Joël Chaskalovic

Joël Chaskalovic, Franck Assous

The aim of this paper is to provide a new perspective on finite element accuracy. Starting from a geometrical reading of the Bramble-Hilbert lemma, we recall the two probabilistic laws we got in previous works that estimate the relative accuracy, considered as a random variable, between two finite elements $P_k$ and $P_m$, ($k < m$). Then, we analyze the asymptotic relation between these two probabilistic laws when the difference $m-k$ goes to infinity. New insights which qualified the relative accuracy in the case of high order finite elements are correspondingly obtained.

Joël Chaskalovic, Franck Assous

The aim of this paper is to provide new perspectives on relative finite element accuracy which is usually based on the asymptotic speed of convergence comparison when the mesh size $h$ goes to zero. Starting from a geometrical reading of the error estimate due to Bramble-Hilbert lemma, we derive two probability distributions that estimate the relative accuracy, considered as a random variable, between two Lagrange finite elements $P_k$ and $P_m$, ($k < m$). We establish mathematical properties of these probabilistic distributions and we get new insights which, among others, show that $P_k$ or $P_m$ is more likely accurate than the other, depending on the value of the mesh size $h$.

Joel Chaskalovic, Franck Assous

The aim of this paper is to provide new perspectives on the relative finite elements accuracy. Starting from a geometrical interpretation of the error estimate which can be deduced from Bramble-Hilbert lemma, we derive a probability law that evaluates the relative accuracy, considered as a random variable, between two finite elements $P_k$ and $P_m$, ($k < m$). We extend this probability law to get a cumulated probabilistic law for two main applications. The first one concerns a family of meshes and the second one is dedicated to a sequence of simplexes which constitute a given mesh. Both of this applications might be relevant for adaptive mesh refinement.

Stéphane Doncieux, Jean Liénard, Benoît Girard et al.

Computational models are of increasing complexity and their behavior may in particular emerge from the interaction of different parts. Studying such models becomes then more and more difficult and there is a need for methods and tools supporting this process. Multi-objective evolutionary algorithms generate a set of trade-off solutions instead of a single optimal solution. The availability of a set of solutions that have the specificity to be optimal relative to carefully chosen objectives allows to perform data mining in order to better understand model features and regularities. We review the corresponding work, propose a unifying framework, and highlight its potential use. Typical questions that such a methodology allows to address are the following: what are the most critical parameters of the model? What are the relations between the parameters and the objectives? What are the typical behaviors of the model? Two examples are provided to illustrate the capabilities of the methodology. The features of a flapping-wing robot are thus evaluated to find out its speed-energy relation, together with the criticality of its parameters. A neurocomputational model of the Basal Ganglia brain nuclei is then considered and its most salient features according to this methodology are presented and discussed.