A unified Eulerian formulation for immersed deformable solids

An Eulerian methodology is formulated to model the dynamics of immersed, deformable solids using a two-way coupled approach implemented within a unified simulation framework for incompressible viscous flow. A neo-Hookean constitutive model for incompressible hyperelastic materials is used to relate solid deformations to solid stresses, and the divergence of the solid stress tensor is added as a source term to the momentum equation. The resulting unified formulation allows a single Poisson equation for pressure to be solved across both the fluid and solid domains using fast solvers. Due to the modular nature of the formulation, material models other than neo-Hookean can be incorporated. The displacement field associated with the solid domain is evolved using a set of scalar functions that form a dynamic grid, while the boundary of the immersed solid is represented by an additional signed distance function. The motion of the dynamic grid relative to the underlying Cartesian grid is used to determine the deformation gradient tensor. A hyperbolic partial differential equation extrapolation scheme, combined with topology-preserving reinitialization of the solid boundary, propagates the dynamic grid into the fluid domain. The material properties and the corresponding solid and fluid stresses are regularized across a finite-width interface using a continuous Heaviside function. For the parameter ranges examined, the formulation produces stable calculations with systematic grid convergence and permits simulations at stiffness and density contrasts extending beyond those considered in the comparison studies. A preliminary three-dimensional implementation is also demonstrated.