1,835 research outputs found
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
The arithmetic of genus two curves with (4,4)-split Jacobians
In this paper we study genus 2 curves whose Jacobians admit a polarized
(4,4)-isogeny to a product of elliptic curves. We consider base fields of
characteristic different from 2 and 3, which we do not assume to be
algebraically closed. We obtain a full classification of all principally
polarized abelian surfaces that can arise from gluing two elliptic curves along
their 4-torsion and we derive the relation their absolute invariants satisfy.
As an intermediate step, we give a general description of Richelot isogenies
between Jacobians of genus 2 curves, where previously only Richelot isogenies
with kernels that are pointwise defined over the base field were considered.
Our main tool is a Galois theoretic characterization of genus 2 curves
admitting multiple Richelot isogenies.Comment: 30 page
Geometric continuity and compatibility conditions for 4-patch surfaces
When considering regularity of surfaces, it is its geometry that is of
interest. Thus, the concept of geometric regularity or geometric continuity of
a specific order is a relevant concept. In this paper we discuss necessary and
sufficient conditions for a 4-patch surface to be geometrically continuous of
order one and two or, in other words, being tangent plane continuous and
curvature continuous respectively. The focus is on the regularity at the point
where the four patches meet and the compatibility conditions that must appear
in this case. In this article the compatibility conditions are proved to be
independent of the patch parametrization, i.e., the compatibility conditions
are universal. In the end of the paper these results are applied to a specific
parametrization such as Bezier representation in order to generalize a 4-patch
surface result by Sarraga.Comment: 25 pages, 6 figure
A Universal Parametrization in B-Spline Curve and Surface Interpolation and Its Performance Evaluation.
The choice of a proper parametrization method is critical in curve and surface fitting using parametric B-splines. Conventional parametrization methods do not work well partly because they are based only on the geometric properties of given data points such as the distances between consecutive data points and the angles between consecutive line segments. The resulting interpolation curves don\u27t look natural and they are often not affine invariant. The conventional parametrization methods don\u27t work well for odd orders k. If a data point is altered, the effect is not limited locally at all with these methods. The localness property with respect to data points is critical in interactive modeling. We present a new parametrization based on the nature of the basis functions called B-splines. It assigns to each data point the parameter value at which the corresponding B-spline N\sb{ik}(t) is maximum. The new method overcomes all four problems mentioned above; (1) It works well for all orders k, (2) it generates affine invariant curves, (3) the resulting curves look more natural, in general, and (4) it has the semi-localness property with respect to data points. The new method is also computationally more efficient and the resulting curve has more regular behavior of the curvature. Fairness evaluation and knot removal are performed on curves obtained from various parametrizations. The results also show that the new parametrization is superior. Fairness is evaluated in terms of total curvature, total length, and curvature plot. The curvature plots are looking natural for the curves obtained from the new parametrization. For the curves obtained from the new method, knot removal is able to provide with the curves which are very close to the original curves. A more efficient and effective method is also presented for knot removal in B-spline curve. A global norm is utilized for approximation unlike other methods which are using some local norms. A geometrical view makes the computation more efficient
Efficient implementation of a van der Waals density functional: Application to double-wall carbon nanotubes
We present an efficient implementation of the van der Waals density
functional of Dion et al [Phys. Rev. Lett. 92, 246401 (2004)], which expresses
the nonlocal correlation energy as a double spacial integral. We factorize the
integration kernel and use fast Fourier transforms to evaluate the
selfconsistent potential, total energy, and atomic forces, in N log(N)
operations. The resulting overhead in total computational cost, over semilocal
functionals, is very moderate for medium and large systems. We apply the method
to calculate the binding energies and the barriers for relative translation and
rotation in double-wall carbon nanotubes.Comment: 4 pages, 1 figure, 1 tabl
On automatic tuning of basis functions in Bezier method
A transition from the fixed basis in Bezier's method to some class of base functions is proposed. A parameter vector of a basis function is introduced as additional information. This achieves a more universal form of presentation and analytical description of geometric objects as compared to the non-uniform rational B-splines (NURBS). This enables control of basis function parameters including control points, their weights and node vectors. This approach can be useful at the final stage of constructing and especially local modification of compound curves and surfaces with required differential and shape properties; it also simplifies solution of geometric problems. In particular, a simple elimination of discontinuities along local spline curves due to automatic tuning of basis functions is demonstrated
H\"older Regularity of Geometric Subdivision Schemes
We present a framework for analyzing non-linear -valued
subdivision schemes which are geometric in the sense that they commute with
similarities in . It admits to establish
-regularity for arbitrary schemes of this type, and
-regularity for an important subset thereof, which includes all
real-valued schemes. Our results are constructive in the sense that they can be
verified explicitly for any scheme and any given set of initial data by a
universal procedure. This procedure can be executed automatically and
rigorously by a computer when using interval arithmetics.Comment: 31 pages, 1 figur
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