21,729 research outputs found
Hierarchical interpolative factorization for elliptic operators: differential equations
This paper introduces the hierarchical interpolative factorization for
elliptic partial differential equations (HIF-DE) in two (2D) and three
dimensions (3D). This factorization takes the form of an approximate
generalized LU/LDL decomposition that facilitates the efficient inversion of
the discretized operator. HIF-DE is based on the multifrontal method but uses
skeletonization on the separator fronts to sparsify the dense frontal matrices
and thus reduce the cost. We conjecture that this strategy yields linear
complexity in 2D and quasilinear complexity in 3D. Estimated linear complexity
in 3D can be achieved by skeletonizing the compressed fronts themselves, which
amounts geometrically to a recursive dimensional reduction scheme. Numerical
experiments support our claims and further demonstrate the performance of our
algorithm as a fast direct solver and preconditioner. MATLAB codes are freely
available.Comment: 37 pages, 13 figures, 12 tables; to appear, Comm. Pure Appl. Math.
arXiv admin note: substantial text overlap with arXiv:1307.266
Hierarchical interpolative factorization for elliptic operators: integral equations
This paper introduces the hierarchical interpolative factorization for
integral equations (HIF-IE) associated with elliptic problems in two and three
dimensions. This factorization takes the form of an approximate generalized LU
decomposition that permits the efficient application of the discretized
operator and its inverse. HIF-IE is based on the recursive skeletonization
algorithm but incorporates a novel combination of two key features: (1) a
matrix factorization framework for sparsifying structured dense matrices and
(2) a recursive dimensional reduction strategy to decrease the cost. Thus,
higher-dimensional problems are effectively mapped to one dimension, and we
conjecture that constructing, applying, and inverting the factorization all
have linear or quasilinear complexity. Numerical experiments support this claim
and further demonstrate the performance of our algorithm as a generalized fast
multipole method, direct solver, and preconditioner. HIF-IE is compatible with
geometric adaptivity and can handle both boundary and volume problems. MATLAB
codes are freely available.Comment: 39 pages, 14 figures, 13 tables; to appear, Comm. Pure Appl. Mat
Effects of fiber/matrix interactions on the properties of graphite/epoxy composites
A state-of-the-art literature review of the interactions between fibers and resin within graphite epoxy composite materials was performed. Emphasis centered on: adhesion theory; wetting characteristics of carbon fiber; load transfer mechanisms; methods to evaluate and measure interfacial bond strengths; environmental influence at the interface; and the effect of the interface/interphase on composite performance, with particular attention to impact toughness. In conjunction with the literature review, efforts were made to design experiments to study the wetting behavior of carbon fibers with various finish variants and their effect on adhesion joint strength. The properties of composites with various fiber finishes were measured and compared to the base-line properties of a control. It was shown that by tailoring the interphase properties, a 30% increase in impact toughness was achieved without loss of mechanical properties at both room and elevated temperatures
Critical evaluation of Jet-A spray combustion using propane chemical kinetics in gas turbine combustion simulated by KIVA-2
Jet-A spray combustion has been evaluated in gas turbine combustion with the use of propane chemical kinetics as the first approximation for the chemical reactions. Here, the numerical solutions are obtained by using the KIVA-2 computer code. The KIVA-2 code is the most developed of the available multidimensional combustion computer programs for application of the in-cylinder combustion dynamics of internal combustion engines. The released version of KIVA-2 assumes that 12 chemical species are present; the code uses an Arrhenius kinetic-controlled combustion model governed by a four-step global chemical reaction and six equilibrium reactions. Researchers efforts involve the addition of Jet-A thermophysical properties and the implementation of detailed reaction mechanisms for propane oxidation. Three different detailed reaction mechanism models are considered. The first model consists of 131 reactions and 45 species. This is considered as the full mechanism which is developed through the study of chemical kinetics of propane combustion in an enclosed chamber. The full mechanism is evaluated by comparing calculated ignition delay times with available shock tube data. However, these detailed reactions occupy too much computer memory and CPU time for the computation. Therefore, it only serves as a benchmark case by which to evaluate other simplified models. Two possible simplified models were tested in the existing computer code KIVA-2 for the same conditions as used with the full mechanism. One model is obtained through a sensitivity analysis using LSENS, the general kinetics and sensitivity analysis program code of D. A. Bittker and K. Radhakrishnan. This model consists of 45 chemical reactions and 27 species. The other model is based on the work published by C. K. Westbrook and F. L. Dryer
Symmetry protected fractional Chern insulators and fractional topological insulators
In this paper we construct fully symmetric wavefunctions for the
spin-polarized fractional Chern insulators (FCI) and time-reversal-invariant
fractional topological insulators (FTI) in two dimensions using the parton
approach. We show that the lattice symmetry gives rise to many different FCI
and FTI phases even with the same filling fraction (and the same
quantized Hall conductance in FCI case). They have different
symmetry-protected topological orders, which are characterized by different
projective symmetry groups. We mainly focus on FCI phases which are realized in
a partially filled band with Chern number one. The low-energy gauge groups of a
generic FCI wavefunctions can be either or
the discrete group , and in the latter case the associated low-energy
physics are described by Chern-Simons-Higgs theories. We use our construction
to compute the ground state degeneracy. Examples of FCI/FTI wavefunctions on
honeycomb lattice and checkerboard lattice are explicitly given. Possible
non-Abelian FCI phases which may be realized in a partially filled band with
Chern number two are discussed. Generic FTI wavefunctions in the absence of
spin conservation are also presented whose low-energy gauge groups can be
either or . The constructed wavefunctions
also set up the framework for future variational Monte Carlo simulations.Comment: 24 pages, 13 figures, published versio
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