Section: New Results

Modeling

Simulation of electrical circuits as nonsmooth dynamical systems

Participants : Vincent Acary, Olivier Bonnefon, Bernard Brogliato.

DC-DC converters are usually difficult to simulate with classical tools like SPICE because of the highly nonlinear behaviour of some components and the frequent occurrence of intrinsically generated switching events.

The simulation of such circuits modelled as nonsmooth systems has been successfully achieved with a clear advantage over several SPICE simulators and a simulator belonging to the hybrid modelling approach [1] [48] .

Spiking neuronal networks dynamics

Participant : Arnaud Tonnelier.

Precise spatiotemporal sequences of spikes are observed in many neural systems and are thought to be involved in the neural processing of sensory stimuli. In [58] we examine the capability of spiking neural networks to propagate stably spatiotemporal sequences of spikes. We derive some analytical results for the wave speed and show that the stability of simple waves is determined by the Schur criteria. The transmission of a sequence of several spikes is related to the existence of stable composite waves, i.e. the existence of stable spatiotemporal periodic traveling waves. We show that the stability of composite waves is related to the roots of a system of multivariate polynomials.

A fundamental aspect that shapes the properties of traveling waves in networks is the underlying lattice-structure of the space . Discreteness has a strong effect on propagating activity patterns and, for instance, anisotropy or propagation failure can be observed. Numerical simulations and analytical calculations have been carried out to characterize more precisely these properties [47] .

Computational Toxicology

Participant : Arnaud Tonnelier.

It is now well recognized that toxicology has entered a new era. Previously mainly based on animal testing, toxicology is now turning to in vitro and in silico experiments. To assess the risk of chemicals but also to gather and to interpret the massive amounts of experimental data generated by modern toxicology, the development of mathematical and computational tools are essential. An important element in risk assessment of chemicals is the human bioaccumulative potential. We developed a predictive tool for human bioaccumulation assesment using a physiologically based toxicokinetic model [28] .

High-order models of mechanical rods

Participants : Florence Bertails-Descoubes, Romain Casati.

Reduced-coordinates models for rods such as the articulated rigid body model or the super-helix model  [50] are able to capture the bending and twisting deformations of thin elastic rods while strictly and robustly avoiding stretching deformations. In this work we are exploring new reduced-coordinates models based on a higher-order geometry. Typically, elements are defined by a polynomial curvature function of the arc length, of degree $d\ge 1$. The main difficulty compared to the super-helix model (where $d=0$) is that the kinematics has no longer a closed form. We have already investigated the clothoidal case ($d=1$) in the 2d case  [51] , relying on Romberg numerical integration, and a general approach in 3d based on power series expansion was formulated in the master thesis of R. Casati, for a single element. R. Casati is currently extending the method to a chain of linked elements as well as to an arbitrary degree $d$ of the curvature function.

Inverse modeling of mechanical rods

Participants : Florence Bertails-Descoubes, Alexandre Derouet-Jourdan.

Controlling the input shape of slender structures such as rods is desirable in many design applications (such as hairstyling, reverse engineering, etc.), but solving the corresponding inverse problem is not straightforward. In [29] we started to extend to 3d our 2d method introduced in [8] for automatically converting a smooth sketched curve into a dynamic curve at stable equilibrium under gravity. The main challenge in 3d amounts to converting an input curve into a continuous piecewise helix. Using a least-squares optimization approach is a natural option, however it may suffer from both robustness and computational issues due to the presence of multiple local minima in the objective function. To overcome these issues, we have recently proposed to reformulate the problem as a geometric interpolation problem. In this new method, only tangents are strictly interpolated while points are displaced in an optimal way so as to lie in a feasible configuration, i.e., a configuration that is compatible with the interpolation by a helix. Our method proves to be much more robust and faster compared to the global optimization approach. We plan to publish these results in 2012.

Multiple impacts modelling

Participants : Bernard Brogliato, Hongjian Zhang, Ngoc-Son Nguyen.

The work consists of studying two systems: the rocking block and tapered chains of balls, using the Darboux-Keller model of multiple impacts previously developed. The objectives are threefold: 1) show that the model predicts well the motion by careful comparisons with experimental data found in the literature, 2) study the system's dynamics and extract critical kinetic angles that allow the engineer to predict the system's gross motion, 3) develop numerical code inside the siconos platform that incorporates the model of multiple impact. Results are in [42] .

Simulating contact with Coulomb friction in fiber assemblies

Participants : Florence Bertails-Descoubes, Gilles Daviet.

We have developed a new frictional contact solver in [21] which is able to robustly and efficiently handle large fiber problems composed of thousands self-contacting rods with exact Coulomb friction. The solver relies on a Gauss-Seidel iterative approach, where each local one-contact solver is based on a hybrid strategy. The solution to the one-contact problem is first searched for using a nonsmooth Newton method based upon a generalized Fischer-Burmeister formulation. This primary solver manages to solve the local problem in 99.9% of the cases. When the solver fails to converge to an acceptable solution, the method switches to a more costly but exact solver, based on the $\alpha$-formulation introduced in [39] . This hybrid strategy experimentally allows us to always find a solution to the local problem, which greatly contributes to improve the robustness of the global solver. We have compared our new solver against other solvers of the literature (e.g., damped Newton solvers relying on the Alart-Curnier function) and observed a noticeable gain, both in terms of robustness and computational efficiency.