On the Shape of Tetrahedra from Bisection

Liu, Anwei; Joe, Barry

doi:10.2307/2153566

Cited by 36 publications

(8 citation statements)

References 12 publications

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“…To do so, we use techniques analogous to finite element methods [see, e.g. 82,103,153]: the phasespace sheet is refined using bisection [see, e.g., 126,15,99,102,12,152, and references therein], that is by cutting when required some simplices into two smaller simplices while preserving the conforming nature of the tessellation at all times. New vertices created during this procedure are placed in such a way that the local representation of the phase-space sheet remains accurate at second order.…”

Section: Introductionmentioning

confidence: 99%

ColDICE: A parallel Vlasov–Poisson solver using moving adaptive simplicial tessellation

Sousbie

Colombi

2016

Journal of Computational Physics

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Resolving numerically Vlasov-Poisson equations for initially cold systems can be reduced to following the evolution of a three-dimensional sheet evolving in six-dimensional phase-space. We describe a public parallel numerical algorithm consisting in representing the phase-space sheet with a conforming, self-adaptive simplicial tessellation of which the vertices follow the Lagrangian equations of motion. The algorithm is implemented both in six-and fourdimensional phase-space. Refinement of the tessellation mesh is performed using the bisection method and a local representation of the phase-space sheet at second order relying on additional tracers created when needed at runtime. In order to preserve in the best way the Hamiltonian nature of the system, refinement is anisotropic and constrained by measurements of local Poincaré invariants. Resolution of Poisson equation is performed using the fast Fourier method on a regular rectangular grid, similarly to particle in cells codes. To compute the density projected onto this grid, the intersection of the tessellation and the grid is calculated using the method of Franklin and Kankanhalli [64,65,66] generalised to linear order. As preliminary tests of the code, we study in four dimensional phase-space the evolution of an initially small patch in a chaotic potential and the cosmological collapse of a fluctuation composed of two sinusoidal waves. We also perform a "warm" dark matter simulation in six-dimensional phase-space that we use to check the parallel scaling of the code. arXiv:1509.07720v1 [physics.comp-ph] 23 Sep 2015 ImplementationSupercomputers featuring large clusters of shared memory nodes are becoming the norm, with a continuing trend of increasing number of cores per node. Taking advantage of such processing power is challenging, especially for problems such as gravitational dynamics that are by essence non-local and for which significant inter-process communication cannot be avoided. Scaling up to (tens of) thousands of cores using pure MPI communication, the standard message passing interface for distributed memory computers, often results in numerous messages being sent all over the network, triggering traffic contentions that almost invariably end up being the limiting performance factors. One way to alleviate this problem consists in using a hybrid approach, combining coarse grained MPI parallelism with local shared-memory multiprocessing via for instance OpenMP. Indeed, MPI parallelism is oftentimes achieved through domain decomposition, each MPI distributed sub-domain communicating preferentially with its direct neighbors, but also potentially with all other sub-domains via all-to-all type communications. The usage of local OpenMP style parallelism allows for larger and less numerous sub-domains. In this case, neighbor-to-neighbor communications, that are often achieved via buffer regions called "ghost layers" locally keeping track of neighboring domains boundaries updates, are therefore reduced, since these regions typically scale like the surface o...

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Section: Introductionmentioning

confidence: 99%

ColDICE: A parallel Vlasov–Poisson solver using moving adaptive simplicial tessellation

Sousbie

Colombi

2016

Journal of Computational Physics

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“…The regular tetrahedron is not an 8-rep-tile but some other special tetrahedra are, one of them found by M.J.M. Hill already in 1895, and two others found in 1994 [13]. Recent results support the conjecture that there are no further 8rep-tile tetrahedra [11].…”

Section: Rep-tilesmentioning

confidence: 84%

Computer Geometry: Rep-Tiles with a Hole

Bandt

Mekhontsev

2019

Math Intelligencer

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“…He conjectured that Hill simplices are the only 3-dimensional reptile simplices. Herman Haverkort recently pointed us to an example of a k-reptile tetrahedron by Liu and Joe [30] which is not Hill, and thus contradicts Hertel's conjecture. In fact, except for the one-parameter family of Hill tetrahedra, two other space-filling tetrahedra described by Sommerville [41] and Goldberg [17] are also k-reptiles for every k = m 3 .…”

Section: Introductionmentioning

confidence: 94%

On the Nonexistence of $k$-Reptile Simplices in $\mathbb R^3$ and $\mathbb R^4$

Kynčl¹,

Patáková²

2017

Electron. J. Combin.

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A d-dimensional simplex S is called a k-reptile (or a k-reptile simplex ) if it can be tiled by k simplices with disjoint interiors that are all mutually congruent and similar to S. For d = 2, triangular k-reptiles exist for all k of the form a 2 , 3a 2 or a 2 + b 2 and they have been completely characterized by Snover, Waiveris, and Williams. On the other hand, the only k-reptile simplices that are known for d ≥ 3, have k = m d , where m is a positive integer. We substantially simplify the proof by Matoušek and the second author that for d = 3, k-reptile tetrahedra can exist only for k = m 3 . We then prove a weaker analogue of this result for d = 4 by showing that four-dimensional k-reptile simplices can exist only for k = m 2 .

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On the Shape of Tetrahedra from Bisection

Cited by 36 publications

References 12 publications

ColDICE: A parallel Vlasov–Poisson solver using moving adaptive simplicial tessellation

ColDICE: A parallel Vlasov–Poisson solver using moving adaptive simplicial tessellation

Computer Geometry: Rep-Tiles with a Hole

On the Nonexistence of $k$-Reptile Simplices in $\mathbb R^3$ and $\mathbb R^4$

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