4,088 research outputs found
A Multiscale Framework for Challenging Discrete Optimization
Current state-of-the-art discrete optimization methods struggle behind when
it comes to challenging contrast-enhancing discrete energies (i.e., favoring
different labels for neighboring variables). This work suggests a multiscale
approach for these challenging problems. Deriving an algebraic representation
allows us to coarsen any pair-wise energy using any interpolation in a
principled algebraic manner. Furthermore, we propose an energy-aware
interpolation operator that efficiently exposes the multiscale landscape of the
energy yielding an effective coarse-to-fine optimization scheme. Results on
challenging contrast-enhancing energies show significant improvement over
state-of-the-art methods.Comment: 5 pages, 1 figure, To appear in NIPS Workshop on Optimization for
Machine Learning (December 2012). Camera-ready version. Fixed typos,
acknowledgements adde
A Variational Formulation of Dissipative Quasicontinuum Methods
Lattice systems and discrete networks with dissipative interactions are
successfully employed as meso-scale models of heterogeneous solids. As the
application scale generally is much larger than that of the discrete links,
physically relevant simulations are computationally expensive. The
QuasiContinuum (QC) method is a multiscale approach that reduces the
computational cost of direct numerical simulations by fully resolving complex
phenomena only in regions of interest while coarsening elsewhere. In previous
work (Beex et al., J. Mech. Phys. Solids 64, 154-169, 2014), the originally
conservative QC methodology was generalized to a virtual-power-based QC
approach that includes local dissipative mechanisms. In this contribution, the
virtual-power-based QC method is reformulated from a variational point of view,
by employing the energy-based variational framework for rate-independent
processes (Mielke and Roub\'i\v{c}ek, Rate-Independent Systems: Theory and
Application, Springer-Verlag, 2015). By construction it is shown that the QC
method with dissipative interactions can be expressed as a minimization problem
of a properly built energy potential, providing solutions equivalent to those
of the virtual-power-based QC formulation. The theoretical considerations are
demonstrated on three simple examples. For them we verify energy consistency,
quantify relative errors in energies, and discuss errors in internal variables
obtained for different meshes and two summation rules.Comment: 38 pages, 21 figures, 4 tables; moderate revision after review, one
example in Section 5.3 adde
eXtended Variational Quasicontinuum Methodology for Lattice Networks with Damage and Crack Propagation
Lattice networks with dissipative interactions are often employed to analyze
materials with discrete micro- or meso-structures, or for a description of
heterogeneous materials which can be modelled discretely. They are, however,
computationally prohibitive for engineering-scale applications. The
(variational) QuasiContinuum (QC) method is a concurrent multiscale approach
that reduces their computational cost by fully resolving the (dissipative)
lattice network in small regions of interest while coarsening elsewhere. When
applied to damageable lattices, moving crack tips can be captured by adaptive
mesh refinement schemes, whereas fully-resolved trails in crack wakes can be
removed by mesh coarsening. In order to address crack propagation efficiently
and accurately, we develop in this contribution the necessary generalizations
of the variational QC methodology. First, a suitable definition of crack paths
in discrete systems is introduced, which allows for their geometrical
representation in terms of the signed distance function. Second, special
function enrichments based on the partition of unity concept are adopted, in
order to capture kinematics in the wakes of crack tips. Third, a summation rule
that reflects the adopted enrichment functions with sufficient degree of
accuracy is developed. Finally, as our standpoint is variational, we discuss
implications of the mesh refinement and coarsening from an energy-consistency
point of view. All theoretical considerations are demonstrated using two
numerical examples for which the resulting reaction forces, energy evolutions,
and crack paths are compared to those of the direct numerical simulations.Comment: 36 pages, 23 figures, 1 table, 2 algorithms; small changes after
review, paper title change
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