Stabilized Reduced Basis Approximation of Incompressible Three-Dimensional Navier-Stokes Equations in Parametrized Deformed Domains

In this work we are interested in the numerical solution of the steady incompressible Navier-Stokes equations for fluid flow in pipes with varying curvatures and cross-sections. We intend to compute a reduced basis approximation of the solution, employing the geometry as a parameter in the reduced basis method. This has previously been done in a spectral element $P_{{ \mathcal{N}}} - P_{{ \mathcal{N}}-2}$ setting in two dimensions for the steady Stokes equations. To compute the necessary basis-functions in the reduced basis method, we propose to use a stabilized P 1−P 1 finite element method for solving the Navier-Stokes equations on different geometries. By employing the same stabilization in the reduced basis approximation, we avoid having to enrich the velocity basis in order to satisfy the inf-sup condition. This reduces the complexity of the reduced basis method for the Navier-Stokes problem, while keeping its good approximation properties. We prove the well posedness of the reduced problem and present numerical results for selected parameter dependent three dimensional pipe

The LifeV library: engineering mathematics beyond the proof of concept

LifeV is a library for the finite element (FE) solution of partial differential equations in one, two, and three dimensions. It is written in C++ and designed to run on diverse parallel architectures, including cloud and high performance computing facilities. In spite of its academic research nature, meaning a library for the development and testing of new methods, one distinguishing feature of LifeV is its use on real world problems and it is intended to provide a tool for many engineering applications. It has been actually used in computational hemodynamics, including cardiac mechanics and fluid-structure interaction problems, in porous media, ice sheets dynamics for both forward and inverse problems. In this paper we give a short overview of the features of LifeV and its coding paradigms on simple problems. The main focus is on the parallel environment which is mainly driven by domain decomposition methods and based on external libraries such as MPI, the Trilinos project, HDF5 and ParMetis. Dedicated to the memory of Fausto Saleri.Comment: Review of the LifeV Finite Element librar

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 estimate

Numerical analysis of axisymmetric flows and methods for fluid-structure interaction arising in blood flow simulation

In this thesis we propose and analyze the numerical methods for the approximation of axisymmetric flows as well as algorithms suitable for the solution of fluid-structure interaction problems. Our investigation is aimed at, but are not restricted to, the simulation of the blood flow dynamics. The first part of this work deals with an axisymmetric fluid model based on three-dimensional incompressible Stokes or Navier–Stokes equations which are solved on a two-dimensional half-section of the domain under consideration. In particular we show optimal a priori error estimates for P1isoP2/P1 axisymmetric finite elements for the steady Stokes equations under the assumption that the domain and the data are axisymmetric and that the data have no angular component. Our analysis is carried out in the framework of weighted Sobolev spaces and takes advantage of a suitably defined Cl´ement type projection operator. We then introduce an axisymmetric formulation of the Navier–Stokes equations in moving domains and, starting from existing results in three-dimensions, we set up an Arbitrary Lagrangian–Eulerian (ALE) formulation and prove some stability results. In the second part, we deal with algorithms for the solution of fluid-structure interaction problems. We introduce the problem in a generic form where the fluid is described by means of incompressible Navier–Stokes equations and the structure by a viscoelastic model. We account for large deformations of the structure and we show how existing algorithms may be improved to reduce the computational time. In particular we show how to use transpiration boundary conditions to approximate the fluid-structure problem in a fixed point strategy. Moreover, in a quasi-Newton strategy we reduce the cost by replacing the Jacobian with inexact Jacobians stemming from reduced physical models for the problem at hand. To speed up the convergence of the Newton algorithm, we also define a dynamic preconditioner and an acceleration scheme which have been successfully tested in haemodynamics simulations in two and three dimensions

