complex vortex-induced vibration with strong added mass effects and turbulent wake flow.

An Arbitrary Lagrangian–Eulerian formulation of a geometrically exact Timoshenko beam running through a tube

When a beam moves through a curved tube, there exists a large number of contacts between the beam and the inner wall of the tube. Usually, it is necessary to use fine Lagrangian meshes to discretize the beam for possible contact area in order to obtain sufficiently accurate results, even if the contact is only once for a very short time. Accordingly, the computation process is very time-consuming due to the large number of degrees of freedom (DOF) of the discretized system. To solve this problem, an Arbitrary Lagrangian–Eulerian (ALE) formulation of a Timoshenko beam based on geometrically exact beam theory, which considers the rotation around the axis of the beam, is presented in this paper. In this formulation, the mesh nodes of the beam are not associated with the material points, which provides the flexibility to mesh the beam freely. For this reason, the ALE mesh nodes can be arranged along the tube and constrained to move just within the corresponding tube cross section. The axial movement of the beam can be described by the material points flowing through the axially constrained mesh nodes. In so doing, only the beam in the high-curvature section of the tube needs to be finely meshed, resulting in a significant reduction of the DOF of the whole system. Several examples are presented and discussed to demonstrate the correctness and efficiency of the proposed method.

On the Application of Acoustic Analogies in the Numerical Simulation of Human Phonation Process

This paper deals with the numerical simulation of the flow induced sound as generated in the human phonation process. The two-dimensional fluid-structure interaction problem is modeled with the aid of linear elastic problem coupled to the incompressible Navier-Stokes equations in the arbitrary Lagrangian-Eulerian form. For calculation of sound sources and its propagation the Lighthill acoustic analogy is used. This approach is compared to the Perturbed Convective Wave Equation method. All partial differential equations, describing the elastic body motion, the fluid flow and the acoustics, are solved by applying the finite element method. In order to overcome numerical instabilities in the Navier-Stokes equations arising due to high Reynolds numbers, the modified streamline-upwind/Petrov-Galerkin stabilization is used. The implemented iterative coupling procedure is described. Numerical results are presented.

A model order reduction method for the simulation of gear contacts based on Arbitrary Lagrangian Eulerian formulation

Gear contact problems are characterized by a large number of possible contact scenarios while only a few of them are simultaneously active at a given time. When applying the model order reduction method in gear contact problems, the addition of a static shape vector for each possibly-loaded boundary Degree of Freedom (DOF) will enable the contact being accurately described but may be computationally expensive when the number of boundary DOF is large. On the other hand, omission of boundary DOF, though might be computationally efficient, will result in a slow convergence. The objective of this paper is to develop a modeling technique based on Arbitrary Lagrangian Eulerian (ALE) formulation to reduce DOF while providing accurate gear meshing contact simulation. Four techniques are adopted to achieve the critical objective of the paper. First, the low-frequency approximation is attained by ignoring fixed boundary normal modes. Second, under the framework of ALE formulation, only the mesh nodes of four engaging tooth-faces are defined as boundary nodes during the whole process resulting in a great reduction of DOF of the system. Then, the calculation of inertial force and Jacobian matrix are simplified by ignoring the inertial forces resulted from deformation. Finally, a four-step high-efficiency contact algorithm is adopted to reduce the number of contact pairs and accelerate the detection process. The performance of the proposed method is demonstrated with four gear contact problems and correlated with commercial nonlinear finite element software.