Volume 13, Issue 1
Spline R-Function and Applications in FEM

Numer. Math. Theor. Meth. Appl., 13 (2020), pp. 150-175.

Published online: 2019-12

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• Abstract

R-function is a widely used tool when considering objects obtained through the Boolean operations start from simple base primitives. However, there is square root operation in the representation. Considering that the use of splines will facilitate the calculations within the CAD system, in this paper, we propose a system of R-functions represented in spline form called Spline R-function (SR). After transforming the function ranges of two base primitives to a new coordinate system, a series of sign constraints following a specific Boolean operation are derived and the spline R-function can be formulated as a piecewise function. Representation of SR in both Bézier form and B-spline form have been given. Among which the Bézier ordinates are determined with the help of the B-net method through setting up a series of relations according to the sign constraints and properties of R-functions. The construction processes for both Boolean intersection and union operations with different smoothness are discussed in detail. Numerical experiments are conducted to show the potential of the proposed spline R-function.

68U07, 65D18, 65D30

dengjs@ustc.edu.cn (Jiansong Deng)

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@Article{NMTMA-13-150, author = {Yang , TianhuiQarariyah , Ammar and Deng , Jiansong}, title = {Spline R-Function and Applications in FEM}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2019}, volume = {13}, number = {1}, pages = {150--175}, abstract = {

R-function is a widely used tool when considering objects obtained through the Boolean operations start from simple base primitives. However, there is square root operation in the representation. Considering that the use of splines will facilitate the calculations within the CAD system, in this paper, we propose a system of R-functions represented in spline form called Spline R-function (SR). After transforming the function ranges of two base primitives to a new coordinate system, a series of sign constraints following a specific Boolean operation are derived and the spline R-function can be formulated as a piecewise function. Representation of SR in both Bézier form and B-spline form have been given. Among which the Bézier ordinates are determined with the help of the B-net method through setting up a series of relations according to the sign constraints and properties of R-functions. The construction processes for both Boolean intersection and union operations with different smoothness are discussed in detail. Numerical experiments are conducted to show the potential of the proposed spline R-function.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2018-0118}, url = {http://global-sci.org/intro/article_detail/nmtma/13435.html} }
TY - JOUR T1 - Spline R-Function and Applications in FEM AU - Yang , Tianhui AU - Qarariyah , Ammar AU - Deng , Jiansong JO - Numerical Mathematics: Theory, Methods and Applications VL - 1 SP - 150 EP - 175 PY - 2019 DA - 2019/12 SN - 13 DO - http://doi.org/10.4208/nmtma.OA-2018-0118 UR - https://global-sci.org/intro/article_detail/nmtma/13435.html KW - R-functions, CSG, Boolean operations, B-splines, B-net method, WEB method. AB -

R-function is a widely used tool when considering objects obtained through the Boolean operations start from simple base primitives. However, there is square root operation in the representation. Considering that the use of splines will facilitate the calculations within the CAD system, in this paper, we propose a system of R-functions represented in spline form called Spline R-function (SR). After transforming the function ranges of two base primitives to a new coordinate system, a series of sign constraints following a specific Boolean operation are derived and the spline R-function can be formulated as a piecewise function. Representation of SR in both Bézier form and B-spline form have been given. Among which the Bézier ordinates are determined with the help of the B-net method through setting up a series of relations according to the sign constraints and properties of R-functions. The construction processes for both Boolean intersection and union operations with different smoothness are discussed in detail. Numerical experiments are conducted to show the potential of the proposed spline R-function.

Tianhui Yang, Ammar Qarariyah & Jiansong Deng. (2019). Spline R-Function and Applications in FEM. Numerical Mathematics: Theory, Methods and Applications. 13 (1). 150-175. doi:10.4208/nmtma.OA-2018-0118
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