## Multiscale modeling using goal-oriented adaptivity and numerical homogenization

##### Abstract

Modeling of engineering objects with complex heterogeneous material
structure at nanoscale level has emerged as an important research problem. In
this research, we are interested in multiscale modeling and analysis of mechanical
properties of the polymer structures created in the Step and Flash Imprint
Lithography (SFIL) process. SFIL is a novel imprint lithography process designed
to transfer circuit patterns for fabricating microchips in low-pressure
and room-temperature environments. Since the smallest features in SFIL are
only a few molecules across, approximating them as a continuum is not completely
accurate. Previous research in this subject has dealt with coupling
discrete models with continuum hyperelasticity models. The modeling of the
post-polymerization step in SFIL involves computing solutions of large nonlinear
energy minimization problems with fast spatial variation in material properties. An equilibrium configuration is found by minimizing the energy of
this heterogeneous polymeric lattice.
Numerical solution of such a molecular statics base model, which is
assumed to describe the microstructure completely, is computationally very
expensive. This is due to the problem size – on the order of millions of degrees
of freedom (DOFs). Rapid variation in material properties, ill-conditioning,
nonlinearity, and non-convexity make this problem even more challenging to
solve.
We devise a method for efficient approximation of the solution. Combining
numerical homogenization, adaptive finite element meshes, and goaloriented
error estimation, we develop a black-box method for efficient solution
of problems with multiple spatial scales. The purpose of this homogenization
method is to reduce the number of DOFs, find locally optimal effective material
properties, and do goal-oriented mesh refinement. In addition, it smoothes
the energy landscape.
Traditionally, a finite element mesh is designed after obtaining material
properties in different regions. The mesh has to resolve material discontinuities
and rapid variations. In our approach, however, we generate a sequence
of coarse meshes (possibly 1-irregular), and homogenize material properties on
each coarse mesh element using a locally posed constrained convex quadratic
optimization problem. This upscaling is done using Moore-Penrose pseudoinverse
of the linearized fine-scale element stiffness matrices, and a material independent
interpolation operator. This requires solution of a continuous-time Lyapunov equation on each element. Using the adjoint solution, we compute
local error estimates in the quantity of interest. The error estimates also drive
the automatic mesh adaptivity algorithm. The results show that this method
uses orders of magnitude fewer degrees of freedom to give fast and approximate
solutions of the original fine-scale problem.
Critical to the computational speed of local homogenization is computing
Moore-Penrose pseudoinverse of rank-deficient matrices without using
Singular Value Decomposition. To this end, we use four algorithms, each
having different desirable features. The algorithms are based on Tikhonov
regularization, sparse QR factorization, a priori knowledge of the null-space
of the matrix, and iterative methods based on proper splittings of matrices.
These algorithms can exploit sparsity and thus are fast.
Although the homogenization method is designed with a specific molecular
statics problem in mind, it is a general method applicable for problems
with a given fine mesh that sufficiently resolves the fine-scale material properties.
We verify the method using a conductivity problem in 2-D, with chessboard
like thermal conductivity pattern, which has a known homogenized
conductivity. We analyze other aspects of the homogenization method, for
example the choice of norm in which we measure local error, optimum coarse
mesh element size for homogenizing SFIL lattices, and the effect of the method
chosen for computing the pseudoinverse.

##### Department

##### Description

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