Vincent Delos

Operations on polytopes: application to tolerance analysis

This article presents numerical methods in order to solve problems of tolerance analysis. A geometric specification, a contact specification and a functional requirement can be respectively characterized by a fmite set of geometric constraints, a finite set of contact constraints and a finite set of functional constraints. Mathematically each constraint formalises a n-face (hyperplan of dimension n) of a n-polytope (1 ≤ n ≤ 6). Thus the relative position between two any surfaces of a mechanism can be calculated with two operations on polytopes: the Minkowski sum and the Intersection. The result is a new polytope: the calculated polytope. The inclusion of the calculated polytope inside the functional polytope indicates if the functional requirement is satisfied or not satisfied. Examples illustrate these numerical methods.

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

Prompted by the development of algorithms for analysing geometric tolerancing, this article describes a method to determine the Minkowski sum for 3-dimensional polytopes. This method is based exclusively on intersection operations on normal cones, using the properties of the normal fan of a Minkowski sum obtained by common refinement of the normal fans of the operands. It can be used to determine from which vertices of the operands the vertices of the Minkowski sum derive. It is also possible to determine to which facets of the operands each facet of the Minkowski sum is oriented. The basic properties of the algorithms can be applied to n-polytopes. First, the main properties of the duality of normal cones and primal cones associated with the vertices of a polytope are described. Next, the properties of normal fans are applied to define the vertices and facets of the Minkowski sum of two polytopes. An algorithm is proposed, which generalises the method. Lastly, there is a discussion of the features of this algorithm, developed using the OpenCascade environment.

Minkowski Sum of Polytopes Defined by Their Vertices

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.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.

Minkowski sum of HV-polytopes in Rn

Minkowski sums cover a wide range of applications in many different fields like algebra, morphing, robotics, mechanical CAD/CAM systems ... This paper deals with sums of polytopes in a n dimensional space provided that both H-representation and V-representation are available i.e. the polytopes are described by both their half-spaces and vertices. The first method uses the polytope normal fans and relies on the ability to intersect dual polyhedral cones. Then we introduce another way of considering Minkowski sums of polytopes based on the primal polyhedral cones attached to each vertex.

SAFlex: A structural alphabet extension to integrate protein structural flexibility and missing data information

In this paper, we describe SAFlex (Structural Alphabet Flexibility), an extension of an existing structural alphabet (HMM-SA), to better explore increasing protein three dimensional structure information by encoding conformations of proteins in case of missing residues or uncertainties. An SA aims to reduce three dimensional conformations of proteins as well as their analysis and comparison complexity by simplifying any conformation in a series of structural letters. Our methodology presents several novelties. Firstly, it can account for the encoding uncertainty by providing a wide range of encoding options: the maximum a posteriori, the marginal posterior distribution, and the effective number of letters at each given position. Secondly, our new algorithm deals with the missing data in the protein structure files (concerning more than 75% of the proteins from the Protein Data Bank) in a rigorous probabilistic framework. Thirdly, SAFlex is able to encode and to build a consensus encoding from different replicates of a single protein such as several homomer chains. This allows localizing structural differences between different chains and detecting structural variability, which is essential for protein flexibility identification. These improvements are illustrated on different proteins, such as the crystal structure of an eukaryotic small heat shock protein. They are promising to explore increasing protein redundancy data and obtain useful quantification of their flexibility.

Model reduction in geometric tolerancing by polytopes

Abstract There are several models used in mechanical design to study the behaviour 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.

Adapting Polytopes Dimension for Managing Degrees of Freedom in Tolerancing Analysis

In tolerancing analysis, geometrical or contact specifications can be represented by polytopes. Due to the degrees of invariance of surfaces and that of freedom of joints, these operand polytopes are originally unbounded in most of the cases (i.e. polyhedra). Homri et al. proposed the introduction of virtual boundaries (called cap half-spaces) over the unbounded displacements of each polyhedron to turn them into 6-polytopes. This decision was motivated by the complexity that operating on polyhedra in R6 supposes. However, that strategy has to face the multiplication of the number of cap half-spaces during the computation of Minkowski sums. In general, the time for computing cap facets is greater than for computing facets representing real limits of bounded displacements. In order to deal with that, this paper proposes the use of the theory of screws to determine the set of displacements that defines the positioning of one surface in relation to another. This set of displacements defines the subspace of R6 in which the polytopes of the respective surfaces have to be projected and operated to avoid calculating facets and vertices along the directions of unbounded displacements. With this new strategy it is possible to decrease the complexity of the Minkowski sums by reducing the dimension of the operands and consequently reducing the computation time. An example illustrates the method and shows the time reduction during the computations.

Counting Regular Expressions in Degenerated Sequences Through Lazy Markov Chain Embedding

Nowadays, Next- Generation Sequencing (NGS) produces huge number of reads which are combined using multiple alignment techniques to produce sequences. During this process, many sequencing errors are corrected, but the resulting sequences nevertheless contain a marginal level of uncertainty in the form of ∼0.1 % or less of degenerated positions (like the letter “N” corresponding to any nucleotide).