Multi-scale computation of plant tissue deformation using models for individual cell behavior

We present a micro-macro method for the simulation of large elastic deformations of plant tissue. At the microscopic level we use a discrete element model to describe the geometrical structure and basic properties of individual plant cells. The macroscopic domain is discretized using standard finite elements, in which the unknown macroscopic material properties (the stress-strain relation) are computed using the microscopic model in small sub-domains, called representative volume elements (RVEs), centered around the macroscopic quadrature points. The boundary conditions for these RVEs are derived from the macroscopic deformation gradient. The computation of the macroscopic stress tensor is based on the definition of virial stress, as defined in molecular dynamics. The anisotropic Eulerian elasticity tensor is estimated using a forward finite difference approximation for the Truesdell rate of the Cauchy stress tensor. We investigate the influence of the size of the RVE and the boundary conditions via numerical experiments. We show that the multi-scale method converges to the solution of the full microscopic simulation, both for globally and adaptively refined finite element meshes and achieves a significant speed-up compared with the full microscopic simulation.