Refinement for a Hybrid Boundary Representation and its Hybrid Volume Completion
The SMAI journal of computational mathematics, Volume S5 (2019) , pp. 3-25.

With the increasing need for volumetric B-spline representations and the lack of methodologies for creating semi-structured volumetric B-spline representations from B-spline Boundary Representations (B-Rep), hybrid approaches combining semi-structured volumetric B-splines and unstructured Bézier tetrahedra have been introduced, including one that transforms a trimmed B-spline B-Rep first to an untrimmed Hybrid B-Rep (HB-Rep) and then to a Hybrid Volume Representation (HV-Rep). Generally, the effect of h-refinement has not been considered over B-spline hybrid representations. Standard refinement approches to tensor product B-splines and subdivision of Bézier triangles and tetrahedra must be adapted to this representation. In this paper, we analyze possible types of h-refinement of the HV-Rep. The revised and trim basis functions for HB- and HV-rep depend on a partition of knot intervals. Therefore, a naive h-refinement can change basis functions in unexpected ways. This paper analyzes the the effect of h-refinement in reducing error as well. Different h-refinement strategies are discussed. We demonstrate their differences and compare their respective consequential trade-offs. Recommendations are also made for different use cases.

Published online: 2020-01-29
DOI: https://doi.org/10.5802/smai-jcm.49
Classification: 65D17
Keywords: h-refinement, Trimmed model, Volume completion
@article{SMAI-JCM_2019__S5__3_0,
     author = {Yang Song and Elaine Cohen},
     title = {Refinement for a Hybrid Boundary Representation and its Hybrid Volume Completion},
     journal = {The SMAI journal of computational mathematics},
     publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
     volume = {S5},
     year = {2019},
     pages = {3-25},
     doi = {10.5802/smai-jcm.49},
     language = {en},
     url = {smai-jcm.centre-mersenne.org/item/SMAI-JCM_2019__S5__3_0/}
}
Song, Yang; Cohen, Elaine. Refinement for a Hybrid Boundary Representation and its Hybrid Volume Completion. The SMAI journal of computational mathematics, Volume S5 (2019) , pp. 3-25. doi : 10.5802/smai-jcm.49. https://smai-jcm.centre-mersenne.org/item/SMAI-JCM_2019__S5__3_0/

[1] H. Al Akhras; T. Elguedj; A. Gravouil; M. Rochette Isogeometric analysis-suitable trivariate NURBS models from standard B-Rep models, Computer Methods in Applied Mechanics and Engineering, Volume 307 (2016), pp. 256-274 | Article | MR 3511715

[2] H. Al Akhras; T. Elguedj; A. Gravouil; M. Rochette Towards an automatic isogeometric analysis suitable trivariate models generation-Application to geometric parametric analysis, Computer Methods in Applied Mechanics and Engineering, Volume 316 (2017), pp. 623-645 | Article | MR 3610114

[3] M. J. Borden; M. A. Scott; J. A. Evans; T. J. R. Hughes Isogeometric finite element data structures based on Bézier extraction of NURBS, International Journal for Numerical Methods in Engineering, Volume 87 (2011) no. 1-5, pp. 15-47 | Article | Zbl 1242.74097

[4] J. Chan; T. Warburton A Short Note on a Bernstein–Bézier Basis for the Pyramid, SIAM J. Sci. Comput., Volume 38 (2016) no. 4, p. A2162-A2172 | Article | Zbl 1342.65167

[5] E. Cohen; R. F. Riesenfeld; G. Elber Geometric modeling with splines : an introduction, A. K. Peters, 2001 | Article | Zbl 0980.65016

[6] J. A. Cottrell; T. J. R. Hughes; Y. Bazilevs Isogeometric Analysis: Toward Integration of CAD and FEA, Wiley Publishing, 2009 | Zbl 1378.65009

[7] L. Engvall; J. A. Evans Isogeometric unstructured tetrahedral and mixed-element Bernstein-Bézier discretizations, Computer Methods in Applied Mechanics and Engineering, Volume 319 (2017), pp. 83-123 | Article

[8] X. Gao; T. Martin; S. Deng; E. Cohen; Z. Deng; G. Chen Structured Volume Decomposition via Generalized Sweeping, Visualization and Computer Graphics, IEEE Transactions on, Volume PP (2015) no. 99, p. 1-1 | Article

