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DPB-Splines: The Decoupled Basis of Patchwork Splines

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Geometric Challenges in Isogeometric Analysis

Part of the book series: Springer INdAM Series ((SINDAMS,volume 49))

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Abstract

We consider partitions of the d-dimensional unit cube into patches with an associated tensor-product spline space for each of them. The spline spaces possess the same multi-degree \(\mathbf {p}=(p_1,\ldots ,p_d)\) and the same maximum smoothness \(C^{\mathbf {p}-\mathbf {1}}\), but the choice of the knots is very flexible. Under certain assumptions, we show how to construct Decoupled Patchwork B-splines (DPB-splines) that span the corresponding patchwork spline space \(\mathbb {P}\). More precisely, we generate a basis for the space \(\mathbb {P}\) formed by all \(C^{\mathbf {p}-\mathbf {1}}\) smooth functions that admit patch-wise representations in the associated spline spaces. Based on the framework of decoupled tensor-product B-splines [31, 32], we obtain a basis that is algebraically complete, forms a convex partition of unity, and preserves the coefficients of the local B-spline representations. Furthermore, we present an adaptive refinement algorithm for surface approximation generating partitions that satisfy the required assumptions and hence can be equipped with a DPB-spline basis.

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Notes

  1. 1.

    i.e., with domain \([0,1]^d\).

  2. 2.

    Note that we use the set-theoretic definition of the support, i.e., for a function \(f:\varOmega \mapsto \mathbb {R}^d\) the support of f is defined as \(\mathrm {supp} f=\{\mathbf {x} \in \varOmega \; : \; f(\mathbf {x})\ne 0\}\).

  3. 3.

    For instance, \(f=f_{x}=f_y=f_{xx}=f_{xy}=f_{yy}=f_{xxy}=f_{xyy}=f_{xxyy}=0\) for \(\mathbf {x}=(x,y)\) and \(\mathbf {s}=(2,2)\).

References

  1. Atri, H.R., Shojaee, S.: Truncated hierarchical B-splines in isogeometric analysis of thin shell structures. Steel Compos. Struct. 26, 171–182 (2018)

    Google Scholar 

  2. Beirão da Veiga, L., Buffa, A., Cho, D., Sangalli, G., Analysis-suitable T-splines are dual-compatible: Comput. Methods Appl. Mech. Eng. 249–252, 42–51 (2012)

    Article  Google Scholar 

  3. Bornemann, P.B., Cirak, F.: A subdivision-based implementation of the hierarchical B-spline finite element method. Comput. Methods Appl. Mech. Eng. 253, 584–598 (2013)

    Article  MathSciNet  Google Scholar 

  4. Bracco, C., Buffa, A., Giannelli, C., Vázquez, R.: Adaptive isogeometric methods with hierarchical splines: an overview. Disc. Contin. Dyn. Syst.—Ser. A 39, 241–262 (2019)

    Article  MathSciNet  Google Scholar 

  5. Bracco, C., Giannelli, C., Großmann, D., Sestini, A.: Adaptive fitting with THB-splines: error analysis and industrial applications. Comput. Aided Geom. Des. 62, 239–252 (2018)

    Article  MathSciNet  Google Scholar 

  6. Buffa, A., Giannelli, C.: Adaptive isogeometric methods with hierarchical splines: error estimator and convergence. Math. Models Methods Appl. Sci. 26, 1–25 (2016)

    Article  MathSciNet  Google Scholar 

  7. Chen, L., de Borst, R.: Adaptive refinement of hierarchical T-splines. Comput. Methods Appl. Mech. Eng. 337, 220–245 (2018)

    Article  MathSciNet  Google Scholar 

  8. Chen, L., de Borst, R.: Locally refined T-splines. Int. J. Numer. Methods Eng. 114, 637–659 (2018)

    Article  MathSciNet  Google Scholar 

  9. Dokken, T., Lyche, T., Pettersen, K.F.: Polynomial splines over locally refined box-partitions. Comput. Aided Geom. Des. 30, 331–356 (2013)

    Article  MathSciNet  Google Scholar 

  10. Engleitner, N., Jüttler, B.: Patchwork B-spline refinement. Comput.-Aided Des. 90, 168–179 (2017)

    Article  MathSciNet  Google Scholar 

  11. Engleitner, N., Jüttler, B.: Lofting with patchwork B-splines. In: Giannelli, C., Speleers, H. (eds.), Advanced methods for geometric modeling and numerical simulation. Springer INdAM Series, vol. 35, pp. 77–98. Springer (2019)

