Articles 1—20 Show more. Arbitrary Lagrangian Eulerian and fluid-structure interaction: Convective linear stability analysis of two-layer coextrusion flow for molten polymers R Valette, P Laure, Y Demay, JF Agassant Journal of non-newtonian fluid mechanics 1 , , Composite reinforcements for optimum performance P Boisse Elsevier , Symbolic computation and equation on the center manifold: Evaluation and modeling of the topographic effect on the spectral response from nadir pointing sensors BN Holben, CO Justice.
Multidomain finite element computations: B 11, , A new three-dimensional mixed finite element for direct numerical simulation of compressible viscoelastic flows with moving free surfaces L Silva, R Valette, P Laure, T Coupez International journal of material forming 5 1 , , Hackbusch, Multigrid methods and applications. Welch, Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface.
Cook, An arbitrary LagrangianEulerian computing method for all flow speeds. Turek, Proposal for numerical benchmarking of fluidstructure interaction between an elastic object and laminar incompressible flow.
Arbitrary Lagrangian–Eulerian and Fluid–Structure Interaction - ISTE
Zimmerman, Lagrangian-Eulerian finite element formulation for incompressible viscous flow. Hwang, Calculations of incompressible flow on a staggered triangle grid, Part I: Heat Transfer B 27, , — Tezduyar, Mesh update strategies in parallel finite element computations of flow problems with moving boundaries and interfaces.
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Arbitrary Lagrangian–Eulerian and Fluid–Structure Interaction
Reggio, A staggered control volume scheme for unstructured triangular grids. Fluids 25, , — Sahin, A stable unstructured finite volume method for parallel large-scale viscoelastic fluid flow calculations. Journal of non-Newtonian Fluid Mechanics , , — Lombard, Geometric conservation law and its application to flow computations on moving grids.
In black the leaflets, in white the fluid domain, and in grey the cells detected for contact. The structural mesh was similar in both models and consisted of quadrilateral or hexahedral solid elements, for 2D and 3D simulations respectively. Due to the requirements imposed for the generation of an adequate fluid mesh in the IB-FSI case, it was necessary to reduce the number of the solid elements in the IB-FSI case to consequently increase the element size.
More details are provided in the following sections. The fluid mesh of the ALE simulation consisted of triangular 2D or tetrahedral 3D elements with a higher cell density in the vicinity of the valve. Due to the ALE formulation, it was possible to obtain a non-uniformly spaced fluid grid, which reduced the overall number of elements for this mesh. To have a satisfactory definition of the initial VOF in the IB method, a homogeneously refined fluid mesh had to be generated. Due to visualization difficulties, the 3D fluid meshes are not reported here. Adequate dimensions of the meshes were chosen for the different set-ups.
The number of elements is listed in Table 1. For the ALE simulations, the initial dimension of the fluid meshes is reported, as the remeshing of the domain was enabled and the number of cells thus varied during the simulation. The leaflets tissue was modelled as linear and elastic Young modulus 1 MPa, Poisson ratio 0. For the comparison purpose of this work we considered these simplifications to be justified, while for a more refined model the elasticity of the wall has to be included and more realistic material models of the soft tissues are necessary [ 9 , 13 , 24 , 25 , 26 ].
In the IB-based simulation, a compressibility factor was added to the fluid to enhance the convergence of the solution [ 8 , 9 , 14 ]. A physiological transvalvular pressure difference curve was applied at the ventricular side of the domain, while the aortic outlet was kept at a reference pressure. Before the loading cycle, the pressure was gradually increased until the pressure of the cycle was reached to provide a good initial condition for the simulation [ 27 ] and to reduce the influence of the compressibility of the flow [ 9 ].
A no-slip condition was imposed on the walls, while the fluid-structure interaction condition was enforced in the leaflets region. To capture the dynamics of the valve motion and to guarantee the contact detection in the structural solver a time step of 0. The time integration scheme used was a first order, implicit method. In the IB-FSI, an automatic and adaptive time-step was selected, and an explicit integration scheme was used. An initial time-step size of 0. The time-step size was then automatically calculated and updated throughout the simulation by the solver.
