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Volumetric T-spline construction using Boolean operations

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Abstract

In this paper, we present a novel algorithm for constructing a volumetric T-spline from B-reps inspired by constructive solid geometry Boolean operations. By solving a harmonic field with proper boundary conditions, the input surface is automatically decomposed into regions that are classified into two groups represented, topologically, by either a cube or a torus. We perform two Boolean operations (union and difference) with the primitives and convert them into polycubes through parametric mapping. With these polycubes, octree subdivision is carried out to obtain a volumetric T-mesh, and sharp features detected from the input model are also preserved. An optimization is then performed to improve the quality of the volumetric T-spline. The obtained T-spline surface is C 2 everywhere except the local region surrounding irregular nodes, where the surface continuity is elevated from C 0 to G 1. Finally, we extract trivariate Bézier elements from the volumetric T-spline and use them directly in isogeometric analysis.

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Notes

  1. Conventionally, a polycube comprises cubes of equal sizes with two neighboring cubes sharing a complete face [2]. In this paper, the “cubes” can be arbitrary hexahedra of different sizes.

References

  1. http://en.wikipedia.org/wiki/constructive_solid_geometry

  2. http://en.wikipedia.org/wiki/polycube

  3. Adams B, Dutré P (2003) Interactive boolean operations on surfel-bounded solids. ACM Trans Graph 22(3):651–656

    Article  Google Scholar 

  4. Aigner M, Heinrich C, Jüttler B, Pilgerstorfer E, Simeon B, Vuong A-V (2009) Swept volume parameterization for isogeometric analysis. In: IMA International conference on mathematics of surfaces XIII:19–44

  5. Bazilevs Y, Akkerman I, Benson DJ, Scovazzi G, Shashkov MJ (2013) Isogeometric analysis of lagrangian hydrodynamics. J Comput Phys 243:224–243

    Article  MathSciNet  Google Scholar 

  6. Bazilevs Y, Calo VM, Cottrell JA, Evans JA, Hughes TJR, Lipton S, Scott MA, Sederberg TW (2010) Isogeometric analysis using T-splines. Comput Methods Appl Mech Eng 199(5-8):229–263

    Article  MathSciNet  MATH  Google Scholar 

  7. Cottrell JA, Hughes TJR, Bazilevs Y (2009) Isogeometric analysis: toward integration of CAD and FEA. Wiley, Chichester

    Book  Google Scholar 

  8. Cottrell JA, Reali A, Bazilevs Y, Hughes TJR (2006) Isogeometric analysis of structural vibrations. Comput Methods Appl Mech Eng 195(41-43):5257–5296

    Article  MathSciNet  MATH  Google Scholar 

  9. Escobar JM, Cascón JM, Rodríguez E, Montenegro R (2011) A new approach to solid modeling with trivariate T-splines based on mesh optimization. Comput Methods Appl Mech Eng 200(45-46):3210–3222

    Article  MATH  Google Scholar 

  10. Evans JA, Bazilevs Y, Babuška I, Hughes TJR (2009) sup-infs, and optimality ratios for the k-version of the isogeometric finite element method. Comput Methods Appl Mech Eng 198(2126):1726–1741

    Article  MATH  Google Scholar 

  11. Floater MS (1997) Parametrization and smooth approximation of surface triangulations. Comput Aided Geom Des 14(3):231–250

    Article  MathSciNet  MATH  Google Scholar 

  12. Goldman R, Lyche T (1993) Knot insertion and deletion algorithms for B-spline curves and surfaces. Society for Industrial and Applied Mathematics–Philadelphia

  13. Gu X, Wang Y, Yau S (2003) Volumetric harmonic map. Commun Inf Syst 3(3):191–202

    MathSciNet  MATH  Google Scholar 

  14. Guerra CAR (2010) Simultaneous untangling and smoothing of hexahedral meshes. Master’s thesis, University PolyTècnica De Catalunya, Barcelona Spain

