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(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.