333 research outputs found
Constructing IGA-suitable planar parameterization from complex CAD boundary by domain partition and global/local optimization
In this paper, we propose a general framework for constructing IGA-suitable
planar B-spline parameterizations from given complex CAD boundaries consisting
of a set of B-spline curves. Instead of forming the computational domain by a
simple boundary, planar domains with high genus and more complex boundary
curves are considered. Firstly, some pre-processing operations including
B\'ezier extraction and subdivision are performed on each boundary curve in
order to generate a high-quality planar parameterization; then a robust planar
domain partition framework is proposed to construct high-quality patch-meshing
results with few singularities from the discrete boundary formed by connecting
the end points of the resulting boundary segments. After the topology
information generation of quadrilateral decomposition, the optimal placement of
interior B\'ezier curves corresponding to the interior edges of the
quadrangulation is constructed by a global optimization method to achieve a
patch-partition with high quality. Finally, after the imposition of
C1=G1-continuity constraints on the interface of neighboring B\'ezier patches
with respect to each quad in the quadrangulation, the high-quality B\'ezier
patch parameterization is obtained by a C1-constrained local optimization
method to achieve uniform and orthogonal iso-parametric structures while
keeping the continuity conditions between patches. The efficiency and
robustness of the proposed method are demonstrated by several examples which
are compared to results obtained by the skeleton-based parameterization
approach
Almost- splines: Biquadratic splines on unstructured quadrilateral meshes and their application to fourth order problems
Isogeometric Analysis generalizes classical finite element analysis and
intends to integrate it with the field of Computer-Aided Design. A central
problem in achieving this objective is the reconstruction of analysis-suitable
models from Computer-Aided Design models, which is in general a non-trivial and
time-consuming task. In this article, we present a novel spline construction,
that enables model reconstruction as well as simulation of high-order PDEs on
the reconstructed models. The proposed almost- are biquadratic splines on
fully unstructured quadrilateral meshes (without restrictions on placements or
number of extraordinary vertices). They are smooth almost everywhere,
that is, at all vertices and across most edges, and in addition almost (i.e.
approximately) smooth across all other edges. Thus, the splines form
-nonconforming analysis-suitable discretization spaces. This is the
lowest-degree unstructured spline construction that can be used to solve
fourth-order problems. The associated spline basis is non-singular and has
several B-spline-like properties (e.g., partition of unity, non-negativity,
local support), the almost- splines are described in an explicit
B\'ezier-extraction-based framework that can be easily implemented. Numerical
tests suggest that the basis is well-conditioned and exhibits optimal
approximation behavior
Flexible G1 Interpolation of Quad Meshes
International audienceTransforming an arbitrary mesh into a smooth G1 surface has been the subject of intensive research works. To get a visual pleasing shape without any imperfection even in the presence of extraordinary mesh vertices is still a challenging problem in particular when interpolation of the mesh vertices is required. We present a new local method, which produces visually smooth shapes while solving the interpolation problem. It consists of combining low degree biquartic BĂ©zier patches with minimum number of pieces per mesh face, assembled together with G1-continuity. All surface control points are given explicitly. The construction is local and free of zero-twists. We further show that within this economical class of surfaces it is however possible to derive a sufficient number of meaningful degrees of freedom so that standard optimization techniques result in high quality surfaces
Gauss images of hyperbolic cusps with convex polyhedral boundary
We prove that a 3--dimensional hyperbolic cusp with convex polyhedral
boundary is uniquely determined by its Gauss image. Furthermore, any spherical
metric on the torus with cone singularities of negative curvature and all
closed contractible geodesics of length greater than is the metric of
the Gauss image of some convex polyhedral cusp. This result is an analog of the
Rivin-Hodgson theorem characterizing compact convex hyperbolic polyhedra in
terms of their Gauss images.
The proof uses a variational method. Namely, a cusp with a given Gauss image
is identified with a critical point of a functional on the space of cusps with
cone-type singularities along a family of half-lines. The functional is shown
to be concave and to attain maximum at an interior point of its domain. As a
byproduct, we prove rigidity statements with respect to the Gauss image for
cusps with or without cone-type singularities.
In a special case, our theorem is equivalent to existence of a circle pattern
on the torus, with prescribed combinatorics and intersection angles. This is
the genus one case of a theorem by Thurston. In fact, our theorem extends
Thurston's theorem in the same way as Rivin-Hodgson's theorem extends Andreev's
theorem on compact convex polyhedra with non-obtuse dihedral angles.
The functional used in the proof is the sum of a volume term and curvature
term. We show that, in the situation of Thurston's theorem, it is the potential
for the combinatorial Ricci flow considered by Chow and Luo.
Our theorem represents the last special case of a general statement about
isometric immersions of compact surfaces.Comment: 55 pages, 17 figure
Manifold-based isogeometric analysis basis functions with prescribed sharp features
We introduce manifold-based basis functions for isogeometric analysis of
surfaces with arbitrary smoothness, prescribed continuous creases and
boundaries. The utility of the manifold-based surface construction techniques
in isogeometric analysis was demonstrated in Majeed and Cirak (CMAME, 2017).
The respective basis functions are derived by combining differential-geometric
manifold techniques with conformal parametrisations and the partition of unity
method. The connectivity of a given unstructured quadrilateral control mesh in
is used to define a set of overlapping charts. Each vertex with
its attached elements is assigned a corresponding conformally parametrised
planar chart domain in , so that a quadrilateral element is
present on four different charts. On the collection of unconnected chart
domains, the partition of unity method is used for approximation. The
transition functions required for navigating between the chart domains are
composed out of conformal maps. The necessary smooth partition of unity, or
blending, functions for the charts are assembled from tensor-product B-spline
pieces and require in contrast to earlier constructions no normalisation.
Creases are introduced across user tagged edges of the control mesh. Planar
chart domains that include creased edges or are adjacent to the domain boundary
require special local polynomial approximants. Three different types of chart
domain geometries are necessary to consider boundaries and arbitrary number and
arrangement of creases. The new chart domain geometries are chosen so that it
becomes trivial to establish local polynomial approximants that are always
continuous across tagged edges. The derived non-rational manifold-based
basis functions are particularly well suited for isogeometric analysis of
Kirchhoff-Love thin shells with kinks
Arbitrary topology meshes in geometric design and vector graphics
Meshes are a powerful means to represent objects and shapes both in 2D and 3D, but the techniques based on meshes can only be used in certain regular settings and restrict their usage. Meshes with an arbitrary topology have many interesting applications in geometric design and (vector) graphics, and can give designers more freedom in designing complex objects. In the first part of the thesis we look at how these meshes can be used in computer aided design to represent objects that consist of multiple regular meshes that are constructed together. Then we extend the B-spline surface technique from the regular setting to work on extraordinary regions in meshes so that multisided B-spline patches are created. In addition, we show how to render multisided objects efficiently, through using the GPU and tessellation. In the second part of the thesis we look at how the gradient mesh vector graphics primitives can be combined with procedural noise functions to create expressive but sparsely defined vector graphic images. We also look at how the gradient mesh can be extended to arbitrary topology variants. Here, we compare existing work with two new formulations of a polygonal gradient mesh. Finally we show how we can turn any image into a vector graphics image in an efficient manner. This vectorisation process automatically extracts important image features and constructs a mesh around it. This automatic pipeline is very efficient and even facilitates interactive image vectorisation
- …