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Register a new userA fictitious domain approach for Fluid-Structure Interactions based on the eXtended Finite Element Method
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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.
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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.
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.
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.
We develop a general framework for construction and analysis of discrete extension operators with application to unfitted finite element approximation of partial differential equations. In unfitted methods so called cut elements intersected by the boundary occur and these elements must in general by stabilized in some way. Discrete extension operators provides such a stabilization by modification of the finite element space close to the boundary. More precisely, the finite element space is extended from the stable interior elements over the boundary in a stable way which also guarantees optimal approximation properties. Our framework is applicable to all standard nodal based finite elements of various order and regularity. We develop an abstract theory for elliptic problems and associated parabolic time dependent partial differential equations and derive a priori error estimates. We finally apply this to some examples of partial differential equations of different order including the interface problems, the biharmonic operator and the sixth order triharmonic operator.
Reduction of computational cost of solutions is a key issue to crack identification or crack propagation problems. One of the solution is to avoid re-meshing the domain when the crack position changes or when the crack extends. To avoid re-meshing, w
e propose a new finite element approach for the numerical simulation of discontinuities of displacements generated by cracks inside elastic media. The approach is based on a fictitious domain method originally developed for Dirichlet conditions for the Poisson problem and for the Stokes problem, which is adapted to the Neumann boundary conditions of crack problems. The crack is represented by level-set functions. Numerical tests are made with a mixed formulation to emphasize the accuracy of the method, as well as its robustness with respect to the geometry enforced by a stabilization technique. In particular an inf-sup condition is theoretically proven for the latter. A realistic simulation with a uniformly pressurized fracture inside a volcano is given for illustrating the applicability of the method.
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