152 research outputs found

    Filling holes under non-linear constraints

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    In this paper we handle the problem of filling the hole in the graphic of a surface by means of a patch that joins the original surface with C1-smoothness and fulfills an additional non-linear geometrical constraint regarding its area or its mean curvature at some points. Furthermore, we develop a technique to estimate the optimum area that the filling patch is expected to have that will allow us to determine optimum filling patches by means of a system of linear and quadratic equations. We present several numerical and graphical examples showing the effectiveness of the proposed method.Funding for open access publishing: Universidad de Granada/CBUANational funds through the FCT - Fundação para a Ciência e a TecnologiaProjects UIDB/00297/2020 and UIDP/00297/2020 (Center for Mathematics and Applications

    Application of spectral/hp element methods to high-order simulation of industrial automotive geometries

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    Flow predictions around cars is a challenge due to massively separated flow and complex flow structures generated. These flow features are usually poorly predicted by present industrial computational fluid dynamics (CFD) codes based on a low fidelity Reynolds averaged Navier Stokes (RANS) approach simulating the mean effects of turbulence. On the other hand, high fidelity approaches resolve turbulent scales but require many more degrees of freedom than classical techniques for an accurate solution. Previous applications have shown that the coupling of the spectral/hp element method and implicit large eddy simulation (iLES) turbulence treatment could be a potential candidate to perform high-fidelity simulations. This work aims at transferring the spectral/hp element technology to the automotive industry in which high Reynolds numbers and complex geometries are typical. Recent developments in stabilisation techniques such as the discontinuous Galerkin kernel spectral vanishing viscosity (SVV) and high-order meshing capabilities open the possibility of the application of the spectral/hp element method to complex cases. The technology is first implemented to an industrial case proposed by McLaren Automotive Limited (MLA) at a realistic Reynolds number of 2,3 million based on the front wheel diameter and is compared to a RANS numerical development tool. Differences in terms of vortical structures arrangement, principally due to the front wheel wake are highlighted. In parallel, a workflow is developed to systematically address similar complex cases. The interaction between h-refinement, related to the size of the elements of the mesh, and p-refinement, corresponding to the polynomial expansion order, is investigated on the SAE notchback body. Two different hp-refinement strategies with similar numbers of degrees of freedom are employed, the first one with a fine mesh and a third-order accurate polynomial expansion and the second one with a coarse mesh and a fifth-order accurate polynomial expansion. Results show that a minimum level of h-refinement is necessary to capture flow features and that p-refinement can subsequently be used to improve their resolution. The final part focuses on wheel rotation modelling. Scale-resolving techniques are intrinsically unsteady and therefore require sophisticated techniques to correctly model rotating wheels. A procedure, built upon an immersed boundary method (IBM) called the smoothed profile method (SPM), is developed to model complex three-dimensional rotating geometries, in particular rim spokes. It is finally applied to an isolated rotating wheel case and results are compared to the moving wall (MW) and the moving reference frame (MRF) modelling techniques. It is concluded that the SPM is in better qualitative agreement with experimental results present in the literature than the two other modelling strategies.Open Acces

    Collapsibility and Z-Compactifications of CAT(0) Cube Complexes

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    We extend the notion of collapsibility to non-compact complexes and prove collapsibility of locally-finite CAT(0) cube complexes. Namely, we construct such a cube complex XX out of nested convex compact subcomplexes {Ci}i=0\{C_i\}_{i=0}^\infty with the properties that X=i=0CiX=\cup_{i=0}^\infty C_i and CiC_i collapses to Ci1C_{i-1} for all i1i\ge 1. We then define bonding maps rir_i between the compacta CiC_i and construct an inverse sequence yielding the inverse limit space lim{Ci,ri}\varprojlim\{C_i,r_i\}. This will provide a new way of Z-compactifying XX. In particular, the process will yield a new Z-boundary, called the cubical boundary

    Filling holes under non-linear constraints

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    Publisher Copyright: © 2023, The Author(s).In this paper we handle the problem of filling the hole in the graphic of a surface by means of a patch that joins the original surface with C1-smoothness and fulfills an additional non-linear geometrical constraint regarding its area or its mean curvature at some points. Furthermore, we develop a technique to estimate the optimum area that the filling patch is expected to have that will allow us to determine optimum filling patches by means of a system of linear and quadratic equations. We present several numerical and graphical examples showing the effectiveness of the proposed method.publishersversionpublishe

