André Garon

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

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

An adaptive finite element method for a two-equation turbulence model in wall-bounded flows

SUMMARY This paper presents an adaptive finite element method for solving incompressible turbulent flows using a k‐ model of turbulence. Solutions are obtained in primitive variables using a highly accurate quadratic finite element on unstructured grids. A projection error estimator is presented that takes into account the relative importance of the errors in velocity, pressure and turbulence variables. The efficiency and convergence rate of the methodology are evaluated by solving problems with known analytical solutions. The method is then applied to turbulent flow over a backward-facing step and predictions are compared with experimental measurements.

NUMERICAL SOLUTION OF PHASE CHANGE PROBLEMS: AN EULERIAN-LAGRANGIAN APPROACH

The problem of solid-liquid phase change is tackled from art Eulerian-Lagrangian kinematic point of view. The theory is first presented for a one-dimensional space and then generalized to a two-dimensional space. The irregular shape of the phase front is treated with generalized curvilinear coordinates. A feature of the resulting finite-difference scheme is that the current field values and front position are solved simultaneously. As a result, numerical solutions show excellent agreement with the analytical solution for the one-dimensional melting problem and with experimental data for a two-dimensional buoyancy-driven phase-change problem.

Asymmetry and transition to turbulence in a smooth axisymmetric constriction

The flow through a smooth axisymmetric constriction (a stenosis in medical applications) of 75% restriction in area is measured using stereoscopic and time-resolved particle image velocimetry (PIV) in the Reynolds number range Re ~ 100–1100. At low Reynolds numbers, steady flow results reveal an asymmetry of the flow downstream of the constriction. The jet emanating from the throat of the nozzle is deflected towards the wall causing the formation of a one-sided recirculation region. The asymmetry results from a Coanda-type wall attachment already observed in symmetric planar sudden expansion flows. When the Reynolds number is increased above the critical value of 400, the separation surface cannot remain attached and an unsteady flow regime begins. Low-frequency axial oscillations of the reattachment point are observed along with a slow swirling motion of the jet. The phenomenon is linked to a periodic discharge of the unstable recirculation region inducing alternating laminar and turbulent flow phases. The resulting flow is highly non-stationary and intermittent. Discrete wavelet transforms are used to discriminate between the large-scale motions of the mean flow and the vortical and turbulent fluctuations. Continuous wavelet transforms reveal the spectral structure of flow disturbances. Temporal measurements of the three velocity components in cross-sections are used with the Taylor hypothesis to qualitatively reconstruct the three-dimensional velocity vector fields, which are validated by comparing with two-dimensional PIV measurements in meridional planes. Visualizations of isosurfaces of the swirling strength criterion allow the identification of the topology of the vortices and highlight the formation and evolution of hairpin-like vortex structures in the flow. Finally, with further increase of the Reynolds number, the flow exhibits less intermittency and becomes stationary for Re ~ 900. Linear stochastic estimation identifies the predominance of vortex rings downstream of the stenosis before breakdown to turbulence.

Asymptotically consistent numerical approximation of hemolysis

In a previous communication, we have proposed a numerical framework for the prediction of in vitro hemolysis indices in the preselection and optimization of medical devices. This numerical methodology is based on a novel interpretation of Giersiepen-Wurzinger blood damage correlation as a volume integration of a damage function over the computational domain. We now propose an improvement of this approach based on a hyperbolic equation of blood damage that is asymptotically consistent. Consequently, while the proposed correction has yet to be proven experimentally, it has the potential to numerically predict more realistic red blood cell destruction in the case of in vitro experiments. We also investigate the appropriate computation of the shear stress scalar of the damage fraction model. Finally, we assess the validity of this consistent approach with an analytical example and with some 3D examples.

MODELING AND DESIGN OF COATED STENTS TO OPTIMIZE THE EFFECT OF THE DOSE

Stents are used in interventional cardiology to keep a diseased vessel open. New stents are coated with a medicinal agent to prevent early reclosure due to the proliferation of smooth muscle cells. It is recognized that it is the dose of the agent that effectively controls the cells in the wall of the vessel. This paper focuses on the effect of the number of struts and the ratio between the coated area of the struts and the targeted area of the vessel on the design problem under set therapeutic bounds on the dose. It introduces mathematical models of the dose for a zero-thickness periodic stent and an asymptotic stent that will playa central role in our analysis. Theoretical and numerical results are presented along with their impact on the design process.

