Finding the "truncated" polynomial that is closest to a function

When implementing regular enough functions (e.g., elementary or special functions) on a computing system, we frequently use polynomial approximations. In most cases, the polynomial that best approximates (for a given distance and in a given interval) a function has coefficients that are not exactly representable with a finite number of bits. And yet, the polynomial approximations that are actually implemented do have coefficients that are represented with a finite - and sometimes small - number of bits: this is due to the finiteness of the floating-point representations (for software implementations), and to the need to have small, hence fast and/or inexpensive, multipliers (for hardware implementations). We then have to consider polynomial approximations for which the degree-$i$ coefficient has at most $m_i$ fractional bits (in other words, it is a rational number with denominator $2^{m_i}$). We provide a general method for finding the best polynomial approximation under this constraint. Then, we suggest refinements than can be used to accelerate our method.Comment: 14 pages, 1 figur

A new adaptive response surface method for reliability analysis

Response surface method is a convenient tool to assess reliability for a wide range of structural mechanical problems. More specifically, adaptive schemes which consist in iteratively refine the experimental design close to the limit state have received much attention. However, it is generally difficult to take into account a lot of variables and to well handle approximation error. The method, proposed in this paper, addresses these points using sparse response surface and a relevant criterion for results accuracy. For this purpose, a response surface is built from an initial Latin Hypercube Sampling (LHS) where the most significant terms are chosen from statistical criteria and cross-validation method. At each step, LHS is refined in a region of interest defined with respect to an importance level on probability density in the design point. Two convergence criteria are used in the procedure: The first one concerns localization of the region and the second one the response surface quality. Finally, a bootstrap method is used to determine the influence of the response error on the estimated probability of failure. This method is applied to several examples and results are discussed

Invariant-based approach to symmetry class detection

In this paper, the problem of the identification of the symmetry class of a given tensor is asked. Contrary to classical approaches which are based on the spectral properties of the linear operator describing the elasticity, our setting is based on the invariants of the irreducible tensors appearing in the harmonic decomposition of the elasticity tensor [Forte-Vianello, 1996]. To that aim we first introduce a geometrical description of the space of elasticity tensors. This framework is used to derive invariant-based conditions that characterize symmetry classes. For low order symmetry classes, such conditions are given on a triplet of quadratic forms extracted from the harmonic decomposition of the elasticity tensor $C$, meanwhile for higher-order classes conditions are provided in terms of elements of $H^{4}$, the higher irreducible space in the decomposition of $C$. Proceeding in such a way some well known conditions appearing in the Mehrabadi-Cowin theorem for the existence of a symmetry plane are retrieved, and a set of algebraic relations on polynomial invariants characterizing the orthotropic, trigonal, tetragonal, transverse isotropic and cubic symmetry classes are provided. Using a genericity assumption on the elasticity tensor under study, an algorithm to identify the symmetry class of a large set of tensors is finally provided.Comment: 32 page

Managing the variability of biomechanical characteristics before the preliminary design stage of a medical device

The very high level of requirements for certification procedures often limit research and development departments to innovate using increments and iterations during the design process for medical devices (MD). Instead of this semi-empirical approach, a structured procedure, a breakthrough innovation should be used when designing an articular MD (prosthesis, implant). The search for concepts can be based on functional analysis and producing behavioural models of the joint in its natural state and/or equipped with the prosthesis. This paper shows how anatomical variables can be managed and integrated using a modular design approach.This study has been realized under the two joint action projects PESSOA 14630YA and PTDC/EME-PME/112977

Phase-resolved heterodyne holographic vibrometry with a strobe local oscillator

We report a demonstration of phase-resolved vibrometry, in which out-of-plane sinusoidal motion is assessed by heterodyne holography. In heterodyne holography, the beam in the reference channel is an optical local oscillator (LO). It is frequency-shifted with respect to the illumination beam to enable frequency conversion within the sensor bandwidth. The proposed scheme introduces a strobe LO, where the reference beam is frequency-shifted and modulated in amplitude, to alleviate the issue of phase retrieval. The strobe LO is both tuned around the first optical modulation side band at the vibration frequency, and modulated in amplitude to freeze selected mechanical vibration states sequentially. The phase map of the vibration can then be derived from the demodulation of successive vibration states

(DH) Noise and Signal scaling factors in Digital Holography in week illumination: relationship with Shot Noise

We have performed off axis heterodyne holography with very weak illumination by recording holograms of the object with and without object illumination in the same acquisition run. We have experimentally studied, how the reconstructed image signal (with illumination) and noise background (without) scale with the holographic acquisition and reconstruction parameters that are the number of frames, and the number of pixels of the reconstruction spatial filter. The first parameter is related to the frequency bandwidth of detection in time, the second one to the bandwidth in space. The signal to background ratio varies roughly like the inverse of the bandwidth in time and space. We have also compared the noise background with the theoretical shot noise background calculated by Monte Carlo simulation. The experimental and Monte Carlo noise background agree very well together