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Abstracts and Talk Materials
Higher Order Geometric Evolution Equations: Theory and Applications from Microfluidics to Image Understanding
March 23 - 26, 2009

Step evolution for crystals of finite size: The ADL case
December 31, 1969

We study the step evolution of crystal structures relaxing toward flat surface when the number of steps is finite. We assume that the mass transport process on the structure's surface is attachment-detachment limited (ADL). We propose a fourth order PDE for the slope of the profile as a function of its height. This PDE is derived from the step equations of motion. The solution is asymptotically self-similar. We prove existence and uniqueness of the self-similar solution in the discrete setting.

Many problems in image processing are addressed via the minimization of a cost functional. The most prominently used optimization technique is gradient-descent, often used due to its simplicity and applicability where other techniques, e.g., those coming from discrete optimization, can not be applied. Yet, gradient-descent suffers from slow convergence, and often to just local minima which highly depend on the initialization and the condition number of the functional Hessian. Newton-type methods, on the other hand, are known to have a faster, quadratic, convergence. In its classical form, the Newton method relies on the L2-type norm to define the descent direction. In this work, we generalize and reformulate this very important optimization method by introducing Newton-type methods based on more general norms. Such norms are introduced both in the descent computation (Newton step), and in the corresponding stabilizing trust-region. This generalization opens up new possibilities in the extraction of the Newton step, including benefits such as mathematical stability and the incorporation of smoothness constraints. We first present the derivation of the modified Newton step in the calculus of variation framework needed for image processing. Then, we demonstrate the method with two common objective functionals: variational image deblurring and geometric active contours for image segmentation. We show that in addition to the fast convergence, norms adapted to the problem at hand yield different and superior results.

I will present an hydrodynamic model for the motility of keratocytes or fibroblasts on substrates in vitro. Cells or fragment of cells have been observed to switch from a stationary round state to a motile and anisotropic crescent-shaped state. Experimentally, a polarization of the actin network occurs in a preferred direction prior to motility and determins the direction of motion. In this talk, I will present first the model for actin flow of Callan-Jones et al for two-dimensionnal cells lying on a substrate with a strong friction. Using Schwarz function techniques, we derive a dynamic equation for the shape contour including the polymerisation-depolymerisation process and show that static circular shapes are stable for enough tension of the lipidic membrane. We extend the model to incorporate the actin cortex whose anisotropy is due to a preferred orientation at the lipidic membrane. To do so, we use the theory of active polar gels of Kruse et al. inspired from the theory of liquid crystals. Since this cortex has a size of order one ten of the cell, we perform a boundary layer analysis. The presence of the cortex is responsible for a modification of the boundary conditions at the cell border. We show that an increase of the motor activity destabilisizes the cell in the tensile case but we also show that a polarization of the whole actin network is necessary to induce a translation motion.

Asymptotic dynamics of attractive-repulsive swarms
December 31, 1969

We classify and predict the asymptotic dynamics of a class of swarming models. The model consists of a conservation equation in one dimension describing the movement of a population density field. The velocity is found by convolving the density with a kernel describing attractive-repulsive social interactions. The kernel's first moment and its limiting behavior at the origin determine whether the population asymptotically spreads, contracts, or reaches steady-state. For the spreading case, the dynamics approach those of the porous medium equation. The widening, compactly-supported population has edges that behave like traveling waves whose speed, density and slope we calculate. For the contracting case, the dynamics of the cumulative density approach those of Burgers' equation. We derive an analytical upper bound for the finite blow-up time after which the solution forms one or more δ-functions.

We model the evolution of human tear film during relaxation (after a blink) using lubrication theory and explore the effects of viscosity, surface tension, gravity and boundary conditions that specify the pressure. The governing nonlinear partial differential equation is solved on an overset grid by a method of lines using finite differences in space and an adaptive second order backward difference formula solver in time. Our two-dimensional simulations, calculated in the Overture framework, display sensitivity in the flow around the boundary to both our choice between two different pressure boundary conditions and to the presence of gravity. The simulations recover features seen in one-dimensional simulations and capture some experimental observations including hydraulic connectivity around the lid margins.

