Multiscale Modeling Using Goal-oriented Adaptivity and Numerical
Homogenization
Chetan Jhurani
Joint work with Leszek F. Demkowicz,
J. Tinsley Oden, Serge Prudhomme, and
Paul T. Bauman
Abstract
Our goal is multiscale modeling and analysis of mechanical properties
of the
polymeric structures created in the Step and Flash Imprint Lithography
(SFIL) process [1]. SFIL is a novel imprint lithography process
designed to
transfer circuit patterns for fabricating microchips in low-pressure
and
room-temperature environments. Polymerization is modeled by a
Monte-Carlo
method that generates the bond structure and distribution of reacting
monomers and other molecules. We are interested in computing
minimum
energy configurations of these lattices.
We have devised a method for efficient approximation of solutions of
such
molecular statics problems posed on a complex lattice-like geometry and
with
fast spatial variation in material properties. We assume that
homogenization by determining optimal effective properties should be
done
locally. This leads us to an algorithm where the Moore-Penrose
pseudoinverse
of the local fine-scale operator is used to obtain a unique
load-dependent
or load-independent local homogenized operator. The purpose of this
technique is to reduce the number of DOFs, find locally optimal
effective
properties, and combine homogenization with mesh refinement.
Locally
homogenized operators are assembled to form the dimensionally reduced
global system.
Critical to the efficiency of the local homogenization is computation
of the
pseudoinverse of the local fine-scale stiffness matrix without using
the
Singular Value Decomposition. This is possible either by using
Tikhonov
regularization of the local operator or by using the knowledge of its
null-space. Currently we are building a general purpose C++ code
based
on PETSc [2] and hp3d [3].
References
----------
[1] M. Colburn, S. C. Johnson, M. D. Stewart, S. Damle, T. C. Bailey,
B. Choi, M.Wedlake, T. B. Michaelson, S. V. Sreenivasan, J. G. Ekerdt,
and C. Grant Willson. Step and Flash Imprint Lithography: A new
approach
to high-resolution patterning. In Y. Vladimirsky, editor, Proceedings
of
SPIE/Emerging Lithographic Technologies III, volume 3676, pages
379-389,
Santa Clara, 3 1999.
[2] S. Balay, W. D. Gropp, L. C. McInnes, and B. F. Smith. PETSc users
manual. Technical Report ANL-95/11 - Revision 2.1.1, Mathematics and
Computer Science Division, Argonne National Laboratory, 2001.
[3] L. Demkowicz, W. Rachowicz, D. Pardo, M. Paszynski, J. Kurtz, and
A. Zdunek. Computing with hp-ADAPTIVE FINITE ELEMENTS: Volume II
Frontiers: Three Dimensional Elliptic and Maxwell Problems with
Applications. Chapman & Hall/CRC Press, 2007.