  15. Hatcher A (1999) Pants decomposition of surfaces. arXiv:math/9906084,

  16. Hughes TJR, Cottrell JA, Bazilevs Y (2005) Isogeometric analysis: CAD, finite elements, NURBS, exact geometry, and mesh refinement. Comput Methods Appl Mech Eng 194:4135–4195

    Article  MathSciNet  MATH  Google Scholar 

  17. Knupp PM (2003) A method for hexahedral mesh shape optimization. Int J Numer Methods Eng 58(2):319–332

    Article  MATH  Google Scholar 

  18. Li B, Li X, Wang K, Qin H (2010) Generalized polycube trivariate splines. In: Shape modeling international conference, p 261–265

  19. Li B, Li X, Wang K, Qin H (2012) Surface mesh to volumetric spline conversion with generalized poly-cubes. IEEE Trans Vis Comput Graph 99:1–14

    Google Scholar 

  20. Li W, Ray N, Lévy B (2006) Automatic and interactive mesh to T-spline conversion. In: Eurographics Symposium on Geometry Processing, p 191–200

  21. Li X, Zheng J, Sederberg TW, Hughes TJR, Scott MA (2012) On linear independence of T-spline blending functions. Comput Aided Geom Des 29(1):63–76

    Article  MathSciNet  MATH  Google Scholar 

  22. Lipton S, Evans JA, Bazilevs Y, Elguedj T, Hughes TJR (2010) Robustness of isogeometric structural discretizations under severe mesh distortion. Comput Methods Appl Mech Eng 199(5-8):357–373

    Article  MATH  Google Scholar 

  23. Liu L, Zhang Y, hughes TJR, Scott MA, Sederberg TW (2013) Volumetric T-spline construction using Boolean operations. In: In 22nd International Meshing Roundtable. Accepted 2013

  24. Mitchell SA, Tautges TJ (1995) Pillowing doublets: refining a mesh to ensure that faces share at most one edge. In: 4th International Meshing Roundtable, p 231–240

  25. Piegl LA, Tiller W (1997) The NURBS Book (monographs in visual communication), 2nd ed. Springer, New York

    Book  Google Scholar 

  26. Qian J, Zhang Y, Wang W, Lewis AC, Qidwai MA, Geltmacher AB (2010) Quality improvement of non-manifold hexahedral meshes for critical feature determination of microstructure materials. Int J Numer Methods Eng 82(11):1406–1423

    MathSciNet  MATH  Google Scholar 

  27. Schillinger D, Dedè L, Scott MA, Evans JA, Borden MJ, Rank E, Hughes TJR (2012) An isogeometric design-through-analysis methodology based on adaptive hierarchical refinement of NURBS, immersed boundary methods, and T-spline CAD surfaces. Comput Methods Appl Mech Eng 249(252):116–150

    Article  Google Scholar 

  28. Scott MA, Borden MJ, Verhoosel CV, Sederberg TW, Hughes TJR (2011) Isogeometric finite element data structures based on Bézier extraction of T-splines. Int J Numer Methods Eng 88(2):126–156

    Article  MathSciNet  MATH  Google Scholar 

  29. Scott MA, Li X, Sederberg TW, Hughes TJR (2012) Local refinement of analysis-suitable T-splines. Comput Methods Appl Mech Eng 213-216(0):206–222

    Article  MathSciNet  Google Scholar 

  30. Scott MA, Simpson RN, Evans JA, Lipton S, Bordas SPA, Hughes TJR, Sederberg TW (2013) Isogeometric boundary element analysis using unstructured T-splines. Comput Methods Appl Mech Eng 254(0):197–221

    Article  MathSciNet  Google Scholar 

  31. Sederberg TW, Cardon DL, Finnigan GT, North NS, Zheng J, Lyche T (2004) T-spline simplification and local refinement. In: ACM SIGGRAPH, p 276–283

