"A multigrid framework for non-smooth rigid body dynamics"

Non-smooth rigid body dynamics (NRBD) have been applied successfully to a vast number of diverse areas, such as computer graphics, granular flow, robot simulation and design, virtual reality, virtual prototyping, and many others. It consists in simulating the dynamics of a group of three-dimensional rigid bodies. These rigid bodies are subjected to collisions and frictional constraints, and possibly to other external forces like gravity and electrostatic forces. In this talk we reformulate the rigid body problem as a non-smooth system of equations and solve this system using a variant of the well known Newton method. We develop an algebraic multigrid algorithm (AMG) to precondition the potentially singular Jacobian matrices at each Newton iteration. The Newton/AMG framework is implemented in C++ for speed, clarity and generality.

"A numerical method for 3d doubly periodic electromagnetic scattering"

Periodic electromagnetic scattering has been of interest since the discovery of the unique transmission properties of photonic crystals. I approach the problem with a boundary integral formulation of Maxwell's equations. I will outline some of the difficulties involved in finding numerical solutions to these integral equations. The most significant of these is that the integrals involved are singular and are therefore difficult to approximate numerically. I do some local analysis to smooth these singularities and then to correct the smoothing errors. The result is a third order numerical method.

"A 3D Motile Rod-Shaped Bacterial Model"

In this talk, we introduce a 3D motile rod-shaped bacterial model with a single polar flagellum which is based on the configuration of a monotrichous type of bacteria such as Pseudomonas aeruginosa. The bacterial cell body is constructed from a set of immersed boundary points and elastic links. The helical flagellum is assumed to be rigid and modeled as a set of discrete points along the helical flagellum and flagellar hook. A set of flagellar forces are applied along this rotating helical curve as the flagellum rotates. An additional set of torque balance forces are applied tangentially on the cell body to drive the counter-revolution of the body and provide torque balance. The fluid flow that drives the model bacterial cell is considered as governed by the incompressible Navier-Stokes equations with a force density contributed from the elasticity of the cell body, the flagellar forces, and the torque balance forces. The immersed boundary method is used to solve the Navier-Stokes equations numerically for the fluid velocity and will be briefly reviewed. Some simulations will be presented following as well as the convergence study of the numerical scheme.

"Conformal curvature flow on S¹ and image processing"

In this talk, I shall describe our recent results on the study of conformal curvature problems on S¹. The results on one dimensional affine curvature flow are obtained as special cases under the general setting. I shall also describe the applications of such study in image processing.

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"A Particle Method for a Fluid Transport Equation"

We consider a model of active fluid transport described by an evolutionary equation, known as the EPDiff equation. The EPDiff equation arises in many scientific applications. In particular, it appears in the nonlinear dynamic of shallow water waves, and coincides, for example, with the Camassa-Holm equation of shallow water in 1-D and 2-D, and with the averaged template matching equation for computer vision in higher dimensions. The EPDiff singular solutions are contact discontinuities, called peakons. The key feature of these contact waves is that they carry momentum; so the wave front interactions they represent are collisions, in which momentum is exchanged. This is very reminiscent to the KdV solitons behavior in 1-D.

We numerically investigate the EPDiff dynamics of contact interactions using particle methods. We also show that the system of ODEs, that describes the evolution of particles, has translational and rotational symmetry. We use this symmetry to reduce the dimensionality of the system, and thus to simplify the theoretical and, especially, numerical analysis of the dynamics of peakons and their interactions in two and more space dimensions.

This is a joint work with Jerrold E. Marsden, California Institute of Technology.

"Solving for a Trans-membrane Protein Structure using SSNMR Data"

Nuclear Magnetic Resonance (NMR) is a spectroscopic technique that involves the study of molecular structures via the interaction of radio-frequency electromagnetic radiation with nuclei immersed in a strong magnetic field. NMR experiments provide typically two kinds of structural constraints; distance constraints using liquid state NMR and orientational constraints using solid state NMR (SSNMR). Membrane protein structures are extremely hard to determine. SSNMR is quite effective in solving membrane protein structures. The mathematical problems that arise while solving for the 3-d atomic structure of such proteins from SSNMR data will be discussed. In particular, tools from linear algebra and differential geometry are used to obtain an initial model of the M2 protein, a trans-membrane protein belonging to Influenza A Virus.

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"Molecular dynamics simulations of live particles"

Special colloquium: Gibson Hall 310 at 3:30pm

"From muscle activation to movement: A combined model of anguilliform swimming"

I will discuss a model of anguilliform (eel-like) swimming.  Most fish swim by passing a wave of curvature along their bodies which develops a thrust from the surrounding fluid that pushes the animal forward.  I will discuss how such waves of curvature are created by considering a simple model for muscle forces stimulated by electrical activity, and coupled to an elastic rod.  This wave of curvature is generated by a wave of electrical activity.  In many animals the wave of curvature travels slower than the wave of electrical activity.   Explanations for this phenomena will suggest themselves in the course of describing the model.

"Analysis and numerical solution for electromagnetic wave propagation"

In this talk we discuss the electromagnetic scattering phenomena in two settings: (1) induced by cavities embedded in a perfectly electrically conducting (PEC) ground plane; (2) by bodies of revolution (BOV). In the first case, we seek to determine the fields scattered by a protruding cavity upon a given incident wave. Our method decomposes the entire solution domain to two sub-domains via an artificial semicircle enclosing the cavity: the infinite upper half plane over the PEC ground plane exterior to the semicircle, and the cavity plus the interior region. The problem is solved exactly in the infinite exterior domain, and numerically in the interior domain. In the second case, we combine the efficiency of the locally corrected Nystrom (LCN) method with the simplicity of a BOV geometry. Specifically, we apply the LCN method to a BOR under plane wave illumination. The BOR geometry allows the 2D surface integral equation to be reduced to a series of 1D integral equations through the use of a Fourier series expansion. The solution of each 1D problem is a mode function in the series expansion of the total current.

"Multi-velocity Viscoelastic Flow in Blood Clotting and other Physiological Systems: Models and Numerics"

Understanding the interactions of moving fluid with distributed viscoelastic networks is an issue that permeates much of physiology. Such interactions are critical in intravascular blood clotting where both the aggregating platelets and the fibrin gel which surrounds them behave as viscoelastic materials, and they both form in and interact with the moving blood. We have developed multiphase models of blood clotting which capture some of these interactions, but these models have limitations that arise because the fluid and the viscoelastic materials move in the same velocity field. 'Multi-velocity' models in which relative motion between fluid and viscoelastic materials is allowed should ease these limitations. We discuss the formulation of such models, and the substantial computational challenges they present. We discuss new robust and efficient methods for the viscosity-dominated limit of the models, as well as our ideas for extending these methods to handle significant inertial and elastic effects.