The management of the contact in the two models was substantially different. The solid-solid contact in the IB-FSI was directly managed by the solver Abaqus via a default contact penalty method [ 20 ]. During contact, in fact, the leaflets of the valve should close completely, and, due to the ALE formulation, this would result in the generation of highly distorted elements and, ultimately, the splitting of the fluid domain and failure of the simulation.
To avoid this phenomenon, it was necessary to preserve a layer of fluid cells in the contact area of the leaflets. The properties of the contact were consistent with the properties of the contact definition of the IB-FSI. A minimum threshold distance between the leaflets was imposed, to preserve a gap during the diastole.
However, on the fluid side, the presence of the gap introduced an unwanted and artificial leakage of the valve when it was in the closed configuration. As we previously demonstrated, valve leakage can be reduced by modifying the permeability of the cells located in the area during the coaptation time [ 28 ]. The hydraulic resistance was increased in these cells, to reduce the overall backflow during diastole. This could be obtained by implementing a specific external function in the fluid solver, which detected the cells of the gap and changed their properties.
This function was activated only during the closed phase of the valve, when the leaflets came closer than a predefined threshold. This technique does not intend to mimic any physiological phenomenon and the permeability coefficients were chosen as a trade-off between the reduction of the backflow and the stability of the solution. The 2D visualization of the contact function is depicted in Fig 2E. The cells marked in grey were selected for contact and a kinematic constraint was imposed to avoid the domain splitting.
The permeability of these cells was modified only during the contact phase to artificially increase their hydraulic resistance to the flow. In this section, we report the comparison between the most significant results of the two techniques in the 2D and 3D case.
The overall 2D flow field and valve kinematics were comparable between the two techniques. The open valve was assumed to be the initial configuration.
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Being already open, the valve did not offer any resistance to the fluid during the opening phase, therefore we report the results of the closing phase only, until early diastole. The flow decelerated when the pressure drop across the valve reversed, and the valve started to close Fig 3A. As visible from Fig 3C , the 2D case was not suitable to simulate the diastolic configuration of the valve.
In diastole, the pressure drop across the valve is significant, and the thin 2D structure was not able to keep the closed configuration, even in presence of the contact condition. For this reason, the 2D simulation was limited to the closing phase of the valve, and only one cardiac systole was simulated.
The initial configuration for the 3D model was assumed to be the closed position, being the prosthetic valve in this configuration during the scanning. The major issue encountered in this comparative work was related to the difficulties in simulating the opening of the 3D valve in the ALE-FSI case. The fluid grid underwent severe deformation in a limited amount of time. Significant remeshing and smoothing algorithms to preserve a good quality grid for the fluid domain was not sufficient to ensure the convergence of the problem.
Due to the limited availability of the ALE-FSI results, the comparison between the two methods was reduced to the early systolic phase. Also, the deformation of the valve leaflets was different: In Fig 4E a layer of higher velocities is detected in the proximity of the walls. This phenomenon was detected in all the time-points of the IB-ALE simulations and is related to a visualization issue of the software it should not have any influence on the results [ 20 ].
At this stage of the cardiac cycle, no blood recirculation in the Valsalva sinuses area was detected, as the leaflets were still in the opening phase. The recirculation zone formation was present in a later phase of the systole, when the AV started the closure phase. The considerations about the pressure distribution of Fig 4 panels from j to l are similar to those for the velocities in Fig 4 , panels d-f.
Furthermore, some variations of the pressure are visible in the area downstream the valve in the IB-FSI. Comparing the rapid valve opening time RVOT of the IB-FSI ms with literature data reported range 45 to 65 ms, both from structural [ 29 — 31 ] and from FSI [ 8 — 10 ] simulations , the time delay previously described in the 2D comparison was even more pronounced in the 3D case. To verify the source of delay, a pure structural simulation was performed on the same geometry.