    Google Scholar 

  12. Engleitner, N., Jüttler, B., Zore, U.: Partially nested hierarchical refinement of bivariate tensor-product splines with highest order smoothness. In: Floater, M. et al. (eds.), Mathematical methods for curves and surfaces. Lecture Notes in Computer Science, vol. 10521, pp. 126–144. Springer (2017)

    Google Scholar 

  13. Evans, E.J., Scott, M.A., Li, X., Thomas, D.C.: Hierarchical T-splines: analysis-suitability, Bézier extraction, and application as an adaptive basis for isogeometric analysis. Comput. Methods Appl. Mech. Eng. 284, 1–20 (2015)

    Article  Google Scholar 

  14. Farin, G., Hoschek, J., Kim, M.S. (eds.): Handbook of Computer Aided Geometric Design. Elsevier (2002)

    Google Scholar 

  15. Forsey, D.R., Bartels, R.H.: Hierarchical B-spline refinement. ACM SIGGRAPH Comput. Graph. 22, 205–212 (1988)

    Article  Google Scholar 

  16. Garau, E.M., Vázquez, R.: Algorithms for the implementation of adaptive isogeometric methods using hierarchical B-splines. Appl. Numer. Math. 123, 58–87 (2018)

    Article  MathSciNet  Google Scholar 

  17. Giannelli, C., Jüttler, B., Kleiss, S., Mantzaflaris, A., Simeon, B., Špeh, J.: THB-splines: an effective mathematical technology for adaptive refinement in geometric design and isogeometric analysis. Comput. Methods Appl. Mech. Eng. 299, 337–365 (2016)

    Article  MathSciNet  Google Scholar 

  18. Giannelli, C., Jüttler, B., Speleers, H.: THB-splines: the truncated basis for hierarchical splines. Comput. Aided Geom. Des. 29, 485–498 (2012)

    Article  MathSciNet  Google Scholar 

  19. Giannelli, C., Jüttler, B., Speleers, H.: Strongly stable bases for adaptively refined multilevel spline spaces. Adv. Comput. Math. 40, 459–490 (2014)

    Article  MathSciNet  Google Scholar 

  20. Giannelli, C., Kanduč, T., Pelosi, F., Speleers, H.: An immersed-isogeometric model: application to linear elasticity and implementation with THBox-splines. J. Comput. Appl. Math. 349, 410–423 (2019)

    Article  MathSciNet  Google Scholar 

  21. Giust, A., Jüttler, B., Mantzaflaris, A.: Local (T)HB-spline projectors via restricted hierarchical spline fitting. Comput. Aided Geom. Des. 80, 101–865 (2020)

    Google Scholar 

  22. Greiner, G., Hormann, K.: Interpolating and approximating scattered 3D-data with hierarchical tensor product B-splines. In: Le Méhauté, A., Rabut, C., Schumaker, L.L. (eds.) Surface Fitting and Multiresolution Methods, pp. 163–172. Vanderbilt University Press, Nashville, TN (1997)

    MATH  Google Scholar 

  23. Kanduč, T., Giannelli, C., Pelosi, F., Speleers, H.: Adaptive isogeometric analysis with hierarchical box splines. Comput. Methods Appl. Mech. Eng. 316, 817–838 (2017)

    Article  MathSciNet  Google Scholar 

  24. Kang, H., Chen, F., Deng, J.: Hierarchical B-splines on regular triangular partitions. Graph. Models 76, 289–300 (2014)

    Article  Google Scholar 

  25. Kang, H., Xu, J., Chen, F., Deng, J.: A new basis for PHT-splines. Graph. Models 82, 149–159 (2015)

    Article  Google Scholar 

  26. Kiss, G., Giannelli, C., Zore, U., Jüttler, B., Großmann, D., Barner, J.: Adaptive CAD model (re-)construction with THB-splines. Graph. Models 76, 273–288 (2014)

    Article  Google Scholar 

  27. Kraft, R.: Adaptive und linear unabhängige Multilevel B-splines und ihre Anwendungen. Ph.D. thesis, University Stuttgart (1998)

    Google Scholar 

  28. Kuru, G., Verhoosel, C.V., van der Zeeb, K.G., van Brummelen, E.H.: Goal-adaptive isogeometric analysis with hierarchical splines. Comput. Methods Appl. Mech. Eng. 270, 270–292 (2014)

