Abstract
We formulate a new class of optimization-based methods for data transfer (remap) of a scalar conserved quantity between two close meshes with the same connectivity. We present the methods in the context of the remap of a mass density field, which preserves global mass (the integral of the density over the computational domain). The key idea is to formulate remap as a global inequality-constrained optimization problem for mass fluxes between neighboring cells. The objective is to minimize the discrepancy between these fluxes and the given high-order target mass fluxes, subject to constraints that enforce physically motivated bounds on the associated primitive variable. In so doing, we separate accuracy considerations, handled by the objective functional, from the enforcement of physical bounds, handled by the constraints. The resulting second-order, conservative, and bound-preserving optimization-based remap (OBR) formulation is applicable to general, unstructured, heterogeneous grids. Under some weak requirements on grid proximity we prove that the OBR algorithm preserves linear fields in one, two and three dimensions. The chapter also examines connections between the OBR and the flux-corrected remap (FCR), which can be interpreted as a modified version of OBR (M-OBR), with the same objective but a smaller feasible set. The feasible set for M-OBR (FCR) is given by simple box constraints derived by using a “worst-case” scenario approach, which may result in loss of linearity preservation and ultimately accuracy for some grid motions. The optimality of the OBR solution means that, given a set of target fluxes and a distance measure, OBR finds the best possible approximations of these fluxes with respect to this measure, which also satisfy the physically motivated bounds. In this sense, OBR can serve as a natural benchmark for evaluating the accuracy of existing and future numerical methods for data transfer with respect to a given class of flux reconstruction methods and flux distance measures. In this context, we perform numerical comparisons between OBR, FCR and iFCR (a version of FCR which utilizes an iterative procedure to enhance the accuracy of FCR numerical fluxes).
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Notes
- 1.
Typically, in a continuous rezone ALE the rezoned grid is close to the Lagrangian but has better geometric quality.
- 2.
In this chapter, we use the set-relational definitions and the corresponding geometric interpretations of ⊂, ⊆, ∪, ∩, ∖ and ∈ interchangeably. Their meaning will be clear from the context. In particular, relations between entities defined on \(\widetilde {K}_{h}({\varOmega})\) and those defined on K h (Ω) only make sense when interpreted geometrically relative to the common domain Ω.
- 3.
In practice, this also means that the integrals in (17) should be approximated by quadratures that are exact for linear functions.
- 4.
The Euclidean distance is used for simplicity. The objective can be defined using any valid distance function (or, equivalently, norm).
- 5.
This guarantees that taking min and max in (24) is well-defined. Otherwise, the correct statement of this result should involve inf and sup.
- 6.
For a complete match with FCR we can set all free variables to 1.
- 7.
Because side nodes can move in different directions swept regions are not simple extrusions of the sides, which can complicate the computation of integrals. Using Green’s theorem, integrals of polynomials over swept regions can be replaced by integrals of higher-degree polynomials over the (lower-dimensional) boundaries of these regions, see Margolin and Shashkov [17], Dukowicz and Kodis [8]. This provides an efficient way to compute the fluxes, regardless of the shape of the swept regions.
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Acknowledgements
All authors acknowledge funding by the DOE Office of Science Advanced Scientific Computing Research (ASCR) Program. PB, DR and GS also acknowledge funding by the NNSA Climate Modeling and Carbon Measurement Project. DR and MS also acknowledge funding by the Advanced Simulation & Computing (ASC) Program.
Our colleagues Dmitri Kuzmin, Richard Liska, Kara Peterson, John Shadid, Pavel Váchal and Joseph Young provided many comments and valuable insights that helped improve this work.
Sandia National Laboratories is a multi-program laboratory operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000. The work of MS was carried out under the auspices of the National Nuclear Security Administration of the U.S. Department of Energy at Los Alamos National Laboratory under Contract No. DE-AC52-06NA25396.
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Bochev, P., Ridzal, D., Scovazzi, G., Shashkov, M. (2012). Constrained-Optimization Based Data Transfer. In: Kuzmin, D., Löhner, R., Turek, S. (eds) Flux-Corrected Transport. Scientific Computation. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-4038-9_10
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