    Derived CC^{\infty}-Geometry I: Foundations

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    This work is the first in a series laying the foundations of derived geometry in the CC^{\infty} setting, and providing tools for the construction and study of moduli spaces of solutions of Partial Differential Equations that arise in differential geometry and mathematical physics. To advertise the advantages of such a theory, we start with a detailed introduction to derived CC^{\infty}-geometry in the context of symplectic topology and compare and contrast with Kuranishi space theory. In the body of this work, we avail ourselves of Lurie's extensive work on abstract structured spaces to define \infty-categories of derived CC^{\infty}-ring and CC^{\infty}-schemes and derived CC^{\infty}-rings and CC^{\infty}-schemes with corners via a universal property in a suitable (,2)(\infty,2)-category of \infty-categories with respect to the ordinary categories of manifolds and manifolds with corners (with morphisms the bb-maps of Melrose in the latter case), and prove many basic structural features about them. Along the way, we establish some derived flatness results for derived CC^{\infty}-rings of independent interest.Comment: 203 pages; comments welcom

    A Generalized Blending Scheme for Arbitrary Order of Continuity

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    In this thesis, new templates and formulas of blending functions, schemes, and algorithms are derived for solving the scattered data interpolation problem. The resulting data fitting scheme interpolates the positions and derivatives of a triangular mesh, and for each triangle of the mesh blends three triangular sub-surfaces, and creates a triangular patch. Similar to some existing schemes, the resulting surface inherits the derivatives of the sub-surfaces on the boundaries. In contrast with existing schemes, the new scheme has additional properties: The order of interpolated derivatives is extended to arbitrary values, and the restrictions of the sub-surfaces are relaxed. Then based on the properties of the new blending functions, an algorithm for constructing smooth triangular surfaces with global geometric continuity is described. The new blending functions and the scheme are then extended to multi-sided faces. The algorithm using these new blending functions accepts data sites formed by multi-sided polygons

    Applications of Special Functions in High Order Finite Element Methods

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    In this thesis, we optimize different parts of high order finite element methods by application of special functions and symbolic computation. In high order finite element methods, orthogonal polynomials like the Jacobi polynomials are deeply rooted. A broad classical theory of these polynomials is known. Moreover, with modern computer algebra software we can extend this knowledge even further. Here, we apply this knowledge and software for different special functions to derive new recursive relations of local matrix entries. This massively optimizes the assembly time of local high order finite element matrices. Furthermore, the introduced algorithm is in optimal complexity. Moreover, we derive new high order dual functions, which result in fast interpolation operators. Lastly, efficient recursive algorithms for hanging node constraint matrices provided by this new dual functions are given

    LIPIcs, Volume 258, SoCG 2023, Complete Volume

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    LIPIcs, Volume 258, SoCG 2023, Complete Volum

    Automatic Generation of Near-Body Structured Grids

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    Numerical grid generation has been a bottleneck in the computational fluid dynamics process for a long time when using the structured overset grids. Many current structured overset grid generation schemes like the hyperbolic grid generation method require significant user interaction to generate good computational grids robustly. Other grid generation schemes like the elliptic grid generation method take a significant amount of time for grid calculation, which is not desirable for computational fluid dynamics. Herein a new grid generation method is presented that combines the hyperbolic grid generation scheme with the elliptic grid generation scheme that uses Poisson’s equation. The new scheme builds upon the strengths of the different techniques by first applying hyperbolic grid generation, which is very fast but sometimes fails in strong concavities, and then using elliptic grid generation to locally fix the problems where hyperbolic grid generation results are not acceptable for computational fluid dynamics calculation. The new technique is demonstrated in various examples that are known to cause problems for either hyperbolic or elliptic grid generation when applied alone. The computational speed of the combined scheme grid generation is also exanimated by comparing the results with hyperbolic and elliptic grid generation. The combined grid generation scheme is further implemented in Engineering Sketch Pad to get useful near-body structure grids based on the geometry of the model. Attributes in Engineering Sketch Pad are used to define the places where the surface and volume grids should be generated, while the tessellations are used to locate and project grid generation results and therefore boost grid generation speed. Three cases are tested to illustrate the implementation of the combined grid generation scheme in Engineering Sketch Pad
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