UD Math Sciences Department
Applied Math Seminar

Spring 2003 Schedule

Want to suggest a speaker? Send email to Pam Cook.

Abstracts

In this talk we present analytical and numerical results of our ongoing research in mathematical modeling of Ductal carcinoma in situ (DCIS). DCIS refers to a specific diagnosis of cancer that is isolated within the breast duct, and has not spread to other parts of the breast. In a recent talk Mary Edgerton of Vandebilt University described two special patterns found in DCIS: one lining up like baby trees and one spreading out evenly. We modify a model proposed by Byrne and Chaplain for the growth of a tumor consisting of live cells (nonnecrotic tumor) to describe the growth inside a rigid cylinder, a model mimicking the growth of a ductal carcinoma. The model is in the form of a free boundary problem. The analysis of stationary solutions of the problem shows that this model has five tumor patterns that mimik some typical patterns of DCIS, including the types described by Edgerton. The analysis shows that there may be two other kinds of patterns that resemble the non-growing tumor. We also show for the solid DCIS case (one-dimensional case) that the stationary solution is unstable.

Classical methods to solve weakly nonlinear oscillator equations include averaging and multiple scales. In recent work with Mudavanhu, the author has shown that the renormalization group method of Chen, Goldenfeld, and Oono has certain advantages. These, however, are shared with an amplitude expansion technique that is more straightforward and has potential for substantial generalization.

I will talk about dynamics of thin liquid films slowly spreading on a substrate. We treat this problem within the framework of lubrication approximation, with particular attention to the role that contact line, where liquid, solid and gas phase meet, plays in determining the dynamics of a spreading film. After explaining our theoretical and computational techniques, I will give an overview of the fluid instabilities in the flow down an inclined plane, and then discuss a possibility of controlling the instability in the flow on a nonuniform substrate. Last part of the talk will concentrate on the issue of a topological change in the fluid configuration, illustrated on the problem of a merge of sesile drops. Some experimental results, obtained in our undergraduate lab at NJIT, will be presented as well. This research is being carried out in collaboration with Javier Diez.

We present a new set of algorithms and methodologies for the numerical solution of problems of scattering by complex bodies in three-dimensional space. These methods, which are based on integral equations, high-order integration, fast Fourier transforms and highly accurate high-frequency methods, can be used in the solution of problems of electromagnetic and acoustic scattering by surfaces and penetrable scatterers --- even in cases in which the scatterers contain geometric singularities such as corners and edges. In all cases the solvers exhibit high-order convergence, they run on low memories and reduced operation counts, and they result in solutions with a high degree of accuracy. In particular, our algorithms can evaluate accurately in a personal computer scattering from hundred-wavelength-long objects by direct solution of integral equations --- a goal, otherwise achievable today only by supercomputing. A new class of high-order surface representation methods will be discussed, which allows for accurate high-order description of surfaces from a given CAD representation. A class of high-order high-frequency methods which we developed recently, finally, are efficient where our direct methods become costly, thus leading to a general and accurate computational methodology which is applicable and accurate for the whole range of frequencies in the electromagnetic spectrum.

Is it possible to tune material parameters so that extremely small structure on a fluid surface can be created with large-scale forcing? Can this be a useful way to make thin threads and thin coatings? Inspired by recent experiments (Cohen & Nagel, PRL, 2002) showing that steady flow past an interface between two layers of viscous liquids can draw out a tendril of fluid (a spout) about 50 times thinner than the smallest lengthscale set by the boundary conditions, we present a simple long-wavelength model of viscous withdrawal. Analytic solution and numerical simulations show the fluid interface develops a steady-state singularity at a critical exterior flow rate for appropriate boundary conditions. Thus, arbitrarily small structures can in theory be made on the fluid surface.

We are concerned with integral equations of scattering. In order to deal with the well-known high frequency problem, we suggest a coupling of two kind of methods that reduce the numerical complexity of iterative solution of these integral equations. The microlocal discretization method introduced by T. Abboud, J.-C. Nédélec and B. Zhou, enables one to reduce efficiently the size of the system considering an approximation of the phase function of the unknown. However, the method needs an expensive precalculation. We suggest the use of the fast multipole method introduced by V. Rokhlin, in order to speed up the precalculation. This work is an original application of the fast multipole method for acceleration of a microlocal discretization method within the new integral formulation written by B. Després. Numerical results obtained for Helmholtz's equation are very satisfying. For Maxwell's equations, they are also quite interesting.

