article

Algorithm 860: SimpleS—an extension of Freudenthal's simplex subdivision

Authors:

Eduardo N. Gonçalves,

Reinaldo M. Palhares,

Ricardo H. C. Takahashi,

Renato C. MesquitaAuthors Info & Claims

ACM Transactions on Mathematical Software (TOMS), Volume 32, Issue 4

Pages 609 - 621

https://doi.org/10.1145/1186785.1186792

Published: 01 December 2006 Publication History

Get Access

Abstract

This article presents a simple efficient algorithm for the subdivision of a d-dimensional simplex in k^d simplices, where k is any positive integer number. The algorithm is an extension of Freudenthal's subdivision method. The proposed algorithm deals with the more general case of k^d subdivision, and is considerably simpler than the RedRefinementND algorithm for implementation of Freudenthal's strategy. The proposed simplex subdivision algorithm is motivated by a problem in the field of robust control theory: the computation of a tight upper bound of a dynamical system performance index by means of a branch-and-bound algorithm.

Supplementary Material

ZIP File (860.zip)

Software for "SimpleS -- an extension of Freudenthal's simplex subdivision"

Download
2.54 KB

References

[1]

Balakrishnan, V., Boyd, S., and Balemi, S. 1991. Computing the minimum stability degree of parameter-dependent linear systems. In Control of Uncertain Dynamic Systems, S. P. Bhattacharyya and L. H. Keel, Eds. CRC Press, Boca Raton, FL, 359--378.

Google Scholar

[2]

Barber, C. B. 1996. The quickhull algorithm for convex hulls. ACM Trans. Math. Softw. 22, 4, 469--483.

Digital Library

Google Scholar

[3]

Bey, J. 2000. Simplicial grid refinement: On freudenthal's algorithm and the optimal number of congruence classes. Numer. Math. 85, 1, 1--29.

Crossref

Google Scholar

[4]

de Berg, M., Kreveld, M. V., Overmas, M., and Schwarzkopt, O. 1998. Computational Geometry: Algorithms and Applications. Springer Verlag, Berlin.

Digital Library

Google Scholar

[5]

Edelsbrunner, H. and Grayson, D. R. 2000. Edgewise subdivision of a simplex. Discrete Comput. Geom. 24, 707--719.

Digital Library

Google Scholar

[6]

Genz, A. and Cools, E. 2003. An adaptive numerical cubature algorithm for simplices. ACM Trans. Math. Softw. 29, 3, 297--308.

Digital Library

Google Scholar

[7]

Gonçalves, E. N., Palhares, R. M., and Takahashi, R. H. C. 2005. Improved optimisation approach to robust H2/H∞ control problem for linear systems. IEE Proceedings Control Theory Applications 152, 2, 171--176.

Crossref

Google Scholar

[8]

Goodman, T. N. T. and Peters, J. 1995. Bézier nets, convexity and subdivision on higher dimensional simplices. Comput. Aided Geom. Des. 12, 53--65.

Digital Library

Google Scholar

Cited By

View all

Souza AOliveira RPeres P(2024) Computation of stable convex combinations of a set of matrices through LMIs with application in control of switched linear systems Systems & Control Letters10.1016/j.sysconle.2024.105833189(105833)Online publication date: Jul-2024
https://doi.org/10.1016/j.sysconle.2024.105833
Gonçalves EOliveira Pda Silva Júnior J(2023) ℋ ∞ robust control of discrete‐time systems based on new linear matrix inequality formulations and evolutionary optimization Optimal Control Applications and Methods10.1002/oca.308745:3(1045-1063)Online publication date: 27-Nov-2023
https://doi.org/10.1002/oca.3087
Petrov MTodorov T(2019)Refinement strategies related to cubic tetrahedral meshesApplied Numerical Mathematics10.1016/j.apnum.2018.11.006137(169-183)Online publication date: Mar-2019
https://doi.org/10.1016/j.apnum.2018.11.006
Show More Cited By

Index Terms

Algorithm 860: SimpleS—an extension of Freudenthal's simplex subdivision
1. Computing methodologies
  1. Computer graphics
    1. Shape modeling
2. Theory of computation
  1. Randomness, geometry and discrete structures

Recommendations

Volume subdivision based hexahedral finite element meshing of domains with interior 2-manifold boundaries
AFRIGRAPH '06: Proceedings of the 4th international conference on Computer graphics, virtual reality, visualisation and interaction in Africa

We present a subdivision based algorithm for multi-resolution Hexahedral meshing. The input is a bounding rectilinear domain with a set of embedded 2-manifold boundaries of arbitrary genus and topology. The algorithm first constructs a simplified ...
Algorithm 872: Parallel 2D constrained Delaunay mesh generation

Delaunay refinement is a widely used method for the construction of guaranteed quality triangular and tetrahedral meshes. We present an algorithm and a software for the parallel constrained Delaunay mesh generation in two dimensions. Our approach is ...
Generating well-shaped d-dimensional Delaunay meshes
Computing and combinatorics

A d-dimensional simplicial mesh is a Delaunay triangulation if the circumsphere of each of its simplices does not contain any vertices inside. A mesh is well shaped if the maximum aspect ratio of all its simplices is bounded from above by a constant. It ...

