189 research outputs found

    Application of Besov spaces to spline approximation

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    Lacunary interpolation by quartic splines on uniform meshes

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    AbstractThe interpolation of a discrete set of data on the interval [0, 1], representing the first and the second derivatives (except at 0) of a smooth function f is investigated via quartic C2-splines. Error bounds in the uniform norm for ∥s(i) − f(i)∥, i=0(1)2, if f ∈ Cl[0, 1], l=3, 5 and (3) ∈ BV[0, 1], together with computational examples will also be presented

    Optimum quadrature formulae and natural splines

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    Geometry considerations for high-order finite-volume methods on structured grids with adaptive mesh refinement

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    2022 Summer.Includes bibliographical references.Computational fluid dynamics (CFD) is an invaluable tool for engineering design. Meshing complex geometries with accuracy and efficiency is vital to a CFD simulation. In particular, using structured grids with adaptive mesh refinement (AMR) will be invaluable to engineering optimization where automation is critical. For high-order (fourth-order and above) finite volume methods (FVMs), discrete representation of complex geometries adds extra challenges. High-order methods are not trivially extended to complex geometries of engineering interest. To accommodate geometric complexity with structured AMR in the context of high-order FVMs, this work aims to develop three new methods. First, a robust method is developed for bounding high-order interpolations between grid levels when using AMR. High-order interpolation is prone to numerical oscillations which can result in unphysical solutions. To overcome this, localized interpolation bounds are enforced while maintaining solution conservation. This method provides great flexibility in how refinement may be used in engineering applications. Second, a mapped multi-block technique is developed, capable of representing moderately complex geometries with structured grids. This method works with high-order FVMs while still enabling AMR and retaining strict solution conservation. This method interfaces with well-established engineering work flows for grid generation and interpolates generalized curvilinear coordinate transformations for each block. Solutions between blocks are then communicated by a generalized interpolation strategy while maintaining a single-valued flux. Finally, an embedded-boundary technique is developed for high-order FVMs. This method is particularly attractive since it automates mesh generation of any complex geometry. However, the algorithms on the resulting meshes require extra attention to achieve both stable and accurate results near boundaries. This is achieved by performing solution reconstructions using a weighted form of high-order interpolation that accounts for boundary geometry. These methods are verified, validated, and tested by complex configurations such as reacting flows in a bluff-body combustor and Stokes flows with complicated geometries. Results demonstrate the new algorithms are effective for solving complex geometries at high-order accuracy with AMR. This study contributes to advance the geometric capability in CFD for efficient and effective engineering applications

    Theory-based Benchmarking of the Blended Force-Based Quasicontinuum Method

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    We formulate an atomistic-to-continuum coupling method based on blending atomistic and continuum forces. Our precise choice of blending mechanism is informed by theoretical predictions. We present a range of numerical experiments studying the accuracy of the scheme, focusing in particular on its stability. These experiments confirm and extend the theoretical predictions, and demonstrate a superior accuracy of B-QCF over energy-based blending schemes.Comment: 25 pages, color figures; some numerical experiments re-don
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