A parallel solver for fluid structure interaction problems with Lagrange multiplier

TL;DR

The paper tackles parallel solution of fluid-structure interaction problems modeled via a fictitious-domain approach with a distributed Lagrange multiplier. It develops a monolithic FE discretization using a $\,\mathcal{Q}_2\,-\mathcal{P}_1\$ fluid pair and a $\mathcal{Q}_1$ solid, with semi-implicit Backward Euler time stepping and a Lagrange-multiplier coupling that yields a saddle-point system solved by parallel GMRES. Two block preconditioners (block-diagonal and block-triangular) are analyzed, with diagonal blocks inverted by a parallel MUMPS solver; numerical results show the block-triangular preconditioner to be robust for both linear and nonlinear solids, while the block-diagonal preconditioner can fail for refined meshes or nonlinear cases, particularly with mesh-intersection coupling. The work highlights the impact of coupling-assembly strategies (exact vs inexact) on sparsity and performance, and demonstrates scalability and accuracy through extensive 2D tests on Linux clusters using the PETSc framework. The findings advance robust parallel FSI solvers for fictitious-domain formulations and set the stage for extensions to more complex multiphysics problems.

Abstract

The aim of this work is to present a parallel solver for a formulation of fluid-structure interaction (FSI) problems which makes use of a distributed Lagrange multiplier in the spirit of the fictitious domain method. The fluid subproblem, consisting of the non-stationary Stokes equations, is discretized in space by $\mathcal{Q}_2$-$\mathcal{P}_1$ finite elements, whereas the structure subproblem, consisting of the linear or finite incompressible elasticity equations, is discretized in space by $\mathcal{Q}_1$ finite elements. A first order semi-implicit finite difference scheme is employed for time discretization. The resulting linear system at each time step is solved by a parallel GMRES solver, accelerated by block diagonal or triangular preconditioners. The parallel implementation is based on the PETSc library. Several numerical tests have been performed on Linux clusters to investigate the effectiveness of the proposed FSI solver.

Figures (8)

• Figure 1: A schematic representation of the geometric aspects of the coupling operations. From the left hand side: a portion of $\mathcal{T}_h^\mathcal{B}$ with a particular element under consideration, its immersed counterpart in the case of the coarse integration with the rule of the vertices, the same immersed element partitioned accordingly with the computation of the intersection.
• Figure 2: Sparsity patterns for the matrix \ref{['eq:matrix']} when $L_f(\mathbf{X}_h^n)$ is computed with the two assembly processes in the case of the linear elastic model described in Section \ref{['sec:linear_model']} discretized with 7846 global dofs. black $nz$ represents the number of nonzero entries of the matrix. The matrix is more dense when the coupling term is assembled with mesh intersection.
• Figure 3: black Simulation of the annulus with linear constitutive law: some snapshots. The structure position is represented in red, while streamlines and color bars refer to the velocity. At the initial condition, the annulus is stretched. When the simulation starts, the structure is released and internal elastic forces bring it back to its resting configuration.
• Figure 4: Test 2, strong scalability in the linear solid model, coupling with mesh intersection. Time evolution of linear iterations, CPU time to assemble the coupling term and to solve the linear system. Run on 32 processors of Shaheen cluster, 478470 dofs and block-tri preconditioner.
• Figure 5: Test 2, strong scalability in the linear solid model, coupling without mesh intersection. Time evolution of linear iterations, CPU time to assemble the coupling term and to solve the linear system. Run on 32 processors of Shaheen cluster, 478470 dofs and block-tri preconditioner.

Theorems & Definitions (6)

• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Remark 5
• Remark 6