Applied Mathematics and Numerical Analysis Seminar

Thursdays, 11:15 am-12:15 pm in Vincent Hall 570

Fall 2004

September 16: Peter Philip   (IMA)   Speaker's Homepage

Title: Numerical Simulation and Control of Sublimation Growth of SiC Bulk Single Crystals: Modeling, Finite Volume Method, Analysis and Results

Abstract:
A transient mathematical model for the sublimation growth of silicon carbide single crystals (SiC) by physical vapor transport is presented. Continuous mixture theory is used to obtain balance equations for energy, mass, and momentum inside the gas phase. In particular, reaction-diffusion equations are deduced. Heat conduction is treated inside solid materials. Heat transport by radiation is modeled via the net radiation method for diffuse-gray radiation to allow for radiative heat transfer between the surfaces of cavities. The model includes the semi-transparency of the single crystal via a band approximation. Induction heating is modeled by an axisymmetric complex-valued magnetic scalar potential that is determined as the solution of an elliptic problem. The resulting heat source distribution is calculated from the magnetic potential. The transient heat problem is discretized in time by the implicit Euler method and in space by the finite volume method. Existence and uniqueness of the discrete solution is shown as well as a maximum principle in a simplified case. A control problem for the optimization of the gradient in the gas phase is considered, as this is relevant to the crystal growth process. Based on a known existence result for the semilinear, but still nonlocal, case, the existence of an optimal solution as well as necessary optimality conditions.
The presented numerical simulations are conducted in an axisymmetric setting. They constitute transient investigations of control parameters affecting the temperature evolution during the heating of the growth apparatus. It is studied how the temperature difference between source and seed, which is highly relevant to the growth process, is related to the measurable temperature difference between bottom and top. Results concerning the time lack between the heating of the surface of the source powder and the heating of its interior are considered. Finally, numerical optimization is used to determine the control parameters frequency, power, and coil position for the radio frequency (RF) induction heating with the objective to minimize a functional, tuning the radial temperature gradient on the single crystal surface as well as the vertical temperature gradient between SiC source and seed, both being crucial for high-quality growth.

September 23: Jesus Carrero   (University of Minnesota)

Title: Hybridized globally divergence-free LDG methods for the Stokes problem

Abstract:
We devise and analyze a new local discontinuous Galerkin (LDG) method for the Stokes equations of incompressible fluid flow. One of the main features of the method is that it has the smallest number of degrees of freedom among all other DG methods for the Stokes problem and yet it is locally conservative and optimally convergent. Moreover, it yields globally divergence-free approximations to the velocity. It is obtained by applying an LDG method to a vorticity-velocity formulation of the Stokes equations and then by hybridizing the resulting formulation.

September 30: NO SEMINAR   (IMA Workshop on Modeling of Soft Matter, 9/27/04 to 10/1/04)

October 7: John Steinhoff   (University of Tennessee Space Institute)   Speaker's Homepage

Title: Wave Confinement: Modeling Short Acoustic Pulses as Nonlinear Solitary Waves

Abstract:
A new computational method, Wave Confinement, is described. The method has been shown to efficiently treat thin acoustic pulses in complex time domain wave equation problems, allowing them to be propagated over arbitrarily long distances with no spreading due to numerical errors. The method involves only a fixed, Eulerian computational grid with no Lagrangian markers. On this grid, the pulses are only 2-3 cells thick, and are treated as a type of weak solution which obey a nonlinear difference equation derived from the wave equation, as opposed to a conventional finite difference approximation. As such, the pulses are, essentially, nonlinear solitary waves that live on the lattice.
An important feature is that, in spite of the non-linearity, the computed pulses can pass through each other with no phase shift or amplitude exchange. This is necessary because the equation that is being simulated is the linear wave equation.

October 14: Gilad Lerman   (University of Minnesota)

Title: Multiscale Curve and Strip Constructions with Applications

Abstract:
We present a fast algorithm for detecting and characterizing a cloud of points that are concentrated around a curve in a D-dimensional Euclidean plane. The algorithm characterizes the cloud data by detecting the underlying curve, separating between a stable set and a deviating set (outliers) and estimating the local variances of the stable set around the underlying curve.
We have adapted this algorithm to analyze DNA array data from ChIP-chip experiments as well as expression profiling microarray data. We use the algorithm for both purposes of normalization and for ranking and identifying enriched sites (or differentially expressed genes). Our methods accommodate the unique characteristics of ChIP-chip data, where the set of immunoprecipitation-enriched segments is large, asymmetric and whose proportion to the whole data varies locally.
We establish some estimates for the performance of our algorithm and exemplify its efficiency with high-dimensional data by applying it to pixel neighborhoods of various images. Here, the ``deviating points'' detected by the algorithm correspond to edges in the original image.
This is a joint work with Joseph McQuown and Bud Mishra. The ChIP-chip analysis is also joint with Alexandre Blais and Brian David Dynlacht.

