Fast Graph Transformation based Direct Solver Algorithm for Regular Three Dimensional Grids

This paper presents a graph-transformation-based multifrontal direct solver with an optimization technique that allows for a significant decrease of time complexity in some multi-scale simulations of the Step and Flash Imprint Lithography (SFIL). The multi-scale simulation consists of a macro-scale linear e lasticity model with thermal expansion coefficient and a nano-scale molecular statics model. The algorithm is exemplified with a photopolimerization simulation that involves densification of a polymer inside a feature followed by shrinkage of the feature after removal of the template. The solver is optimized thanks to a mechanism of reusing sub -domains with similar geometries and similar material properties. The graph transformation formalism is used to describe the algorithm - such an approach helps automatically localize sub-domains that can be reused.