Subtree Reuse in Multi‐Frontal Solvers for Regular Grids in Step‐and‐Flash Imprint Nanolithography Modeling

Abstract

The paper presents multi‐scale modeling of the step‐and‐flash imprint lithography, a modern patterning process, which depends on photopolymerization in order to replicate the topography of a template onto a substrate [Colbum et al., J. Vac. Sci. Technol. B 2001, 19, 6]. Multi‐scale modeling presented in this paper corresponds to densification of the feature inside the template as well as deformation of the feature after removal of the template. Linear elasticity with thermal expansion coefficient, discretized with the finite element method (FEM) [Paszynski et al., IOP Conf. Series: Mat. Sci. Eng. 2010, 10, 012247] is utilized as the macro‐scale model. Molecular statics (MS) with quadratic and Lennard‐Jones potentials is utilized as the nano‐scale model [Paszynski et al., ICES Report, 2005, 05‐38]. Degrees of freedom from the macro‐scale model located on the interface have been identified with particles from nano‐scale domain. In order to improve the performance of the traditional approach, we propose an optimization technique for multi‐frontal direct solvers with constant coeffcients. The technique consists in reuse of sub‐branches of elimination trees over regular cube‐shaped grids built with hexahedral finite elements. To obtain an efficient reuse scheme, we construct an elimination tree in a specific way, so that all frontal matrices at certain level of the tree are identical. It is based on an observation that the solver tree for a uniform, fine 3D finite element grid is very regular, provided that introduction of boundary conditions as well as macro–nano‐scale interface conditions is postponed to the root of the tree (as opposed to applying them at the bottom nodes). Apart from a detailed description of the optimization, we offer a comprehensive estimation of the computational cost and memory usage benefits and showcase its accuracy.