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A locally based construction of analysis-suitable $G^1$ multi-patch spline surfaces (2308.09007v2)

Published 17 Aug 2023 in math.NA and cs.NA

Abstract: Analysis-suitable $G1$ (AS-$G1$) multi-patch spline surfaces [4] are particular $G1$-smooth multi-patch spline surfaces, which are needed to ensure the construction of $C1$-smooth multi-patch spline spaces with optimal polynomial reproduction properties [16]. We present a novel local approach for the design of AS-$G1$ multi-patch spline surfaces, which is based on the use of Lagrange multipliers. The presented method is simple and generates an AS-$G1$ multi-patch spline surface by approximating a given $G1$-smooth but non-AS-$G1$ multi-patch surface. Several numerical examples demonstrate the potential of the proposed technique for the construction of AS-$G1$ multi-patch spline surfaces and show that these surfaces are especially suited for applications in isogeometric analysis by solving the biharmonic problem, a particular fourth order partial differential equation, with optimal rates of convergence over them.

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References (44)
  1. Isogeometric analysis of high order partial differential equations on surfaces. Comput. Methods Appl. Mech. Engrg., 295:446 – 469, 2015.
  2. M. Bercovier and T. Matskewich. Smooth Bézier Surfaces over Unstructured Quadrilateral Meshes. Lecture Notes of the Unione Matematica Italiana, Springer, 2017.
  3. Geometrically smooth spline bases for data fitting and simulation. Computer Aided Geometric Design, 78:101814, 2020.
  4. Adaptive isogeometric methods with C1superscript𝐶1C^{1}italic_C start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT (truncated) hierarchical splines on planar multi-patch domains. Math. Models Methods Appl. Sci., 33(9):1829–1874, 2023.
  5. Iso-geometric analysis based on Catmull-Clark solid subdivision. Computer Graphics Forum, 29(5):1575–1784, 2010.
  6. Subdivision surfaces: a new paradigm for thin-shell finite-element analysis. Int. J. Numer. Meth. Engng, 47(12):2039–2072, 2000.
  7. Analysis-suitable G11{}^{1}start_FLOATSUPERSCRIPT 1 end_FLOATSUPERSCRIPT multi-patch parametrizations for C11{}^{1}start_FLOATSUPERSCRIPT 1 end_FLOATSUPERSCRIPT isogeometric spaces. Comput. Aided Geom. Des., 47:93 – 113, 2016.
  8. Isogeometric analysis with C1superscript𝐶1{C}^{1}italic_C start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT-smooth functions over multi-patch surfaces. Comput. Methods Appl. Mech. Engrg., 403:115706, 2023.
  9. Isogeometric analysis for multi-patch structured Kirchhoff–Love shells. Comput. Methods Appl. Mech. Engrg., 411:116060, 2023.
  10. G. Farin. Curves and Surfaces for Computer-Aided Geometric Design. Academic Press, 1997.
  11. Isogeometric analysis of 2D gradient elasticity. Comput. Mech., 47(3):325–334, 2011.
  12. Isogeometric analysis of the Cahn–Hilliard phase-field model. Comput. Methods Appl. Mech. Engrg., 197(49):4333–4352, 2008.
  13. Isogeometric analysis of Phase–Field models: Application to the Cahn–Hilliard equation. In ECCOMAS Multidisciplinary Jubilee Symposium: New Computational Challenges in Materials, Structures, and Fluids, pages 1–16. Springer Netherlands, 2009.
  14. D. Groisser and J. Peters. Matched Gk𝑘{}^{k}start_FLOATSUPERSCRIPT italic_k end_FLOATSUPERSCRIPT-constructions always yield Ck𝑘{}^{k}start_FLOATSUPERSCRIPT italic_k end_FLOATSUPERSCRIPT-continuous isogeometric elements. Comput. Aided Geom. Des., 34:67 – 72, 2015.
  15. J. Hoschek and D. Lasser. Fundamentals of computer aided geometric design. A K Peters Ltd., Wellesley, MA, 1993.
  16. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput. Methods Appl. Mech. Engrg., 194(39-41):4135–4195, 2005.
  17. Isogeometric analysis with geometrically continuous functions on planar multi-patch geometries. Comput. Methods Appl. Mech. Engrg., 316:209 – 234, 2017.
  18. Triangular bubble spline surfaces. Computer-Aided Design, 43(11):1341–1349, 2011.
  19. Dimension and basis construction for analysis-suitable G11{}^{1}start_FLOATSUPERSCRIPT 1 end_FLOATSUPERSCRIPT two-patch parameterizations. Comput. Aided Geom. Des., 52–53:75 – 89, 2017.