Precursors to splashing on a solid surface
March 25, 2009

A high velocity impact between a liquid droplet and a solid surface produces a splash. Classical work traced the origin of the splash to a thin sheet of liquid ejected near the impact point. Mechanisms of sheet formation have heretofore relied on initial contact of the droplet and the surface. We demonstrate that, neglecting intermolecular forces between the liquid and the solid, the liquid does not contact the solid, and instead spreads on a very thin air film. The interface of the droplet develops a high curvature and emits capillary waves.

We present a multigrid method for solving the dual formulation of the Total Variation-based problem in image restoration. Flat regions of the desired image contribute to the slow convergence of the widely-used Chambolle method. Numerical results confirm that the multigrid method with a modified Chambolle smoother is many orders of magnitude faster than the original Chambolle method.

We consider a Cahn-Hilliard functional with long-range interactions. This functional was introduced as a qualitative way of modeling self-assembly of diblock copolymers. We will consider the phase diagram from the point of view of numerical simulations. We will also describe analytical work, via Gamma convergence, on the asymptotics of the energy in the small volume fraction limit. Our results will be compared with a formal study on the H-1 gradient-flow of the functional, demonstrating separate regimes for coarsening and self-assembly (pattern formation).

This talk will encompass recent work with M. Peletier (Eindhoven), J.F. Williams (SFU), M. Maras (SFU), and with K. Glasner (Arizona).

In this talk we report on some mathematical techniques for modelling evolving geometries at low Reynolds numbers. Two problems will be discussed, both involving free capillary surfaces. The first is a study of organisms swimming in Stokes flows in the presence of free surfaces. An idealized mathematical model is presented whereby the swimmer's interaction with a free capillary surface is captured. The second problem is of industrial importance involving the optimal design of thin optic fibres with microstructure. There is much interest in reducing transmission loss in optic fibres by careful design of the microstucture imparted to a fibre during the drawing process'' in which molten glass is pulled through a casting die. During this process, geometrical changes in the microstucture take place owing to capillary effects resulting in the need to understand a highly nonlinear inverse problem. New ideas for modelling this process will be described.

We will selectively review the application of complex variable methods to moving boundary problems, with specific reference to the Hele-Shaw problem, and slow viscous flow driven by surface tension (in 2D, or quasi-2D). Established theory and results will be discussed, as well as some open questions and new directions.

In this talk, we report some recent works on the diffuse interface models of some interface problems with curvature dependent interfacial energies such as the Helfrich elastic bending energy for vesicle membranes. We discuss various theoretical and computational issues related to the diffuse interface approach and present some simulation results for the deformation of vesicle membranes in a number of environmental conditions.

We perform numerical studies of the Navier–Stokes-αβ equations, which are based on a general framework for fluid-dynamical theories with gradient dependencies. Specifically, we examine the effect of the length scales α and β on the energy spectrum in three-dimensional statistically homogeneous and isotropic turbulent flows in a periodic cubic domain, including the limiting cases of the Navier–Stokes-α and Navier–Stokes equations. A significant increase in the accuracy arises for β < α, but an optimal choice of these scales depends on the grid resolution.

Microfluidic devices without walls have many advantages over channel-based devices. In droplet-based (“digital”) microfluidics, liquids are transported as droplets between parallel plates, rather than as streams. The droplets are created, moved, joined and divided by applying electrical potentials sequentially between electrodes buried beneath a hydrophobic dielectric layer. The resulting device is completely reconfigurable. Samples can be processed in series or simultaneously, each in the same way or through a unique sequence of steps. We have found shown that droplets of a wide range of liquids can be actuated by electrowetting, dielectrophoresis, or a combination of the two. An electromechanical model has been developed that explains the relative ease with which different liquids can be actuated and provides the basis for designing devices and operating conditions for actuating particular liquids. Applications of droplet microfluidics include separations by precipitation, solid phase extraction and liquid-liquid phase transfer. Understanding and controlling these processes represent significant new challenges to the modeling community.

A gradient flow approach to a free boundary droplet model
December 31, 1969

We consider a quasi–stationary free boundary droplet model. This model does not satisfy a comparison principle and can have non unique solutions. Nevertheless it can be seen as a gradient flow on the space of possible supports of the drop. The gradient flow formulation leads to a natural time discretization, which we employ to show the existence of a weak form of viscosity solutions for the model.

