2,004 research outputs found
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
Smooth quasi-developable surfaces bounded by smooth curves
Computing a quasi-developable strip surface bounded by design curves finds
wide industrial applications. Existing methods compute discrete surfaces
composed of developable lines connecting sampling points on input curves which
are not adequate for generating smooth quasi-developable surfaces. We propose
the first method which is capable of exploring the full solution space of
continuous input curves to compute a smooth quasi-developable ruled surface
with as large developability as possible. The resulting surface is exactly
bounded by the input smooth curves and is guaranteed to have no
self-intersections. The main contribution is a variational approach to compute
a continuous mapping of parameters of input curves by minimizing a function
evaluating surface developability. Moreover, we also present an algorithm to
represent a resulting surface as a B-spline surface when input curves are
B-spline curves.Comment: 18 page
A multigrid continuation method for elliptic problems with folds
We introduce a new multigrid continuation method for computing solutions of nonlinear elliptic eigenvalue problems which contain limit points (also called turning points or folds). Our method combines the frozen tau technique of Brandt with pseudo-arc length continuation and correction of the parameter on the coarsest grid. This produces considerable storage savings over direct continuation methods,as well as better initial coarse grid approximations, and avoids complicated algorithms for determining the parameter on finer grids. We provide numerical results for second, fourth and sixth order approximations to the two-parameter, two-dimensional stationary reaction-diffusion problem: Δu+λ exp(u/(1+au)) = 0.
For the higher order interpolations we use bicubic and biquintic splines. The convergence rate is observed to be independent of the occurrence of limit points
Recursive subdivision algorithms for curve and surface design
This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.In this thesis, the author studies recursIve subdivision algorithms for curves and surfaces. Several subdivision algorithms are constructed and investigated. Some graphic examples are also presented.
Inspired by the Chaikin's algorithm and the Catmull-Clark's algorithm, some non-uniform schemes, the non-uniform corner cutting scheme and the recursive subdivision algorithm for non-uniform B-spline curves, are constructed and analysed. The adapted parametrization is introduced to analyse these non-uniform algorithms. In order to solve the surface interpolation problem, the Dyn-Gregory-Levin's 4-point interpolatory scheme is generalized to surfaces and the 10-point interpolatory subdivision scheme for surfaces is formulated. The so-called Butterfly Scheme, which was firstly introduced by Dyn, Gregory Levin in 1988, is just a special case of the scheme. By studying the Cross-Differences of Directional Divided Differences, a matrix approach for analysing uniform subdivision algorithms for surfaces is established and the convergence of the 10-point scheme over both uniform and non-uniform triangular networks is studied. Another algorithm, the subdivision algorithm for uniform bi-quartic B-spline surfaces over arbitrary topology is introduced and investigated. This algorithm is a generalization of Doo-Sabin's and Catmull-Clark's algorithms. It produces uniform Bi-quartic B-spline patches over uniform data. By studying the local subdivision matrix, which is a circulant, the tangent plane and curvature properties of the limit surfaces at the so-called Extraordinary Points are studied in detail.The Chinese Educational Commission and The British Council (SBFSS/1987
A segmentation-free isogeometric extended mortar contact method
This paper presents a new isogeometric mortar contact formulation based on an
extended finite element interpolation to capture physical pressure
discontinuities at the contact boundary. The so called two-half-pass algorithm
is employed, which leads to an unbiased formulation and, when applied to the
mortar setting, has the additional advantage that the mortar coupling term is
no longer present in the contact forces. As a result, the computationally
expensive segmentation at overlapping master-slave element boundaries, usually
required in mortar methods (although often simplified with loss of accuracy),
is not needed from the outset. For the numerical integration of general contact
problems, the so-called refined boundary quadrature is employed, which is based
on adaptive partitioning of contact elements along the contact boundary. The
contact patch test shows that the proposed formulation passes the test without
using either segmentation or refined boundary quadrature. Several numerical
examples are presented to demonstrate the robustness and accuracy of the
proposed formulation.Comment: In this version, we have removed the patch test comparison with the
classical mortar method and removed corresponding statements. They will be
studied in further detail in future work, so that the focus is now entirely
on the new IGA mortar formulatio
A study on spline quasi-interpolation based quadrature rules for the isogeometric Galerkin BEM
Two recently introduced quadrature schemes for weakly singular integrals
[Calabr\`o et al. J. Comput. Appl. Math. 2018] are investigated in the context
of boundary integral equations arising in the isogeometric formulation of
Galerkin Boundary Element Method (BEM). In the first scheme, the regular part
of the integrand is approximated by a suitable quasi--interpolation spline. In
the second scheme the regular part is approximated by a product of two spline
functions. The two schemes are tested and compared against other standard and
novel methods available in literature to evaluate different types of integrals
arising in the Galerkin formulation. Numerical tests reveal that under
reasonable assumptions the second scheme convergences with the optimal order in
the Galerkin method, when performing -refinement, even with a small amount
of quadrature nodes. The quadrature schemes are validated also in numerical
examples to solve 2D Laplace problems with Dirichlet boundary conditions
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