Parallel Preconditioners for Navier-Stokes Equations and Fluid-Structure Interaction Problems

The Navier–Stokes equations play a key role in the modeling of blood flows in the vascular sys- tem. The cost for solving the 3D linear system obtained by Finite Element (FE) discretization of the equations, using tetrahedral unstructured meshes and time advancing finite difference schemes, is very high; to lower the time to solution and to address complex problems, using a parallel framework and developing specific preconditioners become necessary. Important factors to measure parallel performances of a preconditioner are the independence of the number of iterations with respect to the number of parallel tasks (scalability of the precondi- tioner), on the mesh size (optimality), and on the physical parameters (robustness), as well as its properties of strong and weak scalability. We propose a model to explain the effect of nonscalable preconditioners on a parallel iterative solver. Then we propose approximate versions of state of the art preconditioners for the Navier– Stokes equations, namely the Pressure Convection–Diffusion (PCD), Yosida, and SIMPLE preconditioners. We exploit factorizations of the linearized system where inverses are handled using specific core preconditioners such as, e.g., the algebraic additive Schwarz or algebraic multigrid preconditioners. We present new results for the Relaxed Dimensional Factorization (RDF) preconditioner which allows for an automatic parameter tuning. Weak and strong scalability results illustrate the efficiency of our approach using both classical benchmark problems and test cases relevant to hemodynamics simulations, using up to 8192 cores. Then, we extend our preconditioners for the Navier–Stokes equations to Fluid–Structure Interaction problems; we devise preconditioners that rely on physics–specific ad–hoc precon- ditioners for the fluid, the structure and the geometry subproblems. We compare the evolution of the number of iterations to solve the full system with classical methods on a geometry of physiological interest. We also investigate the strong scalability of our FSI solver. Finally, we describe how to develop a C++ flexible Navier–Stokes solver using a policy–based design. In particular we discuss different techniques to access a host class member variables from its compound policies and discuss their main advantages.

MATHICSE Technical Report : FaCSI : A block parallel preconditioner for fluid-structure interaction in hemodynamics

Simone Deparis, Davide Forti, Gwenol Grandperrin, Alfio Quarteroni

Modeling Fluid-Structure Interaction (FSI) in the vascular system is mandatory to reliably compute mechanical indicators in vessels undergoing large deformations. In order to cope with the computational complexity of the coupled 3D FSI problem after discretizations in space and time, a parallel solution is often mandatory. In this paper we propose a new block parallel preconditioner for the coupled linearized FSI system obtained after space and time discretization. We name it FaCSI to indicate that it exploits the factorized form of the linearized FSI matrix, the use of static condensation to formally eliminate the interface degrees of freedom of the fluid equations, and the use of a SIMPLE preconditioner for saddle-point problems. FaCSI is builtupon a block Gauss-Seidel factorization of the FSI Jacobian matrix and it uses ad-hoc preconditioners for each physical component of the coupled problem, namely the fluid, the structure and the geometry. In the fluid subproblem, after operating static condensation of the interface fluid variables, we use a SIMPLE preconditioner on the reduced fluid matrix. Moreover, to efficiently deal with a large number of processes, FaCSI exploits efficient single field preconditioners, e.g., based on domain decomposition or the multigrid method. We measure the parallel performances of FaCSI on a benchmark cylindrical geometry and on a problem of physiological interest, namely the blood flow through a patient-specific femoropopliteal bypass. We analyse the dependence of the number of linear solver iterations on the cores count (scalability of the preconditioner) and on the mesh size (optimality).

MATHICSE2015

FaCSI: A block parallel preconditioner for fluid-structure interaction in hemodynamics

Simone Deparis, Davide Forti, Gwenol Grandperrin, Alfio Quarteroni

Modeling Fluid-Structure Interaction (FSI) in the vascular system is mandatory to reliably compute mechanical indicators in vessels undergoing large deformations. In order to cope with the computational complexity of the coupled 3D FSI problem after discretizations in space and time, a parallel solution is often mandatory. In this paper we propose a new block parallel preconditioner for the coupled linearized FSI system obtained after space and time discretization. We name it FaCSI to indicate that it exploits the Factorized form of the linearized FSI matrix, the use of static Condensation to formally eliminate the interface degrees of freedom of the fluid equations, and the use of a SIMPLE preconditioner for saddle-point problems. FaCSI is built upon a block Gauss-Seidel factorization of the FSI Jacobian matrix and it uses ad-hoc preconditioners for each physical component of the coupled problem, namely the fluid, the structure and the geometry. In the fluid subproblem, after operating static condensation of the interface fluid variables, we use a SIMPLE preconditioner on the reduced fluid matrix. Moreover, to efficiently deal with a large number of processes, FaCSI exploits efficient single field preconditioners, e.g., based on domain decomposition or the multigrid method. We measure the parallel performances of FaCSI on a benchmark cylindrical geometry and on a problem of physiological interest, namely the blood flow through a patient-specific femoropopliteal bypass. We analyze the dependence of the number of linear solver iterations on the cores count (scalability of the preconditioner) and on the mesh size (optimality). (C) 2016 Elsevier Inc. All rights reserved.

Academic Press Inc Elsevier Science2016

Interaction fluide-structure par la méthode des éléments spectraux

Nicolas Bodard

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.

EPFL2006