We continue the study, initiated in Short et al., of criminal activities as described by an agent based model with dynamical target affinities. Here we incorporate effect of law enforcement agents on the spatial distribution and overall level of crime in simulated urban settings. Our focus is on a two–dimensional lattice model of residential burglaries, where each site (target) is characterized by a dynamic attractiveness to burglary and where criminal and law enforcement agents are represented by random walkers. The dynamics of the criminal agents and the attractiveness field are, with certain modifications to be detailed, as described in Short et al. Here the dynamics of enforcement agents are affected by the attractiveness field via a biasing of the walk the detailed rules of which define a deployment strategy. We observe that law enforcement agents, if properly deployed, will in fact reduce the total amount of crime, but their relative effectiveness depends on their numbers, the deployment strategy used, and spatial distribution of criminal activity.

The Kadomtsev-Petviashvili (KP) equation is a two-dimensional dispersive wave equation which was proposed to study the stability of one soliton solution of the KdV equation under the influence of weak transversal perturbations. It is well know that some closed-form solutions can be obtained by  function which have a Wronskian determinant form. It is of interest to study KP with an arbitrary initial condition and see whether the solution converges to any closed-form solution asymptotically. To reveal the answer to this question both numerically and theoretically, we consider different types of initial conditions, including one-line soliton, V-shape wave and cross-shape wave, and investigate the behavior of solutions asymptotically. We provides a detail description of classification on the results.

The challenge of numerical approach comes from the unbounded domain and unvanished solutions in the infinity. In order to do numerical computation on the finite domain, boundary conditions need to be imposed carefully. Due to the non-periodic boundary conditions, the standard spectral method with Fourier methods involving trigonometric polynomials cannot be used. We proposed a new spectral method with a window technique which will make the boundary condition periodic and allow the usage of the classical approach. We demonstrate the robustness and efficiency of our methods through numerous simulations.

On instabilities of finite-size films and rivulets
December 31, 1969

Joint work with J. Diez, A. Gonzalez, and R. Rack.

We discuss the influence of finite size effects on the breakup process involving finite-size films and rivulets. For films, we show that the breakup process due to finite size effects can be related to the so-called nucleation mode of instability of infinite films. We also consider coupling of different modes of instabilities, and the competition between them. Next, we revisit the classical problem of rivulet instability and discuss whether finite size effects may be important in determining relevant breakup mechanisms. We apply our results to rupture of nano-scale metal lines irradiated by repeated laser pulses and discuss relevance of the considered process to self-assembly on nanoscale.

Dynamics of thin liquid films
March 25, 2009

Thin liquid films are important in applications involving lubrication or coating, which arise in both biological and industrial contexts. Recent experiments have uncovered new phenomena that present challenges of modeling, analysis and simulation. These include new wave forms, fingering instabilities, and a variety of driving and control mechanisms. Mathematically the class of problems is interesting because surface tension dominates inertia, leading to fourth order nonlinear parabolic partial differential equations. This talk will include recent developments and open problems in theoretical, experimental and applied aspects of thin liquid films.

Phase-field model of self-assembled copolymer monolayer
December 31, 1969

We develop a phase field model that incorporates the polymer vitrification and diffusion in the self-assembly of polymer blends. Simulation shows the different polymers in the blend cooperate to self-assemble into nanoscale features with varying dimension. The feature dimensions can be tuned by adjusting the blend composition and the surface concentration.

Shape spaces can be endowed with the structure of Riemannian manifolds; this allows one to compute, for example, Euler-Lagrange equations and geodesic distance for such spaces. Until very recently little was known about the actual geometry of shape manifolds; in this talk we summarize results contained in my recent doctoral dissertation, which deals with the computation of curvature for "Landmarks Shape Spaces." Implications on both the qualitative dynamics of geodesics and the statistical analysis on shape manifolds are also discussed.

Dewetting of thin liquid films
December 31, 1969

We present results on various aspects of thin film models for dewetting films involving high order equations and systems of equations. These include results on the rim instability and the shape of the rim where the liquid dewets, as well as the occurence of non-classical shocks for fast dewetting where inertia becomes important.

