SIAM Digital Library
 
 
 

SIAM J. on Applied Mathematics

All Journal content published prior to 1997 is part of LOCUS.

Search Issue | RSS Feeds RSS
Previous Issue

2008

Volume 68, Issue 6, pp. 1503-1806


Electric Discharge Sintering: A Mathematical Model

G. A. Kriegsmann

SIAM J. Appl. Math. 68, pp. 1503-1517 (15 pages)

Online Publication Date: May 28, 2008

Full Text: | Download PDF

Show Abstract
In this paper we mathematically model the densification of metallic powders and the sintering of ceramic powders by electric discharge. The ordinary and partial differential equations governing these processes are the same with the exception of the effective electrical conductivity. This function is a monotonically decreasing (increasing) function of temperature for the metallic (ceramic) powders. We employ asymptotic methods to approximate the solution to these equations in the limit as $\epsilon \rightarrow 0$, where $\epsilon$ is the ratio of the discharge to diffusion time scales. We find on the shortest time scale that the temperature, voltage, and density satisfy a system of nonlinear, coupled ordinary equations. We solve these and find the relationship between the temperature and density, as functions of the input energy. The results on the short or discharge time scale do not take into account diffusion and heat loss into the surrounding medium. These occur on a much longer time scale, which we identify and exploit to deduce a new approximation. On this time scale the capacitor has no more energy to deposit into the powder. The temperature relaxes to that of its surroundings and the density increases to its final value. Our results show the functional relationship between the final density and the initial energy stored in the capacitor, as well as the initial density of the powder.

Numerical Tests of a Phase Field Model with Second Order Accuracy

Gunduz Caginalp, Xinfu Chen, and Christof Eck

SIAM J. Appl. Math. 68, pp. 1518-1534 (17 pages)

Online Publication Date: May 28, 2008

Full Text: | Download PDF

Show Abstract
Numerical computations are performed for a recently derived phase field model for the interface between two phases. The rigorous results indicate that solutions to this new phase field model should converge more rapidly than traditional ones to solutions of the corresponding sharp interface (free boundary) formulation for sufficiently small values of the approximation parameter $\varepsilon $ representing the thickness of the interfacial region. In particular, the distance between the sharp interface of the limiting model and the zero level set of the phase function in the phase field model is of order $\varepsilon^2$ rather than $\varepsilon$. Numerical computations within a three-dimensional spherically symmetric setting compare the computed solutions of this new model with the known exact solutions for the limiting free boundary problem and confirm the second order accuracy predictions of the theory for sufficiently small $\varepsilon$. The sets of parameters include those of succinonitrile used in dendritic experiments.

Time Reversal Focusing of the Initial State for Kirchhoff Plate

Kim Dang Phung and Xu Zhang

SIAM J. Appl. Math. 68, pp. 1535-1556 (22 pages)

Online Publication Date: June 06, 2008

Full Text: | Download PDF

Show Abstract
Consider a Kirchhoff plate $\partial_t^2 u + \Delta^2 u - \partial_t^2 \Delta u = 0$ in $\Omega\times (0,T)$, with boundary data $u=\Delta u=0$ on $\partial\Omega \times (0,T)$ and unknown initial data $u(\cdot,0) = u_0$ and $\partial_t u(\cdot,0) = u_1$ in $\Omega$. We study an inverse problem of determining $(u_0,u_1)$ from an interior observation $u|_{\omega\times(0,T)}$. Here $\Omega$ is a bounded domain, $\omega$ a nonempty open subset of $\Omega$, and $T>0$ a suitable time duration. By means of an iterative time reversal technique, we derive an asymptotic formula of reconstructing $(u_0,u_1)$ approximately with a logarithmical convergence rate for smooth initial data. The convergence becomes uniform and exponential when $(\Omega,\omega,T)$ satisfies the geometric control condition introduced by Bardos, Lebeau, and Rauch.