  32. Sederberg TW, Zheng J, Bakenov A, Nasri A (2003) T-splines and T-NURCCs. ACM Trans Graph 22(3):477–484

    Article  Google Scholar 

  33. Smith JM, Dodgson NA (2007) A topologically robust algorithm for boolean operations on polyhedral shapes using approximate arithmetic. Comput-Aided Des 39(2):149–163

    Article  Google Scholar 

  34. Wang H, He Y, Li X, Gu X, Qin H (2007) Polycube splines. In: Symposium on solid and physical modeling, p 241–251

  35. Wang K, Li X, Li B, Xu H, Qin H (2012) Restricted trivariate polycube splines for volumetric data modeling. IEEE Trans Vis Comput Graph 18:703–716

    Article  Google Scholar 

  36. Wang W, Zhang Y, Liu L, Hughes TJR (2013) Trivariate solid T-spline construction from boundary triangulations with arbitrary genus topology. Comput Aided Des 45:351–360

    Article  MathSciNet  Google Scholar 

  37. Wang W, Zhang Y, Scott MA, Hughes TJR (2011) Converting an unstructured quadrilateral mesh to a standard T-spline surface. Comput Mech 48:477–498

    Article  MathSciNet  MATH  Google Scholar 

  38. Wang W, Zhang Y, Xu G, Hughes TJR (2012) Converting an unstructured quadrilateral/hexahedral mesh to a rational T-spline. Comput Mech 50(1):65–84

    Article  MathSciNet  MATH  Google Scholar 

  39. Xu G, Mourrain B, Duvigneau R, Galligo A (2013) Analysis-suitable volume parameterization of multi-block computational domain in isogeometric applications. Comput-Aided Des 45(2):395–404

    Article  MathSciNet  Google Scholar 

  40. Zhang Y, Bajaj CL, Xu G (2009) Surface smoothing and quality improvement of quadrilateral/hexahedral meshes with geometric flow. Commun Numer Methods Eng 25:1–18

    Article  MathSciNet  Google Scholar 

  41. Zhang Y, Bazilevs Y, Goswami S, Bajaj CL, Hughes TJR (2007) Patient-specific vascular NURBS modeling for isogeometric analysis of blood flow. Comput Methods Appl Mech Eng 196(29-30):2943–2959

    Article  MathSciNet  MATH  Google Scholar 

  42. Zhang Y, Wang W, Hughes TJR (2012) Solid T-spline construction from boundary representations for genus-zero geometry. Comput Methods Appl Mech Eng 249(252):185–197

    Article  MathSciNet  Google Scholar 

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Acknowledgments

The work of L. Liu and Y. Zhang was supported by ONR-YIP award N00014-10-1-0698 and an ONR Grant N00014-08-1-0653. T. J. R. Hughes was supported by ONR Grant N00014-08-1-0992, NSF GOALI CMI-0700807 /0700204, NSF CMMI-1101007 and a SINTEF grant UTA10-000374. A preliminary version of this paper was accepted in the 22th International Meshing Roundtable conference [23].

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Authors and Affiliations

  1. Department of Mechanical Engineering, Carnegie Mellon University, Pittsburgh, PA, 15213, USA

    Lei Liu & Yongjie Zhang

  2. Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, TX, 78712, USA

    Thomas J. R. Hughes

  3. Department of Civil and Environmental Engineering, Brigham Young University, Provo, UT, 84602, USA

    Michael A. Scott

  4. Department of Computer Science, Brigham Young University, Provo, UT, 84602, USA

    Thomas W. Sederberg

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  1. Lei Liu
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  2. Yongjie Zhang
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Correspondence to Yongjie Zhang.

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Liu, L., Zhang, Y., Hughes, T.J.R. et al. Volumetric T-spline construction using Boolean operations. Engineering with Computers 30, 425–439 (2014). https://doi.org/10.1007/s00366-013-0346-6

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