1,646 research outputs found
Convergence of the Lagrange-Galerkin method for the Equations Modelling the Motion of a Fluid-Rigid System
In this paper, we consider a Lagrange-Galerkin scheme to approximate a two dimensional fluid-rigid body problem. The equations of the system are the Navier-Stokes equations in the fluid part, coupled with ordinary differential equations for the dynamics of the rigid body. In this problem, the equations of the fluid are written in a domain whose variation is one of the unknowns. We introduce a numerical method based on the use of characteristics and on finite elements with a fixed mesh. Our main result asserts the convergence of this scheme
Numerical modelling of the Oldroyd-B fluid
The purpose of this thesis is to develop a 3D finite element model of the Oldroyd-B fluid for use in a complex geometry. The model is developed in deal.ii, which is a C++ finite element library. In addition to the standard finite element approach for the momentum equation, the discontinuous Galerkin method is used for the constitutive relation of the fluid model, with the extra stress as the unknown variable. The model developed is verified by using the symmetric “flow over a cylinder” benchmark problem. The effect of using piecewise-constant discontinuous and bilinear discontinuous elements for the extra stress field is investigated. The the results of the scheme are compared to those found in literature. The model is implemented in the solution of a complex problem of blood flow in an arteriovenous fistula, using geometry acquired from MRI data. A resistance boundary condition is used for the outlets. The flow profiles obtained from using both the Newtonian and Oldroyd-B fluids are validated against velocity encoded MRI and also compared to Fluid-Structure Interaction results for Newtonian fluids, from the literature. The effect of using a viscoelastic fluid on the flow profile and wall shear stresses are investigated. The results from this work show that using a viscoelastic fluid, rather than a Newtonian fluid, provides additional details regarding the wall shear stress in the arteriovenous fistula
Rigid body motion in viscous flows using the Finite Element Method
A new model for the numerical simulation of a rigid body moving in a viscous
fluid flow using FEM is presented. One of the most interesting features of this
approach is the small computational effort required to solve the motion of the
rigid body, comparable to a pure fluid solver. The model is based on the idea
of extending the fluid velocity inside the rigid body and solving the flow
equations with a penalization term to enforce rigid motion inside the solid. In
order to get the velocity field in the fluid domain the Navier-Stokes equations
for an incompressible viscous flow are solved using a fractional-step procedure
combined with the two-step Taylor-Galerkin for the fractional linear momentum.
Once the velocity field in the fluid domain is computed, calculation of the
rigid motion is obtained by averaging translation and angular velocities over
the solid. One of the main challenges when dealing with the fluid-solid
interaction is the proper modelling of the interface which separates the solid
moving mass from the viscous fluid. In this work the combination of the level
set technique and the two-step Taylor-Galerkin algorithm for tracking the
fluid-solid interface is proposed. The good properties exhibited by the
two-step Taylor-Galerkin, minimizing oscillations and numerical diffusion, make
this method suitable to accurately advect the solid domain avoiding distortions
at its boundaries, and thus preserving the initial size and shape of the rigid
body. The proposed model has been validated against empirical solutions,
experimental data and numerical simulations found in the literature. In all
tested cases, the numerical results have shown to be accurate, proving the
potential of the proposed model as a valuable tool for the numerical analysis
of the fluid-solid interaction.Comment: Research article; 41 pages, 40 figures, 5 tables, 91 reference
A monolithic fluid-structure interaction formulation for solid and liquid membranes including free-surface contact
A unified fluid-structure interaction (FSI) formulation is presented for
solid, liquid and mixed membranes. Nonlinear finite elements (FE) and the
generalized-alpha scheme are used for the spatial and temporal discretization.
The membrane discretization is based on curvilinear surface elements that can
describe large deformations and rotations, and also provide a straightforward
description for contact. The fluid is described by the incompressible
Navier-Stokes equations, and its discretization is based on stabilized
Petrov-Galerkin FE. The coupling between fluid and structure uses a conforming
sharp interface discretization, and the resulting non-linear FE equations are
solved monolithically within the Newton-Raphson scheme. An arbitrary
Lagrangian-Eulerian formulation is used for the fluid in order to account for
the mesh motion around the structure. The formulation is very general and
admits diverse applications that include contact at free surfaces. This is
demonstrated by two analytical and three numerical examples exhibiting strong
coupling between fluid and structure. The examples include balloon inflation,
droplet rolling and flapping flags. They span a Reynolds-number range from
0.001 to 2000. One of the examples considers the extension to rotation-free
shells using isogeometric FE.Comment: 38 pages, 17 figure
Recent Developments in the Numerics of Nonlinear Hyperbolic Conservation Laws
The development of reliable numerical methods for the simulation of real life problems requires both a fundamental knowledge in the field of numerical analysis and a proper experience in practical applications as well as their mathematical modeling.
Thus, the purpose of the workshop was to bring together experts not only from the field of applied mathematics but also from civil and mechanical engineering working in the area of modern high order methods for the solution of partial differential equations or even approximation theory necessary to improve the accuracy as well as robustness of numerical algorithms
Computing stationary free-surface shapes in microfluidics
A finite-element algorithm for computing free-surface flows driven by
arbitrary body forces is presented. The algorithm is primarily designed for the
microfluidic parameter range where (i) the Reynolds number is small and (ii)
force-driven pressure and flow fields compete with the surface tension for the
shape of a stationary free surface. The free surface shape is represented by
the boundaries of finite elements that move according to the stress applied by
the adjacent fluid. Additionally, the surface tends to minimize its free energy
and by that adapts its curvature to balance the normal stress at the surface.
The numerical approach consists of the iteration of two alternating steps: The
solution of a fluidic problem in a prescribed domain with slip boundary
conditions at the free surface and a consecutive update of the domain driven by
the previously determined pressure and velocity fields. ...Comment: Revised versio
Differential-Algebraic Equations and Beyond: From Smooth to Nonsmooth Constrained Dynamical Systems
The present article presents a summarizing view at differential-algebraic
equations (DAEs) and analyzes how new application fields and corresponding
mathematical models lead to innovations both in theory and in numerical
analysis for this problem class. Recent numerical methods for nonsmooth
dynamical systems subject to unilateral contact and friction illustrate the
topicality of this development.Comment: Preprint of Book Chapte
Numerical analysis of a penalization method for the three-dimensional motion of a rigid body in an incompressible viscous fluid
We present and analyze a penalization method wich extends the the method of
[1] to the case of a rigid body moving freely in an incompressible fluid. The
fluid-solid system is viewed as a single variable density flow with an
interface captured by a level set method. The solid velocity is computed by
averaging at avery time the flow velocity in the solid phase. This velocity is
used to penalize the flow velocity at the fluid-solid interface and to move the
interface. Numerical illustrations are provided to illustrate our convergence
result. A discussion of our result in the light of existing existence results
is also given. [1] Ph. Angot, C.-H. Bruneau and P. Fabrie, A penalization
method to take into account obstacles in incompressible viscous flows, Numer.
Math. 81: 497--520 (1999)Comment: 23 page
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