Multivariate Splines and Their Applications

Lai, Ming-Jun

doi:10.1007/978-0-387-30440-3_345

Cited by 5 publications

(7 citation statements)

References 40 publications

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“…Consider a trivariate quadratic spline space on Δ 1 , Δ 2 , which is a piecewise quadratic polynomial with C 1 continuous at Point P 14 . By the B-net method [16,17], this continuity condition is equivalent to the following equation determined by the Bézier coefficients.…”

Section: The 13-node Pyramid Spline Elementmentioning

confidence: 99%

A 3D pyramid spline element

Chen

Wanji

2011

Acta Mech Sin

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In this paper, a 13-node pyramid spline element is derived by using the tetrahedron volume coordinates and the B-net method, which achieves the second order completeness in Cartesian coordinates. Some appropriate examples were employed to evaluate the performance of the proposed element. The numerical results show that the spline element has much better performance compared with the isoparametric serendipity element Q20 and its degenerate pyramid element P13 especially when mesh is distorted, and it is comparable to the Lagrange element Q27. It has been demonstrated that the spline finite element method is an efficient tool for developing high accuracy elements.

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Section: The 13-node Pyramid Spline Elementmentioning

confidence: 99%

A 3D pyramid spline element

Chen

Wanji

2011

Acta Mech Sin

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“…In mathematics, splines are widely used in finite element methods, as they satisfy certain continuity criteria of piecewise polynomials [18,21]. Li [21] proposed a quadrangle spline element based on triangle area coordinates and the B-net method. These 2D splines are highly accurate and insensitive to mesh distortion [30].…”

Section: Triangular Prism Quadric Interpolationmentioning

confidence: 99%

“…These 2D splines are highly accurate and insensitive to mesh distortion [30]. To upgrade a triangular prism element from 2D elements, internal smoothing can be achieved by constructing a triangular prism spline element by the B-net method [21].…”

Section: Triangular Prism Quadric Interpolationmentioning

confidence: 99%

“…The B-net method is used to create a multivariate spline based on triangles or tetrahedrons. It originated from work on Bernstein polynomials for interpolating barycentric coordinates for triangular areas or the tetrahedral volumes [21].…”

Section: Interpolation Principlementioning

confidence: 99%

“…The triangular prism quadric interpolation is designed by modifying the existing B-net method for tetrahedron domains [7,18,19]. In our research, the triangular prism element is used to express the 3D geometric shapes of geological bodies and structural mechanics, building on these past studies [20,21]. The related tetrahedron and hexahedron interpolation methods have been extensively researched [7,8,12].…”

Section: Introductionmentioning

confidence: 99%

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A Triangular Prism Spatial Interpolation Method for Mapping Geological Property Fields

Cui

et al. 2017

IJGI

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The spatial interpolation of property fields in 3D, such as the temperature, salinity, and organic content of ocean water, is an active area of research in the applied geosciences. Conventional interpolation methods have not adequately addressed anisotropy in these data. Thus, in our research we considered two interpolation methods based on a triangular prism volume element, as a triangular prism structure best represents directivity, to express the anisotropy inherent in geological property fields. A linear triangular prism interpolation is proposed for layered stratum that achieves a complete C 0 continuity based on the volume coordinates of the triangular prism. A triangular prism quadric interpolation (a unit function of a triangular prism spline with 15 nodes) is designed for a smooth transition between adjacent triangular prisms with approximately C 1 continuity, expressing the continuity of the entire model. We designed a specific model which accounts for the different spatial correlations in three dimensions. We evaluated the accuracy of our proposed linear and triangular prism quadric interpolation methods with traditional inverse distance weighting (IDW) and kriging interpolation approaches in comparative experiments. The results show that, in 3D geological modeling, the linear and quadric triangular prism interpolations more accurately represent the changes in the property values of the layered strata than the IDW and kriging interpolation methods. Furthermore, the triangular prism quadric interpolation algorithm with 15 nodes outperforms the other methods. This study of triangular prism interpolation algorithms has implications for the expression of data fields with 3D properties. Moreover, our novel approach will contribute to spatial attribute prediction and representation and is applicable to all 3D geographic information; for example, in studies of atmospheric circulation, ocean circulation, water temperature, salinity, and three-dimensional pollutant diffusion.

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Finite element method by using quartic B-splines

Zhang

2011

Numer. Methods Partial Differential Eq.

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In this article, we discuss a kind of finite element method by using quartic B-splines to solve Dirichlet problem for elliptic equations. Bivariate spline proper subspace of S 2,3 4 ( (2) mn ) satisfying homogeneous boundary conditions on Type-2 triangulations and quadratic B-spline interpolating boundary functions are primarily constructed. Linear and nonlinear elliptic equations are solved by Galerkin quartic B-spline finite element method. Numerical examples are provided to illustrate the proposed method is flexible.

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Multivariate Splines and Their Applications

Cited by 5 publications

References 40 publications

A 3D pyramid spline element

A 3D pyramid spline element

A Triangular Prism Spatial Interpolation Method for Mapping Geological Property Fields

Finite element method by using quartic B-splines

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