Electrical Impedance Tomography by Elastic Deformation

H. Ammari, E. Bonnetier, Y. Capdeboscq, M. Tanter, and M. Fink

SIAM J. Appl. Math. 68, pp. 1557-1573 (17 pages) | Cited 7 times

Online Publication Date: June 06, 2008

Full Text: | Download PDF

Show Abstract
This paper presents a new algorithm for conductivity imaging. Our idea is to extract more information about the conductivity distribution from data that have been enriched by coupling impedance electrical measurements to localized elastic perturbations. Using asymptotics of the fields in the presence of small volume inclusions, we relate the pointwise values of the energy density to the measured data through a nonlinear PDE. Our algorithm is based on this PDE and takes full advantage of the enriched data. We give numerical examples that illustrate the performance and the accuracy of our approach.

Effective Transport Equations and Enhanced Backscattering in Random Waveguides

Josselin Garnier and Knut SØlna

SIAM J. Appl. Math. 68, pp. 1574-1599 (26 pages) | Cited 2 times

Online Publication Date: June 06, 2008

Full Text: | Download PDF

Show Abstract
In this paper we derive a general system of transport equations for the moments of reflected and transmitted mode amplitudes in a randomly perturbed waveguide, in a regime where backscattering is significant. The derivation is based on a limit theorem for the system of coupled differential equations for the mode amplitudes, in the limit where the amplitude of the random fluctuations of the medium is small, the correlation lengths in the transverse and longitudinal directions are of the same order of the wavelength, and the waveguide is long. Using this system we derive several results in specific regimes, including the enhanced backscattering phenomenon for the reflected wave: when an incoming monochromatic wave with a specific incidence angle is present, the mean reflected power has a local maximum in the backward direction twice as large as the mean reflected power in the other directions.

Competitive Exclusion of Microbial Species for a Single Nutrient with Internal Storage

Sze-Bi Hsu and Ting-Hao Hsu

SIAM J. Appl. Math. 68, pp. 1600-1617 (18 pages)

Online Publication Date: June 06, 2008

Full Text: | Download PDF

Show Abstract
We study a chemostat model that describes competition between $n$ microbial species for a single-limited resource based on storage. The model incorporates internal resource storage variables that serve the direct connection between species growth and external resource availability. Mathematical analysis for the global dynamics of the model is carried out by using the fluctuating method. It is shown that the competitive exclusion principle holds for the limiting system of the model. The species with the smallest ambient nutrient concentration wins the competition. We extend the result of competitive exclusion in the paper [H. L. Smith and P. Waltmam, SIAM J. Appl. Math., 54 (1994), pp. 1113–1131] from two species to $n$ species.

Existence, Uniqueness, and a Constructive Solution Algorithm for a Class of Finite Markov Moment Problems

Laurent Gosse and Olof Runborg

SIAM J. Appl. Math. 68, pp. 1618-1640 (23 pages)

Online Publication Date: June 06, 2008

Full Text: | Download PDF

Show Abstract
We consider a class of finite Markov moment problems with an arbitrary number of positive and negative branches. We show criteria for the existence and uniqueness of solutions, and we characterize in detail the nonunique solution families. Moreover, we present a constructive algorithm to solve the moment problems numerically and prove that the algorithm computes the right solution.

Incipient Dynamics of Swelling of Gels

Hang Zhang and M. Carme Calderer

SIAM J. Appl. Math. 68, pp. 1641-1664 (24 pages) | Cited 1 time

Online Publication Date: June 25, 2008

Full Text: | Download PDF

Show Abstract
In this article, we analyze a model of the incipient dynamics of gel swelling and perform numerical simulations. The governing system consists of balance laws for a mixture of nonlinear elastic solid and solvent yielding effective equations for the gel. We discuss the multiscale nature of the problem and identify physically realistic regimes. The mixing mechanism is based on the Flory–Huggins energy. We consider the case that the dissipation mechanism is the solid-solvent friction force. This leads to a system of weakly dissipative nonlinear hyperbolic equations. After addressing the Cauchy problem, we propose physically realistic boundary conditions describing the motion of the swelling boundary. We study the linearized version of the free boundary problem. Numerical simulations of solutions are presented too.