After a brief overview of electrohydrodynamics including Maxwell's electric stress tensor under AC fields where the medium has both conductive and dielectric characteristics, we focus on the problem of electrowetting actuation of sessile drops on a patterned array of electrodes with a thin dielectric coating. For both the case when the drop is electrically grounded from below and when it is floating, we compute the electric field in the vicinity of the drop over a range of frequencies and use the traction derived from the Maxwell stress tensor to calculate the effective electrowetting force on the drop. At low frequencies where the drop behaves like a perfect conductor, the results are compared with previously derived lumped parameter models for the electrowetting force. [Joint work with James Sterling and Maged Ismail.]

Self-similar rupture of thin films with strong slip
March 26, 2009

(Joint work with D. Peschka and A. Muench).

We consider a simple model for line rupture of thin fluid films in which Trouton viscosity and van-der-Waals forces balance. For this model there exists a one-parameter family of second kind self-similar solutions. We establish necessary and sufficient conditions for convergence to any self-similar solution in a certain parameter regime. We also present a conjecture on the domains of attraction of all self-similar solutions which is supported by numerical simulations.

We started with a project where we denoised normals to surfaces, then fit the surface to the normals, which we regarded as solving a 4th order PDE via some kind of splitting. This led to remarkably successful algorithms for L1 tpe minimizations, constrained and unconstrained. These include L1, TV, B1,1, nonlocal TV,... Bregman iteration, in its various incarnations popped up and turned out to be unreasonably effective. I'll discuss this which is joint work with many people.

Thin fluid films with surfactant
December 31, 1969

Thin liquid films driven by surface tension are considered, both when gravity plays a significant role, as on an inclined plane, and when it is less significant, on a horizontal substrate. Motion of the film is modeled in the lubrication approximation by a fourth order system of PDE. In the case of a horizontal substrate, we examine the influence of insoluble surfactant both experimentally and numerically. In the experiments, we visualize surfactant using fluorescence, and its effect on the thin film using a laser. The numerical code tracks the edge of the surfactant as it propagates. We also analyze the stability of a thin film wave traveling down an inclined plane driven by both surfactant and gravity. Numerical results show the propagation of small disturbances, thereby substantiating the analysis. This is joint work with Karen Daniels, Dave Fallest, Rachel Levy and Tom Witelski.

The talk will focus on the numerical approximation of the evolution of a thin viscous films on a curved geometry. Here, the concept of natural time discretization for gradient flows is revisited. This is based on an explicit balance between the energy decay and the corresponding dissipation to be invested. In case of thin films the dissipation is formulated in terms of a transport field, whereas the energy primarily depends on the film profile. The velocity field and the film height are coupled by the underlying transport equation. Hence, one is naturally led to a PDE constraint optimization problem and duality techniques from optimization are applied in the minimization algorithm. For the space discretization a discrete exterior calculus approach is investigated. The method can be generalized to the simulate thin coatings.

Shape optimizer needed
December 31, 1969

Motivation: It is possible to relate the concept of chemical bond to the region of three-dimensional space where the probability to find exactly one pair of electrons is maximal.

Characteristics: - The computation of the probability for a given volume chosen can be time-consuming. It requires the eigenvalues of a matrix having elements computed from integrals over the volume. - The shape derivatives can vary strongly from one part of the delimiting surface to another. - Multiple solutions exist by the nature of the problem. However, the user might have a good intuition of what they are and choose a good starting volume.

Most classical and modern studies of swimming, pumping, and mixing in fluids have considered fluids that are Newtonian. All of these phenomena also take place in fluids that are viscoelastic and at low Reynolds number, and are particularly important to biology and to engineering areas such as microfluidics. I will discuss theoretical studies of the effect of viscoelasticity on low Reynolds number undulatory swimming and peristaltic pumping. I will also discuss an example of how symmetry breaking instabilities in extensional flows of a viscoelastic fluid can lead to new coherent structures and fluid mixing.

Microfluidics enhanced novel materials synthesis
December 31, 1969

The flow of complex fluids in confined geometries produces rich and new phenomena due to the interaction between the intrinsic length-scales of the fluid and the geometric length-scales of the device. In this poster, we will show three examples to illustrate how self-assembly, confinement, and flow can be used to control fluid microstructure and enhance the controlled synthesis of bio-compatible nanomaterials and supramolecular hydrogels.

