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.Comment: 4th Annual International Conference on Computational Mathematics, Computational Geometry and Statistics, Jan 2015, Singapore, Singapor

Procedia CIRP

During the manufacturing process of a product, variability in its parts is unavoidable. Tolerance analysis allows estimating the consequences of the component's deviation of a mechanism on its functionality. Nowadays, it is possible to determine the contribution of each surface and/or contact on the final result in isostatic mechanisms by using the tools already presented in the literature; however, it is still a challenge to do so in over-constrained mechanisms. In previous works, we introduced a method based on prismatic polyhedra to model over-constrained mechanisms. In this paper, several simulations based on the previous approach are performed, varying the tolerances of the surfaces and contacts of the mechanism. The use of statistical methods to analyze the previous simulations’ data is proposed to quantify the contribution of local deviations with respect to the total variation of the mechanism. This analysis determines the most relevant contacts, hence the most critical parts of the mechanism. The process is applied to a pump as an over-constrained case study and uses the prismatic polyhedra method for tolerance analysis

Procedia CIRP

The control of geometrical deviations and form variations throughout the product life-cycle is a fundamental task in geometric dimensioning and tolerancing. As product complexity increases, it has not only become necessary to rely on computers to process geometrical and non-geometrical information from early design stages but also to come up with more realistic shape representation. Most of computer-aided tolerancing (CAT) packages used nowadays are fully integrated to computer-aided design softwares like CATIA and SolidWorks and they allow to model 2D and 3D tolerances stack-up through worst-case or statistical models. These CAT systems are generally available as proprietary commercial software which can sometimes restrict their domain of application and slow the implementation of new paradigms like the Skin Model. The Skin Model is an abstract surface model that represents the interface between a workpiece and its environment whose implementation involves the modeling of finite instances of the Skin Model called Skin Model Shapes (SMS) that encompass different sources of deviations and constitute a non-ideal geometrical model. The aim of this work is to show the first phase of implementation of an integrated open source environment based on PolitoCAT and Salome to model Skin Model Shapes. An Unified Model Language (UML) based logical data model of the integrated system is presented, it is an extended version of current data models for geometric modeling that includes the objects and relationships to manage form variations at different design stages. The work carried out contributes to the conceptualization of Skin Model Shapes model and it constitutes a support on the implementation of SMS in an open source CAT. An example of this integration involving a Skin Model Shapes implementation is shown as an illustration of the functionality of the platform

Canagliflozin and renal outcomes in type 2 diabetes and nephropathy

BACKGROUND Type 2 diabetes mellitus is the leading cause of kidney failure worldwide, but few effective long-term treatments are available. In cardiovascular trials of inhibitors of sodium–glucose cotransporter 2 (SGLT2), exploratory results have suggested that such drugs may improve renal outcomes in patients with type 2 diabetes. METHODS In this double-blind, randomized trial, we assigned patients with type 2 diabetes and albuminuric chronic kidney disease to receive canagliflozin, an oral SGLT2 inhibitor, at a dose of 100 mg daily or placebo. All the patients had an estimated glomerular filtration rate (GFR) of 30 to <90 ml per minute per 1.73 m2 of body-surface area and albuminuria (ratio of albumin [mg] to creatinine [g], >300 to 5000) and were treated with renin–angiotensin system blockade. The primary outcome was a composite of end-stage kidney disease (dialysis, transplantation, or a sustained estimated GFR of <15 ml per minute per 1.73 m2), a doubling of the serum creatinine level, or death from renal or cardiovascular causes. Prespecified secondary outcomes were tested hierarchically. RESULTS The trial was stopped early after a planned interim analysis on the recommendation of the data and safety monitoring committee. At that time, 4401 patients had undergone randomization, with a median follow-up of 2.62 years. The relative risk of the primary outcome was 30% lower in the canagliflozin group than in the placebo group, with event rates of 43.2 and 61.2 per 1000 patient-years, respectively (hazard ratio, 0.70; 95% confidence interval [CI], 0.59 to 0.82; P=0.00001). The relative risk of the renal-specific composite of end-stage kidney disease, a doubling of the creatinine level, or death from renal causes was lower by 34% (hazard ratio, 0.66; 95% CI, 0.53 to 0.81; P<0.001), and the relative risk of end-stage kidney disease was lower by 32% (hazard ratio, 0.68; 95% CI, 0.54 to 0.86; P=0.002). The canagliflozin group also had a lower risk of cardiovascular death, myocardial infarction, or stroke (hazard ratio, 0.80; 95% CI, 0.67 to 0.95; P=0.01) and hospitalization for heart failure (hazard ratio, 0.61; 95% CI, 0.47 to 0.80; P<0.001). There were no significant differences in rates of amputation or fracture. CONCLUSIONS In patients with type 2 diabetes and kidney disease, the risk of kidney failure and cardiovascular events was lower in the canagliflozin group than in the placebo group at a median follow-up of 2.62 years

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

Algorithm to calculate the Minkowski sums of 3-polytopes : application to tolerance analysis

In tolerance analysis, it is necessary to check that the cumulative defect limits specified for the component parts of a product are compliant with the functional requirements expected of the product. Defect limits can be modelled by tolerance zones constructed by offsets on nominal models of parts. Cumulative defect limits can be modelled using a calculated polytope, the result of a set of intersections and Minkowski sums of polytopes. A functional requirement can be qualified by a functional polytope, in other words a target polytope. It is then necessary to verify whether the calculated polytope is included in the functional polytope. The purpose of this article is to determine the Minkowski sum of 3-dimension polytopes and apply this effectively in order to optimise the filling of the functional polytope. This method 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. 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

Applying screw theory for summing sets of constraints in geometric tolerancing

In tolerance analysis, approaches based on sets of constraints (also called convex hull techniques) are able to study simultaneously all the possible extreme configurations of a mechanism when simulating manufacturing defects in its components. The accumulation of these defects can be calculated by summing and intersecting 6-dimensional sets of constraints, i.e. polyhedra. These approaches tend to be time-consuming, however, because of the complexity resulting from manipulating sets in R 6. In this paper, polyhedra are decomposed into a bounded set (a polytope) and an unbounded set (a set of straight lines). The unbounded part of the polyhedra is characterized by the degrees of freedom of the toleranced feature or the joint. Therefore, the decomposition can be performed based on a kinematic analysis of the studied assembly using screw systems. The proposed decomposition is presented for the most common features used in geometric tolerancing. The idea behind this strategy is, instead of summing polyhedra in R 6 , to sum only their underlying polytopes by isolating the unbounded part of the operands. A slider-crank mechanism is used to show the gain in computational time of the proposed method in comparison with the strategy based on complete 6-dimensional sets of constraints