8,992 research outputs found
Adaptive isogeometric analysis with hierarchical box splines
Isogeometric analysis is a recently developed framework based on finite
element analysis, where the simple building blocks in geometry and solution
space are replaced by more complex and geometrically-oriented compounds. Box
splines are an established tool to model complex geometry, and form an
intermediate approach between classical tensor-product B-splines and splines
over triangulations. Local refinement can be achieved by considering
hierarchically nested sequences of box spline spaces. Since box splines do not
offer special elements to impose boundary conditions for the numerical solution
of partial differential equations (PDEs), we discuss a weak treatment of such
boundary conditions. Along the domain boundary, an appropriate domain strip is
introduced to enforce the boundary conditions in a weak sense. The thickness of
the strip is adaptively defined in order to avoid unnecessary computations.
Numerical examples show the optimal convergence rate of box splines and their
hierarchical variants for the solution of PDEs
Characterization of bivariate hierarchical quartic box splines on a three-directional grid
International audienceWe consider the adaptive refinement of bivariate quartic C 2-smooth box spline spaces on the three-directional (type-I) grid G. The polynomial segments of these box splines belong to a certain subspace of the space of quar-tic polynomials, which will be called the space of special quartics. Given a bounded domain Ω ⊂ R 2 and finite sequence (G ℓ) ℓ=0,...,N of dyadically refined grids, we obtain a hierarchical grid by selecting mutually disjoint cells from all levels such that their union covers the entire domain. Using a suitable selection procedure allows to define a basis spanning the hierarchical box spline space. The paper derives a characterization of this space. Under certain mild assumptions on the hierarchical grid, the hierarchical spline space is shown to contain all C 2-smooth functions whose restrictions to the cells of the hierarchical grid are special quartic polynomials. Thus, in this case we can give an affirmative answer to the completeness questions for the hierarchical box spline basis
Completeness characterization of Type-I box splines
We present a completeness characterization of box splines on three-directional triangulations, also called Type-I box spline spaces, based on edge-contact smoothness properties. For any given Type-I box spline, of specific maximum degree and order of global smoothness, our results allow to identify the local linear subspace of polynomials spanned by the box spline translates. We use the global super-smoothness properties of box splines as well as the additional super-smoothness conditions at edges to characterize the spline space spanned by the box spline translates. Subsequently, we prove the completeness of this space space with respect to the local polynomial space induced by the box spline translates. The completeness property allows the construction of hierarchical spaces spanned by the translates of box splines for any polynomial degree on multilevel Type-I grids. We provide a basis for these hierarchical box spline spaces under explicit geometric conditions of the domain
Sum-factorization techniques in Isogeometric Analysis
The fast assembling of stiffness and mass matrices is a key issue in
isogeometric analysis, particularly if the spline degree is increased. We
present two algorithms based on the idea of sum factorization, one for matrix
assembling and one for matrix-free methods, and study the behavior of their
computational complexity in terms of the spline order . Opposed to the
standard approach, these algorithms do not apply the idea element-wise, but
globally or on macro-elements. If this approach is applied to Gauss quadrature,
the computational complexity grows as instead of as
previously achieved.Comment: 34 pages, 8 figure
Linear dependence of bivariate Minimal Support and Locally Refined B-splines over LR-meshes
The focus on locally refined spline spaces has grown rapidly in recent years
due to the need in Isogeoemtric analysis (IgA) of spline spaces with local
adaptivity: a property not offered by the strict regular structure of tensor
product B-spline spaces. However, this flexibility sometimes results in
collections of B-splines spanning the space that are not linearly independent.
In this paper we address the minimal number of B-splines that can form a linear
dependence relation for Minimal Support B-splines (MS B-splines) and for
Locally Refinable B-splines (LR B-splines) on LR-meshes. We show that the
minimal number is six for MS B-splines, and eight for LR B-splines. The risk of
linear dependency is consequently significantly higher for MS B-splines than
for LR B-splines. Further results are established to help detecting collections
of B-splines that are linearly independent
A sparse-grid isogeometric solver
Isogeometric Analysis (IGA) typically adopts tensor-product splines and NURBS
as a basis for the approximation of the solution of PDEs. In this work, we
investigate to which extent IGA solvers can benefit from the so-called
sparse-grids construction in its combination technique form, which was first
introduced in the early 90s in the context of the approximation of
high-dimensional PDEs. The tests that we report show that, in accordance to the
literature, a sparse-grid construction can indeed be useful if the solution of
the PDE at hand is sufficiently smooth. Sparse grids can also be useful in the
case of non-smooth solutions when some a-priori knowledge on the location of
the singularities of the solution can be exploited to devise suitable
non-equispaced meshes. Finally, we remark that sparse grids can be seen as a
simple way to parallelize pre-existing serial IGA solvers in a straightforward
fashion, which can be beneficial in many practical situations.Comment: updated version after revie
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