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Publication# Spectral element approximation of the incompressible Navier-Stokes equations in a moving domain and applications

Résumé

In this thesis we address the numerical approximation of the incompressible Navier-Stokes equations evolving in a moving domain with the spectral element method and high order time integrators. First, we present the spectral element method and the basic tools to perform spectral discretizations of the Galerkin or Galerkin with Numerical Integration (G-NI) type. We cover a large range of possibilities regarding the reference elements, basis functions, interpolation points and quadrature points. In this approach, the integration and differentiation of the polynomial functions is done numerically through the help of suitable point sets. Regarding the differentiation, we present a detailed numerical study of which points should be used to attain better stability (among the choices we present). Second, we introduce the incompressible steady/unsteady Stokes and Navier-Stokes equations and their spectral approximation. In the unsteady case, we introduce a combination of Backward Differentiation Formulas and an extrapolation formula of the same order for the time integration. Once the equations are discretized, a linear system must be solved to obtain the approximate solution. In this context, we consider the solution of the whole system of equations combined with a block type preconditioner. The preconditioner is shown to be optimal in terms of number of iterations used by the GMRES method in the steady case, but not in the unsteady one. Another alternative presented is to use algebraic factorization methods of the Yosida type and decouple the calculation of velocity and pressure. A benchmark is also presented to access the numerical convergence properties of this type of methods in our context. Third, we extend the algorithms developed in the fixed domain case to the Arbitrary Lagrangian Eulerian framework. The issue of defining a high order ALE map is addressed. This allows to construct a computational domain that is described with curved elements. A benchmark using a direct method to solve the linear system or the Yosida-q methods is presented to show the convergence orders of the method proposed. Finally, we apply the developed method with an implicit fully coupled and semi-implicit approach, to solve a fluid-structure interaction problem for a simple 2D hemodynamics example.

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We discuss in this thesis the numerical approximation of fluid-structure interaction (FSI) problems with a particular concern (albeit not exclusive) on hemodynamics applications. Firstly, we model the blood as an incompressible fluid and the artery wall as an elastic structure. To solve the coupled problem, we propose new semi-implicit algorithms based on inexact block-LU factorization of the linear system obtained after the space-time discretization and linearization of the FSI problem. As a result, the fluid velocity is computed separately from the coupled pressure-structure velocity system at each iteration, hence reducing the computational cost. This approach leads to two different families of methods which extend to FSI problems schemes that were previously adopted for pure fluid problems. The algorithms derived from inexact factorization methods are compared with other schemes based on two preconditioners for the FSI system. The first one is the classical Dirichlet-Neumann preconditioner, which has the advantage of modularity (i.e. it allows to reuse existing fluid and structure codes with minimum effort). Unfortunately, its performance is very poor in case of large added-mass effect, as it happens in hemodynamics. Alternatively, we consider a non-modular approach which consists in preconditioning the coupled system with a suitable diagonal scaling combined with an ILUT preconditioner. The system is then solved by a Krylov method. The drawback of this procedure is the loss of modularity. Independently of the preconditioner, the efficiency of semi-implicit algorithms is highlighted. All the methods are tested on two and three-dimensional blood-vessel systems. The algorithm combining the non-modular ILUT preconditioner with Krylov methods proved to be the fastest. However, modular and inexact factorization based methods should not be disregarded because they can considerably benefit from code parallelization, unlike the ILUT-Krylov approach. Finally, we improve the structure model by representing the vessel wall as a linear poroelastic medium. Our non-modular approach and the partitioned procedures arising from a domain decomposition viewpoint are extended to fluid-poroelastic structure interactions. Their numerical performance are analyzed and compared on simplified blood-vessel systems.

We elaborate in this thesis the numerical simulation of the fluid-structure interaction by the spectral element method. To this end we consider the Navier-Stokes equations for a viscous Newtonian incompressible fluid with an elastic solid the movement of which being described by the equations of the dynamics. The arbitrary Lagrangian Eulerian (ALE) formulation is introduced in the fluid governing equations to deal with the structure movement that is described in Lagrangian representation. The geometrical motion in each domain is built up by the mesh deformation. The space-time discretization of the full mathematical model rests upon the spectral element method. The solid is discretized in the space of polynomials of degree N, PN, while the fluid uses the PN - PN-2 approach. The efficiency of the ALE formulation is tested and validated through various applications. The full algorithm is based on a particular case of the staggered method. The simplified case of a solid immersed in a plane channel closes the thesis. It is then possible to draw the conclusions about the pros and cons of the proposed methodology.

Gonçalo Pena, Alfio Quarteroni

In this paper we address the numerical approximation of the incompressible Navier-Stokes equations in a moving domain by the spectral element method and high order time integrators. We present the Arbitrary Lagrangian Eulerian (ALE) formulation of the incompressible Navier-Stokes equations and propose a numerical method based on the following kernels: a Lagrange basis associated with Fekete points in the spectral element method context, BDF time integrators, an ALE map of high degree, and an algebraic linear solver. In particular, the high degree ALE map is appropriate to deal with a computational domain whose boundary is described with curved elements. Finally, we apply the proposed strategy to a test case. (C) 2011 Elsevier B.V. All rights reserved.