Simone Deparis

Parallel Algorithms for Fluid-Structure Interaction Problems in Haemodynamics

The increasing computational load required by most applications and the limits in hardware performances affecting scientific computing contributed in the last decades to the development of parallel software and architectures. In fluid-structure interaction (FSI) for haemodynamic applications, parallelization and scalability are key issues (see [L. Formaggia, A. Quarteroni, and A. Veneziani, eds., Cardiovascular Mathematics: Modeling and Simulation of the Circulatory System, Modeling, Simulation and Applications 1, Springer, Milan, 2009]). In this work we introduce a class of parallel preconditioners for the FSI problem obtained by exploiting the block-structure of the linear system. We stress the possibility of extending the approach to a general linear system with a block-structure, then we provide a bound in the condition number of the preconditioned system in terms of the conditioning of the preconditioned diagonal blocks, and finally we show that the construction and evaluation of the devised preconditioner is modular. The preconditioners are tested on a benchmark three-dimensional (3D) geometry discretized in both a coarse and a fine mesh, as well as on two physiological aorta geometries. The simulations that we have performed show an advantage in using the block preconditioners introduced and confirm our theoretical results.

Physiological simulation of blood flow in the aorta: comparison of hemodynamic indices as predicted by 3-D FSI, 3-D rigid wall and 1-D models

Interest in patient-specific blood-flow circulation modeling has increased substantially in recent years. The availability of clinical data for geometric and elastic properties together with efficient numerical methods has now made model rendering feasible. This work uses 3-D fluid-structure interaction (FSI) to provide physiological simulation resulting in modeling with a high level of detail. Comparisons are made between results using FSI and rigid wall models. The relevance of wall compliance in determining parameters of clinical importance, such as wall shear stress, is discussed together with the significance of differences found in the pressure and flow waveforms when using the 1-D model. Patient-specific geometry of the aorta and its branches was based on MRI angiography data. The arterial wall was created with a variable thickness. The boundary conditions for the fluid domain were pressure waveform at the ascending aorta and flow for each outlet. The waveforms were obtained using a 1-D model validated by in vivo measurements performed on the same person. In order to mimic the mechanical effect of surrounding tissues in the simulation, a stress-displacement relation was applied to the arterial wall. The temporal variation and spatial patterns of wall shear stress are presented in the aortic arch and thoracic aorta together with differences using rigid wall and FSI models. A comparison of the 3-D simulations to the 1-D model shows good reproduction of the pressure and flow waveforms.

Reduced basis method for multi-parameter-dependent steady Navier-Stokes equations: Applications to natural convection in a cavity

This work focuses on the approximation of parametric steady Navier-Stokes equations by the reduced basis method. For a particular instance of the parameters under consideration, we are able to solve the underlying partial differential equations, compute an output, and give sharp error bounds. The computations are split into an offline part, where the values of the parameters are not yet identified, but only given within a range of interest, and an online part, where the problem is solved for an instance of the parameters. The offline part is expensive and is used to build a reduced basis and prepare all the ingredients - mainly matrix-vector and scalar products, but also eigenvalue computations - necessary for the online part, which is fast. We provide a model problem - describing natural convection phenomena in a laterally heated cavity - characterized by three parameters: Grashof and Prandtl numbers and the aspect ratio of the cavity. We show the feasibility and efficiency of the a posteriori error estimation by the natural norm approach considering several test cases by varying two different parameters. The gain in terms of CPU time with respect to a parallel finite element approximation is of three magnitude orders with an acceptable - indeed less than 0.1% - error on the selected outputs.

Weighted Clément operator and application to the finite element discretization of the axisymmetric Stokes problem

We consider the Stokes problem in an axisymmetric three-dimensional domain with data which are axisymmetric and have angular component equal to zero. We observe that the solution is also axisymmetric and the velocity has also zero angular component, hence the solution satisfies a system of equations in the meridian domain. The weak three-dimensional problem reduces to a two-dimensional one with weighted integrals. The latter is discretized by Taylor–Hood type finite elements. A weighted Clément operator is defined and approximation results are proved. This operator is then used to derive the discrete inf–sup condition and optimal a priori error estimates.

Reduced Basis Error Bound Computation of Parameter-Dependent Navier-Stokes Equations by the Natural Norm Approach

This work focuses on the a posteriori error estimation for the reduced basis method applied to partial differential equations with quadratic nonlinearity and affine parameter dependence. We rely on natural norms—local parameter-dependent norms—to provide a sharp and computable lower bound of the inf-sup constant. We prove a formulation of the Brezzi-Rappaz-Raviart existence and uniqueness theorem in the presence of two distinct norms. This allows us to relax the existence condition and to sharpen the field variable error bound. We also provide a robust algorithm to compute the Sobolev embedding constants involved in the error bound and in the inf-sup lower bound computation. We apply our method to a steady natural convection problem in a closed cavity, with a Grashof number varying from 10 to

Implicit Coupling of One-Dimensional and Three-Dimensional Blood Flow Models with Compliant Vessels

10^7

A domain decomposition framework for fluid-structure interaction problems

.

A Rescaled Localized Radial Basis Function Interpolation On Non-Cartesian And Nonconforming Grids

Simulating arterial trees in the cardiovascular system can be made by the help of different models, depending on the outputs of interest and the desired degree of accuracy. In particular, one-dimensional fluid-structure interaction models for arteries are very effective in reproducing the physiological pressure wave propagation and in providing quantities like pressure and velocity, averaged on the cross section of the arterial lumen. In locations where one-dimensional models cannot capture the complete flow dynamics, e.g., in presence of stenoses and aneurysms, three-dimensional coupled fluid-structure interaction models are necessary to evaluate, for instance, critical factors responsible for pathologies which are associated to hemodynamics. In this work we formalize and investigate the geometrical multiscale problem, where heterogeneous fluid-structure interaction models for arteries are implicitly coupled. We introduce new coupling algorithms, describe their implementation and investigate on simple geometries the numerical reflections that occur at the interface between the heterogeneous models. We also simulate on a supercomputer a three-dimensional abdominal aorta under physiological conditions, coupled with up to six one-dimensional models representing the surrounding arterial branches. Finally, we compare CPU times and number of coupling iterations for different algorithms and time discretizations.

Comparisons between reduced order models and full 3D models for fluid-structure interaction problems in haemodynamics

In this note we review some classical algorithms for fluid-structure interaction problems and we propose an alternative viewpoint mutuated from the domain decomposition theory. This approach yields preconditioned Richardson iterations on the Steklov-Poincare nonlinear equation at the fluid-structure interface.

Numerical simulation of left ventricular assist device implantations: Comparing the ascending and the descending aorta cannulations

In this paper we propose a rescaled localized radial basis function (RL-RBF) interpolation method, based on the use of compactly supported radial basis functions. Starting from the classical RBF interpolation technique, we introduce a rescaling that allows for exact interpolation of constant fields between nonconforming meshes without the use of an extra polynomial term. We also present two-dimensional and three-dimensional numerical examples on arbitrary finite element meshes to show that the RL-RBF interpolation leads to accurate results, fast evaluation, and easy parallelization of the algorithm. All the computations are carried out using the open source finite element library LifeV.