We present an adaptive algorithm for time integration of fluid–structure integration problems. The method relies on a fully coupled procedure to solve FSI problems in which a naturally GCL-compliant ALE formulation for the finite-element spatial discretization is used. The main originality of the proposed solution procedure is that time integration is performed using automatic order and stepsize selections (hp-adaptivity) based on the Backward Differentiation Formulas (BDF). The stepsize selection is based on a local error estimate, an error controller and a step rejection mechanism. It guarantees that the solution precision is within the user targeted tolerance. The order selection is based on a stability test and a quarantine mechanism. The selection is performed to ensure that no other methods within the family of 0-stable BDF methods would produce a solution of the targeted precision for a larger stepsize (and thus a lower computational time). To improve efficiency, the time integration procedure also relies on a modified Newton method and a predictor. The time adaptive algorithm behaviors and performances are assessed on the vortex-induced translational and rotational vibrations of a square cylinder and on the wake-induced vibrations of 3 cylinders in an in-line arrangement. The algorithm yields substantial CPU time savings (compared to constant stepsize and order integration) while delivering solutions of prescribed accuracies.

This paper presents a procedure based on the Backward Differentiation Formulas of order 1 to 5 to obtain efficient time integration of the incompressible Navier–Stokes equations. The adaptive algorithm performs both stepsize and order selections to control respectively the solution accuracy and the computational efficiency of the time integration process. The stepsize selection (h-adaptivity) is based on a local error estimate and an error controller to guarantee that the numerical solution accuracy is within a user prescribed tolerance. The order selection (p-adaptivity) relies on the idea that low-accuracy solutions can be computed efficiently by low order time integrators while accurate solutions require high order time integrators to keep computational time low. The selection is based on a stability test that detects growing numerical noise and deems a method of order p stable if there is no method of lower order that delivers the same solution accuracy for a larger stepsize. Hence, it guarantees both that (1) the used method of integration operates inside of its stability region and (2) the time integration procedure is computationally efficient. The proposed time integration procedure also features a time-step rejection and quarantine mechanisms, a modified Newton method with a predictor and dense output techniques to compute solution at off-step points.

The computation of Lagrangian coherent structures is more and more used in fluid mechanics to determine subtle fluid flow structures. We present in this paper a new adaptive method for the efficient computation of Finite Time Lyapunov Exponent (FTLE) from which the coherent Lagrangian structures can be obtained. This new adaptive method considerably reduces the computational burden without any loss of accuracy on the FTLE field.

SummaryThis paper proposes implicit Runge–Kutta (IRK) time integrators to improve the accuracy of a front‐tracking finite‐element method for viscous free‐surface flow predictions. In the front‐tracking approach, the modeling equations must be solved on a moving domain, which is usually performed using an arbitrary Lagrangian–Eulerian (ALE) frame of reference. One of the main difficulties associated with the ALE formulation is related to the accuracy of the time integration procedure. Indeed, most formulations reported in the literature are limited to second‐order accurate time integrators at best. In this paper, we present a finite‐element ALE formulation in which a consistent evaluation of the mesh velocity and its divergence guarantees satisfaction of the discrete geometrical conservation law. More importantly, it also ensures that the high‐order fixed mesh temporal accuracy of time integrators is preserved on deforming grids. It is combined with the use of a family of L‐stable IRK time integrators for the incompressible Navier–Stokes equations to yield high‐order time‐accurate free‐surface simulations. This is demonstrated in the paper using the method of manufactured solution in space and time as recommended in Verification and Validation. In particular, we report up to fifth‐order accuracy in time. The proposed free‐surface front‐tracking approach is then validated against cases of practical interest such as sloshing in a tank, solitary waves propagation, and coupled interaction between a wave and a submerged cylinder. Copyright © 2015 John Wiley & Sons, Ltd.

