We present a new finite element scheme for direct simulations of inertialess particle suspensions in simple shear flows of a Newtonian fluid. The sliding bi-periodic domain concept of Lees and Edwards [J. Phys. C 5 (1972) 1921] has been combined with a standard velocity-pressure formulation of a fictitious-domain/finite-element method by introducing sliding bi-periodic frame constraints and it has been implemented with mortar element methods. Force-free, torque-free rigid body motions of particles are described through rigid-ring constraints and implemented by Lagrangian multipliers only on the particle boundary, which allows easy treatment of boundary-crossing particles. In our formulation, the bulk stress is obtained by simple boundary integrals of Lagrangian multipliers. Concentrating on two-dimensional circular disk particles, we discuss numerical examples of single-, two- and many-particle problems in sliding bi-periodic frames, which can represent an infinite number of such systems because of the bi-periodicity. The accuracy and convergence have been verified via comparison with a boundary-fitted mesh problem for velocities, pressures and velocity gradients. The present formulation is quite well suited for suspensions of micro/nano particles in simple shear flows and can be easily extended to viscoelastic flow problems.
Hwang, W. R., Hulsen, M. A., & Meijer, H. E. H. (2004). Direct simulation of particle suspensions in sliding bi-periodic frames. Journal of Computational Physics, 194(2), 742-772. https://doi.org/10.1016/j.jcp.2003.09.023