Reduced Numerical Approximation of Reduced Fluid-Structure Interaction Problems With Applications in Hemodynamics

This paper deals with fast simulations of the hemodynamics in large arteries by considering a reduced model of the associated fluid-structure interaction problem, which in turn allows an additional reduction in terms of the numerical discretisation. The resulting method is both accurate and computationally cheap. This goal is achieved by means of two levels of reduction: first, we describe the model equations with a reduced mathematical formulation which allows to write the fluid-structure interaction problem as a Navier-Stokes system with non-standard boundary conditions; second, we employ numerical reduction techniques to further and drastically lower the computational costs. The non standard boundary condition is of a generalized Robin type, with a boundary mass and boundary stiffness terms accounting for the arterial wall compliance. The numerical reduction is obtained coupling two well-known techniques: the proper orthogonal decomposition and the reduced basis method, in particular the greedy algorithm. We start by reducing the numerical dimension of the problem at hand with a proper orthogonal decomposition and we measure the system energy with specific norms; this allows to take into account the different orders of magnitude of the state variables, the velocity and the pressure. Then, we introduce a strategy based on a greedy procedure which aims at enriching the reduced discretization space with low offline computational costs. As application, we consider a realistic hemodynamics problem with a perturbation in the boundary conditions and we show the good performances of the reduction techniques presented in the paper. The results obtained with the numerical reduction algorithm are compared with the one obtained by a standard finite element method. The gains obtained in term of CPU time are of three orders of magnitude.

Numerical simulation of free surface flows

Vincent Maronnier, Marco Picasso, Jacques Rappaz

A numerical model is presented for the simulation of complex fluid flows with free surfaces. The unknowns are the velocity and pressure fields in the liquid region, together with a function defining the volume fraction of liquid. Although the mathematical formulation of the model is similar to the volume of fluid (VOF) method, the numerical schemes used to solve the problem are different. A splitting method is used for the time discretization. At each time step, two advection problems and a generalized Stokes problem are to be solved. Two different grids are used for the space discretization. The two advection problems are solved on a fixed, structured grid made out of small rectangular cells, using a forward characteristic method. The generalized Stokes problem is solved using a finite element method on a fixed, unstructured mesh. Numerical results are presented for several test cases: the filling of an S-shaped channel, the filling of a disk with core, the broken dam in a confined domain. (C) 1999 Academic Press.

1999

Numerical simulation of three-dimensional free surface flows

Vincent Maronnier, Marco Picasso, Jacques Rappaz

A numerical model is presented for the simulation of complex fluid flows with free surfaces in three space dimensions. The model described in Maronnier et al. (J. Comput. Phys. 1999; 155(2):439) is extended to three dimensional situations. The mathematical formulation of the model is similar to that of the volume of fluid (VOF) method, but the numerical procedures are different. A splitting method is used for the time discretization. At each time step, two advection problems-one for the predicted velocity field and the other for the volume fraction of liquid-are to be solved. Then, a generalized Stokes problem is solved and the velocity field is corrected. Two different grids are used for the space discretization. The two advection problems are solved on a fixed, structured grid made out of small cubic cells, using a forward characteristic method. The generalized Stokes problem is solved using continuous, piecewise linear stabilized finite elements on a fixed, unstructured mesh of tetrahedrons. The three-dimensional implementation is discussed. Efficient postprocessing algorithms enhance the quality of the numerical solution. A hierarchical data structure reduces memory requirements. Numerical results are presented for complex geometries arising in mold filling. Copyright (C) 2003 John Wiley Sons, Ltd.

2003

Isogeometric Analysis for Reduced Fluid-Structure Interaction Models in Haemodynamic Applications

Anna Tagliabue

Isogeometric analysis (IGA) is a computational methodology recently developed to numerically approximate Partial Differential Equation (PDEs). It is based on the isogeometric paradigm, for which the same basis functions used to represent the geometry are then used to approximate the unknown solution of the PDEs. In the case in which Non-Uniform Rational B-Splines (NURBS) are used as basis functions, their mathematical properties lead to appreciable benefits for the numerical approximation of PDEs, especially for high order PDEs in the standard Galerkin formulation. In this framework, we propose an a priori error estimate, extending existing results limited to second order PDEs. The improvements in both accuracy and efficiency of IGA compared to Finite Element Analysis (FEA), encourage the use of this methodology in the haemodynamic applications. In fact, the simulation of blood flow in arteries requires the numerical approximation of Fluid-Structure Interaction (FSI) problems. In order to account for the deformability of the vessel, the Navier-Stokes equations representing the blood flows, are coupled with structural models describing the mechanical response of the arterial wall. However, the FSI models are complex from both the mathematical and the numerical points of view, leading to high computational costs during the simulations. With the aim of reducing the complexity of the problem and the computational costs of the simulations, reduced FSI models can be considered. A first simplification, based on the assumption of a thin arterial wall structure, consists in considering shell models to describe the mechanical properties of the arterial walls. Moreover, by means of the additional kinematic condition (continuity of velocities) and dynamic condition (balance of contact forces), the structural problem can be rewritten as generalized boundary condition for the fluid problem. This results in a generalized Navier-Stokes problem which can be expressed only in terms of the primitive variables of the fluid equations (velocity and pressure) and in a fixed computational domain. As a consequence, the computational costs of the numerical simulations are significantly reduced. On the other side, the generalized boundary conditions associated to the reduced FSI model could involve high order derivatives, which need to be suitably approximated. With this respect, IGA allows an accurate, straightforward and efficient numerical approximation of the generalized Navier-Stokes equations characterizing the reduced FSI problem. In this work we consider the numerical approximation of reduced FSI models by means of IGA, for which we discuss the numerical results obtained in Haemodynamic applications.

2012

Isogeometric Analysis for Reduced Fluid-Structure Interaction Models in Haemodynamic Applications

Anna Tagliabue

2012