Context: Many engineering problems require to solve PDE on deforming domains to account for the temporal evolution of the domain boundaries. Fluid-Structure Interaction problems and free-surface flows solved by a front-tracking approach are two such examples.Objective: In this paper, we address the numerical solution of the three-dimensional Navier-Stokes equations on deforming domains using the finite-element (FE) method and an Arbitrary Lagrangian Eulerian (ALE) formulation.Method: The proposed formulation incorporates the ALE mapping into the finite-element method in a natural and straightforward manner. It allows the use of any time integrator that can be expressed as a finite-difference in time and maintains the integrators fixed grid convergence rate on ALE deforming grids. Hence, popular time integrators (implicit backward Euler, Gear, BDF, Runge-Kutta, etc.) can be applied directly to moving grid simulations without necessitating any modification or adjustment to the code.Results: Using a manufactured solution, we present thorough time stepsize refinement studies. Results are reported for two families of time-stepping procedures: the implicit Backward Differentiation Formulas (multi-step methods of order 1-5) and the Implicit (Radau IIA) Runge-Kutta methods (one-step methods of order 1,3 and 5).Conclusion: The proposed FE/ALE formulation preserves the fixed-grid orders of accuracy of time-stepping procedures on 3D moving grids. Hence, three-dimensional FE/ALE simulations can be performed with highly accurate time integrators. (C) 2014 Elsevier Ltd. All rights reserved.

This paper presents manufactured solutions (MS) for verification of a fluid-structure interactions code. MS provide benchmark solutions for direct evaluation of the solution errors. The method of manufactured solutions (MMS) is a straight forward and general procedure for generating exact analytical solutions with a sufficiently rich structure to ensure that all terms of the differential equations are exercised in the simulations. When used with systematic grid refinement studies, the MMS provides strong code verification with a theorem-like quality. Manufactured solutions for fluid-structure interaction (FSI) problems with large displacements are presented with sample results from grid convergence studies.

Lagrangian coherent structures (LCS) constitute one of the many tools commonly used to analyze and better explain time dependent flow behavior. They are defined as the ridge curves of the Finite-Time Lyapunov Exponent (FTLE) field, which are usually computed on structured cartesian interpolation grids with a large number of nodes to ensure a proper definition of the field. The main contribution of this work is the introduction of adapted unstructured anisotropic interpolation grids to enhance the definition of the FTLE field, and consequently the precision and quality of extracted LCS lines. Since these lines correspond to the most anisotropic feature of the FTLE field, anisotropic adaptation of the interpolation grid allows the proper concentration of nodes along the LCS lines and keeps the total number of nodes low.

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.

We study an interface problem between a fluid flow, governed by Stokes equations, and a flow in a porous medium, governed by Darcy equations. We consider a weak formulation of the coupled problem which allows to use classical Stokes finite elements in the fluid domain, and standard continuous piecewise polynomials in the porous medium domain. Meshes do not need to match at the interface. The formulation of Stokes equations is standard while a Galerkin least-squares formulation is used for a mixed form of Darcy equations. We prove the well-posedness of the coupled problem for this formulation and the convergence for some finite element approximations. A two-dimensional numerical example is also given.

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.

This paper presents an adaptive Discontinuous-Galerkin finite element method to solve linear advection–reaction equations. The method was developed for the prediction of hemolysis in prosthetic devices, where the flow may include recirculation zones. To ensure the well-posedness of the problem, we solve the transient form of the PDE. Time discretization is achieved with an implicit Euler scheme in order to obtain the steady-state solution rapidly. Linear interpolation functions are used to approximate the hemolysis field. The a posteriori estimation of the error is obtained by solving an advection–reaction equation for the error, which is approximated with quadratic interpolation functions. Then, elementary estimations of the error are computed using a semi-norm developed to achieve the best possible results, even in the presence of recirculation. Code verification – to assess the convergence and conservation properties of the method – is performed with the method of manufactured solutions and with a conservation test using a sharp gaussian function as a reaction term. Excellent results are obtained in both cases. Predictions of hemolysis in a dialysis cannula are presented as an engineering application of our method.

Computational fluid dynamics is extensively used in the design methodology of medical devices. However, for such applications, the predictive capabilities of CFD codes are highly dependent upon geometry, which most of the time is extremely complex, and flow conditions. The study concerns a ventricular assist device (VAD) where the exit flow, generated through a diffuser, is of particular importance for blood damage predictions. The difficulty to predict the flow lies in the fact that the Reynolds number range includes the transition Reynolds number of the separated diffuser flow as well as the critical Reynolds number of pipe flows. In order to choose the appropriate CFD methodology in terms of flow hypothesis and turbulence model, an experimental setup of the diffuser was built to run PIV velocity measurements and to analyze the flow pattern with the influence of Reynolds number. The flow is described with mean and variance values of the in-plane velocity components and timeresolved results are used to visualize the development of unsteady phenomena introduced in the diffuser separated region. An optimal filter is also used to remove noise in measured velocity vector fields.