A Mathematical Model for the Control and Eradication of a Wood Boring Beetle Infestation

Stephen A. Gourley and Xingfu Zou

SIAM J. Appl. Math. 68, pp. 1665-1687 (23 pages)

Online Publication Date: June 25, 2008

Full Text: | Download PDF

Show Abstract
We propose a mathematical model for an infestation of a wooded area by a beetle species in which the larva develop deep in the wood of living trees. Due to the difficulties of detection, we presume that only a certain proportion of infested trees will be detected and that detection, if it happens, will occur only after some delay, which could be long. An infested tree once detected is immediately cut down and burned. The model is stage structured and contains a second time delay, which is the development time of the beetle from egg to adult. There is a delicate interplay between the two time delays due to the possibility in one case for a larva to mature even in a tree destined for destruction. We present conditions sufficient for infestation eradication and discuss the significance of the conditions, particularly in terms of the proportion of infested trees that need to be detected and removed. If the infestation is successfully eradicated, there are always a number of trees that completely escape infestation, and we compute lower bounds and an approximation for this number. Finally, we present the results of some numerical simulations.

The Frederiks Effect and Related Phenomena in Ferronematic Materials

V. I. Zadorozhnii, T. J. Sluckin, V. Yu. Reshetnyak, and K. S. Thomas

SIAM J. Appl. Math. 68, pp. 1688-1716 (29 pages) | Cited 2 times

Online Publication Date: July 02, 2008

Full Text: | Download PDF

Show Abstract
Using continuum and statistical mechanical theories, we study the switching properties of a ferronematic in a nematic liquid crystal cell subject to homeotropic boundary conditions at the cell and particle walls. An external magnetic field normal to the cell plane is also imposed. At low fields we find thresholdless switching of the nematic director, consistent with experimental data. At higher fields, there are three regimes, depending on the strength of the anchoring interaction between the director and the ferroparticle orientation. For low anchoring strengths, there is an inverse Frederiks effect, and the nematic reorientation reduces and then disappears continuously at a critical magnetic field. At intermediate fields, the degree of reorientation reduces at high fields but remains finite. For high fields, however, the director switching saturates. The dimensionless temperature scale in the problem involves the temperature, the mean nematic elastic constant, the colloidal density, and the cell dimension. If this quantity is sufficiently low, then high magnetic fields can cause magnetic segregation. The segregation order parameter is coupled to the director distortion, and this can change the inverse Frederiks transition into a first order transition, leading to bistability in an intermediate field regime. These features are perturbed but not changed structurally by the effect of a small bias magnetic field ($<10$ Oe) normal to the unperturbed director. Subject to suitable choice of parameters, the theory is also quantitatively consistent with the results of the classic experiment of Chen and Amer in 1983.

Models of Virulent Phage Growth with Application to Phage Therapy

Hal L. Smith

SIAM J. Appl. Math. 68, pp. 1717-1737 (21 pages) | Cited 1 time

Online Publication Date: July 02, 2008

Full Text: | Download PDF

Show Abstract
We modify existing models of bacteriophage growth on an exponentially growing bacterial population by including (1) density dependent phage attack rates and (2) loss to phage due to adsorption to both infected and uninfected bacteria. The effects of these modifications on key pharmacokinetic parameters associated with phage therapy are examined. More general phage growth models are explored which account for infection-age of bacteria, bacteria-phage complex formation, and decoupling phage progeny release from host cell lysis.

