Joseph Benzaken

Joseph Benzaken, Alireza Doostan, John A. Evans

In this paper, we present a novel tolerance allocation algorithm for the assessment and control of geometric variation on system performance that is applicable to any system of partial differential equations. In particular, we parameterize the geometric domain of the system in terms of design parameters and subsequently measure the effect of design parameter variation on system performance. A surrogate model via a tensor representation is constructed to map the design parameter variation to the system performance. A set of optimization problems over this surrogate model restricted to nested hyperrectangles represents the effect of prescribing design tolerances, where the maximizer of this restricted function depicts the worst-case member, i.e. the worst-case design. Moreover, the loci of these tolerance hyperrectangles with maximizers attaining, but not surpassing, the performance constraint represents the boundary to the feasible region of allocatable tolerances. Every tolerance in this domain is measured through a user-specified, weighted norm which is informed by design considerations such as cost and manufacturability. The boundary of the feasible set is elucidated as an immersed manifold of codimension one, over which a suite of optimization routines exist and are employed to efficiently determine an optimal feasible tolerance with respect to the specified measure. Examples of this algorithm are presented with applications to a plate with a hole described by two design parameters, a plate with a hole described by six design parameters, and an L-Bracket described by seventeen design parameters.

Christopher Coley, Joseph Benzaken, John A. Evans

In this paper, we present a geometric multigrid methodology for the solution of matrix systems associated with isogeometric compatible discretizations of the generalized Stokes and Oseen problems. The methodology provably yields a pointwise divergence-free velocity field independent of the number of pre-smoothing steps, post-smoothing steps, grid levels, or cycles in a V-cycle implementation. The methodology relies upon Scwharz-style smoothers in conjunction with specially defined overlapping subdomains that respect the underlying topological structure of the generalized Stokes and Oseen problems. Numerical results in both two- and three-dimensions demonstrate the robustness of the methodology through the invariance of convergence rates with respect to grid resolution and flow parameters for the generalized Stokes problem as well as the generalized Oseen problem provided it is not advection-dominated.