1 / 18

Hierarchical Finite Element Mesh Refinement

Hierarchical Finite Element Mesh Refinement. Petr Krysl* Eitan Grinspun, Peter Schröder. *Structural Engineering Department, University of California, San Diego Computer Science Department, California Institute of Technology. Adaptive Approximations. Adjust spatial resolution by:

napoleon
Télécharger la présentation

Hierarchical Finite Element Mesh Refinement

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Hierarchical Finite Element Mesh Refinement Petr Krysl* Eitan Grinspun, Peter Schröder *Structural Engineering Department, University of California, San DiegoComputer Science Department, California Institute of Technology

  2. Adaptive Approximations Adjust spatial resolution by: • Remeshing • Local refinement (Adaptive Mesh Refinement)Split the finite elements, ensure compatibility via • Constraints • Lagrangian multipliers or penalty methods • Irregular splitting of neighboring elements Major implementation effort!

  3. Refinement for Subdivision • State-of-the-art refinement not applicable to subdivision surfaces. • Refinement should take advantage of the multiresolutionnature of subdivision surfaces. Subdivision surface: overlap of two basis functions.

  4. Conceptual Hierarchy • Infinite globally-refined sequence • Mesh is globally refined to form and so on… • Strict nesting of

  5. Refinement Equation • Refinement relation • Refined basis of • Any linearly independent set of basis functions chosen from with

  6. Adapted basis 1 • Quasi-hierarchical basis: • Some basis functions are removed: Nodes associated with active basis functions

  7. Adapted basis 2 • True hierarchical basis • Details are added to coarser functions: Nodes associated with active basis functions

  8. Multi-level approximation • Approximation of a function on multiple mesh levels • Literal interpretation of the refinement equation has a big advantage: genericity. = set of refined basis functions on level m

  9. CHARMS • Refinement equation:Naturally conforming, dimension and order independent. • Multiresolution: • True hierarchical basis: Functions N(j+1) add details. • Quasi-hierarchical basis: Functions N(j+1) replace N(j). • Adaptation: • Refinement/coarsening intrinsic (prolongation and restriction). ConformingHierarchicalAdaptiveRefinementMethodS

  10. CHARMS vs common AMR Original basis on quadrilateral mesh Adapted basis on a refined mesh Common AMR w/ constraints CHARMS Quasi-hierarchical basis True hierarchical basis Level 0 Level 1

  11. Refinement for Subdivision … • CHARMS apply to subdivision surfaces without any change. • The multiresolution character of subdivision surfaces is taken advantage of quite naturally.

  12. Algorithms • Field transfer: • prolongation, restriction operators. • Integration: • single level vs. multiple-level. • Algorithms: • independent of order, dimensions: generic; • easy to program, easy to debug. • Multiscale approximation: • hierarchical and multiresolution (quasi-hierarchical) basis; • multigrid solvers.

  13. 2D Example True hierarchical basis. Poisson equation with homogeneous Dirichlet bc. Hierarchy of basis function sets; Red balls: the active functions. Solution painted on the integration cells. Quasi hierarchical basis.

  14. 3D Example Solution painted on the integration cells 3-level grid (true hierarchical)

  15. Heat diffusion: Hierarchical Solution displayedon the integrationcells True hierarchical basis; Adaptive step 2: 5,000 degrees of freedom (~3,000 hierarchical) Grid hierarchy Level 1 Level 2 Level 3

  16. Heat diffusion: Quasi-hier. Solution displayedon the integrationcells Quasi-hierarchical basis; Adaptive step 2: 3,900 degrees of freedom Grid hierarchy Level 1 Level 2 Level 3

  17. Highlights • Easy implementation: • The adaptivity code was debugged in 1D. It then took a little over two hours to implement 2D and 3D mesh refinement: Clear evidence of the generic nature of the approach. • Expanded options: • True hierarchical basis and multiresolution basis implemented by the same code: It takes two lines of code to switch between those two bases.

  18. Onwards to … • Theoretical underpinnings. • Links to AVI’s, model reduction, wavelets, ... • Multiresolution solvers. • Countless applications.

More Related