basis refinement. The model reduction and control optimization stages could be repeated after an adaptive refinement of the Galerkin discretization, using the performance evaluation of the optimized controller to design an error indicator. This could potentially create multi-resolution reduced-models with very few inputs and outputs at the coarsest level.
Novel Error Estimators In our implementation we adopted well-established a posteriori error indicators.
Since these give per-element errors, and we need to make (de)activation decisions per-function, we must con- vert: we lump the element error onto the basis functions, e.g., the error associated toφi(x)isR
Ωφi(x)γ(x)dx, whereγ(x)is the piecewise constant error density, computed via equidistribution of elemental error over the elemental subdomain. This appears to be an effective technique, and it has the benefit that we may adopt any element-based error measure. However, we would be more satisfied with a posteriori error estimators which focus directly on the basis functions, i.e., estimate the change in error due to (de)activating a particular basis function. Later, it would be very desirable to build a mathematical framework for transforming per-element derivations into per-function derivations.
Links to Hierarchical and Multigrid Preconditioners Preconditioning techniques are critical for accel- erating the convergence of iterative solvers. Recently Green demonstrated the preconditioning of problems which use Cirak’s subdivision elements [Cirak et al. 2000a]; Green exploits the multi-resolution of subdivi- sion discretizations to great effect, reporting encouraging results for the preconditioning of an elliptic PDE based on the Kirchoff-Love formulation of thin shells [Green et al. 2002]. Since multi-resolution discretiza- tions lie at the heart of our method, the necessary structure exists to accommodate multigrid [Bank et al. 1988, Briggs et al. 2000] or wavelet [Cohen and Masson 1997] preconditioning following the example presented by Green.
Advances in Interpolated Unrefinement Unrefinement is inherently lossy. It requires projection onto a smaller approximation space. In general, this leads to undesirable artifacts in simulation applications, in particular temporal discontinuities in the configuration at the point in time that unrefinement occurs. One way to resolve this is to permit unrefinement only when the current approximated solution is contained in the smaller, unrefined trial space. This is lossless, but performance crippling —we might never be allowed to unrefine!— in general we expect there to be some (small but non-zero) error during refinement.
To remedy this, our implementation uses interpolated unrefinement. A basis function is smoothly deac- tivated: the deactivation ofφi(x)is separated into two steps: (a) the solver no longer considers as unknown the associated coefficient,ui, and (b) the coefficient is set to zero, i.e.,ui →0. In immediate (nonsmooth) deactivation, both of these steps are performed instantaneously. In smooth deactivation, (a) the unknown, ui, is (as always) immediately removed from the solver’s reach, i.e., it is considered a prescribed boundary condition, and (b) the value of the coefficient is prescribed by a smoothly decaying function over some finite time interval. e.g.,ui(t)→0ast→t0+ ∆tdd, wheret0is the time at which deactivation commences and
∆tddis the deactivation delay. In our implementation we chose a simple linear decay, leading to continuity over time of a configuration but not its velocity. This removed the visual “blips” from the animation; how- ever keen observers note that the discontinuity in velocity is noticeable and undesirable—human observers are naturally trained to recognize sudden changes in velocity since these are associated with impulses. To that end, it may be desirable to use decay curves which have zero initial and final time derivatives, such that continuity of velocity is preserved during interpolated unrefinement. In contrast, we believe that continuity of acceleration is not equally important, since many physical systems have discontinuous accelerations.
Links to Advected Bases and Traveling Wavelets Adaptive basis refinement may be of benefit in the simulation of turbulent fluid flows. These problems are characterized by advection phenomena, i.e., the solution’s features are moving over the domain with some velocity. Consider that an advection operator, to first order, locally translates pieces of the solution from one region of spatial domain to another. There is beauty in approximating the advected solution using advected basis functions, i.e., let the advection operator be a map from one approximation space not onto itself, rather to an advected space. This space is formed as follows. Place particles at the center of the original approximation space. Advect the particles, and build a set of advected basis functions centered at the advected particles. This approach is inspired by earlier work on traveling wavelets [Perrier and Basdevant 1991].
Links to Tagged Subdivision: Multi-nesting Relations An attractive attribute of subdivision schemes is their support for tagged meshes, which associate descriptive tags to each mesh entity, e.g., sharp crease edge, corner vertex, flat face (see, e.g., Biermann’s use of tags with Loop subdivision [Biermann et al. 2000]).
Tags modify the subdivision stencil in the vicinity of the tagged entity. For example, sharp crease tags modify the subdivision stencil to prevent smoothing (or otherwise propagating information) across the two surface patches incident to the tagged edge. This gives significant control over the shape of the limit function, most often over its smoothness. Sharp creases, for example, induce discontinuous derivatives across the associated patch boundaries. Recently DeRose introduced more general semi-sharp creases, which under the control of a real-valued sharpness index induce arbitrarily large but bounded derivatives across patch boundaries [DeRose et al. 1998].
The possibility of decorating a mesh with tags creates an expanded family of subdivision scaling functions associated to every (local) permutation of tags. With the introduction of parameterized tags (e.g., sharpness indices) an infinite set of scaling functions (corresponding to different values of the parameter) associates to a single mesh entity. Every spaceV(p)is now richer; subdivision theory guarantees that the nesting relation still holds, i.e.,V(p) ⊂ V(p+1). Furthermore, we may be able to generalize the nesting relation, defining a multi-dimensional nesting (multi-nesting) relation: let the spaceV(p,s) be spanned by all level-pscaling functions associated to meshes with arbitrary sharpness indices< s. We view this space as a point in a two dimensional space-of-spaces, i.e.,V(p,s)∈ V, with a discrete first dimension (p∈Z∗) and continuous second
dimension (s∈R, s≥0). ThenV(p,s)is nested in its first dimension, i.e.,V(p,s)⊂V(p+1,s), as well as its second, i.e.,V(p,s)⊂V(p,t), s < t. Refinement may naturally extend the current approximation space along any of the nesting directions, e.g., adding finer functions, or adding sharper functions.
Simulations of crushing, buckling and wrinkling benefit from adaptive approaches because they have phenomenological singularities. When the only approach to capturing these singularities is to make the dis- cretization locally finer (i.e., so-calledh-refinement [Eriksson et al. 1996]), the result is the introduction of numerous extremely fine scaling functions. While adaptivity avoids globally switching to a finer discretiza- tion, there is still a computational penalty near sharp (less smooth) features of the solution. Our hope is that the ability to introduce better-suited (e.g., sharper) scaling functions fitted to the features of the solution may significantly reduce the need for exceedingly-fine discretizations.
Links to Tagged Subdivision: Scaling Functionals Alternatively, mesh tag parameters (such as the sharp- ness index) may be viewed as parameters of a scaling functional, i.e., the trial space consists of all functions Pφi(ui, si)(x), where the scaling functionals,φi(ui, si) ∈ H(Ω), are linear in their first argument (e.g., displacement) but not their second (e.g., sharpness). In general, the functional may take several sharp- ness arguments, i.e., φi(ui, si,1, si,2, . . . , si,N), corresponding to theN edges within its support. In this approach there are multiple coefficients associated to a single scaling functional. Of all formulations, it may be that the variational form is best-suited for dealing with this setting. It remains to be seen whether this is a useful construction; for both applications of tagged subdivision, we are inspired by the success of p-refinement [Eriksson et al. 1996], ridgelets [Cand`es 1998] and curvelets [Cand`es and Donoho 2000].