(August 24) Prof. Dr. Eberhard Baensch:
A phase transition problem related to steel production -
A severe problem in steel production is distortion of the processed material. This distortion originates from mechanical stresses due to inhomgenieties on micro and macro scale. Not only chemical
impurities but also different states of the crystalline structure,
so called phases, are responsible for these inhomogenieties.
In this talk a model consisting of a system of partial and ordinary
differential equations to describe the process is presented. Numerically
this system is discretized in time by a semi-implicit
Euler scheme and in
space by a finite element method. (August 31) Prof. Qing Nie:
Microstructural evolution in anisotropic elastic media
-
The pattern of multi-phase microstructures in materials is a key variable in setting its macroscopic mechanical properties. In this talk, the diffusional evolution of microstructure is considered for a two-dimensional inhomogeneous system, which consists
of arbitrarily shaped precipitates embedded in a matrix. The system is studied through a sharp interface model and
a diffusive interface model. Efficient numerical methods for solving
the corresponding equations along with some physical results
will be discussed. (September 7) Dr. Rainer Grauer:
Singular structures in fluids, plasmas and nonlinear optics -
The formation of singular structures in incompressible hydrodynamic
and related systems is still a controversial issue. Up to now,
numerical simulations do not show a coherent picture, the main reason
being the insufficient resolution due to limited computing resources.
Here we investigate such problems using adaptive mesh refinement where
each grid is integrated using an upwind-scheme combined with a second
order projection method. Our findings are that the magnetohydrodynamic
equations in two and three dimensions show only exponential growth of
vorticity and current density (an effective resolution of 40963mesh points could be reached). This depletion of nonlinearity resultsfrom the formation of current sheets. The growth of vorticity and
current density is consistent with a scaling ansatz for the stream
functions. The simulations for the three dimensional incompressible
Euler equations indicate the formation of a finite time singularity
(here, an effective resolution of 20483 mesh points was achieved). In addition, we present results concerning the question of
multisplitting in media with anisotropic dispersion. Simulations
corresponding to 81922 × 12288 will be presented.
From a numerical point of view we discuss the details of the AMR
implementation and our impressions on different integration methods
(including recent central schemes) with emphasis on future projects
concerning magnetic reconnection. (September 14) Prof. Gabriel Caloz:
Magnetic susceptibility artifacts in MRI -
This is a work made in collaboration with the medicine department of the University of Rennes 1. It deals with
- Modelling aspects related to MRI
- Computation of magnetic field perturbations
- Numerical algorithm to reconstruct images
(September 21) Prof. Dan Anderson:
A Phase-Field Model of Convection with Solidification -
I will describe a phase-field model for the solidification of a pure material in which effects of fluid flow are incorporated.
The phase-field methodology offers an alternative to the classical
sharp-interface formulation of this free-boundary problem.In this model, a phase-field variable is used to distinguish the solid and liquid phases. The material density, which often
plays the role of order parameter in diffuse-interface models
of a fluid near its critical point, is taken here to be a prescribed function
of the phase-field parameter. We model the solid phase as an extremely
viscous liquid and the formalism of irreversible thermodynamics is used to derive the governing equations. I will discuss the model, analyses of the equations in the sharp-interface limit, and calculations in some simplified
geometries. (September 28) Prof. Jie Shen:
Some New Developments on Projection Methods
for Incompressible Flows -
We will start with a review of the projection methods, and then
introduce the rotational form the pressure-correction projection
method and a new class of velocity-correction projection methods.
We will show that the rotational forms of the pressure-correction and velocity-correction schemes provide consistent Neumann
boundary condition for the pressure approximation, and consequently,
lead to improved error estimates. Furthermore, we will show that
the schemes proposed by Orszag, Isreali & Deville (1986) and
Karniadakis, Isreali & Orszag (1991) can be reintepreted as rotational forms of our velocity-correction projection methods. (October 5) Prof. Jian-Guo Liu:
Energy and Helicity Preserving Schemes for Hydrodynamic
and Magnetohydrodynamic Flows with Coordinate Symmetry -
In the computational design of direct numerical simulation of
incompressible flows, it is very desirable to have exact numerical
conservation of physically conserved quantities such as the energy and
the helicity. The conservation of quadratic moments not only provides
a diagnostic check, it also guarantees that the numerical scheme is
nonlinearly stable and free from excess numerical viscosity. Previous results date back to Arakawa's pioneering work on a
numerical scheme that conserves energy and enstrophy for 2d flows.
