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.
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.
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