Fixed mesh approximation of ordinary differential equations with impulses

When trains of impulse controls are present on the right-hand side of a system of ordinary differential equations, the solution is no longer smooth and contains jumps which can accumulate at several points in the time interval. In technological and physical systems the sum of the absolute value of all the impulses is finite and hence the total variation of the solution is finite. So the solution at best belongs to the space BV of vector functions with bounded variation. Unless variable node methods are used, the loss of smoothness of the solution would a priori make higher-order methods over a fixed mesh inactractive. Indeed in general the order of -convergence is and the nodal rate is . However in the linear case -convergence rate remains but the nodal rate can go up to by using one-step or multistep scheme with a nodal rate up to when the solution belongs to . Proofs are given of error estimates and several numerical experiments confirm the optimality of the estimates. Received March 15, 1996 / Revised version received January 3, 1997     

Evolution equations for shapes and geometries

The first object of this paper is to introduce a new evolution equation for the characteristic function of the boundary Γ of a Lipschitzian domain Ω in the N-dimensional Euclidean space under the influence of a smooth time-dependent velocity field. The originality of this equation is that the evolution takes place in an L^p-space with respect to the (N − 1)-Hausdorff measure. A second more speculative objective is to discuss how that equation can be relaxed to rougher velocity fields via some weak formulation. A candidate is presented and some of the technical difficulties and open issues are discussed. Continuity results in several metric topologies are also presented. The paper also specializes the results on the evolution of the oriented distance function to initial sets with zero N-dimensional Lebesgue measure.     

Cationic polymerization of dienes VII. New electron donors in the polymerization of 1,3-pentadiene initiated by aluminum trichloride in non-polar solvent

The aim of this research was to investigate new bulky electron donors (EDs) generating hindered active species in the cationic polymerization of 1,3-pentadiene initiated by AlCl₃ in pentane, in order to avoid or strongly reduce the reaction between the active species and the double bonds of the polymer which are responsible for side reactions. At room temperature, the polymerization in the presence of new ED, such as OPh₂, N(PhBr)₃, NPh₃ and SPh₂, allowed to obtain higher conversions and lower insoluble fractions than without electron donor. The formation of a complex ED/AlCl₃ was shown for each electron donor. However, in the case of NPh₃ and SPh₂, variations of the polymer microstructure demonstrated an interaction between active species and these EDs. Similar results were obtained at lower temperature (−10 °C). The beneficial effect of the presence of electron donors such as NPh₃ and SPh₂ demonstrated the validity of the concept of sterically hindered active species, but the polymerization was still uncontrolled.     

Hereditary differential systems with constant delays. II. A class of affine systems and the adjoint problem

In this paper we (i) specialize some of the results of Delfour and Mitter (J. Differential Equations, 12, 1972, 213–235) to a class of representable affine hereditary differential systems, (ii) introduce the hereditary adjoint system, and (iii) give an integral representation of solutions.     

Shape sensitivity analysis via a penalization method

Summary The object of this paper is the development of a penalization technique to compute the shape derivative of cost functionals where the state is the solution of a non-linear equation and/or a linear variational inequality. This type of problem is frequently encountered in Shape Sensitivity Analysis.

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Résumé Cet article présente le calcul des dérivées de forme de fonctionnelles définies sur un domaine géométrique par une méthode de pénalisation. On suppose que l'état est la solution d'une équation non-linéaire ou d'une inéquation linéaire. Ce type de problème est fréquemment rencontré en analyse de sensitivité des formes.

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This research was supported in part by the National Sciences and Engineering Council of Canada Operating Grant A-8730 and a FCAR Grant from the « Ministère de l'Education du Québec ».     

Optimal design of a minimum weight thermal diffuser with constraint on the output thermal power flux

The object of this paper is the development of efficient mathematical and numerical tools to find the optimal shape of a minimum-weight thermal diffuser with a priori specifications on the inward thermal power flux (TPF) and a bound on the outward TPF. The present problem arises in connection with the use of high-power solid state devices in future communications satellites. In a space application the thermal power must ultimately be dissipated to the environment by using heatpipes and correspondingly large radiating areas. However, heatpipes can accept only a limited TPF from a source. Hence we have the requirement of a minimum-weight thermal diffuser with a uniform bound on the outward TPF. Shape optimal design and finite elements methods are used. Complete numerical results are provided.This research was supported by the Natural Sciences and Engineering Research Council of Canada, Strategic Grants G-0573 and G-0654 (Communications) and Operating Grant A-8730.     

Error analysis of mixed finite elements for cylindrical shells

In this paper, based on the Naghdi shell model, we analyze the uniform convergence of mixed finite element methods for cylindrical shell problems using macroelement techniques. We show that Taylor–Hood elements p ₂-P ₁ and P ₁ iso P ₂ are locking free elements for the model problems. Optimal error estimates are presented with these elements. This revised version was published online in June 2006 with corrections to the Cover Date.     

State theory of linear hereditary differential systems

In this paper we present a state theory for a class of linear functional differential equations of the retarded type considered by Delfour and Mitter (J. Differential Equations, 18 1975, 18–28) with initial functions in the product space M^p = X × L^p(?b, 0; X). Roughly speaking, the state at time t is a piece of trajectory defined over an interval [t ? b, t] for a fixed b > 0. From a study of the properties of the state in M^p an operational differential equation, the so-called state equation, is derived in order to describe its evolution. An adjoint state equation is also introduced for the adjoint state and the connection between solutions of the hereditary adjoint system and those of the adjoint state equation is established. All this provides the appropriate framework for the solution and the numerical approximation of the associated linear-quadratic optimal control and filtering problems.     

Interaction between ultra short pulses and a dense scattering medium by Monte Carlo simulation: consideration of particle size effect

With an enough short-pulse incident to an individual particle, elementary scattering modes can be observed: internal or external reflection, refraction and diffraction. Simulation of pulse propagation in dense scattering medium is usually computed for large observation time, so that time delays of pulse interaction with the particles are negligible compared to propagation times between particle. A Monte Carlo method is proposed to compute the propagation of an incident 100 fs laser pulse in dense medium taking into account time-dependent scattering characteristics of particle: observation time of scattered light is less than 5000 fs. Two extreme cases are exemplified: predominance of direct and single-scattered photons appears in a thin time window for small particles (1 μm). On the contrary multiple scattering is always predominant and scrambles the transmitted signal for large particles (100 μm).