15,200 research outputs found
Counterexample-Guided Polynomial Loop Invariant Generation by Lagrange Interpolation
We apply multivariate Lagrange interpolation to synthesize polynomial
quantitative loop invariants for probabilistic programs. We reduce the
computation of an quantitative loop invariant to solving constraints over
program variables and unknown coefficients. Lagrange interpolation allows us to
find constraints with less unknown coefficients. Counterexample-guided
refinement furthermore generates linear constraints that pinpoint the desired
quantitative invariants. We evaluate our technique by several case studies with
polynomial quantitative loop invariants in the experiments
Macro-element interpolation on tensor product meshes
A general theory for obtaining anisotropic interpolation error estimates for
macro-element interpolation is developed revealing general construction
principles. We apply this theory to interpolation operators on a macro type of
biquadratic finite elements on rectangle grids which can be viewed as a
rectangular version of the Powell-Sabin element. This theory also shows
how interpolation on the Bogner-Fox-Schmidt finite element space (or higher
order generalizations) can be analyzed in a unified framework. Moreover we
discuss a modification of Scott-Zhang type giving optimal error estimates under
the regularity required without imposing quasi uniformity on the family of
macro-element meshes used. We introduce and analyze an anisotropic
macro-element interpolation operator, which is the tensor product of
one-dimensional macro interpolation and Lagrange interpolation.
These results are used to approximate the solution of a singularly perturbed
reaction-diffusion problem on a Shishkin mesh that features highly anisotropic
elements. Hereby we obtain an approximation whose normal derivative is
continuous along certain edges of the mesh, enabling a more sophisticated
analysis of a continuous interior penalty method in another paper
The Argyris isogeometric space on unstructured multi-patch planar domains
Multi-patch spline parametrizations are used in geometric design and
isogeometric analysis to represent complex domains. We deal with a particular
class of planar multi-patch spline parametrizations called
analysis-suitable (AS-) multi-patch parametrizations (Collin,
Sangalli, Takacs; CAGD, 2016). This class of parametrizations has to satisfy
specific geometric continuity constraints, and is of importance since it allows
to construct, on the multi-patch domain, isogeometric spaces with optimal
approximation properties. It was demonstrated in (Kapl, Sangalli, Takacs; CAD,
2018) that AS- multi-patch parametrizations are suitable for modeling
complex planar multi-patch domains.
In this work, we construct a basis, and an associated dual basis, for a
specific isogeometric spline space over a given AS-
multi-patch parametrization. We call the space the Argyris
isogeometric space, since it is across interfaces and at all
vertices and generalizes the idea of Argyris finite elements to tensor-product
splines. The considered space is a subspace of the entire
isogeometric space , which maintains the reproduction
properties of traces and normal derivatives along the interfaces. Moreover, it
reproduces all derivatives up to second order at the vertices. In contrast to
, the dimension of does not depend on the domain
parametrization, and admits a basis and dual basis which possess
a simple explicit representation and local support.
We conclude the paper with some numerical experiments, which exhibit the
optimal approximation order of the Argyris isogeometric space and
demonstrate the applicability of our approach for isogeometric analysis
A family of quadrilateral finite elements
We present a novel family of quadrilateral finite elements, which
define global spaces over a general quadrilateral mesh with vertices of
arbitrary valency. The elements extend the construction by (Brenner and Sung,
J. Sci. Comput., 2005), which is based on polynomial elements of tensor-product
degree , to all degrees . Thus, we call the family of
finite elements Brenner-Sung quadrilaterals. The proposed quadrilateral
can be seen as a special case of the Argyris isogeometric element of (Kapl,
Sangalli and Takacs, CAGD, 2019). The quadrilateral elements possess similar
degrees of freedom as the classical Argyris triangles. Just as for the Argyris
triangle, we additionally impose continuity at the vertices. In this
paper we focus on the lower degree cases, that may be desirable for their lower
computational cost and better conditioning of the basis: We consider indeed the
polynomial quadrilateral of (bi-)degree~, and the polynomial degrees
and by employing a splitting into or polynomial
pieces, respectively.
The proposed elements reproduce polynomials of total degree . We show that
the space provides optimal approximation order. Due to the interpolation
properties, the error bounds are local on each element. In addition, we
describe the construction of a simple, local basis and give for
explicit formulas for the B\'{e}zier or B-spline coefficients of the basis
functions. Numerical experiments by solving the biharmonic equation demonstrate
the potential of the proposed quadrilateral finite element for the
numerical analysis of fourth order problems, also indicating that (for )
the proposed element performs comparable or in general even better than the
Argyris triangle with respect to the number of degrees of freedom
Polynomial-based non-uniform interpolatory subdivision with features control
Starting from a well-known construction of polynomial-based interpolatory 4-point schemes, in this paper we present
an original affine combination of quadratic polynomial samples that leads to a non-uniform 4-point scheme with edge
parameters. This blending-type formulation is then further generalized to provide a powerful subdivision algorithm
that combines the fairing curve of a non-uniform refinement with the advantages of a shape-controlled interpolation
method and an arbitrary point insertion rule. The result is a non-uniform interpolatory 4-point scheme that is unique
in combining a number of distinctive properties. In fact it generates visually-pleasing limit curves where special
features ranging from cusps and flat edges to point/edge tension effects may be included without creating undesired
undulations. Moreover such a scheme is capable of inserting new points at any positions of existing intervals, so that
the most convenient parameter values may be chosen as well as the intervals for insertion.
Such a fully flexible curve scheme is a fundamental step towards the construction of high-quality interpolatory subdivision surfaces with features control
A general approach to transforming finite elements
The use of a reference element on which a finite element basis is constructed
once and mapped to each cell in a mesh greatly expedites the structure and
efficiency of finite element codes. However, many famous finite elements such
as Hermite, Morley, Argyris, and Bell, do not possess the kind of equivalence
needed to work with a reference element in the standard way. This paper gives a
generalizated approach to mapping bases for such finite elements by means of
studying relationships between the finite element nodes under push-forward.Comment: 28 page
Near-best quartic spline quasi-interpolants on type-6 tetrahedral partitions of bounded domains
In this paper, we present new quasi-interpolating spline schemes defined on
3D bounded domains, based on trivariate quartic box splines on type-6
tetrahedral partitions and with approximation order four. Such methods can be
used for the reconstruction of gridded volume data. More precisely, we propose
near-best quasi-interpolants, i.e. with coefficient functionals obtained by
imposing the exactness of the quasi-interpolants on the space of polynomials of
total degree three and minimizing an upper bound for their infinity norm. In
case of bounded domains the main problem consists in the construction of the
coefficient functionals associated with boundary generators (i.e. generators
with supports not completely inside the domain), so that the functionals
involve data points inside or on the boundary of the domain.
We give norm and error estimates and we present some numerical tests,
illustrating the approximation properties of the proposed quasi-interpolants,
and comparisons with other known spline methods. Some applications with real
world volume data are also provided.Comment: In the new version of the paper, we have done some minor revisions
with respect to the previous version, CALCOLO, Published online: 10 October
201
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