A method for computing curved meshes via the linear elasticity analogy, application to fluid dynamics problems

Abgrall, Rémi; Dobrzynski, Cécile; Froehly, Algiane

doi:10.1002/fld.3932

Cited by 37 publications

(38 citation statements)

References 21 publications

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“…Hierarchical methods proceed to curve edges first, then faces, then elements [Ziel et al 2017] although behavior on complex 3D domains has not been demonstrated. Arguably, the most successful family of curved meshing methods rely on a mechanical analogy: these optimization-based methods use an initial straight-edge mesh as a reference domain and minimize a non-linear distortion using FEM between the reference domain and the parametric elements, thus effectively computing an elastostatic solution where the distortion they target defines a potential energy of deformation [Abgrall et al 2012;Johnen et al 2013]. However, the non-linearity of the potential is often an obstacle to finding a good minimum, and starting from a straight-edge mesh of the domain unnecessarily adds distortion if this mesh is not of high quality.…”

Section: Previous Workmentioning

confidence: 99%

“…Our use of a reference domain with unit elements is quite different from any of the other methods anchored in continuum mechanics such as [Abgrall et al 2012;Bargteil and Cohen 2014;Johnen et al 2013;Persson and Peraire 2009] as they all use non-unit straightedge triangulations as reference, which unnecessarily biases the notion of isotropy based on the quality of the tetrahedron mesh they start from. Instead, our interpretation phrases the optimization as a minimization of distortion with respect to a perfect mesh.…”

Section: Odt As Elastostaticsmentioning

confidence: 99%

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Curved optimal delaunay triangulation

et al. 2018

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Meshes with curvilinear elements hold the appealing promise of enhanced geometric flexibility and higher-order numerical accuracy compared to their commonly-used straight-edge counterparts. However, the generation of curved meshes remains a computationally expensive endeavor with current meshing approaches: high-order parametric elements are notoriously difficult to conform to a given boundary geometry, and enforcing a smooth and non-degenerate Jacobian everywhere brings additional numerical difficulties to the meshing of complex domains. In this paper, we propose an extension of Optimal Delaunay Triangulations (ODT) to curved and graded isotropic meshes. By exploiting a continuum mechanics interpretation of ODT instead of the usual approximation theoretical foundations, we formulate a very robust geometry and topology optimization of Bézier meshes based on a new simple functional promoting isotropic and uniform Jacobians throughout the domain. We demonstrate that our resulting curved meshes can adapt to complex domains with high precision even for a small count of elements thanks to the added flexibility afforded by more control points and higher order basis functions.

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Section: Previous Workmentioning

confidence: 99%

Section: Odt As Elastostaticsmentioning

confidence: 99%

Curved optimal delaunay triangulation

et al. 2018

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“…Most of them involve the ratio of minimum to maximum Jacobian in each element [8,10,11,12,13,14], the ratio of the minimum Jacobian to the Jacobian of the corresponding straight-sided element [1] or the integral of the Jacobian divided by the Jacobian of the straight-sided element [15].…”

Section: Detection Of Invaliditiesmentioning

confidence: 99%

“…A more robust method may be obtained by using non-uniform material properties, that is by using stiffer material in small elements near curved boundaries. More successful strategies have been proposed, including a nonlinear mechanics formulation [14] and an iterative method working on the Bézier control points of the elements instead of the Lagrangian nodes [12].…”

Section: Regularization Of Invalid Meshesmentioning

confidence: 99%

The Generation of Valid Curvilinear Meshes

Geuzaine

Johnen

Lambrechts

et al. 2015

Notes on Numerical Fluid Mechanics and Multidisciplinary Design

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Abstract. It is now well-known that a curvilinear discretization of the geometry is most often required to benefit from the computational efficiency of high-order numerical schemes in simulations. In this article, we explain how appropriate curvilinear meshes can be generated. We pay particular attention to the problem of invalid (tangled) mesh parts created by curving the domain boundaries. An efficient technique that computes provable bounds on the element Jacobian determinant is used to characterize the mesh validity, and we describe fast and robust techniques to regularize the mesh. The methods presented in this article are thoroughly discussed in Ref. [1,2], and implemented in the free mesh generation software Gmsh [3,4].

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“…The initial approach proposed in [58] used a non-linear neoHookean constitutive model. Several attempts to reduce the computational cost of this approach have been proposed based on a linear elastic analogy, see [1,77]. It is clear that when large deformations are induced to produce the deformed curvilinear high-order mesh, a linear elastic model can result in non-valid elements due to the violation of the hypothesis of small deformations.…”

Section: Introductionmentioning

confidence: 99%

A unified approach for a posteriori high-order curved mesh generation using solid mechanics

2016

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The paper presents a unified approach for the a posteriori generation of arbitrary high-order curvilinear meshes via a solid mechanics analogy. The approach encompasses a variety of methodologies, ranging from the popular incremental linear elastic approach to very sophisticated non-linear elasticity. In addition, an intermediate consistent incrementally linearised approach is also presented and applied for the first time in this context. Utilising a consistent derivation from energy principles, a theoretical comparison of the various approaches is presented which enables a detailed discussion regarding the material characterisation (calibration) employed for the different solid mechanics formulations. Five independent quality measures are proposed and their relations with existing quality indicators, used in the context of a posteriori mesh generation, are discussed. Finally, a comprehensive range of numerical examples, both in two and three dimensions, including challenging geometries of interest to the solids, fluids and electromagnetics communities, are shown in order to illustrate and thoroughly compare the performance of the different methodologies. This comparison considers the influence of material parameters and number of load increments on the quality of the generated high-order mesh, overall computational cost and, crucially, the approximation properties of the resulting mesh

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A method for computing curved meshes via the linear elasticity analogy, application to fluid dynamics problems

Cited by 37 publications

References 21 publications

Curved optimal delaunay triangulation

Curved optimal delaunay triangulation

The Generation of Valid Curvilinear Meshes

A unified approach for a posteriori high-order curved mesh generation using solid mechanics

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