The motion of a viscous drop on a substrate is a fundamental problem with rich applications. Here, we present a generic numerical method based on front-tracking to solve the Navier--Stokes equations, including contact line motion within a two-phase region. Both viscous and inertia effects are included. A Navier slip boundary condition is used to relax the contact line singularity. Under the proper conditions, the interface in the neighborhood of the spreading edge can become unstable and form a ring of fluid, and dry spot could form at the center. Investigations are also made on the effects of an applied flow field on the shape of the drop.

I will give an overview on the usefulness of configuration spaces in robotics coordination and control. These problems necessitate a topological investigation. I will detail a class of configuration spaces which are particularly useful for systems involving self-reconfiguration.

In the last decade significant progress has taken place within non-equilibrium thermodynamics resulting in several formulations for coupled transport phenomena that allowed for an extension and a more systematic application of continuum mechanics to continua endowed with an internal microstructure. Recently, the consistency between two of those approaches (one and two-generator formulations) has been extensively confirmed for a wide range of polymer models. In this presentation we develop gradually the general approach by emphasizing the close connection that exists between one of the new formalisms, that involving a generalized bracket, and the more traditional approach of modeling dissipative phenomena through Linear Irreversible Thermodynamics (LIT). The new approach is shown to involve a merging of LIT (and its nonlinear generalizations) with the most general Poisson-bracket description of conservative dynamic. Then, we overview illustrative examples of the implementation of the bracket non-equilibrium thermodynamic description to the study of polymer systems under flow. New phenomena are observed due to the coupling between various transport and relaxation processes with important consequences as far as the understanding of the nonlinear dynamics and the rheology of polymer systems.

Accurate determination of velocity statistics in sedimentation is a long-standing problem in multiphase fluid dynamics. Batchelor (1972) calculated the average velocity of a dilute collection of solid particles sedimenting in a fluid at low Reynolds number; but Caflisch & Luke (1985) showed that Batchelor's assumptions imply velocity variances scaling with the container size. This scaling has been the source of significant controversy, with simulations generally agreeing with the scaling and experiments generally indicating the presence of another "universal" length scale. We present a theory for the velocity fluctuations that unifies the experimental results and numerical simulations, the central idea being that fluctuations may decrease below those of independent distributions because of the development of a vertical stratification in the cell. We show through numerical simulations, scaling arguments, theoretical calculations, and experimental observations that a very small stratification changes the characteristics of the density and velocity fluctuations. Quantitative estimates then agree with the simulations and previous experiments. The velocity fluctuations in sedimentation are therefore not universal, but instead depend on both cell shape and the developing stratification. This talk represents joint work with Michael Brenner and with multiple members of David Weitz's experimental group at Harvard.

In the first half of my talk, I will discuss approximate symmetries and approximate first integrals (via approximate version of Noether's theorem) of the deterministic dynamical systems with a small parameter in such a way that it will remain in touch with normal form theory, hence, bifurcations. Applications to examples arising from astrophysics and nonlinear vibrations will also be mentioned. During the second half of the talk, I will introduce a new definition of symmetry and quasi-symmetry for the Ito and Stratonovich dynamical systems. It will be shown that they do satisfy deterministic (not involving Wiener terms) system of partial differential equations. Theorems which yield the construction of conserved quantities from (quasi) symmetries without recourse to Noether's theorem will also be mentioned. An application to N-species stochastic Lotka-Volterra system will be discussed.