The 3D fluid and plasma flow simulation is much more difficult due to
the singular nature of such flows arising from vortex stretching and
reconnection. To date, there is no numerical method preserving both
helicity and energy for 3D flows. However, in many physical problems
such as the pipe flow, axisymmetric flow, flow on a sphere, and flow
within a torus, an coordinate symmetry can be assumed. In this case,
we can introduce a stream vector whose only component is in the
symmetry direction and the flow is effectively two dimensional. We
developed a class of efficient numerical methods for incompressible
flows and plasma in this generalized vorticity-stream formulation.
The divergence free constraint for both fluid velocity and magnetic
field is thus trivially satisfied. We then give a general recipe to
treat the nonlinear terms (convection term and the geometric source
term) in such a way that both energy and helicity are conserved
numerically. A second order scheme based on central differencing is
illustrated as an example. The scheme can be easily generalized to
higher order finite difference, finite element and spectral methods in
this setting. Moreover, mesh refinement near the boundary can also be
built into the equation without extra cost. In practical
implementations of the scheme, we also resolved the difficulty
associated with the pole singularity and severe time stepping
stability constraint near the pole due to cluster of grid points.
For the general 3D MHD system, we developed a class of MAC-Yee
scheme preserving both energy and cross helicity (we are not able to
preserve the magnetic helicity). We expect this class of methods will
have many applications in fluid and plasma flow simulations,
particularly in large time direct numerical simulation and numerical
search for possible singularity formation. (October 19) Ed Seidel :
Solving Einstein's Equations on the Grid: Using Supercomputers to Collapse
Gravitational Waves, Collide Black Holes -
Einstein's equations of general relativity govern such exotic phenomena as black holes, neutron stars,
and gravitational waves. Unfortunately they are among the most complex in physics, and require very
large scale computational power --- which we are just on the verge of achieving --- to solve. I will
motivate and describe these equations, and the worldwide effort to develop advanced computational
tools to solve them in their full generality for the first time since they were written down nearly a century
ago. I will focus on applications of these tools to the study of black hole collisions, considered to be
promising sources of observable gravitational waves that may soon be seen for the first time by the
worldwide network of gravitational wave detectors (LIGO, VIRGO, GEO, and others) currently under
construction. I will show movies of large scale simulations of black hole formation and black hole
collisions. I will discuss the new scientific research possibilities opened up by the emergence of high
performance computing as a tool for basic physics research, and the impact of the study of Einstein's
equations on high performance computing technology. Finally, I will describe the emerging "Grid" of
networked computational resources, and the techniques being developed to exploit it for scientific
computing. (November 2) Luc Tartar:
Microlocal measures and their use in partial differential equations
-
Microlocal methods have been introduced in the early 70s, independently by SATO and by LarsHÖRMANDER, I believe. These initial ideas are not adapted to solving questions related to the
partial differential equations from Continuum Mechanics or Physics, despite the use of catch words like
``propagation of singularities''. What is really being proved in
HÖRMANDER's approach are results of
propagation of microlocal regularity, and that is hardly a relevant question in Continuum Mechanics or
Physics; another defect, shared by the theory of pseudo-differential operators, is that one assumes
that coefficients of the partial differential equations are infinitely differentiable; another defect is that
systems of partial differential equations are not considered. I was led to introducing microlocal
measures for a different question of Homogenization (and I give to this term a much more generalmeaning than most other people do), and therefore I called these new objects H-measures. Then, I
tried to use them for problems of propagations of oscillations and concentration effects, which are
important questions for Continuum Mechanics or Physics. It worked, and it gave a new point of view onold things like the Geometrical Optics approximation for the wave equation; my results are not like the
classical approach using an amplitude and a phase, as H-measures do not see any phase and
describe limits when frequency tends to infinity, but the classical approach can at best show that there
are solutions of the wave equations for which the energy almost propagates along light rays (away from
caustics), while my results says that all oscillating solutions have this property (and no problem with
caustics). The method can be used for systems, and it also gives a qualitative answer to a few
puzzling facts (for mathematicians) about some physicists' computations; one of them is that somecomputations done in a periodic framework can be used with success in a nonperiodic setting; another
one is the classical puzzle of Quantum Mechanics about particles which sometimes are waves, and it
suggests a different point of view, where there are only waves satisfying some partial differential
