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Register a new userA fictitious domain finite element method for simulations of fluid-structure interactions: The Navier-Stokes equations coupled with a moving solid
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The paper extends a stabilized fictitious domain finite element method initially developed for the Stokes problem to the incompressible Navier-Stokes equations coupled with a moving solid. This method presents the advantage to predict an optimal approximation of the normal stress tensor at the interface. The dynamics of the solid is governed by the Newtons laws and the interface between the fluid and the structure is materialized by a level-set which cuts the elements of the mesh. An algorithm is proposed in order to treat the time evolution of the geometry and numerical results are presented on a classical benchmark of the motion of a disk falling in a channel.
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In this work we develop a fictitious domain method for the Stokes problem which allows computations in domains whose boundaries do not depend on the mesh. The method is based on the ideas of Xfem and has been first introduced for the Poisson problem. The fluid part is treated by a mixed finite element method, and a Dirichlet condition is imposed by a Lagrange multiplier on an immersed structure localized by a level-set function. A stabilization technique is carried out in order to get the convergence for this multiplier. The latter represents the forces that the fluid applies on the structure. The aim is to perform fluid-structure simulations for which these forces have a central role. We illustrate the capacities of the method by extending it to the incompressible Navier-Stokes equations coupled with a moving rigid solid.
In the present work, we propose to extend to the Stokes problem a fictitious domain approach inspired by eXtended Finite Element Method and studied for Poisson problem in [Renard]. The method allows computations in domains whose boundaries do not match. A mixed finite element method is used for fluid flow. The interface between the fluid and the structure is localized by a level-set function. Dirichlet boundary conditions are taken into account using Lagrange multiplier. A stabilization term is introduced to improve the approximation of the normal trace of the Cauchy stress tensor at the interface and avoid the inf-sup condition between the spaces for velocity and the Lagrange multiplier. Convergence analysis is given and several numerical tests are performed to illustrate the capabilities of the method.
In this article we study a mixed finite element formulation for solving the Stokes problem with general surface forces that induce a jump of the normal trace of the stress tensor, on an interface that splits the domain into two subdomains. Equality of velocities is assumed at the interface. The interface conditions are taken into account with multipliers. A suitable Lagrangian functional is introduced for deriving a weak formulation of the problem. A specificity of this work is the consideration of the interface with a fictitious domain approach. The latter is inspired by the XFEM approach in the sense that cut-off functions are used, but it is simpler to implement since no enrichment by singular functions is provided. In that context, getting convergence for the dual variables defined on the interface is non-trivial. For that purpose, an augmented Lagrangian technique stabilizes the convergence of the multipliers, which is important because their value would determine the dynamics of the interface in an unsteady framework. Theoretical analysis is provided, where we show that a discrete inf-sup condition, independent of the mesh size, is satisfied for the stabilized formulation. This guarantees optimal convergence rates, that we observe with numerical tests. The capacity of the method is demonstrated with robustness tests, and with an unsteady model tested for deformations of the interface that correspond to ellipsoidal shapes in dimension 2.
In this paper, we analyse a Vector Penalty Projection Scheme (see [1]) to treat the displacement of a moving body in incompressible viscous flows in the case where the interaction of the fluid on the body can be neglected. The presence of the obstacle inside the computational domain is treated with a penalization method introducing a parameter $eta$. We show the stability of the scheme and that the pressure and velocity converge towards a limit when the penalty parameter $epsilon$, which induces a small divergence and the time step $delta$t tend to zero with a proportionality constraint $epsilon$ = $lambda$$delta$t. Finally, when $eta$ goes to 0, we show that the problem admits a weak limit which is a weak solution of the Navier-Stokes equations with no-sleep condition on the solid boundary. R{e}sum{e} Dans ce travail nous analysons un sch{e}ma de projection vectorielle (voir [1]) pour traiter le d{e}placement dun corps solide dans un fluide visqueux incompressible dans le cas o` u linteraction du fluide sur le solide est n{e}gligeable. La pr{e}sence de lobstacle dans le domaine solide est mod{e}lis{e}e par une m{e}thode de p{e}nalisation. Nous montrons la stabilit{e} du sch{e}ma et la convergence des variables vitesse-pression vers une limite quand le param etre $epsilon$ qui assure une faible divergence et le pas de temps $delta$t tendent vers 0 avec une contrainte de proportionalit{e} $epsilon$ = $lambda$$delta$t. Finalement nous montrons que leprob{`i} eme converge au sens faible vers une solution des equations de Navier-Stokes avec une condition aux limites de non glissement sur lafront{`i} ere immerg{e}e quand le param etre de p{e}nalisation $eta$ tend vers 0.
This paper constructs and analyzes a boundary correction finite element method for the Stokes problem based on the Scott-Vogelius pair on Clough-Tocher splits. The velocity space consists of continuous piecewise quadratic polynomials, and the pressure space consists of piecewise linear polynomials without continuity constraints. A Lagrange multiplier space that consists of continuous piecewise quadratic polynomials with respect to boundary partition is introduced to enforce boundary conditions as well as to mitigate the lack of pressure-robustness. We prove several inf-sup conditions, leading to the well-posedness of the method. In addition, we show that the method converges with optimal order and the velocity approximation is divergence free.
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