345 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
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
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
Unstructured spline spaces for isogeometric analysis based on spline manifolds
Based on spline manifolds we introduce and study a mathematical framework for
analysis-suitable unstructured B-spline spaces. In this setting the parameter
domain has a manifold structure, which allows for the definition of function
spaces that have a tensor-product structure locally, but not globally. This
includes configurations such as B-splines over multi-patch domains with
extraordinary points, analysis-suitable unstructured T-splines, or more general
constructions. Within this framework, we generalize the concept of
dual-compatible B-splines, which was originally developed for structured
T-splines. This allows us to prove the key properties that are needed for
isogeometric analysis, such as linear independence and optimal approximation
properties for -refined meshes
Volumetric Untrimming: Precise decomposition of trimmed trivariates into tensor products
3D objects, modeled using Computer Aided Geometric Design tools, are
traditionally represented using a boundary representation (B-rep), and
typically use spline functions to parameterize these boundary surfaces.
However, recent development in physical analysis, in isogeometric analysis
(IGA) in specific, necessitates a volumetric parametrization of the interior of
the object. IGA is performed directly by integrating over the spline spaces of
the volumetric spline representation of the object. Typically, tensor-product
B-spline trivariates are used to parameterize the volumetric domain. A general
3D object, that can be modeled in contemporary B-rep CAD tools, is typically
represented using trimmed B-spline surfaces. In order to capture the generality
of the contemporary B-rep modeling space, while supporting IGA needs, Massarwi
and Elber (2016) proposed the use of trimmed trivariates volumetric elements.
However, the use of trimmed geometry makes the integration process more
difficult since integration over trimmed B-spline basis functions is a highly
challenging task. In this work, we propose an algorithm that precisely
decomposes a trimmed B-spline trivariate into a set of (singular only on the
boundary) tensor-product B-spline trivariates, that can be utilized to simplify
the integration process in IGA. The trimmed B-spline trivariate is first
subdivided into a set of trimmed B\'ezier trivariates, at all its internal
knots. Then, each trimmed B\'ezier trivariate, is decomposed into a set of
mutually exclusive tensor-product B-spline trivariates, that precisely cover
the entire trimmed domain. This process, denoted untrimming, can be performed
in either the Euclidean space or the parametric space of the trivariate. We
present examples on complex trimmed trivariates' based geometry, and we
demonstrate the effectiveness of the method by applying IGA over the
(untrimmed) results.Comment: 18 pages, 32 figures. Contribution accepted in International
Conference on Geometric Modeling and Processing (GMP 2019
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