Goal-oriented model adaptivity for viscous incompressible flows

Abstract

In van Opstal et al. (Comput Mech 50:779-788, 2012) airbag inflation simulations were performed where the flow was approximated by Stokes flow. Inside the intricately folded initial geometry the Stokes assumption is argued to hold. This linearity assumption leads to a boundary-integral representation, the key to bypassing mesh generation and remeshing. It therefore enables very large displacements with near-contact. However, such a coarse assumption cannot hold throughout the domain, where it breaks down one needs to revert to the original model. The present work formalizes this idea. A model adaptive approach is proposed, in which the coarse model (a Stokes boundary-integral equation) is locally replaced by the original high-fidelity model (Navier-Stokes) based on a-posteriori estimates of the error in a quantity of interest. This adaptive modeling framework aims at taking away the burden and heuristics of manually partitioning the domain while providing new insight into the physics. We elucidate how challenges pertaining to model disparity can be addressed. Essentially, the solution in the interior of the B T. M. van Opstal coarse model domain is reconstructed as a post-processing step. We furthermore present a two-dimensional numerical experiments to show that the error estimator is reliable.

Model adaptivity for goal-oriented inference using adjoints

“…We instead use the alternative method described in [35], decomposing the error estimate into contributions from locally supported basis functions rather than elements. Recall the error estimate…”

“…In contrast, concurrent methods (also called hybrid methods) simultaneously solve the higher-and lower-fidelity models in different parts of the domain. Applications include computational mechanics [20,30], porous media flow [5,34], and fluid dynamics [3,15,17,24,35,39]. We focus on concurrent methods of combining models, which have desirable features: in the case where the high-fidelity model is nonlinear, replacing it with a linear lower-fidelity model in most of the domain can reduce the number of iterative solves needed; when the high-fidelity model has a fine resolution and/or many parameters, replacing it with a lower-fidelity model can reduce the number of degrees of freedom of the mixed-fidelity model.…”

Worst-case multi-objective error estimation and adaptivity

“…If the worst-case multi-objective error estimate is to be applied in an adaptiverefinement process, then a decomposition of the error into basis-function contributions must be introduced, cf. Equation (43). Denoting by {ψ i } a basis of V h/2 0,∂Ω\ω , there exist coefficients {σ i } such that:…”

“…In these approaches, the error indicator is applied to systematically decide between a simple coarse model and a complex sophisticated model, in such a manner that the sophisticated model is only applied in regions of the domain that contribute most significantly to the objective functional under consideration. For examples of goal-oriented model adaptivity, we refer to [29] for application of goal-oriented model adaptivity to heterogeneous materials, to [3] for goal-oriented atomistic/continuum adaptivity in solid materials, and to [43] for goal-oriented adaptivity between a boundary-integral formulation of the Stokes equations and a PDE formulation of the Navier-Stokes equations.…”

Error estimation and adaptive moment hierarchies for goal-oriented approximations of the Boltzmann equation

“…Applications in which rarefaction effects play a significant role are multitudinous, including gas flow problems involving large mean free paths in high-altitude flows and hypobaric applications such as chemical vapor deposition; see [8,9] and references therein for further examples. Moreover, the perpetual trend toward miniaturization in science and technology renders accurate descriptions of fluid flows in the transitional molecular/continuum regime of fundamental technological relevance, for instance, in nanoscale applications, micro-channel flows refer to [34] for application of goal-oriented model adaptivity to heterogeneous materials, to [35] for goaloriented atomistic/continuum adaptivity in solid materials, and to [36] for goal-oriented adaptivity between the Stokes equations and the Navier-Stokes equations. The Galerkin form of moment methods enables the construction of accurate a-posteriori error estimates, while the hierarchical structure provides an intrinsic mode of refinement.…”