October 21: Chiu Yen Kao   (IMA)   Speaker's Homepage

Title: Fast Sweeping Methods for Static Hamilton-Jacobi Equations

Abstract:
Hamilton-Jacobi equations arise in many applications such as geometrical optics, crystal growth, path planning, and seismology. Viscosity solutions of these nonlinear differential equations usually develop singularities in their derivatives. In this talk, we will present several fast sweeping methods which are based on the Godunov Hamiltonian or the Lax-Friedrichs Hamiltonian to approximate the viscosity solution of convex or arbitrary static Hamilton-Jacobi equations in any number of spatial dimensions. We solve for the value of a specific grid point in terms of its neighbors, so that a Gauss-Seidel type nonlinear iterative method can be utilized. Furthermore, by incorporating a group-wise causality principle into the Gauss-Seidel iteration by following a finite group of characteristics, we have an easy-to-implement, sweeping-type, and fast convergent numerical method. For the sweeping methods based on Lax-Friedrichs Hamiltonian, unlike other methods based on the Godunov numerical Hamiltonian, some computational boundary conditions are needed in the implementation. We give a simple recipe which enforces a version of discrete min-max principle. Some convergence analysis is done for the one-dimensional eikonal equation. Extensive 2-D and 3-D numerical examples illustrate the efficiency and accuracy of the new approaches.

October 28: NO SEMINAR   (IMA Workshop on Singularities in Materials, 10/25/04 to 10/29/04)

November 4: Sonia Mogilevskaya   (University of Minnesota)   Speaker's Homepage

Title: Computational Modeling of Fiber-Reinforced and Particulate Composite Materials

Abstract:
This presentation discusses the development of a computational basis for modeling micro- and macroscopic behavior of composite materials. A significant feature of the work is the capability to directly simulate the microstructure of micro-porous and composite materials by numerous non-overlapping inclusions (fibers or particles) with interphases and holes (pores) of arbitrary size.
In our model the fibers are idealized as uniform, parallel, infinite circular cylindrical inclusions (two-dimensional case) and the particles are assumed to be spherical. In general, the fibers and particles can be distributed randomly, they can all have different elastic properties, and the bond between them and the surrounding material matrix can be imperfect. For example, the inclusions may be connected to the material matrix through arbitrarily thin interphase layers.
Our approach allows one to incorporate in a model the effect of a free boundary as well as some nonlinear effects due to localized slip and/or separation of the interfaces between the fibers and the material matrix, and cracking inside the matrix or reinforcements. Time-dependent effects due to transient heat conduction with thermal stresses or to viscoelastic behavior of the material can be also considered.
Computational realization of the model includes the use of fast methods (based on fast multipole acceleration) that make it possible to study detailed interactions among thousands of particles (or fibers) and pores. The method allows for accurate calculation of the displacement, stress, and temperature fields anywhere within the material, including the inclusions and interphases. The overall properties of an equivalent homogeneous material can be found directly from the properties of microstructural elements.
Computer simulations provide a means to test and compare major effective-medium theories and will allow us to make more realistic evaluations of the microscopic behavior and overall properties of micro-porous and composite materials than is currently feasible. Such simulations could enhance understanding of failure mechanisms in materials and provide valuable information necessary to design composite materials with specified physical properties.

November 11: L. Pamela Cook   (University of Delaware)   Speaker's Homepage

Title: The human eye tear film: Modeling and analysis

Abstract:
Problem formulation, modeling, and resultant fits to data are presented of a single layer thin complex fluid film representing the human eye tear film. The model parameters are fit to viscometric shear data of extracted tear fluid and the resultant system is analyzed under drainage conditions. The lubrication approximation is used and its validity is analyzed. Asymptotic, analytic and computational results are presented. The work is important for understanding the behavior of the tear film in both normal and "dry" eyes.

*** SPECIAL DATE, TIME AND ROOM (joint with PDE seminar) ***

MONDAY November 15, 4:30 - 5:30 in Vincent Hall 16: Avner Friedman   (Ohio State University)   Speaker's Homepage

Title: Bifurcation and asymptotic stability of free boundary problems

Abstract:
I shall consider several examples of free boundary problems, with spherical free boundary. The examples include electrically charged liquid droplets, and several tumor models. I will state results on existence of bifurcation branches of solutions which break the symmetric structure of the spherical solutions. I shall also discuss the asymptotic stability of the spherical and non-spherical solutions.