We derive a time-dependent exact solution of the free surface problem for the Navier-Stokes equations that describes the planar extensional motion of a viscous sheet driven by inertia. The linear stability of the exact solution to one- and two-dimensional symmetric perturbations is examined in the inviscid and viscous limits within the framework of the long-wave or slender body approximation. Both transient growth and long-time asymptotic stability are considered. For one-dimensional perturbations in the axial direction, viscous and inviscid sheets are asymptotically marginally stable, though depending on the Reynolds and Weber numbers transient growth can have an important effect. For one-dimensional perturbations in the transverse direction, inviscid sheets are asymptotically unstable to perturbations of all wavelengths. For two-dimensional perturbations, inviscid sheets are unstable to perturbations of all wavelengths with the transient dynamics controlled by axial perturbations and the long-time dynamics controlled by transverse perturbations. The asymptotic stability of viscous sheets to one-dimensional transverse perturbations and to two-dimensional perturbations depends on the capillary number (Ca); in both cases, the sheet is unstable to longwave transverse perturbations for any finite Ca. This work is in collaboration with Thomas P. Witelski.

Effect of boundary conditions on mixing efficiency
December 31, 1969

We consider the mixing of fluid by chaotic advection. Many well-studied examples may be modeled by a class of dynamical systems known as linked-twist maps. The mathematical discipline of ergodic theory studies concepts such as mixing which will be familiar to experimentalists. New analytical results for linked-twist maps suggest mixing rates similar to those observed experimentally and numerically.

Eulerian indicators for predicting mixing efficiency
December 31, 1969

Mixing is inherently a Lagrangian phenomenon, a property of the movement of fluid particles. Many different methods exist for measuring, quantifying and predicting the quality of a mixing process, all involving evolution of individual trajectories. We propose indicative tools which are formulated using only Eulerian information, and illustrate their use briefly on a variety of different model mixers.

Numerical simulation of moving interface problems often requires the solution of elliptic PDEs involving coefficients that can be discontinuous and sources that are singular. Since the interface is moving, it is advantageous to solve the problem on a fixed Eulerian grid which does not conform to the interface as it moves. We propose an intuitive new method which acheives second order accurate results in L-infinity on a fixed cartesian grid with embedded interfaces. The method is largely independent of the geometry and the interface can be represented either as an arbitrary (closed) segmented curve or a levelset. The problem is formulated as a variational constrained minimization problem which preserves a symmetric positive definite discretization.

Two-phase gas-liquid flows are important in a variety of heat transfer systems, such as in the on-chip cooling of microelectromechanical devices up to the infrastructure of safety systems in nuclear power plants. We focus on the case of two-layer flows in inclined channels, where a gas and a liquid, immiscibly separated by a sharp interface with large surface tension, flow in opposite directions. The liquid is driven by gravity while the gas flows due to an imposed pressure gradient. For disturbance wavelengths that are much longer than the channel thickness, a fourth-order nonlinear equation which describes the evolution of the separating interfacial shape is found that is coupled to an elliptic equation for the pressure, whose solution provides a constraint to the dynamics of the flow. We survey the impact of these different constraints on the solutions, and extend the analysis to include incompressibility effects. This work was a collaboration with T.M. Segin and L. Kondic.

Joint work with Z. Borden, H. Grandjean, L. Kondic, and A.E. Hosoi.

Experiments with glycerol-water thin films flowing down an inclined plane reveal a localized instability that is primarily three-dimensional. These transient structures, referred to as "dimples", appear initially as nearly isotropic depressions on the interface. A linear stability analysis of a binary mixture model in which barodiffusive effects dominate over thermophoresis (i.e. the Soret effect) reveals unstable modes when the components of the mixture have different bulk densities and surface tensions. This instability occurs when Fickian diffusion and Taylor dispersion effects are small, and is driven by solutalcapillary stresses arising from gradients in concentration of one component, across the depth of the film. Qualitative comparison between the experiments and the linear stability results over a wide range of parameters is presented.