    Article  MathSciNet  Google Scholar 

  29. Li, X., Deng, J., Chen, F.: Polynomial splines over general T-meshes. Vis. Comput. 26, 277–286 (2010)

    Article  Google Scholar 

  30. Li, X., Zheng, J., Sederberg, T.W., Hughes, T.J.R., Scott, M.A.: On linear independence of T-spline blending functions. Comput. Aided Geom. Des. 29, 63–76 (2012)

    Article  MathSciNet  Google Scholar 

  31. Mokriš, D., Jüttler, B.: TDHB-splines: the truncated decoupled basis of hierarchical tensor-product splines. Comput. Aided Geom. Des. 31, 531–544 (2014)

    Article  MathSciNet  Google Scholar 

  32. Mokriš, D., Jüttler, B., Giannelli, C.: On the completeness of hierarchical tensor-product B-splines. J. Comput. Appl. Math. 271, 53–70 (2014)

    Article  MathSciNet  Google Scholar 

  33. Morgenstern, P., Peterseim, D.: Analysis-suitable adaptive T-mesh refinement with linear complexity. Comput. Aided Geom. Des. 34, 50–66 (2015)

    Article  MathSciNet  Google Scholar 

  34. Schillinger, D., Dedé, L., Scott, M.A., Evans, J.A., Borden, M.J., Rank, E., Hughes, T.J.R.: 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 (2012)

    Article  MathSciNet  Google Scholar 

  35. Sederberg, T.W., Zheng, J., Bakenov, A., Nasri, A.: T-splines and T-NURCCS. ACM Trans. Graph. 22, 477–484 (2003)

    Article  Google Scholar 

  36. Speleers, H.: Hierarchical spline spaces: Quasi-interpolants and local approximation estimates. Adv. Comput. Math. 43, 235–255 (2017)

    Article  MathSciNet  Google Scholar 

  37. Speleers, H., Dierckx, P., Vandewalle, S.: Quasi-hierarchical Powell-Sabin B-splines. Comput. Aided Geom. Des. 26, 174–191 (2009)

    Article  MathSciNet  Google Scholar 

  38. Speleers, H., Manni, C.: Effortless quasi-interpolation in hierarchical spaces. Numerische Mathematik 132, 155–184 (2016)

    Article  MathSciNet  Google Scholar 

  39. Wei, X., Zhang, Y., Hughes, T.J.R., Scott, M.A.: Extended truncated hierarchical Catmull-Clark subdivision. Comput. Methods Appl. Mech. Eng. 299, 316–336 (2016)

    Article  MathSciNet  Google Scholar 

  40. Yvart, A., Hahmann, S., Bonneau, G.P.: Hierarchical triangular splines. ACM Trans. Graph. 24, 1374–1391 (2005)

    Article  Google Scholar 

  41. Zhang, J., Li, X.: Local refinement for analysis-suitable++ T-splines. Comput. Methods Appl. Mech. Eng. 342, 32–45 (2018)

    Article  MathSciNet  Google Scholar 

  42. Zore, U., Jüttler, B.: Adaptively refined multilevel spline spaces from generating systems. Comput. Aided Geom. Des. 31, 545–566 (2014)

    Article  MathSciNet  Google Scholar 

  43. Zore, U., Jüttler, B., Kosinka, J.: On the linear independence of truncated hierarchical generating systems. J. Comput. Appl. Math. 306, 200–216 (2016)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The financial support of the FWF (NFN S117 “Geometry + Simulation” and Doctoral Program W1214 “Computational Mathematics”) and of the ERC (GA no. 694515) is gratefully acknowledged. Special thanks go to the geometry department at MTU Aero Engines for providing the data of the turbine blade example.

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Authors and Affiliations

  1. Institute of Applied Geometry, Johannes Kepler University, Linz, Austria

    Nora Hemelmayr & Bert Jüttler

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  1. Nora Hemelmayr
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  2. Bert Jüttler
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Correspondence to Bert Jüttler .

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Editors and Affiliations

  1. Department of Mathematics, University of Rome Tor Vergata, Rome, Italy

    Carla Manni

  2. Department of Mathematics, University of Rome Tor Vergata, Rome, Italy

    Hendrik Speleers

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Hemelmayr, N., Jüttler, B. (2022). DPB-Splines: The Decoupled Basis of Patchwork Splines. In: Manni, C., Speleers, H. (eds) Geometric Challenges in Isogeometric Analysis. Springer INdAM Series, vol 49. Springer, Cham. https://doi.org/10.1007/978-3-030-92313-6_3

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