The model was consistent with the IB-FSI simulations in terms of geometry, mesh, material and element type.
Arbitrary Lagrangian Eulerian and Fluid-Structure Interaction : Numerical Simulation
In the structural case the loading pressure curves were directly applied on the leaflets surface. The obtained RVOT in this case was about 80ms, significantly closer to the expected value but remaining outside the physiological range for this parameter. The colour scale indicates the magnitude of the displacement calculated from the initial configuration. For the sake of completeness, in Fig 5 right panel we report the displacement of the valve during systole up to the beginning of diastole obtained with the IB-FSI [ 32 ].
The flow field results are omitted for conciseness reasons. This was also verified by following the displacements of the central part of the aortic leaflet nodulus of Arantio. A residual central opening was visible. To investigate the unexpected asymmetric motion of the valve, a separate test on the diastolic phase was performed, where the valve was kept in the closed position and the pressure on the aortic side was gradually raised from the reference pressure to a value of mmHg, while the ventricular pressure was kept at the reference pressure.
By analysing the definition and the evolution of the VOF in this set-up, a loss of void Eulerian cells in which no fluid is defined, in white in Fig 6 was detected on the ventricular side of the valve. By increasing the transvalvular pressure Fig 6 , panels b and c , the empty cells propagate in the ventricular portion of the fluid domain.
This effect was noticed in both the simulation of the diastole and in the complete cycle. In the latter, the loss of void was not symmetric, therefore causing the asymmetry of the leaflets kinematics shown in Fig 5. This phenomenon made the diastolic results unreliable, reason why the simulation was stopped at the beginning of the diastolic phase, and the simulation of one single cardiac systole was performed.
This phenomenon had no physical meaning, it was a numerical issue related to the model discretization. There was a clear advantage in using the IB method in terms of computing time. Despite the use of a much larger computational grid for the Eulerian case Table 1 , the simulation resulted to be much faster than the ALE case, as there was no need of remeshing and no issues related to a highly deformed mesh.
In addition, the in-house coupling code might require further optimization to deal with a complex scenario as the described problem. In Table 2 the required time is reported for each case. In the 2D case, the comparison of the two models provided similar results, despite the presence of some time delay in the IB-FSI case. The kinematics of the valve was comparable to the literature data [ 15 , 33 , 34 ] and the discretization technique does not introduce significant differences between the ALE and the IB case.
Also, the ALE fluid solver is capable to automatically manage the smoothing and the remeshing of the computational mesh. A simplified but similar comparison can be found in Dos Santos et al. Due to the simplified model and its 2D nature, it was not possible to simulate the diastolic phase: A transvalvular pressure difference was imposed between both sides of the leaflets. Two different cases were tested to check the potential impact of the definition of the contact properties: In all the tested cases the structure reversed in the ventricular side of the tube during diastole.
Therefore, we could conclude that the buckling of the structure was mainly related to the two dimensionality of the problems, and not on the fluid grid discretization or the contact function used. We hypothesize that this phenomenon was no shortcoming of the numerical codes, but rather due to the fact that a closed valve in the 2D configuration was physically not possible under the assumed boundary conditions and leaflet properties, which would lead to valve prolapse.
Even though in theory the ALE-FSI would be preferable to obtain results that are precise and in which the surface of interest is sharply defined [ 6 ], in our experience this approach was significantly limited by the large deformations of the fluid grid, which were the cause of the failure of the simulation.
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The mesh motion algorithms tested were the spring-based model and the diffusion model. The required parameters for the two algorithms were chosen according to the characteristic dimensions of the fluid mesh. Several tests were performed to increase the mesh density and decrease the time-step size to avoid the excessive deformation of the mesh, leading to cells with negative volume. As the chosen grid was unstructured and the motion was complex and not known a priori but defined by the interaction of the two domains, it was not possible to calculate the displacement of the structure prior the calculation and adjust the time-step size and dimensions of the fluid grid accordingly [ 35 ].