Finite element solvers are a basic component of simulation applications; they are common in computer graphics, engineering, and medical simulations. Although adaptive solvers can be of great value in reducing the often high computational cost of simulations they are not employed broadly. Indeed, building adaptive solvers can be a daunting task especially for 3D finite elements. In this paper we are introducing a new approach to produce conforming, hierarchical, adaptive refinement methods (CHARMS). The basic principle of our approach is to refine basis functions, not elements. This removes a number of implementation headaches associated with other approaches and is a general technique independent of domain dimension (here 2D and 3D), element type (e.g., triangle, quad, tetrahedron, hexahedron), and basis function order (piecewise linear, higher order B-splines, Loop subdivision, etc.). The(un-)refinement algorithms are simple and req! uire little in terms of data structure support. We demonstrate the versatility of our new approach through 2D and 3D examples, including medical applications and thin-shell animations.

References in zbMATH (referenced in 27 articles , 2 standard articles )

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  1. Garau, Eduardo M.; Vázquez, Rafael: Algorithms for the implementation of adaptive isogeometric methods using hierarchical B-splines (2018)
  2. Zore, Urška; Jüttler, Bert; Kosinka, Jiří: On the linear independence of truncated hierarchical generating systems (2016)
  3. Evans, E. J.; Scott, M. A.; Li, Xin; Thomas, D. C.: Hierarchical T-splines: analysis-suitability, Bézier extraction, and application as an adaptive basis for isogeometric analysis (2015)
  4. Jiang, Wen; Dolbow, John E.: Adaptive refinement of hierarchical B-spline finite elements with an efficient data transfer algorithm (2015)
  5. Thomas, D. C.; Scott, M. A.; Evans, J. A.; Tew, K.; Evans, E. J.: Bézier projection: a unified approach for local projection and quadrature-free refinement and coarsening of NURBS and T-splines with particular application to isogeometric design and analysis (2015)
  6. Bargteil, Adam W.; Cohen, Elaine: Animation of deformable bodies with quadratic Bézier finite elements (2014)
  7. Rumpf, Martin; Wardetzky, Max: Geometry processing from an elastic perspective (2014)
  8. Bornemann, P. B.; Cirak, F.: A subdivision-based implementation of the hierarchical b-spline finite element method (2013)
  9. Quraishi, S. M.; Sandeep, K.: Multiscale modeling of beam and plates using customized second-generation wavelets (2013)
  10. Vetter, Roman; Stoop, Norbert; Jenni, Thomas; Wittel, Falk K.; Herrmann, Hans J.: Subdivision shell elements with anisotropic growth (2013)
  11. Long, Quan; Bornemann, P. Burkhard; Cirak, Fehmi: Shear-flexible subdivision shells (2012)
  12. Schillinger, Dominik; Dedè, Luca; Scott, Michael A.; Evans, John A.; Borden, Michael J.; Rank, Ernst; Hughes, Thomas J. R.: An isogeometric design-through-analysis methodology based on adaptive hierarchical refinement of NURBS, immersed boundary methods, and T-spline CAD surfaces (2012)
  13. Mazur, David R.: Combinatorics. A guided tour (2010)
  14. Speleers, Hendrik; Dierckx, Paul; Vandewalle, Stefan: On the local approximation power of quasi-hierarchical Powell-Sabin splines (2010)
  15. Boyer, Franck; Lapuerta, Celine; Minjeaud, Sebastian; Piar, Bruno: A local adaptive refinement method with multigrid preconditionning illustrated by multiphase flow simulations. (2009)
  16. Fabris, Francesco: Shannon information theory and molecular biology (2009)
  17. Speleers, Hendrik; Dierckx, Paul; Vandewalle, Stefan: Quasi-hierarchical Powell-Sabin B-splines (2009)
  18. Tabarraei, A.; Sukumar, N.: Extended finite element method on polygonal and quadtree meshes (2008)
  19. Tabarraei, A.; Sukumar, N.: Adaptive computations using material forces and residual-based error estimators on quadtree meshes (2007)
  20. Huang, Jin; Shi, Xiaohan; Liu, Xinguo; Zhou, Kun; Guo, Baining; Bao, Hujun: Geometrically based potential energy for simulating deformable objects (2006) ioport

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