Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2205.06670

Rectangular mesh contour generation algorithm for finite differences calculus

Pedro Zaffalon da Silva·Neyva Maria Lopes Romeiro·Iury Pereira de Souza·Paulo Laerte Natti·Eliandro Rodrigues Cirilo
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this work, a 2D contour generation algorithm is proposed for irregular regions. The contour of the physical domain is approximated by mesh segments using the known coordinates of the contour. For this purpose, the algorithm uses a repeating structure that analyzes the known irregular contour coordinates to approximate the physical domain contour by mesh segments. To this end, the algorithm calculates the slope of the line defined by the known point of the irregular contours and the neighboring vertices. In this way, the algorithm calculates the points of the line and its distance to the closest known nodes of the mesh, allowing to obtain the points of the approximate contour. This process is repeated until the approximate contour is obtained. Therefore, this approximate contour generation algorithm, from known nodes of a mesh, is suitable for describing meshes involving geometries with irregular contours and for calculating finite differences in numerical simulations. The contour is evaluated through three geometries, the difference between the areas delimited by the given contour and the approximate contour, the number of nodes and the number of internal points. It can be seen that the increase in geometry complexity implies the need for a greater number of nodes in the contour, generating more refined meshes that allow reaching differences in areas below 2%.

Explore related subjects

cs.CGcs.NAmath.NA

Keep this discovery

BibTeXRIS

Pedro Zaffalon da Silva, Neyva Maria Lopes Romeiro, Iury Pereira de Souza, Paulo Laerte Natti, Eliandro Rodrigues Cirilo. 2022-05-10. Rectangular mesh contour generation algorithm for finite differences calculus. https://doi.org/10.22533/at.ed.8702118084

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Almost Linear Universal Point Sets for Planar Graphs

A point set is universal for planar graphs on $n$ vertices if every such graph has a straight-line drawing without crossings whose vertices belong to the set. We construct universal point sets of size $n^{1+o(1)}$, improving the previous quadratic upper bound. Our construction uses the reduction of Bannister, Cheng, Devanny, and Eppstein from universal point sets to superpatterns for $213$-avoiding permutations. We represent these permutations by ordered rooted forests and construct a small family of intervals containing every such forest. The result follows from a straightforward bound on the size of the family of intervals. GPT-6 Astra assisted in developing the construction and proof.

cs.CG↗
arXiv2026

Some results on Archdeacon's conjecture for rotation systems

A rotation system on $n$ elements assigns to each element a cyclic order of the other $n-1$ elements. A four-element subset is non-planar if its induced rotation system cannot be realized by a crossing-free drawing of $K_4$. As a combinatorial strengthening of Hill's conjecture on the crossing number of the complete graph, Archdeacon conjectured that every rotation system on $n$ elements has at least $H(n)=\frac{1}{4} \lfloor\frac {n}{2}\rfloor \lfloor\frac{n-1}{2}\rfloor \lfloor\frac{n-2}{2}\rfloor \lfloor\frac{n-3}{2}\rfloor$ non-planar four-element subsets. We computationally verify Archdeacon's conjecture for $n\leq 10$ and show that every extremal rotation system in these orders is realizable by a simple drawing. With computer assistance, we prove that every rotation system on $n$ elements has at least $(8/9 - o(1)) H(n)$ non-planar four-element subsets. We also present a proof by hand for a weaker lower bound of $(2/3-o(1)) H(n)$. Finally, extending recent work of Felsner on antipodal pairs in drawings, we show that Archdeacon's conjecture holds for antipodally shellable rotation systems.

cs.CG↗
arXiv2026

The Hyperbolic Surface Distance, Diameter, and Dirichlet Problems

Despite the prominence of hyperbolic surfaces in mathematics, basic algorithmic questions about them, even computing the distance between two points, have remained open, leaving many features of these surfaces inaccessible. The classical machinery assumes a polyhedral structure absent on a smooth surface. We remove these obstacles. We begin with an efficient $O(g^2)$ algorithm for the distance between two points, where $g$ is the genus of the surface. Building on it, we obtain an $O(g^2 \log g)$ method for answering distance queries from a fixed source and, as a consequence, for recentering a Dirichlet domain around an arbitrary point. This understanding of distances on the surface then lets us approximate the diameter to within any $\eps$ in time $O(g^3 \log g / \eps^2)$. We further show that the diameter, a single real number encoding a great deal about the surface, is exactly computable. Its hyperbolic cosine is an algebraic number over the field encoding the coefficients of the hyperbolic isometries defining the surface.

cs.CG↗