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

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  State-of-the-art refinement not applicable to subdivision surfaces.  Refinement should take advantage of the multiresolution nature 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

Adapted basis 1  Quasi-hierarchical basis:  Some basis functions are removed: Nodes associated with active basis functions

Adapted basis 2  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 C onforming H ierarchica l A daptive R efinement M ethod S  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 mesh Common AMR w/ constraints True hierarchical basis Quasi-hierarchical basis

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 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 1Level 2Level 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 1Level 2Level 3 Solution displayed on the integration cells Grid hierarchy Quasi-hierarchical basis; Adaptive step 2: 3,900 degrees of freedom

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.