[9] L. Liu; Y. Zhang; T. J. R. Hughes; M. A. Scott; T. W. Sederberg Volumetric T-spline construction using Boolean operations, Engineering with Computers, Volume 30 (2014) no. 4, pp. 425-439 | Article

[10] T. Martin; E. Cohen Volumetric parameterization of complex objects by respecting multiple materials, Computers & Graphics, Volume 34 (2010) no. 3, pp. 187-197 (Shape Modelling International (SMI) Conference 2010) | Article

[11] T. Martin; E. Cohen; R. M. Kirby Mixed-element volume completion from NURBS surfaces, Computers & Graphics, Volume 36 (2012) no. 5, pp. 548-554 (Shape Modeling International (SMI) Conference 2012) | Article

[12] B. Marussig; T. J. R. Hughes A Review of Trimming in Isogeometric Analysis: Challenges, Data Exchange and Simulation Aspects, Archives of Computational Methods in Engineering, Volume 25 (2018) no. 4, pp. 1059-1127 | Article | MR 3867694

[13] J. O’Rourke; G. Tewari The structure of optimal partitions of orthogonal polygons into fat rectangles, Computational Geometry, Volume 28 (2004) no. 1, pp. 49-71 (14th Canadian Conference on Computational Geometry-CCCG02) | Article | MR 2070713 | Zbl 1116.90107

[14] T. W. Sederberg; G. T. Finnigan; X. Li; H. Lin; H. Ipson Watertight Trimmed NURBS, ACM SIGGRAPH 2008 Papers (2008), p. 79:1-79:8

[15] J. Shen; J. Kosinka; M. A. Sabin; N. A. Dodgson Conversion of trimmed NURBS surfaces to Catmull-Clark subdivision surfaces, Computer Aided Geometric Design, Volume 31 (2014), pp. 486-498 (Recent Trends in Theoretical and Applied Geometry) | Article | MR 3268223 | Zbl 1364.65046

[16] J. Shen; J. Kosinka; M. A. Sabin; N. A. Dodgson Converting a CAD model into a non-uniform subdivision surface, Computer Aided Geometric Design, Volume 48 (2016), pp. 17-35 | Article | MR 3561691 | Zbl 1366.65032

[17] Y. Song; E. Cohen Creating Hybrid B-Reps and Hybrid Volume Completions from Trimmed B-Spline B-Reps (2019) (Technical report)

[18] Y. Song; E. Cohen Making Trimmed B-Spline B-Reps Watertight With a Hybrid Representation, International Design Engineering Technical Conferences and Computers and Information in Engineering Conference (2019) (to appear)

[19] S. Xia; X. Qian Isogeometric analysis with Bézier tetrahedra, Computer Methods in Applied Mechanics and Engineering, Volume 316 (2017), pp. 782-816 (Special Issue on Isogeometric Analysis: Progress and Challenges) | Article

[20] S. Xia; X. Wang; X. Qian Continuity and convergence in rational triangular Bézier spline based isogeometric analysis, Computer Methods in Applied Mechanics and Engineering, Volume 297 (2015), pp. 292-324 | Article | Zbl 1425.65188

[21] G. Xu; Y. Jin; Z. Xiao; Q. Wu; B. Mourrain; T. Rabczuk Exact conversion from Bézier tetrahedra to Bézier hexahedra, Computer Aided Geometric Design, Volume 62 (2018), pp. 154-165 | Article | Zbl 06892785

[22] G. Xu; B. Mourrain; R. Duvigneau; A. Galligo Analysis-suitable volume parameterization of multi-block computational domain in isogeometric applications, Computer-Aided Design (2013), pp. 395-404 | Article | MR 3041214

[23] G. Xu; B. Mourrain; R. Duvigneau; A. Galligo Constructing analysis-suitable parameterization of computational domain from CAD boundary by variational harmonic method, J. Comput. Phys., Volume 252 (2013), pp. 275-289 | Article | MR 3101507 | Zbl 1349.65079

[24] S. Zeng; E. Cohen Hybrid Volume Completion with Higher-order BéZier Elements, Comput. Aided Geom. Des., Volume 35 (2015) no. C, pp. 180-191 | Article | Zbl 1417.65100

[25] Y. Zhang; W. Wang; T. J. R. Hughes Solid T-spline construction from boundary representations for genus-zero geometry, Computer Methods in Applied Mechanics and Engineering, Volume 249-252 (2012), pp. 185-197 (Higher Order Finite Element and Isogeometric Methods) | Article | MR 3003070 | Zbl 1348.65057