Comments

Information & Contributors

Information

Published In

cover image ACM Transactions on Mathematical Software

ACM Transactions on Mathematical Software Volume 32, Issue 4

December 2006

145 pages

ISSN:0098-3500

EISSN:1557-7295

DOI:10.1145/1186785

Issue’s Table of Contents

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 01 December 2006

Published in TOMS Volume 32, Issue 4

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Badges

Author Tags

Qualifiers

Article

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

15
Total Citations
View Citations
459
Total Downloads

Downloads (Last 12 months)10
Downloads (Last 6 weeks)0

Reflects downloads up to 22 Sep 2024

Other Metrics

View Author Metrics

Citations

Cited By

View all

Souza AOliveira RPeres P(2024) Computation of stable convex combinations of a set of matrices through LMIs with application in control of switched linear systems Systems & Control Letters10.1016/j.sysconle.2024.105833189(105833)Online publication date: Jul-2024
https://doi.org/10.1016/j.sysconle.2024.105833
Gonçalves EOliveira Pda Silva Júnior J(2023) ℋ ∞ robust control of discrete‐time systems based on new linear matrix inequality formulations and evolutionary optimization Optimal Control Applications and Methods10.1002/oca.308745:3(1045-1063)Online publication date: 27-Nov-2023
https://doi.org/10.1002/oca.3087
Petrov MTodorov T(2019)Refinement strategies related to cubic tetrahedral meshesApplied Numerical Mathematics10.1016/j.apnum.2018.11.006137(169-183)Online publication date: Mar-2019
https://doi.org/10.1016/j.apnum.2018.11.006
Bachur WGonalves ERamrez JBatista L(2017)A multiobjective robust controller synthesis approach aided by multicriteria decision analysisApplied Soft Computing10.1016/j.asoc.2017.06.02760:C(374-386)Online publication date: 1-Nov-2017
https://dl.acm.org/doi/10.1016/j.asoc.2017.06.027
Xuan ZPapadopoulos P(2016)Computing an upper bound on contact stress with surrogate dualityComputational Mechanics10.1007/s00466-016-1287-x58:1(171-183)Online publication date: 1-Jul-2016
https://dl.acm.org/doi/10.1007/s00466-016-1287-x
Siqueira MSilva LGoncalves EPalhares RTakahashi R(2014)Robust decoupling control synthesis2014 American Control Conference10.1109/ACC.2014.6858972(4280-4285)Online publication date: Jun-2014
https://doi.org/10.1109/ACC.2014.6858972
Paulavičius RŽilinskas JPaulavičius RŽilinskas J(2013)Simplicial Partitions in Global OptimizationSimplicial Global Optimization10.1007/978-1-4614-9093-7_1(1-19)Online publication date: 15-Sep-2013
https://doi.org/10.1007/978-1-4614-9093-7_1
Gonçalves BGonçalves EPalhares RTakahashi R(2012)Robust decoupling PI controllers for multi-loop control2012 IEEE 51st IEEE Conference on Decision and Control (CDC)10.1109/CDC.2012.6426606(1530-1535)Online publication date: Dec-2012
https://doi.org/10.1109/CDC.2012.6426606
Neumüller MSteinbach O(2012)Refinement of flexible space–time finite element meshes and discontinuous Galerkin methodsComputing and Visualization in Science10.1007/s00791-012-0174-z14:5(189-205)Online publication date: 18-Apr-2012
https://doi.org/10.1007/s00791-012-0174-z
Gonçalves EBachur WPalhares RTakahashi R(2011) Robust ℋ 2 /ℋ ∞ /reference model dynamic output-feedback control synthesis International Journal of Control10.1080/00207179.2011.63323284:12(2067-2080)Online publication date: Dec-2011
https://doi.org/10.1080/00207179.2011.633232
Show More Cited By

View Options

Get Access

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Cited By

Index Terms

Recommendations

Volume subdivision based hexahedral finite element meshing of domains with interior 2-manifold boundaries

Algorithm 872: Parallel 2D constrained Delaunay mesh generation

Generating well-shaped d-dimensional Delaunay meshes