Computing Eigenelements of Real Symmetric Matrices via Optimization

In certain circumstances, it is advantageous to use an optimization approach in order to solve the generalized eigenproblem, Ax = Bx, where A and B are real symmetric matrices and B is positive definite. In particular, this is the case when the matrices A and B are very large the computational cost, prohibitive, of solving, with high accuracy, systems of equations involving these matrices. Usually, the optimization approach involves optimizing the Rayleigh quotient.We first propose alternative objective functions to solve the (generalized) eigenproblem via (unconstrained) optimization, and we describe the variational properties of these functions.We then introduce some optimization algorithms (based on one of these formulations) designed to compute the largest eigenpair. According to preliminary numerical experiments, this work could lead the way to practical methods for computing the largest eigenpair of a (very) large symmetric matrix (pair).     

Broadband sound generation by confined pulsating jets in a mechanical model of the human larynx

Experiments were performed to study the production of broadband sound in confined pulsating jets through orifices with a time-varying area. The goal was to better understand broadband sound generation at the human glottis during voicing. The broadband component was extracted from measured sound signals by the elimination of the periodic component through ensemble averaging. Comparisons were made between the probability density functions of the broadband sound in pulsating jets and of comparable stationary jets. The results indicate that the quasi-steady approximation may be valid for the broadband component when the turbulence is well established and the turbulence kinetic energy is comparatively large. A wavelet analysis of the broadband sound showed that random sound production was modulated at the driving frequency. Two distinct sound production peaks were observed during one cycle, presumably associated firstly with jet formation and secondly with flow deceleration during orifice closing. Most high-frequency sound was produced during the closing phase. Deviations from quasi-steady behavior were observed. As the driving frequency increased, sound production during the opening phase was reduced, possibly due to the shorter time available for turbulence to develop. These results may be useful for better quality voice synthesis.     

A Generic Global Optimization Algorithm for the Chemical and Phase Equilibrium Problem

This paper addresses the problem of finding the number, K, of phases present at equilibrium and their composition, in a chemical mixture of n _s substances. This corresponds to the global minimum of the Gibbs free energy of the system, subject to constraints representing m _b independent conserved quantities, where m _b=n _s when no reaction is possible and m _b n _e +1 when reaction is possible and n _e is the number of elements present. After surveying previous work in the field and pointing out the main issues, we extend the necessary and sufficient condition for global optimality based on the reaction tangent-plane criterion, to the case involving different thermodynamical models (multiple phase classes). We then present an algorithmic approach that reduces this global optimization problem (involving a search space of m _b(n _s-1) dimensions) to a finite sequence of local optimization steps inK(n _s-1) -space, K m _b, and global optimization steps in (n _s-1)-space. The global step uses the tangent-plane criterion to determine whether the current solution is optimal, and, if it is not, it finds an improved feasible solution either with the same number of phases or with one added phase. The global step also determines what class of phase (e.g. liquid or vapour) is to be added, if any phase is to be added. Given a local minimization procedure returning a Kuhn–Tucker point and a global optimization procedure (for a lower-dimensional search space) returning a global minimum, the algorithm is proved to converge to a global minimum in a finite number of the above local and global steps. The theory is supported by encouraging computational results.     

A method for estimating the parameters of electrodynamic drivers in thermoacoustic coolers

The electroacoustic efficiency of high-power actuators used in thermoacoustic coolers may be estimated using a linear model involving a combination of six parameters. A method to identify these equivalent driver parameters from measured total electrical impedance and velocity-voltage transfer function data was developed. A commercially available, moving-magnet driver coupled to a functional thermoacoustic cooler was used to demonstrate the procedure experimentally. The method, based on linear electrical circuit theory, allowed for the possible frequency and amplitude dependence of the driver parameters to be estimated. The results demonstrated that driver parameters measured in vacuo using this method can be used to predict the driver efficiency and performance for operating conditions which may be encountered under load.     

