Stable subdivision of 4D polytopes

Petrov, Miroslav S.; Todorov, Todor D.

doi:10.1007/s11075-017-0454-2

Cited by 10 publications

(9 citation statements)

References 21 publications

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“…where V (G) is the set of all vertices of the polytope G. The polytopes H i are ten identical tesseract base hyperpyramids with apex the penteract center. Following Petrov and Todorov [8] we divide the tesseract T i optimally with respect to the measure of degeneracy. Let T be the partition technique [8] subdividing T i uniformly into 24 tesseract corners C i j , i = 1, 2, ..., 10 and j = 1, 2, ..., 24.…”

Section: The Cell Centered Penteract Divisionmentioning

confidence: 99%

“…Following Petrov and Todorov [8] we divide the tesseract T i optimally with respect to the measure of degeneracy. Let T be the partition technique [8] subdividing T i uniformly into 24 tesseract corners C i j , i = 1, 2, ..., 10 and j = 1, 2, ..., 24. Despite all 24 elements C i j , j = 1, 2, ..., 24 are from the same class we distinguish two kind of pentatopes belonging to T T i : (i) tesseract corners that satisfy a i+32 ∈ V (C i j ); (ii) elements that a i+32 ∈ V (C i j ).…”

Section: The Cell Centered Penteract Divisionmentioning

confidence: 99%

“…The distance from the penteract center to each cell is the same. Moreover, all elements of T ∂ P are from the same class and have the same volume [8]. It remains only to apply the formula for volume of a hexateron [6] volK = h.volB 5 ,…”

Section: The Cell Centered Penteract Divisionmentioning

confidence: 99%

“…The finite element multigrid method for solving multidimensional boundary and eigenvalue problems needs stable subdivisions of simply connected bounded domains in R n . The subdivision of the n-dimensional cube plays an important role in dividing canonical domains [8,9]. Freudental's subdivision of the cube in combination with the red refinement strategy is widely used by researchers devoted to multigrid methods.…”

Section: Introductionmentioning

confidence: 99%

“…Many authors [2] have used combinatorial approaches to obtain stable subdivisions of the cube. Todorov [9] and Petrov and Todorov [8] that the Freudental partition of the cube is not optimal with respect to the measure of degeneracy in the 3-dimensional and 4-dimensional cases. Moreover, the optimal refinement strategies in these cases can not be obtained by combinatorial approaches.…”

Section: Introductionmentioning

confidence: 99%

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A penteract partition by means of the optimal subdivision of cells

Petrov¹,

Todorov²

2019

Sci Pub Univ Novi Pazar Ser A

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Freudental's algorithm obtained way back in early forties have been traditionally used for simplicial triangulating of the hypercube. The main advantage of this algorithm is that it only generates one congruence class. Unfortunately, Freudental's algorithm is not optimal with respect to the measure of degeneracy. The multigrid methods require the degeneracy measure to be as small as possible. The minimal subdivision in the 3-dimensional case and the uniform tesseract corner subdivision in the 4-dimensional case are optimal in regards the measure of degeneracy and multigrid applications. The question about the optimal refinement strategy in more dimensional cases is still an open problem. This paper deals with a penteract subdivision with degeneracy measure much better than one obtained by the Freudental algorithm.

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Section: The Cell Centered Penteract Divisionmentioning

confidence: 99%

Section: The Cell Centered Penteract Divisionmentioning

confidence: 99%

Section: The Cell Centered Penteract Divisionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%