Fall 2002 Schedule

Abstracts

In order to design and control micro-fluidic networks and surface tension driven devices, we must have accurate but small models of the underlying physics. I will describe our modeling efforts in this direction for channel filling and electro-wetting flows. The channel filling problem is important for lab-on-a-chip applications where we want to ensure that chemical reaction A takes place before chemical reaction B, and so we must fill the network in the right order. We will phrase the quasi-steady filling problem as a geometrically constrained energy minimization problem and will show how we can simulate the filling of thousands of channels in seconds. Extensions to the dynamic case will be discussed. Finally, I will show how the model may be used to design multi-channel networks that will fill in the desired order. (This work is joint with Nanostream Inc.) The electro-wetting modeling effort is in collaboration with CJ Kim and Robin Garrell. Here applied voltages are used to change the surface tension locally, and this enables one to move, split, merge, and mix micro-fluids. Again, we will phrase the problem in an energetic framework. By using scaling arguments and solutions of the shape dependent electro-static problem, we will show how sessile drops change shape with applied voltage and material variations. In particular, we will explain why the liquid drop stops changing shape past some critical applied voltage (this is known as contact angle saturation). At the close of the seminar, I will outline our vision and initial results for micro-fluidic particle control. In particular, I will discuss the type of models and sensors required for practical real-time control.

For a smooth function f supported in a region D in R^n, we consider the question of recovering f from a knowledge of its mean values over a family of spheres centered on a part or the whole boundary of D. We prove a uniqueness result when D is convex, provide an inversion algorithm, and provide an inversion formula when D is a ball. The last result also has important implications for the regularity of traces of solutions of Initial Value Problems for hyperbolic PDE for compactly supported initial data.

Classic calculus of variations theory offers us a conjugate point test, ideally suited for numerical implementation, for determining the stability of a critical point. With the appropriate functional-analytic viewpoint, this test may be extended to problems not handled by the classic theory, such as contact problems. This idea will be illustrated in a relatively simple case, that of an elastic rod buckling in the presence of an external field designed to model a "soft" wall. Here the second variation operator is an integrodifferential operator, but a conjugate point test may still be developed, and it lends itself easily to numerical implementation. By studying the dependence of the rod stability on soft-wall strength parameters, we may make predictions about the stability of rod configurations contacting a true "hard" wall.

We construct a microscale model for a a rigid particle suspension in a viscous fluid that includes Maxwell electrostatic forces. This enables us, via homogenization techniques, to characterize the properties the material exhibits at the macroscopic scale. The change in the effective constitutive equations is due to the highly oscillating electrostatic forces. It is confimed that in some rheological experiments the behavior of these suspensions is similar to the Bingham materials. The material properties are determined by both hydrodynamic and electrostatic interparticle interactions. Interfacial effects due to hydration are also considered. The understanding of the underlying physics governing the electrorheological response can contribute to the engineering of better ER fluids and can be useful in building controllable systems.

We consider heterogeneous reactions in which one reactant is mobile (free to diffuse in a given domain) while the other is immobile, and is confined to some portion of the boundary of the given domain. If the concentration of the mobile reactant is assumed to obey a linear diffusion equation, the problem can in principle be reduced to an integral equation or integrodifferential equation on the boundary, which can then be reformulated in several ways. I will demonstrate how to do this for some simple geometries, and then show how to asymptotically approximate the nonlinear Volterra type integral equations that result for these cases by differential equations in the limits of both large and small Damkohler number. The Damkohler number is the limit of a transport (here diffusive) time scale to a reaction time scale. The motivating application is immunocolloid labeling, which is a technique for visualizing molecular-scale features of cells under an electron microscope. The technique uses composite labeling particles consisting of nanometer scale metal particles conjugated to biomolecules which bind to the features of interest. The work is joint with R. Albrecht (UW-Madison, Animal Sciences), and P. Milewski (UW-Madison Mathematics).

Active processes are a distinguishing characteristic of life. Fascinating examples include molecular motors, true nanoengines that convert chemical energy to mechanical work. These proteins underlie a wide array of processes in cells and tissues, such as contraction of smooth and skeletal muscle, cell division, intracellular trafficking, and endo- and exocytosis. While unprecedented insight into the mechanics of single motor molecules and the protein polymers they walk along on has been made, an understanding of the dynamic properties of such active systems remains to be established.