Sound generation by steady flow through glottis-shaped orifices

Although the signature of human voice is mostly tonal, it also includes a significant broadband component. Quadrupolelike sources due to turbulence in the region downstream of the glottis, and dipolelike sources due to the force applied by the vocal folds onto the surrounding fluid are the two primary broadband sound generating mechanisms. In this study, experiments were conducted to characterize the broadband sound emissions of confined stationary jets through rubber orifices formed to imitate the approximate shape of the human glottis at different stages during one cycle of vocal fold vibrations. The radiated sound pressure spectra downstream of the orifices were measured for varying flow rates, orifice shapes, and gas mixtures. The nondimensional sound pressure spectra were decomposed into the product of three functions: a source function F, a radiation efficiency function M, and an acoustic response function G. The results show that, as for circular jets, the quadrupole source contributions dominated for straight and convergent orifices. For divergent jets, whistling tonal sounds were emitted at low flow rates. At high flow rates for the same geometry, dipole contributions dominated the sound radiated by free jets. However, possible source-load acoustic feedback may have hampered accurate source identification in confined flows.     

Instantaneous orifice discharge coefficient of a physical, driven model of the human larynx

The quasisteady approximation is often made in the study of phonatory aerodynamics to facilitate the modeling of time-varying air flows through the self-oscillating vocal folds. The unsteady, pulsating flow is approximated by a sequence of steady flows through representative configurations of the vocal folds at rest. Previous studies have discussed the accuracy of this approximation for a range of orifice geometries, and flow conditions. The purpose of the present study was to further evaluate the quasisteady approximation experimentally using an improved procedure, from a direct comparison between the discharge coefficients of steady jets through fixed orifices and unsteady jets through modulated orifices of identical shape, area, and transglottal pressures at a given time. Life-scale convergent and divergent glottis-shaped rubber orifices were used in a rigid-walled tube and a low Mach number flow representative of human phonation. It was found that the quasisteady approximation is valid during 70% of the duty cycle, when the Reynolds number was above 3000, for a frequency of oscillations of 100 Hz. The steady form of Bernoulli's equation along a streamline, and Bernoulli's flow obstruction theory were found to be reasonably accurate for the unsteady flows. These models break down at low Reynolds numbers, near the beginning and the end of the duty cycle, due to viscous effects and to the influence of flow displaced by the motion of the walls.     

Discontinuous piecewise linear optimization

A theoretical framework and a practical algorithm are presented to solve discontinuous piecewise linear optimization problems dealing with functions for which theridges are known. A penalty approach allows one to consider such problems subject to a wide range of constraints involving piecewise linear functions. Although the theory is expounded in detail in the special case of discontinuous piecewiselinear functions, it is straightforwardly extendable, using standard nonlinear programming techniques, tononlinear (discontinuous piecewise differentiable) functions.The descent algorithm which is elaborated uses active-set and projected gradient approaches. It is a generalization of the ideas used by Conn to deal with nonsmoothness in thel ₁ exact penalty function, and it is based on the notion ofdecomposition of a function into a smooth and a nonsmooth part. The constrained case is reduced to the unconstrained minimization of a (piecewise linear)l ₁ exact penalty function. We also discuss how the algorithm is modified when it encounters degenerate points. Preliminary numerical results are presented: the algorithm is applied to discontinuous optimization problems from models in industrial engineering. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.Supported by the Natural Sciences and Engineering Council of Canada and the Centre de Recherches Mathématiques, Université de Montréal.This research was supported in part by the Advanced Research Projects Agency of the Department of Defense and was monitored by the Air Force Office of Scientific Research under Contract No. F49620-91-C-0079. The United States Government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright notation hereon.     

Characterization of a hierarchical network of hyaluronic acid/gelatin composite for use as a smart injectable biomaterial

Hybrid HA/Ge hydrogel particles are embedded in a secondary HA network to improve their structural integrity. The internal microstructure of the particles is imaged through TEM. CSLM is used to identify the location of the Ge molecules in the microgels. Through indentation tests, the Young's modulus of the individual particles is found to be 22 ± 2.5 kPa. The overall shear modulus of the composite is 75 ± 15 Pa at 1 Hz. The mechanical properties of the substrate are found to be viable for cell adhesion. The particles' diameter at pH = 8 is twice that at pH = 5. The pH sensitivity is found to be appropriate for smart drug delivery. Based on their mechanical and structural properties, HA-Ge hierarchical materials may be well suited for use as injectable biomaterials for tissue reconstruction.