equations (of hyperbolic type) but the limit for infinite frequency leads to an ordinary differential equationfor a microlocal measure, which can be interpreted as describing some kind of ``particles''. The same
objects have been introduced independently by Patrick GERARD, for a different reason. For variants
using one characteristic length, I will describe my approach and the independent idea of Patrick
GERARD (semi-classical measures), together with the later work, and mistake, of Pierre-Louis LIONS
and Thierry PAUL. A lot remains to be done, and possible developments will be mentioned. (November 9) Prof. Weimin Han:
Error Analysis of a Meshfree Method -
Recently, a next generation of numerical methods, collectively called meshfree methods or meshless methods, have attracted more and more researchers in computational sciences and engineering. The goal of the research on the meshfree methods is to modify the internal structure of the traditional finite element method to make it more flexible, versatile and robust. In this talk,
we provide a theoretical analysis of a meshfree method--the reproducing kernel particle method. We derive rigorously
error estimates for meshfree interpolants and meshfree solutions,
both for smooth problems and singular problems. Numerical examples are
presented confirming the theoretical predictions. (November 15) Prof. Eitan Tadmor:
High Resolution Methods
with Unresolved Small Scales -
Spontaneous evolution of different scales leads to challenging
difficulties of stable computations with unresolved small scales. We
discuss how modern algorithms address these difficulties: detection of
edges, high-resolution reconstruction of piecewise-smooth data between
edges, and the interplay between the theory and computational aspects of
high-resolution in low regularity spaces. We focus our attention on two particular examples. We discuss
non-oscillatory central schemes for computing piecewise smooth solutions
of hyperbolic conservation laws, Hamilton-Jacobi equations and related
nonlinear problems. The high-resolution of these locally based algorithms
is gained by coupling nonlinear edge detectors, and turning the piecewise
polynomials reconstructions into the direction of smoothness. The second
part is an example for global methods. Here we discuss the reconstruction
of piecewise smooth data from (pseudo-) spectral global information. To
avoid spurious Gibbs oscillations and to regain the superior exponential
accuracy we proceed in two separate steps: a detection procedure which
identifies (the location and amplitude of finitely many) edges, followed
by a family of spectrally accurate mollifiers which recover the data
between those edges. We demonstrate applications from CFD (-- formation of
shocks), geometrical optics, MHD problems, image processing and more. (November 16) Tim Schulze:
Convection in dendritic layers of a solidifying alloy -
The well-studied morphological instability of solidification fronts
leads to the formation of layers of dendrites which are referred to as``mushy zones.'' Solute gradients within mushy zones can subsequently
lead to convection in the melt and the coupling of these instabilities
can lead to additional macroscopic morphological change. I will briefly
review the fundamentals of solidification and morphological
instability, describe a bulk model for convection in mushy zones and
present some numerical solutions based on that model. The development
of dendrite-free channels, called ``chimneys,'' is of particular
interest. The modeling and numerical resolution of chimneys presents
a number challenges that will be discussed in some detail. (November 30) Richard Braun:
Insoluble Surfactant Models for a Vertical Draining Free Film -
A sequence of models have been constructed to study the evolution of a vertically-oriented thin liquid free film draining under gravity when there is an insoluble surfactant with finite surface viscosity on its free surface.
An introduction and some one-dimensional results will be briefly
presented, and the model for a two-dimensional film will be discussed. Lubrication theory for this free film results in four coupled nonlinear PDEs describing the time-evolution of the free surface shape, the surface velocities and the surfactant transport at leading order. The draining film is assumed to terminate on a one-dimensional static meniscus. Numerical experiments are performed to understand the stability of the system to perturbations across the film. After spatial discretization with finite differences,
the resulting differential-algebraic system is solved using DASPK
(courtesy of Petzold et al). The limit of large surface viscosities recovers the tangentially-immobile film and is also found to have a stabilizing influence on transverse perturbations due to their energy dissipating effect.
An instability is seen in the mobile film case; this is caused by a competition between gravity and the Marangoni effect. The instability is closely related to that calculated by Miller and coworkers in nonaxisymmetric film drainage in a ring. This work is supported by the NSF and Dow Corning. (December 7) John Weeks:
Current Induced Step Dynamics and Pattern Formation on Crystal Surfaces -
We review experimental and theoretical work on two dimensional step patterns
that form on crystal surfaces in response
to nonequilibrium driving forces. Steps are viewed as a set of non crossing interacting lines. The derivation and solution
of effective differential equations describing their coupled motion will be discussed.
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