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  • 1.
    Ålund, Oskar
    et al.
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Iaccarino, Gianluca
    Department of Mechanical Engineering and Institute for Computational Mathematical Engineering, Stanford University, Stanford, California, USA.
    Nordström, Jan
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Learning to Differentiate2020Report (Other academic)
    Abstract [en]

    Artificial neural networks together with associated computational libraries provide a powerful framework for constructing both classification and regression algorithms. In this paper we use neural networks to design linear and non-linear discrete differential operators. We show that neural network based operators can be used to construct stable discretizations of initial boundary-value problems by ensuring that the operators satisfy a discrete analogue of integration-byparts known as summation-by-parts. Furthermore we demonstrate the benefits of building the summation-by-parts property into the network by weight restriction, rather than enforcing it through a regularizer. We conclude that, if possible, known structural elements of an operation are best implemented as innate—rather than learned—properties of the network. The strategy developed in this work also opens the door for constructing stable differential operators on general meshes.

  • 2.
    Ålund, Oskar
    et al.
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Nordström, Jan
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    A stable, high order accurate and efficient hybrid method for flow calculations in complex geometries2018In: 2018 AIAA Aerospace Sciences Meeting, AIAA SciTech Forum, (AIAA 2018-1096), American Institute of Aeronautics and Astronautics, 2018, no 210059, p. 1-9Conference paper (Refereed)
    Abstract [en]

    The suitability of a discretization method is highly dependent on the shape of the domain. Finite difference schemes are typically efficient, but struggle with complex geometry, while finite element methods are expensive but well suited for complex geometries. In this paper we propose a provably stable hybrid method for a 2D advection–diffusion problem, using a class of inner product compatible projection operators to couple the non-conforming grids that arise due to varying the discretization method throughout the domain.

  • 3.
    Ålund, Oskar
    et al.
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Nordström, Jan
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Encapsulated high order difference operators on curvilinear non-conforming grids2019In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 385, p. 209-224Article in journal (Refereed)
    Abstract [en]

    Constructing stable difference schemes on complex geometries is an arduous task. Even fairly simple partial differential equations end up very convoluted in their discretized form, making them difficult to implement and manage. Spatial discretizations using so called summation-by-parts operators have mitigated this issue to some extent, particularly on rectangular domains, making it possible to formulate stable discretizations in a compact and understandable manner. However, the simplicity of these formulations is lost for curvilinear grids, where the standard procedure is to transform the grid to a rectangular one, and change the structure of the original equation. In this paper we reinterpret the grid transformation as a transformation of the summation-by-parts operators. This results in operators acting directly on the curvilinear grid. Together with previous developments in the field of nonconforming grid couplings we can formulate simple, implementable, and provably stable schemes on general nonconforming curvilinear grids. The theory is applicable to methods on summation-by-parts form, including finite differences, discontinuous Galerkin spectral element, finite volume, and flux reconstruction methods. Time dependent advection–diffusion simulations corroborate the theoretical development.

    The full text will be freely available from 2021-02-26 12:23
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