Olivier Gibaru

C1 and C2-continuous polynomial parametric Lp splines (p≥1)

Abstract We introduce L p ( 1 ⩽ p ⩽ ∞ ) piecewise polynomial parametric splines of degree 3 or 5 which smoothly interpolate data points. The coefficients of these polynomial splines are calculated by minimizing the L p norm of the second derivative. It is demonstrated that the C 1 -continuous cubic (resp. G 1 -continuous cubic and C 2 -continuous quintic) L p polynomial splines are unique for 1 p ∞ . The focus is mainly on L 1 polynomial splines which preserve the shape of data even with abrupt changes in direction or spacing. When there are several candidates we introduce a tension parameter which selects from the set of solutions the smallest in norm. The optimisation uses a nonlinear functional; therefore we discretize it so as to obtain a linear problem and we give an upper bound to the error. Computational examples using a primal affine (interior point) algorithm are presented. We will also show how interesting it can be to be able to change the partition associated to the data points in order to obtain convexity properties when possible. Moreover, we will show that the solution is not unvarying by rotation in the L 1 -case and we propose an alternate approach with a local change of coordinates on each segment [ P i , P i + 1 ] . We show that this method can be applied to any kind of norm for the minimization problem.

On the cubic L 1 spline interpolant to the Heaviside function

We prove that the univariate interpolating cubic L1 spline to the Heaviside function at three sites to the left of the jump and three sites to the right of the jump entirely agrees with the Heaviside function except in the middle interval where it is the interpolating cubic with zero slopes at the end point. This shows that there is no oscillation near the discontinuous point i.e. no Gibbs’ phenomenon.

A rectangular G m -continuous filling surface patch and some improvements at corners

Abstract We propose a method for filling a four-sided hole that interpolates four connected boundary curves in a net of patches. The surfaces defining these given boundary curves can be of different kinds with their suitable representations: Bezier–de Casteljau polynomial patches, (SBR) rational patches, or other parametric surfaces. Pole-functions are introduced as they extend the Gregory square-functions. The pole-functions result from appropriate combinations of derivatives up to order m (m⩾1) at points belonging to the boundary of [0,1] 2 of functions that define the surfaces to be joined. The proposed filling surface is a combination of these pole-functions. It has the G m geometric continuity join property and is in one piece. We shall improve the quality of the interior filling patch by requiring that the filling surface should become continuously differentiable at corners. This strategy turns out to be significant. Some examples of G 1 , G 2 and G 3 -continuous filling surfaces illustrate this construction.

A generic decision support tool for lot-sizing and scheduling problems with setup and due dates

Decision support tools are essential to help in the management of industrial systems at different levels: strategic, to design the system; tactical to plan activities or assign resources; operational to schedule activities. In this paper, we present a generic and modular decision support tool to solve different planning, assignment, scheduling or lot-sizing problems. To the best of our knowledge, such generic tool does not exist. The methodology is illustrated by solving a real world lot-sizing and scheduling problem from a plastic injection company.

Blowing up: application to

Summary. We propose a method for filling a n-sided hole,

G^2

5\leq n\leq 8

-continuous 8-sided filling patch

, that interpolates n connected boundary curves of a given net of patches. This method allows the joining with patches defined in many different ways. A new class of blowing up pole-functions is introduced in order to build a G

Blowing-up method. Application to G2-continuous filling surfaces in CAD

^{2}

Solving a Discrete Lot Sizing and Scheduling Problem with Unrelated Parallel Machines and Sequence Dependent Setup Using a Generic Decision Support Tool

-continuous n-sided filling surface. This filling surface is in one piece, image of

Détermination des images d'un point de base d'une surface rationnelle tensorielle définie par des vecteurs massiques

[0,1]^{2}