Shape Sensitivity Analysis of Fluid-Structure Interaction Problems

This paper presents a general monolithic formulation for sensitivity analysis of the steady interaction of a viscous incompressible o w with an elastic structure undergoing large displacements (geometric non-linearities). This is a direct extension of our previous work on value parameter sensitivity of such problems. 1 The coupled set of equations is solved in a direct implicit manner using a Newton-Raphson adaptive nite element method. A pseudo-solid formulation is used to manage the deformations of the uid domain. The formulation uses uid velocity, pressure, and pseudo-solid displacements as unknowns in the o w domain and displacements in the structural components. The adaptive formulation is veried on a problem with a closed form solution. It is then applied to sensitivity analysis of three elastic plates placed in a channel o w. Sensitivities are used for fast evaluation of nearby problems (i.e. for nearby values of the parameters or geometric characteristics) and for cascading uncertainty through the Computational Fluid Dynamics/Computational Structural Dynamics code to yield uncertainty estimates of the deformed plates shape.

Interpolation‐free space–time remeshing for the Burgers Equation

We present a mesh management strategy for use with discontinuous-Galerkin space-time finite element formulations of flow problems. We propose a refinement technique for simplex-type meshes which requires no interpolation between grids at slab interfaces. The strategy uses an a posteriori spatial error estimator to tag refinement or coarsening. Orientation of element edges along flow characteristics is accomplished by nodal displacement, and by a new diagonal-swapping technique to correct for the effects of misalignment due to h-refinement. The swapping procedure realigns the mesh to improve the effectiveness of the h-adaptive process. Results are presented for the Burgers Equation using large time steps on a problem which exhibits merging and steepening fronts

Sensitivity Analysis of Unsteady Fluid-Structure Interaction Problems

´This paper presents a general monolithic formulation for sensitivity analysis of the unsteady interaction of a viscous incompressible flow with an elastic structure undergoing large displacements (geometric non-linearities). This is a direct extension of our previous work on value parameter sensitivity of such problems. 1,2 The coupled set of equations is solved in a direct implicit manner using a Newton-Raphson finite element method. A pseudo-solid formulation is used to manage the deformations of the fluid domain. The formulation uses fluid velocity, pressure, and pseudo-solid displacements as unknowns in the flow domain and displacements in the structural components. The finite element method is verified on a problem with a closed form solution. It is then applied to sensitivity analysis of an elastic plate placed in a channel flow. Sensitivities are used for fast evaluation of nearby problems (i.e. for nearby values of the parameters or geometric characteristics) I. Introduction This paper presents a formulation suitable for simulating the interaction between an incompressible flow and a structure undergoing large displacements and for computing its sensitivities with respect to parameters of interest. We assume existence and uniqueness of the solution. Previous works have been published on sensitivity analysis of Fluid-Structure Interactions (FSI) 3‐7 but not with the continuous sensitivity equation (CSE).

Novel, Remotely Controlled, Artificial Urinary Sphincter: A Retro-Compatible Device

Implantation of artificial urinary sphincters (AUSs) is considered to be the gold standard treatment in severe cases of stress urinary incontinence. The functioning of these implants is purely hydromechanical, as they apply constant pressure around the bulbous urethra. Common reasons for device revision are insufficient cuff pressure and, in contrast, urethral atrophy secondary to this constant pressure. Furthermore, functioning requires some dexterity, limiting their implantation in some patients. We present in this study a novel electronic AUS that offers the possibility of remotely controlling the sphincter rapidly and without mechanical effort. The implants embedded software can also be updated remotely. Its design eliminates the manual pump, making implantation easier in men and women. Furthermore, it is compatible with already-implanted AUS and can be employed for treating other sphincter deficiencies. The device has been tested on a custom test bed and on pig bladders in vitro. Different occlusive cuff pressure ranges were employed and acceptable performance was obtained. Design challenges and results are reported and discussed here.