20 research outputs found
A locally based construction of analysis-suitable multi-patch spline surfaces
Analysis-suitable (AS-) multi-patch spline surfaces [4] are
particular -smooth multi-patch spline surfaces, which are needed to ensure
the construction of -smooth multi-patch spline spaces with optimal
polynomial reproduction properties [16]. We present a novel local approach for
the design of AS- multi-patch spline surfaces, which is based on the use
of Lagrange multipliers. The presented method is simple and generates an
AS- multi-patch spline surface by approximating a given -smooth but
non-AS- multi-patch surface. Several numerical examples demonstrate the
potential of the proposed technique for the construction of AS-
multi-patch spline surfaces and show that these surfaces are especially suited
for applications in isogeometric analysis by solving the biharmonic problem, a
particular fourth order partial differential equation, over them
A comparison of smooth basis constructions for isogeometric analysis
In order to perform isogeometric analysis with increased smoothness on
complex domains, trimming, variational coupling or unstructured spline methods
can be used. The latter two classes of methods require a multi-patch
segmentation of the domain, and provide continuous bases along patch
interfaces. In the context of shell modeling, variational methods are widely
used, whereas the application of unstructured spline methods on shell problems
is rather scarce. In this paper, we therefore provide a qualitative and a
quantitative comparison of a selection of unstructured spline constructions, in
particular the D-Patch, Almost-, Analysis-Suitable and the
Approximate constructions. Using this comparison, we aim to provide
insight into the selection of methods for practical problems, as well as
directions for future research. In the qualitative comparison, the properties
of each method are evaluated and compared. In the quantitative comparison, a
selection of numerical examples is used to highlight different advantages and
disadvantages of each method. In the latter, comparison with weak coupling
methods such as Nitsche's method or penalty methods is made as well. In brief,
it is concluded that the Approximate and Analysis-Suitable converge
optimally in the analysis of a bi-harmonic problem, without the need of special
refinement procedures. Furthermore, these methods provide accurate stress
fields. On the other hand, the Almost- and D-Patch provide relatively easy
construction on complex geometries. The Almost- method does not have
limitations on the valence of boundary vertices, unlike the D-Patch, but is
only applicable to biquadratic local bases. Following from these conclusions,
future research directions are proposed, for example towards making the
Approximate and Analysis-Suitable applicable to more complex
geometries
Adaptive isogeometric methods with (truncated) hierarchical splines on planar multi-patch domains
Isogeometric analysis is a powerful paradigm which exploits the high
smoothness of splines for the numerical solution of high order partial
differential equations. However, the tensor-product structure of standard
multivariate B-spline models is not well suited for the representation of
complex geometries, and to maintain high continuity on general domains special
constructions on multi-patch geometries must be used. In this paper we focus on
adaptive isogeometric methods with hierarchical splines, and extend the
construction of isogeometric spline spaces on multi-patch planar domains
to the hierarchical setting. We introduce a new abstract framework for the
definition of hierarchical splines, which replaces the hypothesis of local
linear independence for the basis of each level by a weaker assumption. We also
develop a refinement algorithm that guarantees that the assumption is fulfilled
by splines on certain suitably graded hierarchical multi-patch mesh
configurations, and prove that it has linear complexity. The performance of the
adaptive method is tested by solving the Poisson and the biharmonic problems
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
CAD Aspects on Isogeometric Analysis and Hybrid Domains
This thesis is the result of a Ph.D. program in Alto Apprendistato carried
out at the Dipartimento di Informatica - Scienza e Ingegneria (DISI) of the
University of Bologna and at the company devDept Software.
With regard to the professional side of my Individual Training Project, I
developed technical and scientific skills in 3D geometry of curves and surfaces, CAD, and Finite Element Analysis (FEA). Regarding the academic side, I investigated CAD aspects in the field of Isogeometric Analysis (IGA) on both single and hybrid multipatch physical domains.
Simulations are performed in classical FEA systems, which require the conversion of designs, made by CAD systems, into finite element meshes. IGA
is a new approach that aims to unify the worlds of CAD and FEA by using
the same geometry for analysis as what is used for modeling. That is, the
same set of basis functions are adopted both to describe the computational
geometry in the CAD tool, and to span the solution space for FEA.
The traditional FEA pipeline works on meshes and the most advanced IGA
systems work on NURBS or T-spline geometries. Hybrid geometric models
(i.e., models in which mesh and NURBS entities coexist), are an emergent
way to represent a solid object, but in most CAD systems mesh and NURBS
geometries cannot interact with each other, and conversions to a common
representation are often needed.
In this thesis, we investigate how IGA can be applied on 2D and 3D hybrid
models made by both mesh and NURBS entities without requiring laborious
and time consuming conversion processes