Hierarchical bases of spline spaces with highest order smoothness over hierarchical T-subdivisions

Wu, Meng; Xu, Jinlan; Wang, Ruimin; Yang, Zhouwang

doi:10.1016/j.cagd.2012.03.024

Cited by 11 publications

(16 citation statements)

References 9 publications

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“…In this part, we present two numerical examples in isogeometric analysis. The splines we used to compare are biquadratic tensor-product B-splines defined on uniform tensor-product meshes and splines in [30]. Both examples show that to reach a given accuracy, few degrees of freedom are needed using spline bases in this paper than using the biquadratic tensor-product B-splines defined on uniform tensor-product meshes.…”

Section: Isogeometric Analysismentioning

confidence: 99%

“…It is noted that [29] gave a general dimension formula of S(m, n, m − 1, n − 1, T ) over a special T-mesh ((m, n)-subdivided T-mesh). Hierarchical bases of S(m, n, m − 1, n − 1, T ) over (m, n)-subdivided T-mesh were constructed in [30]. However, hierarchical bases in [30] did not have the properties of non-negativity and partition of unity.…”

Section: Introductionmentioning

confidence: 99%

“…Hierarchical bases of S(m, n, m − 1, n − 1, T ) over (m, n)-subdivided T-mesh were constructed in [30]. However, hierarchical bases in [30] did not have the properties of non-negativity and partition of unity. [34] gave a dimension formula of S(3, 3, 2, 2, T ) over a more general T-mesh than that in [29], but also a special hierarchical T-mesh.…”

Section: Introductionmentioning

confidence: 99%

“…As unit partition is an important property for geometry modeling, a new basis with the property of unit partition is given out by an elementary transformation of the previous basis functions. Compared with spline bases in [30], the new basis has good properties of nonnegativity, partition of unity, and smaller local support which may result in a more stable stiffness matrix in numerical solutions of PDE. Also, these properties will facilitate their applications in geometric modeling.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Bases of Biquadratic Polynomial Spline Spaces over Hierarchical T-Meshes

Deng¹,

Zeng²,

Wu³

et al. 2017

JCM

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Basis functions of biquadratic polynomial spline spaces over hierarchical T-meshes are constructed. The basis functions are all tensor-product B-splines, which are linearly independent, nonnegative and complete. To make basis functions more efficient for geometric modeling, we also give out a new basis with the property of unit partition. Two preliminary applications are given to demonstrate that the new basis is efficient. Mathematics subject classification: 65D07

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Section: Isogeometric Analysismentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Bases of Biquadratic Polynomial Spline Spaces over Hierarchical T-Meshes

Deng¹,

Zeng²,

Wu³

et al. 2017

JCM

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“…The corresponding application can be found in [46,71]. However, in this case, one must assume the reduced regularity of basis [11] or fulfil a certain constraint on admissible mesh configuration [74]. In [47], the adaptive procedure in based on the recovery-based error estimator, in which discontinuous enrichment functions are added to the IgA approximation.…”

Section: Introductionmentioning

confidence: 99%

Functional approach to the error control in adaptive IgA schemes for elliptic boundary value problems

Matculevich

2018

Journal of Computational and Applied Mathematics

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This work presents a numerical study of functional type a posteriori error estimates for IgA approximation schemes in the context of elliptic boundary-value problems. Along with the detailed discussion of the most crucial properties of such estimates, we present the algorithm of a reliable solution approximation together with the scheme of an efficient a posteriori error bound generation. In this approach, we take advantage of B-(THB-) spline's high smoothness for the auxiliary vector function reconstruction, which, at the same time, allows to use much coarser meshes and decrease the number of unknowns substantially. The most representative numerical results, obtained during a systematic testing of error estimates, are presented in the second part of the paper. The efficiency of the obtained error bounds is analysed from both the error estimation (indication) and the computational expenses points of view. Several examples illustrate that functional error estimates (alternatively referred to as the majorants and minorants of deviation from an exact solution) perform a much sharper error control than, for instance, residual-based error estimates. Simultaneously, assembling and solving routines for an auxiliary variables reconstruction, which generate the majorant (or minorant) of an error, can be executed several times faster than the routines for a primal unknown.

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Strongly stable bases for adaptively refined multilevel spline spaces

2013

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The problem of constructing a normalized hierarchical basis for adaptively refined spline spaces is addressed. Multilevel representations are defined in terms of a hierarchy of basis functions, reflecting different levels of refinement. When the hierarchical model is constructed by considering an underlying sequence of bases Γℓ=0,…,N−1 with properties analogous to classical tensor-product B-splines, we can define a set of locally supported basis functions that form a partition of unity and possess the property of coefficient preservation, i.e., they preserve the coefficients of functions represented with respect to one of the bases Γℓ. Our construction relies on a certain truncation procedure, which eliminates the contributions of functions from finer levels in the hierarchy to coarser level ones. Consequently, the support of the original basis functions defined on coarse grids is possibly reduced according to finer levels in the hierarchy. This truncation mechanism not only decreases the overlapping of basis supports, but it also guarantees strong stability of the construction. In addition to presenting the theory for the general framework, we apply it to hierarchically refined tensor-product spline spaces, under certain reasonable assumptions on the given knot configuration

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Hierarchical bases of spline spaces with highest order smoothness over hierarchical T-subdivisions

Cited by 11 publications

References 9 publications

Bases of Biquadratic Polynomial Spline Spaces over Hierarchical T-Meshes

Bases of Biquadratic Polynomial Spline Spaces over Hierarchical T-Meshes

Functional approach to the error control in adaptive IgA schemes for elliptic boundary value problems

Strongly stable bases for adaptively refined multilevel spline spaces

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