203 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
A subdivision-based implementation of non-uniform local refinement with THB-splines
Paper accepted for 15th IMA International Conference on Mathematics on Surfaces, 2017. Abstract: Local refinement of spline basis functions is an important process for spline approximation and local feature modelling in computer aided design (CAD). This paper develops an efficient local refinement method for non-uniform and general degree THB-splines(Truncated hierarchical B-splines). A non-uniform subdivision algorithm is improved to efficiently subdivide a single non-uniform B-spline basis function. The subdivision scheme is then applied to locally hierarchically refine non-uniform B-spline basis functions. The refined basis functions are non-uniform and satisfy the properties of linear independence, partition of unity and are locally supported. The refined basis functions are suitable for spline approximation and numerical analysis. The implementation makes it possible for hierarchical approximation to use the same non-uniform B-spline basis functions as existing modelling tools have used. The improved subdivision algorithm is faster than classic knot insertion. The non-uniform THB-spline approximation is shown to be more accurate than uniform low degree hierarchical local refinement when applied to two classical approximation problems
Suitably graded THB-spline refinement and coarsening: Towards an adaptive isogeometric analysis of additive manufacturing processes
In the present work we introduce a complete set of algorithms to efficiently
perform adaptive refinement and coarsening by exploiting truncated hierarchical
B-splines (THB-splines) defined on suitably graded isogeometric meshes, that
are called admissible mesh configurations. We apply the proposed algorithms to
two-dimensional linear heat transfer problems with localized moving heat
source, as simplified models for additive manufacturing applications. We first
verify the accuracy of the admissible adaptive scheme with respect to an
overkilled solution, for then comparing our results with similar schemes which
consider different refinement and coarsening algorithms, with or without taking
into account grading parameters. This study shows that the THB-spline
admissible solution delivers an optimal discretization for what concerns not
only the accuracy of the approximation, but also the (reduced) number of
degrees of freedom per time step. In the last example we investigate the
capability of the algorithms to approximate the thermal history of the problem
for a more complicated source path. The comparison with uniform and
non-admissible hierarchical meshes demonstrates that also in this case our
adaptive scheme returns the desired accuracy while strongly improving the
computational efficiency.Comment: 20 pages, 12 figure
Bivariate hierarchical Hermite spline quasi--interpolation
Spline quasi-interpolation (QI) is a general and powerful approach for the
construction of low cost and accurate approximations of a given function. In
order to provide an efficient adaptive approximation scheme in the bivariate
setting, we consider quasi-interpolation in hierarchical spline spaces. In
particular, we study and experiment the features of the hierarchical extension
of the tensor-product formulation of the Hermite BS quasi-interpolation scheme.
The convergence properties of this hierarchical operator, suitably defined in
terms of truncated hierarchical B-spline bases, are analyzed. A selection of
numerical examples is presented to compare the performances of the hierarchical
and tensor-product versions of the scheme
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
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