Algorithm to calculate the Minkowski sums of 3-polytopes based on normal fans

Teissandier, Denis; Delos, Vincent

doi:10.1016/j.cad.2011.06.016

Cited by 30 publications

(16 citation statements)

References 19 publications

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“…This can be problematic as obtaining the double description can turn out to be impossible in high dimensions, see [4] where Fukuda uses both vertices and edges. Reference [6] works in R 3 in a dual space where it intersects dual cones attached to the vertices, and it can be considered as the dual version of property 6 where the intersection is computed with primal cones. It actually implements Weibel's approach described in [9].…”

Section: Resultsmentioning

confidence: 99%

Minkowski Sum of Polytopes Defined by Their Vertices

Delos¹,

Teissandier²

2015

JAMP

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Minkowski sums are of theoretical interest and have applications in fields related to industrial backgrounds. In this paper we focus on the specific case of summing polytopes as we want to solve the tolerance analysis problem described in [1]. Our approach is based on the use of linear programming and is solvable in polynomial time. The algorithm we developed can be implemented and parallelized in a very easy way.

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Section: Resultsmentioning

confidence: 99%

Minkowski Sum of Polytopes Defined by Their Vertices

Delos¹,

Teissandier²

2015

JAMP

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“…How- [12] ever, they do not provide the H-description of the calculated polytope, required in tolerance analysis for computing subsequent intersections. In [20] and [21] algorithms are presented for summing HV-polytopes (i.e. polytopes defined by both their H-representation and V-representation) in R n taking advantage of the dual property of polytopes.…”

Section: Introductionmentioning

confidence: 99%

Model reduction in geometric tolerancing by polytopes

Delos

Arroyave-Tobón

Teissandier

2018

Computer-Aided Design

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There are several models used in mechanical design to study the behavior of mechanical systems involving geometric variations. By simulating defects with sets of constraints it is possible to study simultaneously all the configurations of mechanisms, whether over-constrained or not. Using this method, the accumulation of defects is calculated by summing sets of constraints derived from features (toleranced surfaces and joints) in the tolerance chain. These sets are usually unbounded objects (R 6-polyhedra, 3 parameters for the small rotation, 3 for the small translation), due to the unbounded displacements associated with the degrees of freedom of features. For computational and algorithmic reasons, cap facets are introduced into the operand polyhedra to obtain bounded objects (R 6-polytopes) and facilitate computations. However, the consequence is an increase in the complexity of the models due to the multiplication of caps during the computations. In response to this situation, we formalized and tested a method for controlling the effects of cap facets. Based on the combinatorial properties of polytopes, we propose to trace the operand faces during the different operations. An industrial case is solved and discussed in order to show the significant gain in computational time when applying the new method. This example has been chosen to be as general as possible to illustrate the genericity of the method.

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“…A polytope is used to define all the positions of a surface within a tolerance zone (geometric polytope) and all the relative positions between two surfaces potentially in contact (contact polytope). By applying operations (Minkowski sums [3], [4] and intersections) to geometric and contact polytopes it is then possible to characterise the relative position between the rotor and the stator in a turbine. These operations are deduced from the topological structure of the turbine defined by a contact graph for one connected component [5].…”

Section: Introductionmentioning

confidence: 99%

Variational tolerancing analysis taking thermomechanical strains into account: Application to a high pressure turbine

Pierre¹,

Teissandier²,

Nadeau³

2014

Mechanism and Machine Theory

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is an open access repository that collects the work of Arts et Métiers ParisTech researchers and makes it freely available over the web where possible. Abstract:The aim of this study is to propose a variational method of tolerancing analysis using a multiphysical approach. This method is based on operations on polytopes (Minkowski sum and intersection) and can be used to validate geometric specifications, contact specifications and thermomechanical specifications. The first part describes how thermomechanical strains are integrated into a tolerancing analysis tool, based on operations on polytopes. In the second part correlations are defined between two turbine performance criteria, leakage section and risk of touching, and two geometric conditions respectively. In the third part, the influence of design choices is described, in particular the influence of the shape of the parts and the behaviour of the joints on the thermomechanical operating regime of the turbine. Two turbine architectures are considered in relation to the same two performance criteria, and lastly the main turbine architecture results are discussed and future developments described.

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Algorithm to calculate the Minkowski sums of 3-polytopes based on normal fans

Cited by 30 publications

References 19 publications

Minkowski Sum of Polytopes Defined by Their Vertices

Minkowski Sum of Polytopes Defined by Their Vertices

Model reduction in geometric tolerancing by polytopes

Variational tolerancing analysis taking thermomechanical strains into account: Application to a high pressure turbine

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