A projection semi‐implicit scheme for the coupling of an elastic structure with an incompressible fluid

Abstract

International audienceWe address the numerical simulation of fluid-structure systems involving an incompressible viscous fluid. This issue is particularly difficult to face when the fluid added-mass acting on the structure is strong, as it happens in hemodynamics for example. Indeed, several works have shown that, in such situations, implicit coupling seems to be necessary in order to avoid numerical instabilities. Although significant improvements have been achieved during the last years, solving implicit coupling often exhibits a prohibitive computational cost. In this work, we introduce a semi-implicit coupling scheme which remains stable for a reasonable range of the discretization parameters. The first idea consists in treating implicitly the added-mass effect, whereas the other contributions (geometrical non-linearities, viscous and convective effects) are treated explicitly. The second idea, relies on the fact that this kind of explicit-implicit splitting can be naturally performed using a Chorin-Temam projection scheme in the fluid. We prove (conditional) stability of the scheme for a fully discrete formulation. Several numerical experiments point out the efficiency of the present scheme compared to several implicit approaches

Stabilized explicit coupling for fluid–structure interaction using Nitsche's method

“…As a result, our stability result is independent of the dissipative features of the fluid and solid time discretization schemes. This is a significant progress with respect to the stability result stated in [12], for a semi-implicit coupling scheme, whose proof purely depends on the dissipative properties of the solid time discretization scheme (a leap-frog scheme). On the other hand, as regards the fluid timediscretization, one can use, for instance, a neutrally stable second order scheme.…”

“…[24,30,7,12]). Extensions to the non-linear case of some of the coupling strategies introduced in this paper are provided in paragraph §5.…”

“…Such procedures enable the design of efficient fluid-structure solution algorithms, while keeping state-of-the-art methods and software in each sub-domain (see e.g. [31,11,27,17,13,14,12]). …”

“…In summary, the fluid-structure Nitsche's formulation (4) can be reformulated, in a partitioned fashion, as two sub-problems (structure and fluid) coupled through Robin, (12), and Dirichlet-Nitsche, (13), transmission conditions. This is in contrast with the traditional Dirichlet-Neumann formulation, where only forces are transferred from the fluid to the solid (see e.g.…”

“…Such an approach leads to a fully coupled problem at each time step, the solution of which often requires a huge computational effort. Recent advances suggest the use of semi-implicit coupling schemes [12] (see also [34]), which involve a simplified fully coupled problem. Note that a critical added-mass effect can also compromise the efficiency of the iterative procedure used for solving the implicit component of the coupling scheme (see [7,1]).…”