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High-order DLM-ALE discretizations with robust operator preconditioning for fluid-rigid-body interaction

Qi Xin, Shihua Gong, Lingyue Shen, Pinjing Wen, Yumiao Zhang, Yan Chen, Jiarui Han, Jinchao Xu

TL;DR

This work develops a high-order fitted-mesh DLM–ALE framework for fluid–rigid-body interaction motivated by DLD microfluidics. It combines a monolithic DLM variational formulation with isoparametric Taylor–Hood discretization and a partitioned IMEX Runge–Kutta time integrator that updates the mesh explicitly while solving the FSI subsystem implicitly. A rigorous well-posedness analysis yields a robust block preconditioner for the resulting saddle-point systems, with the preconditioner’s performance shown to be insensitive to discretization and physical parameters. Numerical experiments across curved flows, shear, sedimentation, and DLD-like geometries demonstrate high-order convergence and efficient solver behavior, supporting the approach’s potential for device-scale design and optimization.

Abstract

Motivated by the design of deterministic lateral displacement (DLD) microfluidic devices, we develop a high-order numerical framework for fluid-rigid-body interaction on fitted moving meshes. Rigid-body motion is enforced by a distributed Lagrange multiplier (DLM) formulation, while the moving fluid domain is treated by an arbitrary Lagrangian-Eulerian (ALE) mapping. In space, we use isoparametric Taylor-Hood elements to achieve high-order accuracy and to represent curved boundaries and the fluid-particle interface. In time, we employ a high-order partitioned Runge-Kutta strategy in which the mesh motion is advanced explicitly and the coupled physical fields are advanced implicitly, yielding high-order accuracy for the particle trajectory. The fully coupled system is linearized into a generalized Stokes problem subject to distributed constraints of incompressibility and rigid-body motion. We establish well-posedness of this generalized Stokes formulation at both the continuous and discrete levels, providing the stability foundation for operator preconditioning that is robust with respect to key physical and discretization parameters. Numerical experiments on representative benchmarks, including a DLD case, demonstrate high-order convergence for the fluid solution and rigid-body dynamics, as well as robust iterative convergence of the proposed preconditioners.

High-order DLM-ALE discretizations with robust operator preconditioning for fluid-rigid-body interaction

TL;DR

This work develops a high-order fitted-mesh DLM–ALE framework for fluid–rigid-body interaction motivated by DLD microfluidics. It combines a monolithic DLM variational formulation with isoparametric Taylor–Hood discretization and a partitioned IMEX Runge–Kutta time integrator that updates the mesh explicitly while solving the FSI subsystem implicitly. A rigorous well-posedness analysis yields a robust block preconditioner for the resulting saddle-point systems, with the preconditioner’s performance shown to be insensitive to discretization and physical parameters. Numerical experiments across curved flows, shear, sedimentation, and DLD-like geometries demonstrate high-order convergence and efficient solver behavior, supporting the approach’s potential for device-scale design and optimization.

Abstract

Motivated by the design of deterministic lateral displacement (DLD) microfluidic devices, we develop a high-order numerical framework for fluid-rigid-body interaction on fitted moving meshes. Rigid-body motion is enforced by a distributed Lagrange multiplier (DLM) formulation, while the moving fluid domain is treated by an arbitrary Lagrangian-Eulerian (ALE) mapping. In space, we use isoparametric Taylor-Hood elements to achieve high-order accuracy and to represent curved boundaries and the fluid-particle interface. In time, we employ a high-order partitioned Runge-Kutta strategy in which the mesh motion is advanced explicitly and the coupled physical fields are advanced implicitly, yielding high-order accuracy for the particle trajectory. The fully coupled system is linearized into a generalized Stokes problem subject to distributed constraints of incompressibility and rigid-body motion. We establish well-posedness of this generalized Stokes formulation at both the continuous and discrete levels, providing the stability foundation for operator preconditioning that is robust with respect to key physical and discretization parameters. Numerical experiments on representative benchmarks, including a DLD case, demonstrate high-order convergence for the fluid solution and rigid-body dynamics, as well as robust iterative convergence of the proposed preconditioners.
Paper Structure (25 sections, 9 theorems, 120 equations, 11 figures, 6 tables, 2 algorithms)

This paper contains 25 sections, 9 theorems, 120 equations, 11 figures, 6 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Mathematical model
  3. Variational Formulation
  4. Monolithic DLM Formulation
  5. Derivation of the Monolithic DLM Formulation
  6. Discretisation schemes
  7. Formulation in reference configuration
  8. Spatial discretisation
  9. Construction of the domain mapping
  10. Temporal discretisation
  11. Forward-Backward Euler:
  12. Two-stage second-order scheme:
  13. First-Order Semi-Implicit Scheme
  14. Second-Order Semi-Implicit Scheme
  15. Well-posedness of the linearized system and the robust preconditioner
  16. ...and 10 more sections

Key Result

Theorem 5.1

There exist positive constants $\alpha$, $C_a$ and $C_b$ such that the following coercivity and continuity conditions hold for all ${\mathbf{u}} \in V_h$, ${\mathbf{U}} \in T_M$, $\boldsymbol{\omega} \in R_M$, $q \in Q_h$, and $\boldsymbol{\mu} \in \boldsymbol{\Lambda}_h$: Moreover, the following LBB condition holds:

Figures (11)

  • Figure 1: Mesh reconstruction and interpolation of the velocity field.
  • Figure 2: Geometry and boundary conditions for flow around a cylinder.
  • Figure 3: Convergence of the velocity for flow around a cylinder.
  • Figure 4: Geometry and boundary conditions for the shear flow problem.
  • Figure 5: Convergence of the angular velocity error in shear flow.
  • ...and 6 more figures

Theorems & Definitions (24)

  • Remark 2.1: Non-dimensionalization
  • Remark 3.1
  • Remark 4.1
  • Remark 4.2
  • Remark 4.3
  • Remark 4.4
  • Remark 4.5
  • Remark 4.6
  • Theorem 5.1
  • Corollary 5.2
  • ...and 14 more