We take advantage of the mechanochemical activity of the single-headed molecular motor protein myosin S1 to study active dynamics and response in reconstituted actin polymer networks. Using diffusing wave spectroscopy to measure the high-frequency viscoelastic response and relaxation mechanisms of actomyosin, we find a new scaling law for the shear modulus amplitude that is consistent with random perturbations along the filaments. In addition, persistence length measurements of single polymers indicate that motor activity leads to an increase in the effective temperature for tangential motion. Lastly, the timescales of our measurements provide critical insight into the rapid conformational changes underlying force-generation in motor proteins.

Real biological neurons display complex behavior suggesting that there may be significant computation going on at the single nerve level. We will give an overview of cable theory for dendrites that will include simple geometric non-uniformities, cable models with clusters of spines, and as time permits, introduce problems associated with non-uniform density of ion channels.

On going developments in the area of micromachining and nanotechnology has enabled the research and development of variety of chemical, biological, mechanical and optical sensors and actuators. Micro-electro-mechanical systems (MEMS) being a mature technology, it is now possible to use MEMS based structures to probe and control variety of phenomena that occur at nanoscale. An insight into the nature of nanoscale and molecular phenomena will enable us to develop sophisticated sensors with high sensitivity, stability and selectivity. This talk will focus on the development and utilization of microhotplate platforms as chemical sensing and thermal actuator devices. Originally developed at the National Institute of Standards and Technology (NIST), the microhotplates are finding use in variety of areas such as gas sensors for molecular recognition, nanoparticle research, microfluidics, and biological sensor applications. The advantages of using a microhotplate stems from its small size, rapid thermal time constants, high temperature capability (~500(C) and ability to fabricate multiple element arrays on a single chip. Patterned with metal oxide thin film, these microhotplates can be used as chemical sensing devices. The use and control of metal nanoparticles as seed layers for controlling the microstructure of tin oxide during CVD growth process will be discussed. By selective deposition of different types of metal nanoparticles as seed layers on microhoplate arrays, it is possible to develop a range of different types of devices that could be used with suitable olfactory signal processing techniques in order to identify a variety of gases. Further, it is possible to use the microhotplates as time resolved thermal platforms for conducting combinatorial research into variety of chemical and biological materials. One new area that is currently being researched at the Delaware MEMS and nanosystems laboratory is the use of microhotplate as thermal actuators for driving nanoscale and nanotube based sensors. The talk will conclude with discussions about the ongoing developments in microhotplate research and its application towards chemical sensing, nanoscale thermal actuation and the future of microhotplate technology. Thermal and mechanical modeling of microhotplate suspended structures is critical for the success of this technology. Mathematical modeling and simulation presents a powerful approach towards understanding the mechanical and thermal issues involved in microhotplates.

Since their original invention in the 1960s, methods for solving electromagnetic scattering problems on a digital computer have advanced profoundly. Modern computational electromagnetics methods can solve scattering problems with computational complexities of O(NlogN), and O(N) memory requirements, where N is the number of unknowns in the problem. Indeed, even dense matrix problems of more than ten million unknowns have been solved. Nonetheless, there is a gaping hole in the computational electromagnetics toolbox. Electromagnetic problems can be solved in either the time or frequency domains, and using either integral equation or differential equation methods. While numerical techniques exist for solving time-domain differential equations (the finite difference time domain method), frequency domain differential equations (the finite element method), and frequency domain integral equations (the method of moments), no generally accepted method currently exists for solving time domain integral equations. Computational methods for the solution of time domain integral equations have been largely ignored for forty years because the most obvious methods for their solution lead to unstable results. In this talk, we will examine the roots of these instabilities, and propose a new set of temporal basis functions that solves the problem. In particular, the temporal basis functions are band-limited and thus prevent high-frequency errors from corrupting the solution. Unfortunately, the proposed basis functions are noncausal, so methods for producing a causal system of equations will be discussed. It will also be argued that low frequency instabilities result from a null space in some of the equations of electromagnetics, and a method will be presented for mitigating this problem. Numerical results will demonstrate that the solutions obtained by the proposed method converge exponentially fast (in the amount of computation done) to frequency domain solutions of the same problem based on the same spatial discretization.

Previous years

• 2001-2002