"Reduced Basis Simulation"

Molecular dynamics (MD) simulation provides a powerful tool to study molecular motion. However, MD can be computationally expensive and can require large amounts of data storage. In this presentation we will describe a method for performing molecular simulations with respect to a reduced coordinate space. Given a standard MD trajectory we use principal component analysis to identify k dominant characteristics of a trajectory and construct a k-dimensional (k-D) representation of the atomic coordinates with respect to these k characteristics. Using this model we define equations of motion and perform simulations with respect to the constructed k-D representation.
We apply our method to test molecules and compare the simulations to standard MD simulations of the molecules. Our method allows us to efficiently simulate test molecules by reducing the storage and the computational requirements. The results indicate that the molecular activity with respect to our simulation method is comparable to that observed in the standard MD simulations of these molecules.

"Mathematical Modeling of Cartilage Regeneration"

Articular cartilage is a connective tissue that lines the surface of bones in diarthrodial joints (hips, shoulders, and knees). Cartilage covers each end of the bone, and serves to protect these surfaces from impact stresses, and minimize friction and wear in the joint. A structural extracellular matrix (ECM), consisting largely of water, as well as collagen, glycosaminoglycan (GAG), and other proteins, surrounds the cells in cartilage (chondrocytes). Aging and osteoarthritis can lead to degeneration of cartilage ECM, leading to holes or defects and, ultimately, complete tissue degradation resulting in painful bone-on-bone contact necessitating joint replacement. Cartilage has a limited capacity for growth and repair of large defects; therefore biomaterials for defect-filling are being studied. Hydrogels, which are superabsorbent natural or synthetic polymers with flexibility similar to tissues, are being explored to provide a 3-d scaffold for cartilage regeneration. Upon injection into a defect, hydrogels will slowly degrade as cell proliferation and biosynthesis result in turnover of the gel scaffold. A reaction-diffusion model of cartilage regeneration will be presented for an in vitro experiment performed on a cylindrical tissue explant.

"Accounting for Temperature-Dependent Sex Determination in Crocodilians Using Delay Differential Equations"

The crocodilia have multiple interesting characteristics that affect their population dynamics. They are among several reptile species which exhibit temperature-dependent sex determination (TSD) in which the temperature of egg incubation determines the sex of the hatchlings. Their life parameters, specifically birth and death rates, exhibit strong age-dependence. We develop delay-differential equation (DDE) models describing the evolution of a crocodilian population. In using the delay formulation, we are able to account for both the TSD and the age-dependence of the life parameters while maintaining some analytical tractability. In our single-delay model we also find an equilibrium point and prove its local asymptotic stability. We numerically solve the different models and investigate the effects of multiple delays on the age structure of the population as well as the sex ratio of the population. For all models we obtain very strong agreement with the age structure of crocodilian population data as reported in Smith and Webb (Aust. Wild. Res. 12, 541–554, 1985). We also obtain reasonable values for the sex ratio of the simulated population.  This is joint work with Tenecia Plummer, David Uminsky, Cinthia Vega, Clare Wickman and Michael Zawoiski.

"A Fractional Step θ-method for Fluid Flow Problems"

The accurate numerical approximation of viscoelastic fluid flow poses two difficulties: the large number of unknowns in the approximating algebraic system (corresponding to velocity, pressure, and stress), and the different mathematical types of the modeling equations. Specifically, the viscoelastic modeling equations have a hyperbolic constitutive equation coupled to a parabolic conservation of momentum equation. An appealing approximation approach is to use a fractional step θ-method. The θ-method is an operator splitting technique that may be used to decouple mathematical equations of different types as well as separate the updates of distinct modeling equation variables when modeling mixed systems of partial differential equations.

In this talk a fractional step θ-method is described, and its analysis outlined for both the time dependent convection-diffusion equation and the time dependent equations of viscoelastic fluid flow using the Johnson-Segalman constitutive model. Numerical computations supporting the theoretical results and demonstrating the θ-method will also be presented.

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Fast Spectral Methods for Cylindrical Geometries with Application to Incompressible Flow
In the first part of the talk, a new spectral-Galerkin approach for polar and cylindrical geometries is introduced and analyzed. The pole singularity is treated naturally through an appropriate variational formulation. Clustering of collocation points near the pole, a problem common to the spectral-Galerkin algorithms in the literature, is prevented through a change of variable in the radial direction. The method is very efficient and can be easily adopted to solve a wide range of problems.

In the second part, the coupling between a bulk vortical flow and a surfactant-influenced interface in a open cylinder driven by the rotation of the bottom disk is investigated. The governing equations are developed and analyzed. The fast spectral methods mentioned above are used to obtained the numerical solution. It is found that the base axisymmetric flow is unstable to three-dimensional perturbations for sufficiently large rotation rates.

Nonlinear Phenomena in Acoustics: Traveling Waves, Bifurcations, and Singular Surfaces

An analytical study of finite-amplitude, homentropic acoustic waves in a fluid that saturates a rigid porous medium is presented. The exact traveling wave solution (TWS) is obtained in terms of the Lambert W-function. It is shown that the acceleration profile assumes the form of an asymmetric diffusive soliton and that a jump discontinuity in the acceleration forms as the TWS's propagation speed approaches the sound speed of the fluid. Additionally, a connection between this TWS and results from singular surface theory is established and the findings of the present study are compared/contrasted with those for the thermoviscous, nonporous case. [Work supported by ONR/NRL funding (PE 061153N).]

"A Method for Double-Mirror Catadiopttric Sensor Design"

For many applications such as surveillance,  photography, and robot navigation, it is required that the camera have a wide field of view. Traditional approaches to solve this problem include using a rotating camera, stitching images, complex lenses or multiple cameras. Catadioptric sensors are devices consisting of reflective surfaces (catoptrics) and a camera (dioptrics), that address the problem of wide-angle imaging. We discuss the family of double-mirror catadioptric sensors and  propose a method to design a  double-mirror catadioptric sensor that will satisfy a given world-to-sensor mapping via numerically solving a system of PDEs