17,094 research outputs found
Addressing Integration Error for Polygonal Finite Elements Through Polynomial Projections: A Patch Test Connection
Polygonal finite elements generally do not pass the patch test as a result of
quadrature error in the evaluation of weak form integrals. In this work, we
examine the consequences of lack of polynomial consistency and show that it can
lead to a deterioration of convergence of the finite element solutions. We
propose a general remedy, inspired by techniques in the recent literature of
mimetic finite differences, for restoring consistency and thereby ensuring the
satisfaction of the patch test and recovering optimal rates of convergence. The
proposed approach, based on polynomial projections of the basis functions,
allows for the use of moderate number of integration points and brings the
computational cost of polygonal finite elements closer to that of the commonly
used linear triangles and bilinear quadrilaterals. Numerical studies of a
two-dimensional scalar diffusion problem accompany the theoretical
considerations
Improved method for finding optimal formulae for bilinear maps in a finite field
In 2012, Barbulescu, Detrey, Estibals and Zimmermann proposed a new framework
to exhaustively search for optimal formulae for evaluating bilinear maps, such
as Strassen or Karatsuba formulae. The main contribution of this work is a new
criterion to aggressively prune useless branches in the exhaustive search, thus
leading to the computation of new optimal formulae, in particular for the short
product modulo X 5 and the circulant product modulo (X 5 -- 1). Moreover , we
are able to prove that there is essentially only one optimal decomposition of
the product of 3 x 2 by 2 x 3 matrices up to the action of some group of
automorphisms
Sum-factorization techniques in Isogeometric Analysis
The fast assembling of stiffness and mass matrices is a key issue in
isogeometric analysis, particularly if the spline degree is increased. We
present two algorithms based on the idea of sum factorization, one for matrix
assembling and one for matrix-free methods, and study the behavior of their
computational complexity in terms of the spline order . Opposed to the
standard approach, these algorithms do not apply the idea element-wise, but
globally or on macro-elements. If this approach is applied to Gauss quadrature,
the computational complexity grows as instead of as
previously achieved.Comment: 34 pages, 8 figure
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