Abstract
We propose a finite element method for simulating one-dimensional solid models with finite thickness and finite length that move and experience large deformations while immersed in generalized Newtonian fluids. The method is oriented towards applications involving microscopic devices or organisms in the soft-bio-matter realm. By considering that the strain energy of the solid may explicitly depend on time, we incorporate a mechanism for active response. The solids are modeled as Cosserat rods, a detailed formulation being provided for the planar non-shearable case. The discretization adopts one-dimensional Hermite elements for the rod and two-dimensional low-order Lagrange elements for the fluid’s velocity and pressure. The fluid mesh is boundary-fitted, with remeshing at each time step. Several time marching schemes are studied, of which a semi-implicit scheme emerges as most effective. The method is demonstrated in very challenging examples: the roll-up of a rod to circular shape and later sudden release, the interaction of a soft rod with a fluid jet and the active self-locomotion of a sperm-like rod. The article includes a detailed description of a code that implements the method in the Firedrake library.
Highlights
One-dimensional solid models are widely used in macroscopic structural analysis
We propose a finite element method for simulating one-dimensional solid models moving and experiencing large deformations while immersed in generalized Newtonian fluids
The solids are modeled as Cosserat rods, a detailed formulation being provided for the special case of a planar non-shearable rod
Summary
One-dimensional solid models (strings, cables, trusses, bars, beams, filaments, rods, etc.) are widely used in macroscopic structural analysis. The reader is referred to [14, 15] for further details in the context of similar formulations in the three-dimensional setting Another peculiarity of soft-bio-matter problems is that, as happens in aerodynamics, the effect of the surrounding fluid on the solid bodies is much more consequential than energy dissipation. It should be mentioned that the complexity of micro-organisms’ shape and movement is simpler to address using boundary element and other singularity-based methods They have been developed and exploited in a quite extensive body of previous work (see [32, 33, 34, 35, 36, 37, 38] and references therein), but they have limitations that make a finite element method for the fluid to be interesting. Singularity-based methods do not compute a sparse representation of the bulk velocity of the fluid, which requires a (costly) additional reconstruction step before the transport calculations
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