Hypergraph Grammar based Adaptive Linear Computational Cost Projection Solvers for Two and Three Dimensional Modeling of Brain

In this paper we present a hypergraph grammar model for transformation of two and three dimensional grids. The hypergraph grammar concerns the proces of generation of uniform grids with two or three dimensional rectangular or hexahedral elements, followed by the proces of h refinements, namely breaking selected elements into four or eight son elements, in two or three dimensions, respectively. The hypergraph grammar presented in this paper expresses also the two solver algorithms. The first one is the projection based interpolation solver algorithm used for computing H1 or L2 projections of MRI scan of human head, in two and three dimensions. The second one is the multi-frontal direct solver utilized in the loop of the Euler scheme for solving the non-stationary problem modeling the three dimensional heat transport in the human head generated by the cellphone usage.