We give a convergent expansion of solutions of the two-dimensional, incompressible Navier-Stokes equations which generalizes the Helmholtz-Kirchhoff point vortex model to systematically include the effects of both viscosity and finite core size. The evolution of each vortex is represented by a system of coupled ordinary differential equations for the location of its center, and for the coefficients in the expansion of the vortex with respect to a basis of Hermite functions. The differential equations for the evolution of the moments contain only quadratic nonlinearities and we give explicit combinatorial formulas for the coefficients of these terms. We also show that in the limit of vanishing viscosity and core size we recover the classical Helmholtz-Kirchhoff point vortex model.

A family of high-order geometric and potential driving evolution equations was introduced and applied to image analysis and biomolecular surface formation. Coupled geometric PDEs were introduced for image edge detection.

Motivated by the dewetting of viscous thin films on hydrophobic substrates, we study models for the coarsening dynamics of interacting localized structures in one dimension. For the thin films problem, lubrication theory yields a Cahn-Hilliard-type governing PDE which describes spinodal dewetting and the subsequent formation of arrays of metastable fluid droplets. The evolution for the masses and positions of the droplets can be reduced to a coarsening dynamical system (CDS) consisting of a set of coupled ODEs and deletion rules. Previous studies have established that the number of drops will follow a statistical scaling law, N(t)=O(t-2/5). We derive a Lifshitz-Slyozov-Wagner-type (LSW) continuous model for the drop size distribution and compare it with discrete models derived from the CDS. Large deviations from self-similar LSW dynamics are examined on short- to moderate-times and are shown to conform to bounds given by Kohn and Otto. Insight can be applied to similar models in image processing and other problems in materials science. Joint work with M.B. Gratton (Northwestern Applied Math).

We discuss the use of "geometric" (i.e. formulated exclusively in terms of a curve's arclength parameter) Sobolev metrics to devise new gradient flows of curves. We refer to the resulting evolving contours as "Sobolev Active Contours". An interesting property of Sobolev gradient flows is that they stabilize many gradient descent processes that are unstable when formulated in the more traditional L2 sense. Furthermore, the order of the gradient flow partial differential equation is reduced when employing the Sobolev metric rather than L2. This greatly facilitates numerical implementation methods since higher order PDE's are replaced by lower order integral-differential PDE's to minimize the exact same geometric energy functional. The fourth order L2 gradient flow for the elastic energy of a curve, for example, is substituted by a second order Sobolev gradient flow for the same energy. In this talk we give some background on Sobolev active contours, show some applications using energy regularizers normally connected with fourth order flows, and present some recent results in visual tracking.

Joint work with Ganesh Sundaramoorthi, Andrea Mennucci, Guillermo Sapiro, and Stefano Soatto.

Head-on impact of liquid drops
December 31, 1969

When two point particles collide, the outcome is governed entirely by energy and momentum conservation, with no dependence on the detailed interaction potential. Here we use a Volume of Fluid (VOF) simulation to examine what happens in the analogous case when two liquid drops collide. At low speeds, the liquid drops rebounce elastically, just as seen for point particles. At high speeds, however, a liquid sheet is ejected along the impact plane. When ambient gas pressure is low, both simple estimates and simulation show that the ejection is dominated by inertial effects. This idea enables us to collapse the pressure variation within the liquid drop at early times. In addition we find that surface tension effects are confined to the rim of the expanding sheet and acts primarily to slow the radial expansion.

Memory as vibration in a disconnecting air bubble
December 31, 1969

Focusing a finite amount of energy dynamically into a vanishingly small amount of material requires that the initial condition be perfectly symmetric. In reality, imperfections are always present and cut-off the approach towards the focusing singularity. The disconnection of an underwater bubble provides a simple example of this competition between asymmetry and focusing. We use a combination of theory, simulation and experiments to show that the dynamics near disconnection contradicts the prevailing view that the disconnection dynamics converges towards a universal, cylindrically-symmetric singularity. Instead an initial asymmetry in the shape of the bubble neck excites vibrations that persist until disconnection. We argue that such memory-encoding vibrations may arise whenever initial asymmetries perturb the approach towards a singularity whose dynamics has an integrable form.

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