Petr Krysl*Eitan Grinspun, Peter

Schröder

Hierarchical Finite

Element Mesh

Refinement*Structural Engineering Department,

University of California, San DiegoComputer Science Department,

California Institute of Technology

Adaptive Approximations Adjust spatial resolution by:

1. Remeshing2. 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 SubdivisionState-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

sequenceMesh is globally refined to form and so on…

Strict nesting of

)0(M )1(M

)1(M)2(M)3(M

)0(M

)(nM

Refinement EquationRefinement relation

Refined basis of Any linearly independent set of basis functions chosen from with

)(on basis iMB

][span][span *BB

]supp[

)1()1()(

)(

)()(nkNj

nk

nkj

nk xNxN

*B*B

)(iM

Adapted basis 1Quasi-hierarchical basis:

Some basis functions are removed:

)1(M)2(M)3(M

)0(M

Nodes associated with active basis functions

Adapted basis 2True hierarchical basis

Details are added to coarser functions:

)1(M)2(M)3(M

)0(M

Nodes associated with active basis functions

Multi-level approximationApproximation of a function

on multiple mesh levels

Literal interpretation of the refinement equation has a big advantage: genericity.

)()()...0( )()()(

mj

m Rj

mj

mh vxNxv

m

)(mR= set of refined basis functions on level m

CHARMS

Conforming

Hierarchical

Adaptive

Refinement

MethodS

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).

CHARMS vs common AMR

CHARMS

Level 0

Level 1

Original basis on quadrilateral mesh

Adapted basis on a refined meshCommon AMR w/ constraints

True hierarchical

basis

Quasi-hierarchicalbasis

Refinement for SubdivisionCHARMS apply to subdivision

surfaces without any change.The multiresolution character

of subdivision surfaces is taken advantage of quite naturally.

…

AlgorithmsField 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

Hierarchy of basis function sets;Red balls: the active functions.

Solution painted on the integration cells.

Poisson equation with homogeneous Dirichlet bc.

Quasi hierarchical basis.

True hierarchical basis.

3D Example

3-level grid(true hierarchical)

Solution painted on the integration cells

Heat diffusion: Hierarchical

Level 1 Level 2 Level 3

Solution displayed on the integration cells

Grid hierarchy

True hierarchical basis; Adaptive step 2:

5,000 degrees of freedom(~3,000 hierarchical)

Heat diffusion: Quasi-hier.

Level 1 Level 2 Level 3

Solution displayed on the integration cells

Grid hierarchy

Quasi-hierarchical basis; Adaptive step 2:

3,900 degrees of freedom

HighlightsEasy 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.