Coarse implicit time integration of a cellular scale particle model for plant tissue deformation

We describe a multiscale method to simulate the deformation of plant tissue. At the cellular scale we use a combination of Smoothed Particle Hydrodynamics (SPH) and discrete elements to model the geometrical structure and basic properties of individual plant cells. At the coarse level, the material is described by the standard continuum approach without explicitly constructing a constitutive equation. Instead, the coarse scale finite element model uses simulations with the fine (cellular) scale model in small subdomains, called Representative Volume Elements (RVEs), to determine the necessary coarse scale variables; such as the stress and the elasticity and viscosity tensors. We present an implicit time integration scheme for the coarse finite element model allowing much larger time steps than possible with explicit methods. Computation of the Cauchy stress from an RVE is straightforward by volume averaging over the RVE. In this work, we use forward finite differencing of the objective Truesdell stress rate to estimate both the fourth order elasticity and viscosity tensors. These tensors are then used to construct the coarse scale stiffness and damping matrices, required for implicit integration.