Space-time reduced basis methods for parametrized unsteady Stokes equations

In this work, we analyse space-time reduced basis methods for the efficient numerical simulation of hemodynamics in arteries. The classical formulation of the reduced basis (RB) method features dimensionality reduction in space, while finite differences schemes are employed for the time integration of the resulting ordinary differential equation (ODE). Space-time reduced basis (ST-RB) methods extend the dimensionality reduction paradigm to the temporal dimension, projecting the full-order problem onto a low-dimensional spatio-temporal subspace. Our goal is to investigate the application of ST-RB methods to the unsteady incompressible Stokes equations, with a particular focus on stability. High-fidelity simulations are performed using the Finite Element (FE) method and BDF2 as time marching scheme. We consider two different ST-RB methods. In the first one - called ST-GRB - space-time model order reduction is achieved by means of a Galerkin projection; a spatio-temporal velocity basis enrichment procedure is introduced to guarantee stability. The second method - called ST-PGRB - is characterized by a Petrov--Galerkin projection, stemming from a suitable minimization of the FOM residual, that allows to automatically attain stability. The classical RB method - denoted as SRB-TFO - serves as a baseline for the theoretical development. Numerical tests have been conducted on an idealized symmetric bifurcation geometry and on the patient-specific one of a femoropopliteal bypass. The results show that both ST-RB methods provide accurate approximations of the high-fidelity solutions, while considerably reducing the computational cost. In particular, the ST-PGRB method exhibits the best performance, as it features a better computational efficiency while retaining accuracies in accordance with theoretical expectations.Comment: 30 pages (25 + 5 in appendix), 4 figures, 4 tables. To appear on SIAM Journal on Scientific Computing (SISC

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

When modelling the cardiovascular system, the effect of the vessel wall on the blood flow has great relevance. Arterial vessels are complex living tissues and three-dimensional specific models have been proposed to represent their behaviour. The numerical simulation of the 3D-3D Fluid-Structure Interaction (FSI) coupled problem has high computational costs in terms of required time and memory storage. Even if many possible solutions have been explored to speed up the resolution of such problem, we are far from having a 3D-3D FSI model that can be solved quickly. In 3D-3D FSI models two of the main sources of complexity are represented by the domain motion and the coupling between the fluid and the structural part. Nevertheless, in many cases, we are interested in the blood flow dynamics in compliant vessels, whereas the displacement of the domain is small and the structure dynamics is less relevant. In these situations, techniques to reduce the complexity of the problem can be used. One consists in using transpiration conditions for the fluid model as surrogate for the wall displacement, thus allowing problem's solution on a fixed domain. Another strategy consists in modelling the arterial wall as a thin membrane under specific assumptions (Figueroa et al., 2006, Nobile and Vergara, 2008) instead of using a more realistic (but more computationally intensive) 3D elastodynamic model. Using this strategy the dynamics of the vessel motion is embedded in the equation for the blood flow. Combining the transpiration conditions with the membrane model assumption, we obtain an attractive formulation, in fact, instead of solving two different models on two moving physical domains, we solve only a Navier-Stokes system in a fixed fluid domain where the structure model is integrated as a generalized Robin condition. In this paper, we present a general formulation in the boundary conditions which is independent of the time discretization scheme choice and on the stress-strain constitutive relation adopted for the vessel wall structure. Our aim is, first, to write a formulation of a reduced order model with zero order transpiration conditions for a generic time discretization scheme, then to compare a 3D-3D PSI model and a reduced FSI one in two realistic patient-specific cases: a femoropopliteal bypass and an aorta. In particular, we are interested in comparing the wall shear stresses, in fact this quantity can be used as a risk factor for some pathologies such as atherosclerosis or thrombogenesis. More in general we want to assess the accuracy and the computational convenience to use simpler formulations based on reduced order models. In particular, we show that, in the case of small displacements, using a 3D-3D PSI linear elastic model or the correspondent reduced order one yields many similar results. (c) 2013 Elsevier B.V. All rights reserved

A rescaled method for RBF approximation

In the recent paper [8], a new method to compute stable kernel-based interpolants has been presented. This \textit{rescaled interpolation} method combines the standard kernel interpolation with a properly defined rescaling operation, which smooths the oscillations of the interpolant. Although promising, this procedure lacks a systematic theoretical investigation. Through our analysis, this novel method can be understood as standard kernel interpolation by means of a properly rescaled kernel. This point of view allow us to consider its error and stability properties