13 research outputs found
Construction of analysis-suitable planar multi-patch parameterizations
Isogeometric analysis allows to define shape functions of global
continuity (or of higher continuity) over multi-patch geometries. The
construction of such -smooth isogeometric functions is a non-trivial
task and requires particular multi-patch parameterizations, so-called
analysis-suitable (in short, AS-) parameterizations, to ensure
that the resulting isogeometric spaces possess optimal approximation
properties, cf. [7]. In this work, we show through examples that it is possible
to construct AS- multi-patch parameterizations of planar domains, given
their boundary. More precisely, given a generic multi-patch geometry, we
generate an AS- multi-patch parameterization possessing the same
boundary, the same vertices and the same first derivatives at the vertices, and
which is as close as possible to this initial geometry. Our algorithm is based
on a quadratic optimization problem with linear side constraints. Numerical
tests also confirm that isogeometric spaces over AS- multi-patch
parameterized domains converge optimally under mesh refinement, while for
generic parameterizations the convergence order is severely reduced
Exact conversion from BĂ©zier tetrahedra to BĂ©zier hexahedra
International audienceModeling and computing of trivariate parametric volumes is an important research topic in the field of three-dimensional isogeo-metric analysis. In this paper, we propose two kinds of exact conversion approaches from BĂ©zier tetrahedra to BĂ©zier hexahedra with the same degree by reparametrization technique. In the first method, a BĂ©zier tetrahedron is converted into a degenerate BĂ©zier hexahedron, and in the second approach, a non-degenerate BĂ©zier tetrahedron is converted into four non-degenerate BĂ©zier hexahedra. For the proposed methods, explicit formulas are given to compute the control points of the resulting tensor-product BĂ©zier hexahedra. Furthermore, in the second method, we prove that tetrahedral spline solids with C k-continuity can be converted into a set of tensor-product BĂ©zier volumes with G k-continuity. The proposed methods can be used for the volumetric data exchange problems between different trivariate spline representations in CAD/CAE. Several experimental results are presented to show the effectiveness of the proposed methods
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
-smooth isogeometric spline functions of general degree over planar mixed meshes: The case of two quadratic mesh elements
Splines over triangulations and splines over quadrangulations (tensor product
splines) are two common ways to extend bivariate polynomials to splines.
However, combination of both approaches leads to splines defined over mixed
triangle and quadrilateral meshes using the isogeometric approach. Mixed meshes
are especially useful for representing complicated geometries obtained e.g.
from trimming. As (bi)-linearly parameterized mesh elements are not flexible
enough to cover smooth domains, we focus in this work on the case of planar
mixed meshes parameterized by (bi)-quadratic geometry mappings. In particular
we study in detail the space of -smooth isogeometric spline functions of
general polynomial degree over two such mixed mesh elements. We present the
theoretical framework to analyze the smoothness conditions over the common
interface for all possible configurations of mesh elements. This comprises the
investigation of the dimension as well as the construction of a basis of the
corresponding -smooth isogeometric spline space over the domain described
by two elements. Several examples of interest are presented in detail
Conversion of B-rep CAD models into globally G<sup>1</sup> triangular splines
Existing techniques that convert B-rep (boundary representation) patches into Clough-Tocher splines guarantee watertight, that is C0, conversion results across B-rep edges. In contrast, our approach ensures global tangent-plane, that is G1, continuity of the converted B-rep CAD models. We achieve this by careful boundary curve and normal vector management, and by converting the input models into Shirman-SĂ©quin macro-elements near their (trimmed) B-rep edges. We propose several different variants and compare them with respect to their locality, visual quality, and difference with the input B-rep CAD model. Although the same global G1 continuity can also be achieved by conversion techniques based on subdivision surfaces, our approach uses triangular splines and thus enjoys full compatibility with CAD
Geometrically smooth spline bases for data fitting and simulation
International audienceGiven a topological complex with glueing data along edges shared by adjacent faces, we study the associated space of geometrically smooth spline functions that satisfy differentiability properties across shared edges. We present new and efficient constructions of basis functions of the space of -spline functions on quadrangular meshes, which are tensor product b-spline functions on each quadrangle and with b-spline transition maps across the shared edges. This new strategy for constructing basis functions is based on a local analysis of the edge functions, and does not depend on the global topology of . We show that the separability of the space of splines across an edge allows to determine the dimension and a basis of the space of splines on .This leads to explicit and effective constructions of basis functions attached to the vertices, edges and faces of .This basis construction has important applications in geometric modeling and simulation. We illustrate it by the fitting of point clouds by splines on quadrangular meshes of complex topology and in Isogeometric Analysis methods for the solution of diffusion equations. The ingredients are detailed and experimentation results showing the behavior of the method are presented