Mathematics of Materials and Macromolecules: Multiple
Scales, Disorder, and Singularities, September 2004 - June 2005
March 28-30, 2005
Biographies and Lecture
Gerard Awanou (IMA, University of Minnesota)
Trivariate Spline Approximations of 3D Navier-Stokes
The numerical analysis of the Navier-Stokes equations is very
important. Among many reasons it plays an important role in the study of the
cardiovascular system. For example, blood flow in large arteries can be modeled
by the Navier-Stokes equations.
There are many computational methods available in the literature
for the numerical solution of the 3D Navier-Stokes equations. New and more
efficient methods are being developed to increase the power of computational
flow simulations. To achieve significant improvements for the quality of computer
simulations for real-life problems is not only dependent on the continuously
increasing computing power, but also the approximation power of the numerical
We propose to use trivariate spline functions for the numerical
solution of 3D Navier-Stokes equations. This approach also captures the smoothness
of the solution inside the domain.
Dimitrova (Virginia Bioinformatics Institute (0477), Virginia
A Graph-theoretic Method for the Discretization of Gene Expression
The paper introduces a method for the discretization of experimental data
into a finite number of states. While it is of interest in various fields, this
method is particularly useful in bioinformatics for reverse engineering of gene
regulatory networks built from gene expression data. Many of these applications
require discrete data, but gene expression measurements are continuous. Statistical
methods for discretization are not applicable due to the prohibitive cost of
obtaining sample sets of sufficient size. We have developed a new method of
discretizing the variables of a network into the same optimal number of states
while at the same time maintaining high information content. We employ a graph-theoretic
clustering method to affect the discretization of gene expression measurements.
Our C++ program takes as an input one or more time series of gene expression
data and discretizes these values into a number of states that best fits the
data. The method is being validated by incorporating it into the recently published
computational algebra approach to the reverse engineering of gene regulatory
networks by Laubenbacher and Stigler.
Qiang Du (Department
of Mathematics, Pennsylvania State University)
Phase Field Modeling and Simulation of Cell Membranes
Recently, we have produced a series of works on the phase field modeling and
simulation of vesicle bio-membranes formed by lipid bilayers. We have considered
both the shape deformation of vesicles minimizing the elastic bending energy
with volume and surface area constraints and those moving in an incompressible
viscous fluid. Rigorous mathematical analysis have been carried out along with
extensive numerical experiments. We have also developed useful computational
techniques for detecting the topological changes within a broad phase field
- A Phase Field Approach in the Numerical Study of the Elastic Bending Energy
for Vesicle Membranes, Q. Du, C. Liu and X. Wang, J. Computational Physics,
198, pp450-468, 2004
- Retrieving topological information for phase field models, Q. Du, C. Liu
and X. Wang, 2004, to appear in SIAM J. Appl. Math
- Phase field modeling of the spontaneous curvature effect in cell membranes,
Q. Du C. Liu, R. Ryham and X. Wang, 2005, to appear in CPAA
- A phase field formulation of the Willmore problem. Q. Du, C. Liu, R. Ryham
and X. Wang, 2005, to appear in Nonlinearity
Changfeng Gui (Department of
Mathematics, University of Connecticut)
Level Set Evolution without Re-initialization: A New
Level set methods have been a powerful tool and extensively used
in image processing and computer vision. Periodically reinitializing the level
set function to a signed distance function during the curve evolution is a critical
numerical remedy to maintain stable curve evolution and ensure usable results.
However, most of the level set methods are plagued with such questions as when
and how to reinitialize the level set function to a signed distance function.
I will present a variational level set formulation of active contours
without reinitialization. The evolution of the level set function is the gradient
flow of an energy functional. We define an energy functional with two terms:
the external energy term and the internal energy term. The external energy characterizes
the conformity of the zero level set and the object boundaries, while the internal
energy penalizes the deviation of the level set function from a signed distance
function. As a result of introducing such internal energy term, the reinitialization
procedure is eliminated, and the level set function converges very well in the
whole image domain, with zero level set converging to the desired object boundaries.
Moreover, the proposed variational level set formulation has at least the following
practical advantages over the traditional level set formulations. Firstly, significantly
larger time step can be used in our method for numerically solving the evolution
PDE, and therefore the evolution is faster than the traditional level set methods.
Secondly, the curve evolution is less sensitive to the location of the initial
contour. Thirdly, the evolving curve can be stopped at quite weak object boundaries.
Our method has been successfully applied to synthetic and real images of different
modalities, including optical, MR, and ultrasound images.
Viet Ha Hoang
(Department of Applied Mathematics and Theoretical Physics, CMS, United Kingdom
University of Cambridge)
High-dimensional Finite Elements for Elliptic Problems with Multiple
Joint work with Christoph Schwab.
Elliptic homogenization problems in a d dimensional domain
with n+1 separated scales are reduced to elliptic one-scale problems in dimension
(n+1)d. These one-scale problems are discretized by a sparse tensor product
finite element method (FEM). We prove that this sparse FEM has accuracy, work
and memory requirement comparable to standard FEM for single scale problems
in while it gives numerical
approximations of the correct homogenized limit as well as of all first order
correctors, throughout the physical domain with performance independent of the
physical problem's scale parameters. Numerical examples for model diffusion
problems with two and three scales confirm our results.
