Iterative solutions of multimodels for incompressible flows

Summary: For the simulation of two-dimensional incompressible flows, the Navier-Stokes equations can be coupled with models of reduced complexity, such as Oseen and Stokes models. This approach is briefly reviewed here. Then we propose a numerical solution of the coupled problem by subdomain splitting. We give numerical results obtained by finite element and spectral element approximations on a couple of test problems of physical interest.

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

Claudia Maria Colciago, Simone Deparis

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.

2018

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

A greedy non-intrusive reduced order model for fluid dynamics

Jan Sickmann Hesthaven

We introduce a greedy non-intrusive reduced order method(ROM) for parameterized time-dependent problems with an emphasis on problems in fluid dynamics. The non-intrusive ROM is based on two-level proper orthogonal decomposition(POD) to extract temporal and spatial reduced basis from a set of candidates and a radial basis function(RBF) model to be used to approximate the coecients of the reduced model. And instead of adopting uniform or random sampling strategies, the candidates are determined using an adaptive greedy approach to minimize the overall computational online cost. Numerical studies are presented for a two-dimensional diusion problem and a driven cavity problem governed by incompressible Navier-Stokes equations. The results demonstrate that the greedy non-intrusive ROM predicts the flow field accurately and efficiently.

American Institute of Aeronautics and Astronautics2017

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

Claudia Maria Colciago, Simone Deparis

2018