Transport-Based Imaging in Random Media

Guillaume Bal and Kui Ren

SIAM J. Appl. Math. 68, pp. 1738-1762 (25 pages)

Online Publication Date: July 03, 2008

Full Text: | Download PDF

Show Abstract
This paper generalizes well-established derivations of the radiative transfer equation from first principles to model the energy density of time-dependent and monochromatic high frequency waves propagating in a random medium composed of localized scatterers. The correlation length of the random scatterers is small compared to the overall distance of propagation so that ensemble averaging may take place. The correlation length may be either comparable to the typical wavelength in the system (the weak-coupling regime) or larger than the wavelength (the low-density regime). The paper also considers the detection and imaging of inclusions buried in highly scattering random media. In such multiple scattering environments, the coherent wave fields may be too weak to be used for imaging purposes. We thus propose to model the inclusions as parameters in the macroscopic radiative transfer equations and consider the imaging problem as an inverse transport problem. Numerical simulations address the domain of validity of the radiative transfer equation and of the imaging method. Wave propagation is solved by using a Foldy–Lax framework, and the forward and inverse transport problems are solved by using a Monte Carlo method. Since the inverse transport problem is ill-posed, the buried inclusions are parameterized by a small number of degrees of freedom, typically their position and a few geometric properties.

Time-Local Dissipative Formulation and Stable Numerical Schemes for a Class of Integrodifferential Wave Equations

C. Casenave and E. Montseny

SIAM J. Appl. Math. 68, pp. 1763-1782 (20 pages) | Cited 1 time

Online Publication Date: July 16, 2008

Full Text: | Download PDF

Show Abstract
We consider integrodifferential equations of the abstract form $\mathbf{H} (\partial_{t})\Phi=\mathbf{G}(\nabla)\Phi+f$, where $\mathbf{H}(\partial_{t})$ is a diagonal convolution operator and $\mathbf{G}(\nabla)$ is a linear anti-selfadjoint differential operator. On the basis of an original approach devoted to integral causal operators, we propose and study a time-local augmented formulation under the form of a Cauchy problem $\partial_{t}\Psi= \mathcal{A}\Psi+\mathcal{B}f$ such that $\Phi=\mathcal{C}\Psi$. We show that under a suitable hypothesis on the symbol $\mathbf{H}(p)$, this new formulation is dissipative in the sense of a natural energy functional. We then establish the stability of numerical schemes built from this time-local formulation, thanks to the dissipation of appropriate discrete energies. Finally, the efficiency of these schemes is highlighted by concrete numerical results relating to a model recently proposed for 1D acoustic waves in porous media.

An Interaction Theory for Scattering by Defects in Arrays

I. Thompson and C. M. Linton

SIAM J. Appl. Math. 68, pp. 1783-1806 (24 pages) | Cited 1 time

Online Publication Date: July 16, 2008

Full Text: | Download PDF

Show Abstract
Wave scattering by an array of bodies that is periodic except for a finite number of missing or irregular elements is considered. The field is decomposed into contributions from a set of canonical problems, which are solved using a modified array scanning method. The resulting interaction theory for defects is very efficient and can be used to construct the field in a large number of different situations. Numerical results are presented for several cases, and particular attention is paid to the amplitude with which surface waves are excited along the array. We also show how other approaches can be incorporated into the theory so as to increase the range of problems that can be solved.
Close

Your Recent History Recent History Help

Recently Viewed

  1. Scattering by a Semi-Infinite Periodic Array and the Excitation of Surface Waves
  2. Multiphase Image Segmentation via Modica–Mortola Phase Transition
  3. A Mixture Theory for the Genesis of Residual Stresses in Growing Tissues I: A General Formulation
  4. Distance Functions and Geodesics on Submanifolds of $\R^d$ and Point Clouds
  5. Motion and Homogenization of Vortices in Anisotropic Type II Superconductors
© SIAM By using SIAM Publications Online you agree to abide by the Terms and Conditions of Use. Banner art adapted from a figure by Hinke M. Osinga and Bernd Krauskopf (University of Bristol, UK).
close