A Unified-Field Monolithic Fictitious Domain-Finite Element Method for Fluid-Structure-Contact Interactions and Applications to Deterministic Lateral Displacement Problems

TL;DR

The paper presents a unified-field, monolithic fictitious-domain finite element method (UFMFD-FEM) for fluid-structure-contact interactions (FSCI), enabling accurate simulation of moving interfaces and collisions without Dirac delta or Lagrange multiplier formulations. By employing two overlapped body-unfitted meshes and a stabilized mixed FE framework on a fixed Eulerian fluid mesh with an updated-Lagrangian structure, the method achieves robust coupling between fluid and structure under large deformations and contact with a wall. The authors derive both weak and IBM-based strong forms, discretize using stabilized $P_1/P_1$ or $P_2/P_1$ elements, and apply Newton and fixed-point iterations to handle nonlinearities and collisions, respectively. Numerical experiments on benchmark FSI problems and a DLD microchip scenario show accurate convergence, correct particle migration modes, and good agreement with experimental data, highlighting the method's potential for design optimization in microfluidic and biofluidic applications.

Abstract

Based upon two overlapped, body-unfitted meshes, a type of unified-field monolithic fictitious domain-finite element method (UFMFD-FEM) is developed in this paper for moving interface problems of dynamic fluid-structure interactions (FSI) accompanying with high-contrast physical coefficients across the interface and contacting collisions between the structure and fluidic channel wall when the structure is immersed in the fluid. In particular, the proposed novel numerical method consists of a monolithic, stabilized mixed finite element method within the frame of fictitious domain/immersed boundary method (IBM) for generic fluid-structure-contact interaction (FSCI) problems in the Eulerian-updated Lagrangian description, while involving the no-slip type of interface conditions on the fluid-structure interface, and the repulsive contact force on the structural surface when the immersed structure contacts the fluidic channel wall. The developed UFMFD-FEM for FSI or FSCI problems can deal with the structural motion with large rotational and translational displacements and/or large deformation in an accurate and efficient fashion, which are first validated by two benchmark FSI problems and one FSCI model problem, then by experimental results of a realistic FSCI scenario -- the microfluidic deterministic lateral displacement (DLD) problem that is applied to isolate circulating tumor cells (CTCs) from blood cells in the blood fluid through a cascaded filter DLD microchip in practice, where a particulate fluid with the pillar obstacles effect in the fluidic channel, i.e., the effects of fluid-structure interaction and structure collision, play significant roles to sort particles (cells) of different sizes with tilted pillar arrays.

Figures (12)

• Figure 1: Illustrations of ( Left:) the immersed case of FSI; ( Middle:) the back-to-back case of FSI; ( Right:) two overlapped, body-unfitted meshes.
• Figure 2: A schematic domain of FSI.
• Figure 3: Schematic illustrations of two types of structural boundary conditions: (a) no collision occurs and thus interface conditions are applied to the (blue) fluid-structure interface $\Gamma^t$; (b) the collision occurs and generates a repulsive contact force on the (red) contacting surface, $\Gamma_C^t$, between the structure and the fluidic channel wall, where the light blue area represents the fluid domain $\Omega_f^t$, the yellow areas are the immersed structures of different shapes, $\Omega_s^t$, and, the solid black lines denote the fluidic channel wall $\partial\Omega$.
• Figure 5: The simple shear flow-particle interaction problem: the computational domain (left); the velocity field of the case $\kappa=0.2$ (middle); the angular speed $|\omega|$ versus the confined ratio $\kappa$ (right).
• Figure 6: The plane Poiseuille flow-particle interaction problem: (a) the computational domain with the length $L=1$; (b) the total velocity field at the equilibrium position in Test #5; (c) the rotating velocity field surrounding the particle at the equilibrium position in Test #5 after an average (translational) velocity of the particle is deducted from the total velocity field; (d) lateral migrations of a circular particle in different tests; (e) lateral migrations of a circular particle with different mesh sizes in Test #5.

Theorems & Definitions (4)

• Remark 2.1
• Remark 4.1
• Proposition 4.2
• Remark 5.1