Delaunay Hodge star

Hirani, Anil N.; Kalyanaraman, Kaushik; VanderZee, Evan

doi:10.1016/j.cad.2012.10.038

Cited by 30 publications

(20 citation statements)

References 9 publications

(9 reference statements)

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“…The simplicial meshes considered here are Delaunay meshes, with an extra requirement only for the N-simplices with a face on ∂Ω, the domain boundary. Previous investigation [20] showed that in order to correctly represent the discrete Hodge star operator, the mesh interior N-simplices have to be pairwise Delaunay, while the N-simplicial elements with a face on the domain boundary have to be one-sided (i.e., with respect to the face of the N-simplex on the domain boundary, both the simplex circumcenter and its apex have to be on the same side). Such Delaunay meshes can easily be generated using commercial or open source mesh generators.…”

Section: The Domain Discretizationmentioning

confidence: 99%

Discrete exterior calculus discretization of incompressible Navier–Stokes equations over surface simplicial meshes

Mohamed

Hirani

Samtaney

2016

Journal of Computational Physics

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A conservative discretization of incompressible Navier-Stokes equations is developed based on discrete exterior calculus (DEC). A distinguishing feature of our method is the use of an algebraic discretization of the interior product operator and a combinatorial discretization of the wedge product. The governing equations are first rewritten using the exterior calculus notation, replacing vector calculus differential operators by the exterior derivative, Hodge star and wedge product operators. The discretization is then carried out by substituting with the corresponding discrete operators based on the DEC framework. Numerical experiments for flows over surfaces reveal a second order accuracy for the developed scheme when using structured-triangular meshes, and first order accuracy for otherwise unstructured meshes. By construction, the method is conservative in that both mass and vorticity are conserved up to machine precision. The relative error in kinetic energy for inviscid flow test cases converges in a second order fashion with both the mesh size and the time step.

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Section: The Domain Discretizationmentioning

confidence: 99%

Discrete exterior calculus discretization of incompressible Navier–Stokes equations over surface simplicial meshes

Mohamed

Hirani

Samtaney

2016

Journal of Computational Physics

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“…Note that this choice of signs matches the traditional sign convention for circumcentric duals (see [Hirani et al 2013] for a recent exposition). Finally one can compute the signed area a * i of a dual cell * i as the sum of the signed areas of triangles formed by vertex i and each surrounding dual edge, resulting in:…”

Section: Weighted Triangulationsmentioning

confidence: 87%

“…Orthogonal dual meshes are most commonly constructed by connecting neighboring triangle circumcenters . However, this choice of dual mesh is only appropriate for so-called pairwise-Delaunay triangulations (see, e.g., [Dyer and Schaefer 2009;Hirani et al 2013]), while most triangulations require combinatorial alterations for this dual to be well formed [Fisher et al 2007]. This construc- tion is thus often too restrictive for the demands of many graphics applications such as the generation of well-centered meshes [ VanderZee et al 2010;Mullen et al 2011] or the construction of discrete Laplacian operators with only positive coefficients [Wardetzky et al 2007;Vouga et al 2012].…”

Section: Related Workmentioning

confidence: 99%

Weighted Triangulations for Geometry Processing

et al. 2014

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In this paper, we investigate the use of weighted triangulations as discrete, augmented approximations of surfaces for digital geometry processing. By incorporating a scalar weight per mesh vertex, we introduce a new notion of discrete metric that defines an orthogonal dual structure for arbitrary triangle meshes and thus extends weighted Delaunay triangulations to surface meshes. We also present alternative characterizations of this primal-dual structure (through combinations of angles, areas, and lengths) and, in the process, uncover closed-form expressions of mesh energies that were previously known in implicit form only. Finally, we demonstrate how weighted triangulations provide a faster and more robust approach to a series of geometry processing applications, including the generation of well-centered meshes, self-supporting surfaces, and sphere packing.

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“…In particular, it has the drawback that, like in covolume method [117], it does not handle non-acute triangulation. Amelioration and alternatives can be found in literature [116,221,227,[233][234][235][236]. For example, as presented in [221], on can choose a dual mesh based on the barycenter or on the incenter, instead of the circumcenter, to remove the angle condition.…”

Section: Discrete Hodge Star and Codifferential Operatorsmentioning

confidence: 99%

A review of some geometric integrators

Razafindralandy

Hamdouni

Chhay

2018

Adv. Model. and Simul. in Eng. Sci.

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Some of the most important geometric integrators for both ordinary and partial differential equations are reviewed and illustrated with examples in mechanics. The class of Hamiltonian differential systems is recalled and its symplectic structure is highlighted. The associated natural geometric integrators, known as symplectic integrators, are then presented. In particular, their ability to numerically reproduce first integrals with a bounded error over a long time interval is shown. The extension to partial differential Hamiltonian systems and to multisymplectic integrators is presented afterwards. Next, the class of Lagrangian systems is described. It is highlighted that the variational structure carries both the dynamics (Euler-Lagrange equations) and the conservation laws (Noether's theorem). Integrators preserving the variational structure are constructed by mimicking the calculus of variation at the discrete level. We show that this approach leads to numerical schemes which preserve exactly the energy of the system. After that, the Lie group of local symmetries of partial differential equations is recalled. A construction of Lie-symmetry-preserving numerical scheme is then exposed. This is done via the moving frame method. Applications to Burgers equation are shown. The last part is devoted to the Discrete Exterior Calculus, which is a structure-preserving integrator based on differential geometry and exterior calculus. The efficiency of the approach is demonstrated on fluid flow problems with a passive scalar advection.

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Delaunay Hodge star

Cited by 30 publications

References 9 publications

Discrete exterior calculus discretization of incompressible Navier–Stokes equations over surface simplicial meshes

Discrete exterior calculus discretization of incompressible Navier–Stokes equations over surface simplicial meshes

Weighted Triangulations for Geometry Processing

A review of some geometric integrators

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