  20. Construction of analysis-suitable G11{}^{1}start_FLOATSUPERSCRIPT 1 end_FLOATSUPERSCRIPT planar multi-patch parameterizations. Comput.-Aided Des., 97:41–55, 2018.
  21. An isogeometric C11{}^{1}start_FLOATSUPERSCRIPT 1 end_FLOATSUPERSCRIPT subspace on unstructured multi-patch planar domains. Comput. Aided Geom. Des., 69:55–75, 2019.
  22. Isogeometric analysis with geometrically continuous functions on two-patch geometries. Comput. Math. Appl., 70(7):1518 – 1538, 2015.
  23. Generalizing bicubic splines for modeling and IGA with irregular layout. Comput.-Aided Des., 70:23 – 35, 2016.
  24. K. Karčiauskas and J. Peters. Refinable G1superscript𝐺1{G}^{1}italic_G start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT functions on G1superscript𝐺1{G}^{1}italic_G start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT free-form surfaces. Comput. Aided Geom. Des., 54:61–73, 2017.
  25. K. Karčiauskas and J. Peters. Refinable bi-quartics for design and analysis. Comput.-Aided Des., pages 204–214, 2018.
  26. Isogeometric analysis of the Cahn–Hilliard equation – a convergence study. Journal of Computational Physics, 305:360–371, 2016.
  27. The bending strip method for isogeometric analysis of Kirchhoff-Love shell structures comprised of multiple patches. Comput. Methods Appl. Mech. Engrg., 199(35):2403–2416, 2010.
  28. Isogeometric shell analysis with Kirchhoff-Love elements. Comput. Methods Appl. Mech. Engrg., 198(49):3902–3914, 2009.
  29. Isogeometric analysis of first and second strain gradient elasticity. Comput. Mech., 61(3):351–363, 2018.
  30. G1superscript𝐺1{G}^{1}italic_G start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT spline functions for point cloud fitting. Applied Mathematics and Computation, 460:128279, 2024.
  31. Dimension and bases for geometrically continuous splines on surfaces of arbitrary topology. Comput. Aided Geom. Des., 45:108 – 133, 2016.
  32. A comparative study of several classical, discrete differential and isogeometric methods for solving poisson’s equation on the disk. Axioms, 3(2):280–299, 2014.
  33. C1superscript𝐶1{C}^{1}italic_C start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT finite elements on non-tensor-product 2d and 3d manifolds. Applied Mathematics and Computation, 272:148 – 158, 2016.
  34. T. Nguyen and J. Peters. Refinable C1superscript𝐶1{C}^{1}italic_C start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT spline elements for irregular quad layout. Comput. Aided Geom. Des., 43:123 – 130, 2016.
  35. Variational formulation and isogeometric analysis for fourth-order boundary value problems of gradient-elastic bar and plane strain/stress problems. Comput. Methods Appl. Mech. Engrg., 308:182–211, 2016.
  36. G1superscript𝐺1{G}^{1}italic_G start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT-smooth planar parameterization of complex domains for isogeometric analysis. Comput. Methods Appl. Mech. Engrg., 417:116330, 2023.
  37. J. Peters. Parametrizing singularly to enclose data points by a smooth parametric surface. In Proceedings of graphics interface, volume 91, 1991.
  38. J. Peters. Geometric continuity. In Handbook of computer aided geometric design, pages 193–227. North-Holland, Amsterdam, 2002.
  39. U. Reif. Biquadratic G-spline surfaces. Comput. Aided Geom. Des., 12(2):193–205, 1995.
  40. U. Reif. A refinable space of smooth spline surfaces of arbitrary topological genus. Journal of Approximation Theory, 90(2):174–199, 1997.
  41. Isogeometric shell analysis with NURBS compatible subdivision surfaces. Applied Mathematics and Computation, 272:139–147, 2016.
  42. Smooth cubic spline spaces on unstructured quadrilateral meshes with particular emphasis on extraordinary points: Geometric design and isogeometric analysis considerations. Comput. Methods Appl. Mech. Engrg., 327:411–458, 2017.
  43. Isogeometric analysis using G-spline surfaces with arbitrary unstructured quadrilateral layout. Comput. Methods Appl. Mech. Engrg., 408:115965, 2023.
  44. Subdivision surfaces with isogeometric analysis adapted refinement weights. Computer-Aided Design, 102:104–114, 2018.
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