The Finite Element Immersed Boundary Method with Distributed Lagrange Multiplier

Abstract

Abstract. We introduce a new formulation for the finite element immersed boundary method which makes use of a distributed Lagrange multiplier. We prove that a full discretization of our model, based on a semi-implicit time advancing scheme, is unconditionally stable with respect to the time step size.

Numerical Study of a Monolithic Fluid–Structure Formulation

“…Even more so with immersed boundary methods (IBM) [24] [11], although the convergence analysis is more advanced [3].…”

On a fictitious domain method with distributed Lagrange multiplier for interface problems

“…The importance of the present formulation lies in its applicability in the framework of the Immersed Boundary Method where an analogous formulation is found in the case of a variational implementation of the solid movement. We refer the interested reader to [4] where this application will be made clearer; see also [3] for some details. In [1] some numerical formulations for a 1D interface problem have been presented.…”

Numerical Study of a 3D Eulerian Monolithic Formulation for Incompressible Fluid-Structures Systems

“…Spatial discretization can be done with the most popular finite element for fluids: the Lagrangian triangular elements of degree 2 for the space V h of velocities and displacements and Lagrangian triangular elements of degree 1 for the pressure space Q h , provided that the pressure is different in the structure and the fluid because the pressure is discontinuous at the interface σ; therefore Q h is the space of piecewise linear functions on the triangulations, continuous in Ω n+1 r , r = s, f . Appropriate couples like {V h , Q h } are chosen to satisfy the inf-sup condition to avoid checker-board oscillations (see for example [8]). A small penalization with parameter must be added to impose uniqueness of the pressure when L 2 0 is replaced by Q h ≈ L 2 .…”

“…The immersed boundary method (IBM) proposed by Charles Peskin [2] and analyzed in [8] is efficient for shells in fluid but is more difficult to implement for thick structures [9]. Particles in suspension in a fluid have been simulated by IBM-like methods but mostly for hard particles (see for instance [10,11]).…”