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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:

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hierarchical finite element mesh refinement

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

adaptive approximations
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!

refinement for subdivision
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.

conceptual hierarchy
Conceptual Hierarchy
  • Infinite globally-refined sequence
    • Mesh is globally refined to form and so on…
    • Strict nesting of
refinement equation
Refinement Equation
  • Refinement relation
  • Refined basis of
    • Any linearly independent set of basis functions chosen from with
adapted basis 1
Adapted basis 1
  • Quasi-hierarchical basis:
    • Some basis functions are removed:

Nodes associated with active basis functions

adapted basis 2
Adapted basis 2
  • True hierarchical basis
    • Details are added to coarser functions:

Nodes associated with active basis functions

multi level approximation
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

charms
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

charms vs common amr
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

refinement for subdivision1
Refinement for Subdivision

  • CHARMS apply to subdivision surfaces without any change.
  • The multiresolution character of subdivision surfaces is taken advantage of quite naturally.
algorithms
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.
2d example
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.

3d example
3D Example

Solution painted on the integration cells

3-level grid

(true hierarchical)

heat diffusion hierarchical
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

heat diffusion quasi hier
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

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