A numerical study of several finite element approximations for the primal mixed formulation of the Darcy equations is presented. In all cases the pressure is approximated by continuous piecewise-linear functions. The difference between each scheme is in the choice of the finite element approximation space for the velocity. Numerical tests confirm the theoretical convergence of some of these schemes and we investigate the convergence properties of schemes for which theoretical results are not available. Numerical results for some 2D problems suggest that some of the new schemes provide better convergence properties for the velocity. Copyright (C) 2006 John Wiley & Sons, Ltd.

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 play a central role in our analysis. Theoretical and numerical results are presented along with their impact on the design process.

We present several aspects of a general monolithic formulation for the steady-state interaction of a viscous incompressible flow with an elastic structure undergoing large displacements. The problem is solved in a direct fully-coupled manner by a Newton-Raphson adaptive 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 adaptive formulation is verified on a problem with a closed form solution. Its capabilities are then demonstrated on different fluid-structure configurations ranging from aeronautical to biomedical fields.

The behaviour and accuracy of the stable mixed space-time finite element methods for the incompressible Navier-Stokes equations in a two-dimensional bounded domain are investigated in this work. The mixed method is based on the recently developed space-time mini element, which consists of piecewise linear functions for the pressure and of piecewise linear functions enriched with a bubble function for the velocity. This element is stable in the sense that the underlying pair of discrete spaces for velocity and pressure satisfies the so-called 'inf-sup' condition. We assess the behaviour and accuracy of the underlying mixed approximation when compared with the stabilized Galerkin/least-squares space-time method for some two-dimensional problems. Both methods are based on the time-discontinuous Galerkin method with the use of simplex-type meshes. Copyright (C) 2002 John Wiley Sons, Ltd.

A space–time finite element method for the incompressible Navier–Stokes equations in a bounded domain in ℝ d (with d =2 or 3) is presented. The method is based on the time‐discontinuous Galerkin method with the use of simplex‐type meshes together with the requirement that the space–time finite element discretization for the velocity and the pressure satisfy the inf–sup stability condition of Brezzi and Babuška. The finite element discretization for the pressure consists of piecewise linear functions, while piecewise linear functions enriched with a bubble function are used for the velocity. The stability proof and numerical results for some two‐dimensional problems are presented. Copyright © 2001 John Wiley & Sons, Ltd.

A numerical study is presented for the laminar natural convection in a differentially heated horizontal bare and finned rhombic annulus filled with air. An adaptive finite-element method is adopted to predict the complex recirculating buoyancy driven flows. Simulations were conducted for cavity widths Eg ranging from 0.25 to 0.875, for two different fin configurations with fin lengths l varying from 0.3 to 0.7 and Rayleigh numbers Ra ranging from 10(3) to 10(7). Results indicate that the heat transfer is maximized for a narrow cavity Eg = 0.25 with fin length l = 0.7 (configuration 1). In this case, conduction prevails over convection. On the other hand, when convection is dominant, the heat transfer is maximum for Eg = 0.875 and l = 0.7. The equivalent thermal conductivity k(equ) was correlated in terms of Ra for all geometries via k(equ) = CRan. (C) 1999 Elsevier Science Ltd. All rights reserved.

Une étude numérique de la convection naturelle laminaire dans une cavité rhombique horizontale remplie d’air avec et sans ailette a été réalisée. Une méthode d’éléments finis adaptative est retenue pour prédire les écoulements convectifs complexes. Des simulations ont été effectuées pour des largeurs de cavité Eg variant de 0.25 à 0.875, deux configurations d’ailettes dont la longueur va de 0.3 à 0.7 et des nombres de Rayleigh s’échelonnant entre 103 et 107. Les résultats démontrent que le transfert de chaleur est maximal dans une cavité étroite de largeur Eg=0.25 munie d’ailettes de longueur l=0.7 (configuration 1). Dans ce cas, la conduction domine la convection. Quand, d’autre part, la convection domine, le transfert de chaleur est maximisé avec Eg=0.875 et l=0.7. La conductivité équivalente kequ a été corrélée ainsi en fonction du nombre de Rayleigh pour toutes les géométries étudiées: kequ=CRan.

In this paper 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 how 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. (C) 1997 John Wiley & Sons, Ltd.