Large‐scale eigenvalue calculations for stability analysis of steady flows on massively parallel computers

This paper presents an approach for determining the linear stability of steady states of partial differential equations (PDEs) on massively parallel computers. Linearizing the transient behavior around a steady state solution leads to an eigenvalue problem. The eigenvalues with the largest real part are calculated using Arnoldi's iteration driven by a novel implementation of the Cayley transformation. The Cayley transformation requires the solution of a linear system at each Arnoldi iteration. This is done iteratively so that the algorithm scales with problem size. A representative model problem of three‐dimensional incompressible flow and heat transfer in a rotating disk reactor is used to analyze the effect of algorithmic parameters on the performance of the eigenvalue algorithm. Successful calculations of leading eigenvalues for matrix systems of order up to 4 million were performed, identifying the critical Grashof number for a Hopf bifurcation. Copyright © 2001 John Wiley & Sons, Ltd.     

Real-time increase in depth of field of an uncooled thermal camera using several phase-mask technologies

Imaging systems that combine a phase mask in the pupil and digital postprocessing may have better performance than conventional ones. We have built such a system to enhance the depth of field of an uncooled thermal camera. The phase masks are binary, their structures are optimized thanks to an image quality criterion, and they have been realized with three different technologies that give equivalent results. The deconvolution postprocessing is performed in real time with a graphics processing unit. A significant increase of the depth of field of a factor 3 has been obtained.     

Force flux and the peridynamic stress tensor

The peridynamic model is a framework for continuum mechanics based on the idea that pairs of particles exert forces on each other across a finite distance. The equation of motion in the peridynamic model is an integro-differential equation. In this paper, a notion of a peridynamic stress tensor derived from nonlocal interactions is defined. At any point in the body, this stress tensor is obtained from the forces within peridynamic bonds that geometrically go through the point. The peridynamic equation of motion can be expressed in terms of this stress tensor, and the result is formally identical to the Cauchy equation of motion in the classical model, even though the classical model is a local theory. We also establish that this stress tensor field is unique in a certain function space compatible with finite element approximations.     

Translation of Walter Noll’s “Derivation of the Fundamental Equations of Continuum Thermodynamics from Statistical Mechanics”

This article represents a translation of the original paper, “Die Herleitung der Grundgleichungen der Thermomechanik der Kontinua aus der Statistischen Mechanik”, which was written by Walter Noll and appeared in the Journal of Rational Mechanics and Analysis 4 (1955), 627–646. In the original paper, Noll addressed and analyzed the seminal paper of Irving & Kirkwood, published five years earlier, on “The statistical mechanical theory of transport processes. IV, The Equations of Hydrodynamics.” Noll gave new interpretations and provided a firm setting for ideas advanced by Irving & Kirkwood that clearly and directly related to the basic principles of continuum mechanics. This translation aims to expose the important contribution of Noll to a wider community of researchers at a time when the atomistic modeling of material behavior is being advanced. Noll’s use of elementary mathematics to discover physical effects, to explain physical concepts, and to draw conclusions of a physical nature is exhibited. Noll’s paper emerged from a report that he presented in a seminar at Indiana University in the summer of 1954. The seminar was organized by Clifford Truesdell, whose inspiration Noll gratefully acknowledged.     

On stabilized finite element methods for the Stokes problem in the small time step limit

Recent studies indicate that consistently stabilized methods for unsteady incompressible flows, obtained by a method of lines approach may experience difficulty when the time step is small relative to the spatial grid size. Using as a model problem the unsteady Stokes equations, we show that the semi‐discrete pressure operator associated with such methods is not uniformly coercive. We prove that for sufficiently large (relative to the square of the spatial grid size) time steps, implicit time discretizations contribute terms that stabilize this operator. However, we also prove that if the time step is sufficiently small, then the fully discrete problem necessarily leads to unstable pressure approximations. The semi‐discrete pressure operator studied in the paper also arises in pressure‐projection methods, thereby making our results potentially useful in other settings. Copyright © 2006 John Wiley & Sons, Ltd.     

Efficient parametric interactions in a low loss GaInP photonic crystal waveguide

We describe time domain characterizations of dynamic four-wave mixing in a low loss modified W1 GaInP photonic crystal waveguide. Using 32 ps wide pump pulses with peak powers of up to 1.1 W we achieved a very large conversion efficiency of -6.8 dB as well as a 1.3 dB parametric gain experienced by a weak CW probe signal. Time domain simulations confirm quantitatively all the measured results.     

The Effect of the Ill-posed Problem on Quantitative Error Assessment in Digital Image Correlation

Experimental Mechanics - This work explores the effect of the ill-posed problem on uncertainty quantification for motion estimation using digital image correlation (DIC) (Sutton et al. [2009]). We...     

Analysis of the Volume-Constrained Peridynamic Navier Equation of Linear Elasticity

Well-posedness results for the state-based peridynamic nonlocal continuum model of solid mechanics are established with the help of a nonlocal vector calculus. The peridynamic strain energy density for an elastic constitutively linear anisotropic heterogeneous solid is expressed in terms of the field operators of that calculus, after which a variational principle for the equilibrium state is defined. The peridynamic Navier equilibrium equation is then derived as the first-order necessary conditions and are shown to reduce, for the case of homogeneous materials, to the classical Navier equation as the extent of nonlocal interactions vanishes. Then, for certain peridynamic constitutive relations, the peridynamic energy space is shown to be equivalent to the space of square-integrable functions; this result leads to well-posedness results for volume-constrained problems of both the Dirichlet and Neumann types. Using standard results, well-posedness is also established for the time-dependent peridynamic equation of motion.     

Convergence of Peridynamics to Classical Elasticity Theory

The peridynamic model of solid mechanics is a nonlocal theory containing a length scale. It is based on direct interactions between points in a continuum separated from each other by a finite distance. The maximum interaction distance provides a length scale for the material model. This paper addresses the question of whether the peridynamic model for an elastic material reproduces the classical local model as this length scale goes to zero. We show that if the motion, constitutive model, and any nonhomogeneities are sufficiently smooth, then the peridynamic stress tensor converges in this limit to a Piola-Kirchhoff stress tensor that is a function only of the local deformation gradient tensor, as in the classical theory. This limiting Piola-Kirchhoff stress tensor field is differentiable, and its divergence represents the force density due to internal forces. The limiting, or collapsed, stress-strain model satisfies the conditions in the classical theory for angular momentum balance, isotropy, objectivity, and hyperelasticity, provided the original peridynamic constitutive model satisfies the appropriate conditions.      

Passive coherent beam combining of quantum-cascade lasers with a Dammann grating

An external cavity using a binary phase grating has been developed to achieve coherent combining of five quantum-cascade lasers emitting at 4.65 μm. The grating phase profile is designed to combine five beams of equal intensities into a single beam with a good efficiency (~75%). The performances of this cavity concerning output power, stability, combining efficiency and beam quality are detailed. We report a CW combining efficiency of 66% corresponding to an output power of ~0.5 W with a good beam quality (M(2)<1.6).