137 research outputs found
Generalized Finite Element Systems for smooth differential forms and Stokes problem
We provide both a general framework for discretizing de Rham sequences of
differential forms of high regularity, and some examples of finite element
spaces that fit in the framework. The general framework is an extension of the
previously introduced notion of Finite Element Systems, and the examples
include conforming mixed finite elements for Stokes' equation. In dimension 2
we detail four low order finite element complexes and one infinite family of
highorder finite element complexes. In dimension 3 we define one low order
complex, which may be branched into Whitney forms at a chosen index. Stokes
pairs with continuous or discontinuous pressure are provided in arbitrary
dimension. The finite element spaces all consist of composite polynomials. The
framework guarantees some nice properties of the spaces, in particular the
existence of commuting interpolators. It also shows that some of the examples
are minimal spaces.Comment: v1: 27 pages. v2: 34 pages. Numerous details added. v3: 44 pages. 8
figures and several comments adde
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Multivariate Splines and Algebraic Geometry
Multivariate splines are effective tools in numerical analysis and approximation theory. Despite an extensive literature on the subject, there remain open questions in finding their dimension, constructing local bases, and determining their approximation power. Much of what is currently known was developed by numerical analysts, using classical methods, in particular the so-called Bernstein-B´ezier techniques. Due to their many interesting structural properties, splines have become of keen interest to researchers in commutative and homological algebra and algebraic geometry. Unfortunately, these communities have not collaborated much. The purpose of the half-size workshop is to intensify the interaction between the different groups by bringing them together. This could lead to essential breakthroughs on several of the above problems
B-spline-like bases for cubics on the Powell-Sabin 12-split
For spaces of constant, linear, and quadratic splines of maximal smoothness
on the Powell-Sabin 12-split of a triangle, the so-called S-bases were recently
introduced. These are simplex spline bases with B-spline-like properties on the
12-split of a single triangle, which are tied together across triangles in a
B\'ezier-like manner.
In this paper we give a formal definition of an S-basis in terms of certain
basic properties. We proceed to investigate the existence of S-bases for the
aforementioned spaces and additionally the cubic case, resulting in an
exhaustive list. From their nature as simplex splines, we derive simple
differentiation and recurrence formulas to other S-bases. We establish a
Marsden identity that gives rise to various quasi-interpolants and domain
points forming an intuitive control net, in terms of which conditions for
-, -, and -smoothness are derived
Quadratic Spline Quasi-interpolants on Powell-Sabin Partitions
2004-16International audienceIn this paper we address the problem of constructing quasi-interpolants in the space of quadratic Powell-Sabin splines on nonuniform triangulations. Quasi-interpolants of optimal approximation order are proposed and numerical tests are presented
Splines for damage and fracture in solids
This thesis addresses different aspects of numerical fracture mechanics and spline technology for analysis.
An energy-based arc-length control for physically non-linear problems is proposed. It switches between an internal energy-based and a dissipation-based arc-length method. The arc-length control allows to trace an equilibrium path with multiple snap-through and/or snap-back phenomena and only requires two parameters.
Phase field models for brittle and cohesive fracture are numerically assessed. The impact of different parameters and boundary conditions on the phase field model for brittle fracture is investigated. It is demonstrated that Γ-convergence is not attained numerically for the phase field model for brittle fracture and that the phase field model for cohesive fracture does not pass a two-dimensional patch test when using an unstructured mesh.
The properties of the Bézier extraction operator for T-splines are exploited for the determination of linear dependencies, partition of unity properties, nesting behaviour and local refinement. Unstructured T-spline meshes with extraordinary points are modified such that the blending functions fulfil the partition of unity property and possess a higher continuity.
Bézier extraction for Powell-Sabin B-splines is introduced. Different spline technologies are compared when solving Kirchhoff-Love plate theory on a disc with simply supported and clamped boundary conditions.
Powell-Sabin B-splines are utilised for smeared and discrete approaches to fracture. Due to the higher continuity of Powell-Sabin B-splines, the implicit fourth order gradient damage model for quasi-brittle materials can be solved and stresses can be computed directly at the crack tip when considering the cohesive zone method
BPX-Preconditioning for isogeometric analysis
We consider elliptic PDEs (partial differential equations) in the framework of isogeometric analysis, i.e., we treat the physical domain by means of a B-spline or Nurbs mapping which we assume to be regular. The numerical solution of the PDE is computed by means of tensor product B-splines mapped onto the physical domain. We construct additive multilevel preconditioners and show that they are asymptotically optimal, i.e., the spectral condition number of the resulting preconditioned stiffness matrix is independent of . Together with a nested iteration scheme, this enables an iterative solution scheme of optimal linear complexity. The theoretical results are substantiated by numerical examples in two and three space dimensions
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
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