Hybrid vertex-midline modelling of elongated plant organs.
We present a multiscale method for the simulation of large viscoelastic deformations and show its applicability to biological tissue such as plant tissue. At the microscopic level we use a particle method to model the geometrical structure and basic properties of individual cells. The cell fluid, modeled as a viscoelastic fluid by means of Smoothed Particle Hydrodynamics (SPH) is enclosed in an elastic cell wall, modeled by discrete elements. The macroscopic equation and stress tensor are derived from the SPH model by means of the Generalized Mathematical Homogenization technique. The macroscopic domain is discretized using standard finite elements, where the stress tensor is evaluated from microscopic simulations in small sub-domains, called Representative Volume Elements (RVEs). Our emphasis is on reconstructing the microscopic state inside the RVE for given macroscopic deformation and velocity gradient. We propose a scheme to initialize the RVE consistently not only with the macroscopic variables, but also with the microscopic dynamics.