Hierarchical Finite Element Mesh Refinement

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# Hierarchical Finite Element Mesh Refinement - PowerPoint PPT Presentation

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

Petr Krysl*

Eitan Grinspun, Peter Schröder

*Structural Engineering Department,

University of California, San DiegoComputer Science Department,

California Institute of Technology

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

Nodes associated with active basis functions

• True hierarchical basis
• Details are added to coarser functions:

Nodes associated with active basis functions

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
• 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).
• Refinement/coarsening intrinsic (prolongation and restriction).

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 Subdivision

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

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

Solution painted on the integration cells

3-level grid

(true hierarchical)

Heat diffusion: Hierarchical

Solution displayedon the integrationcells

True hierarchical basis;

5,000 degrees of freedom

(~3,000 hierarchical)

Grid hierarchy

Level 1

Level 2

Level 3

Heat diffusion: Quasi-hier.

Solution displayedon the integrationcells

Quasi-hierarchical basis;

3,900 degrees of freedom

Grid hierarchy

Level 1

Level 2

Level 3

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