November 18: NO SEMINAR   (IMA Workshop on Future Challenges in Multiscale Modeling and Simulation, 11/18/04 to 11/20/04)

November 25: NO SEMINAR   (Thanksgiving)

December 2: Matthias Gobbert   (University of Maryland, Baltimore County)   Speaker's Homepage

Title: Parallel Simulations of the Linear Boltzmann Equation for Models in Microelectronics Manufacturing

Abstract:
Production steps in the manufacturing of microelectronic devices involve gas flow at a wide range of pressures. We develop a kinetic transport and reaction model (KTRM) based on a system of time-dependent linear Boltzmann equations. These kinetic equations have the property that velocity appears as an independent variable, in addition to position and time. A deterministic numerical solution for realistic three-dimensional application problems requires the discretization of the three-dimensional velocity space, the three-dimensional position space, and time. We design a spectral Galerkin method to discretize the velocity space by specially chosen basis functions. The basis functions in the expansion lead to a system of hyperbolic conservation laws with constant diagonal coefficient matrices for each of the linear Boltzmann equations. These systems of conservation laws are solved using the discontinuous Galerkin finite element method. As an application example, we simulate chemical vapor deposition at the feature scale in two and three spatial dimensions and analyze the effect of pressure. Finally, we present parallel performance results which indicate that the implementation of the method possesses excellent scalability on a Beowulf cluster with a high-performance Myrinet interconnect.

December 9: Jamylle Carter   (University of Minnesota)   Speaker's Homepage

Title: A Dual Method for Total Variation-Based Image Restoration

Abstract:
This talk will describe a computational method for the inverse problem of edge-preserving image restoration. Total Variation (TV) regularization removes noise from an image while retaining its edges. We solve an equivalent dual version of the TV problem. Images restored using this dual approach will have crisp edges (discontinuities), whereas images recovered under earlier primal methods may contain blurred edges. Joint work with Tony F. Chan, Pep Mulet, and Lieven Vandenberghe.

Spring 2005

January 20: Catalin Turc   (University of Minnesota)   Speaker's Homepage

Title: TBA

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January 27: Peter Sternberg   (Indiana University)   Speaker's Homepage

Title: Elliptic Variational Problems on Constricted Networks with Applications to Ginzburg-Landau Theory

Abstract:
I will analyze variational problems set on a network of thin constricted tubes. In the asymptotic regime where the tubes collapse to a graph, one can identify a one dimensional variational problem with interesting natural boundary conditions at the nodes which in particular encourage jumps in the (limit of) minimizers across the various branches of the graph. An application to tunneling across weak links in a superconductor will be discussed.

February 3: Xiantao Li   (IMA)   Speaker's Homepage

Title: A multiscale model for the dynamics of solids

Abstract:
We present a multiscale method for coupling atomistic and continuum models of solids. Both models are formulated in the form of physical conservation laws, and the coupling is achieved through balancing the fluxes. Error estimate is provided, which is of great help in choosing the size of the macro grid and microscale system. Finally we shall show some applications including phase transformation, and dynamic fracture mechanics.

February 10: Kumar Tamma   (University of Minnesota)   Speaker's Homepage

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February 17: TBA   (TBA)   Speaker's Homepage

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February 24: Hongjie Dong   (University of Minnesota)   Speaker's Homepage

Title: On the Rate of Convergence of Finite-difference Approximations for Bellman Equations with Constant Coefficients

Abstract:
We consider elliptic Bellman equations with coefficients independent of the variable x. Error bounds for certain types of finite-difference schemes are obtained. These estimates are sharper than those of earlier results.

March 3: Yassine Boubendir   (University of Minnesota)   Speaker's Homepage

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March 10: TBA   (TBA)   Speaker's Homepage

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March 17: NO SEMINAR   (Spring Break)

March 24: TBA   (TBA)   Speaker's Homepage

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March 31: TBA   (TBA)   Speaker's Homepage

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April 7: TBA   (TBA)   Speaker's Homepage

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April 14: NO SEMINAR   (IMA Workshop on Atomic Motion to Macroscopic Models: The Problem of Disparate Temporal and Spatial Scales in Matter, 4/11/05 to 4/15/05)

April 21: TBA   (TBA)   Speaker's Homepage

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April 28: TBA   (TBA)   Speaker's Homepage

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May 5: NO SEMINAR   (IMA Workshop on Experiments in Physical Biology, Part I, 5/2/05 to 5/6/05)