Fluid-structure interaction for vascular flows: From supercomputers to laptops

Several models exist for the simulation of vascular flows; they span from simple circuit models to full three-dimensional ones that take into account detailed features of the blood and of the arterialwall. Eachmodel comeswith both benefits and drawbacks, the main denominator being a compromise between detailed resolution requirements versus computational time. We first present a fluid-structure interaction computationalmodelwhere both the fluid and the structure are three dimensional. In particular, the fluid includes modeling of large eddies by the variationalmultiscalemethod. After time and space discretizations carried out by finite differences and finite elements, respectively, we set up a parallel solver based ondomain decomposition and a FaCSI preconditioner. These simulations allow one to capture details of the flow dynamics and of the structure deformation even in the transitional regime characterizing hemodynamics in the aorta. It takes roughly 10 hours to complete a simulation of one heartbeat with 35 million degrees of freedom on 2048 cores. We then reduce both the model and its numerical complexity. The structural model is simplified to a two-dimensional membrane located at the fluid-structure interface and the fluid computational domain is fixed. For a fixed geometry andmesh, these assumptions allow one to apply proper orthogonal decomposition and generate a space discretization which has only a few dozen degrees of freedom. It is then possible to perform the simulation of one heartbeat on a laptop in less than one second. Themodeling and numerical reduction therefore allows a dramatic reduction of computational time. However, the price to pay comes, on the one hand, in terms of the preparation of a reduced basis specific to the patient and the geometry of the vessel and, on the other hand, with a detriment of certain quantities of interest. For example, when using a finite element discretization with 9 million degrees of freedom, the offline part takes about 12 hours on 720 cores for the example provided in this work; in this case, the flow profiles in the aorta are pretty close to the full three-dimensional

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

Analysis of Fluid-Structure Interaction by Means of Dynamic Unstructured Meshes

Pénélope Leyland

This paper presents a computational analysis on forced vibration fluid-structure interaction in compressible flow regimes. A so-called staggered approach is pursued where the fluid and structure are integrated in time by distinct solvers. Their interaction is then taken into account by a coupling algorithm. The unsteady fluid motion is simulated by means of an explicit time-accurate solver. For the fluid-structure interaction problems which are considered here the effects due to the viscosity can be neglected. The fluid is hence modeled by the Euler equations for compressible inviscid flow. Unstructured grids are used to discretise the fluid domain. These grids are particularly suited to simulate unsteady flows over complex geometries by their capacity of being dynamically refined and unrefined. Dynamic mesh adaptation is used to enhance the computational precision with minimal CPU and memory constraints. Fluid-structure interaction involves moving boundaries. Therefore the Arbitrary Lagrange Euler method (ALE-method) is adopted to solve the Euler equations on a moving domain. The deformation of the mesh is controlled by means of a spring analogy in conjunction with a boundary correction to circumvent the principle of Saint Venant. To take advantage of the differences between fluid and structure time scales, the fluid calculation is subcycled within the structural time step. Numerical results are presented for large rotation, pitching oscillation and aeroelastic motion of the NACA0012 airfoil. The boundary deformation is validated by comparing the numerical solution for a flat plate under supersonic flow with the analytical solution.

1998

Model Order Reduction by geometrical parametrization for shape optimization in computational fluid dynamics

Andrea Manzoni, Gianluigi Rozza

Shape Optimization problems governed by partial differential equations result from many applications in computational fluid dynamics; they involve the repetitive evaluation of outputs expressed as functionals of the field variables and usually imply big computational efforts. For this reason looking for computational efficiency in numerical methods and algorithms is mandatory. The interplay between scientific computing and new reduction strategies is crucial in applications of great complexity. In order to achieve an efficient model order reduction, reduced basis methods built upon a high-fidelity ``truth'' finite element approximation -- and combined with suitable geometrical parametrization techniques for efficient shape description -- can be introduced, thus decreasing both the computational effort and the geometrical complexity. Starting from an excursus on classical approaches -- such as local boundary variation and shape boundary parametrization -- we focus on more efficient parametrization techniques which are well suited for a combination with a reduced basis approach, such as the one based on affine mapping (even automatic), nonaffine mapping (coupled with a suitable empirical interpolation technique for better numerical performances) and free-form deformations. We thus describe (and compare) the principal features of these parametrization techniques by showing some applications dealing with shape optimization of parametrized configurations in viscous flows,and discussing computational advantages and efficiency obtained by geometrical and computational model order reduction.

2010