Time integration schemes for fluid-structure interaction problems: non fitted FEMs for immersed thin structures

TL;DR

This paper analyzes and compares three time integration schemes for fluid-structure interaction problems using unfitted finite element methods, demonstrating stability and convergence properties for each scheme, with a focus on a promising splitting algorithm.

Contribution

It introduces and analyzes three novel time integration algorithms for unfitted fluid-structure interaction, including a fully discrete monolithic scheme and two partitioned schemes with stability and convergence proofs.

Findings

Algorithm 1 is well-posed, unconditionally stable, and convergent.

Algorithm 2 is conditionally stable for all extrapolations.

Algorithm 3 is unconditionally stable for certain extrapolations and converges optimally.

Abstract

We analyse three time integration schemes for unfitted methods in fluid structure interaction. In Alghorithm 1 we propose a fully discrete monolithic algorithm with P1 P1 stabilized finite elements for the fluid problem; for this alghorithm we prove well-posedness, unconditional stability and convergence in the case of linearized problem (see Propositions 2.4.2, 2.4.3 and Theorem 3.3.7, respectively). The analysis optimal convergence rates as expected from the Euler scheme, and the supposed regularity of the solution to the continuous problem. Moreover we introduce two algorithms that allow for a partitioning of the coupled problem by exploiting an explicit-implicit treatment of the transmission conditions. Algorithm 2 represents, essentially, a simplification of Algorithm 1 since it simply treat the solid elastic forces in explicit form using the displacement and velocities of the structure evaluated in the previous time steps. Instead, Algorithm 3, is really a splitting algorithm that involves the solution of two staggered problems. It splits the forces that solid transfers to fluid in two contributions: the inertial contribution that is treated in implicit form and the elastic contribution that is treated in explicit form. We perform the stability analysis for both the schemes in Theorems 4.3.1 and 4.3.3. Algorithm 2 results conditionally stable for all the extrapolations considered, instead Algorithm 3 is unconditionally stable, for extrapolations of order zero and one, and conditionally stable for the extrapolation of order two. Since Algorithm 3 is the most promising, we perform the convergence analysis in the linearized case (see Theorem 4.4.2) obtaining results in line with those of the monolithic case, in particular the splitting introduced preserves optimal conevegence rates.