Frederic Legoll (IMA, University of Minnesota)
Analysis of a Prototypical Multiscale Method Coupling
Atomistic and Continuum Mechanics
In order to describe a solid which deforms smoothly in some region,
but non smoothly in some other region, many multiscale methods have been recently
proposed, that aim at coupling an atomistic model (discrete mechanics) with
a macroscopic model (continuum mechanics). We present here a theoretical analysis
for such a coupling in a one-dimensional setting. We study both the general
case of a convex energy and a specific example of a nonconvex energy, the Lennard-Jones
In the latter situation, we prove that the discretization needs
to account in an adequate way for the coexistence of a discrete model and a
continuous one. Otherwise, spurious discretization effects may appear. We also
consider the effect of the discretization of the continuum model on the behaviour
of the coupled model.
This work is joint with
(Paris 6) and
Melvin Leok (Department
of Mathematics, University of Michigan)
Generalized Galerkin Variational Integrators: Lie Group, Multiscale
and Spectral Methods
Geometric mechanics involves the study of Lagrangian and Hamiltonian mechanics
using geometric and symmetry techniques. Computational algorithms obtained from
a discrete Hamilton's principle yield a discrete analogue of Lagrangian mechanics,
and they exhibit excellent structure-preserving properties that can be ascribed
to their variational derivation.
We propose a natural generalization of discrete variational mechanics, whereby
the discrete action, as opposed to the discrete Lagrangian, is the fundamental
object. This is achieved by appropriately choosing a finite dimensional function
space to approximate sections of the configuration bundle and numerical quadrature
techniques to approximate the action integral.
We will discuss how this general framework allows us to recover high-order
Galerkin variational integrators, asynchronous variational integrators, and
symplectic-energy-momentum integrators. In addition, we will also introduce
generalizations such as high-order symplectic-energy-momentum integrators, Lie
group integrators, high-order Euler-Poincare integrators, multiscale variational
integrators, and pseudospectral variational integrators.
This framework will be illustrated by an application of Lie group variational
integrators to rigid body dynamics wherein the discrete trajectory evolves in
the space of 3x3 matrices, while automatically staying on the rotation group,
without the use of local coordinates, constraints, or reprojection.
This is joint work with Taeyoung Lee and Harris
(IMA, University of Minnesota) http://www.ima.umn.edu/~philip/homepage/
Numerical Simulation of Heat Transfer in Materials with
Anisotropic Thermal Conductivity: A Finite Volume Scheme to Handle Complex Geometries
A finite volume scheme suitable for nonlinear heat transfer in
materials with anisotropic thermal conductivity is formulated, focussing on
the difficulties arising from the discretization of complex domains which are
typical in the simulation of industrially relevant processes. The discretization
is based on unstructured constrained Delaunay triangulations of the domain.
For simplicity, it is assumed that the thermal conductivity tensor has vanishing
off-diagonal entries and that the anisotropy is independent of the temperature.
Numerical simulations verify the accuracy of the method in two test cases where
a closed-form solution is available. Further results demonstrate the effectiveness
of the method in computing the heat transfer in a complex growth apparatus used
in crystal growth.
Jie Shen (Department of Mathematics, Purdue
University ) http://www.math.purdue.edu/~shen
Numerical Simulations of Drop Pinching Using a Phase-Field
The dynamics and pinch-off phenomena of a liquid filament is studied
by using a phase field model which describes the motion of mixtures of two incompressible
fluids. An efficient and accurate numerical scheme is presented and implemented
for the coupled nonlinear system of Navier-Stokes equations and Allen-Cahn phase
equation. Detailed numerical simulations for a Newtonian fluid filament falling
into another ambient Newtonian fluid are carried out. The dynamical scaling
behavior and the pinch-off behavior, as well as the formation of the consequent
satellite droplets are investigated.
(Department of Electrical & Computer Engineering , University of Akron)
A New Finite-Difference Calculus and Its Applications
The generic Taylor approximation in standard Finite-Difference
(FD) methods often fails to capture with sufficient accuracy many important
features of the solution, for example: discontinuities at material interfaces;
boundary layers; dominant dipole components near polarized or magnetized spherical
particles; electrostatic double layers around colloidal particles, and much
more. A new FD calculus of Flexible Local Approximation MEthods (FLAME) replaces
the Taylor polynomials with any desired local basis functions, such as exponentials,
cylindrical or spherical harmonics, plane waves, and so on. The accuracy of
the solution is improved, both quantitatively and qualitatively.
Although the method usually operates on regular Cartesian grids,
it is in some cases much more accurate than the Finite Element Method with its
complex meshes. While one motivation is to minimize the notorious 'staircase'
effect at curved and slanted interface boundaries, the new approach has much
broader applications and implications.
Illustrative examples include 3-point schemes of arbitrarily high
order for the Schrödinger equation and for a 1D singular equation; fields
of finite-size charged, polarized and/or magnetized colloidal particles in free
space or in a solvent with or without salt; scattering of electromagnetic waves;
plasmon resonances; wave propagation in a photonic crystal. Furthermore, many
existing FD schemes are revealed as natural particular cases of the FLAME calculus:
low- and high-order Taylor schemes, the Collatz "Mehrstellen" schemes, flux
balance / control volume schemes, and a few more special techniques.
Biographies and Lecture Abstracts