(January 29) Professor Tien-Mo Shih:
Modularized Navier-Stokes Codes
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My talk is not intended to present new technical ideas regarding numerical
schemes. More than 50% of the content will be somewhat philosophical. The central theme of my talk is related to distributing among worldwide
scientific/engineering (S/E) communities modules that are: 1. easily understood, replaced, improved, inserted, or deleted,
2. of 2nd-order accuracy,
Consequently, S/E users of these modules can readily build up their own
personal computer codes (PCC) simply linking these modules, implementing their
own parametric values, grid generations, and boundary conditions.
The main differences between a commercial computer code (CCC) and a
modularized PCC will be discussed. They are: degree of understanding, cost,
customization, and user group.
(February 5) Professor Florian Potra:
Multibody dynamics with contacts and friction -
The field of multi-body dynamics simulation is expected to have a
major impact on the design of complex mechanical systems, such as
robots and assembly line manipulators. Finding realistic models for
impact and friction is very important for accurate simulation. Using
the rigid body hypothesis substantially simplifies the model and
therefore reduces the complesity of the governing equations.
Unfortunately, it has been known for over a century that there are
examples of rigid multi-body systems with Coulomb friction which have
no mathematical solution in the classical sense. Various authors have
proposed different settings in which the rigid multi-body system
problem with Coulomb friction has a generalized solution either by
allowing for impulsive forces (i.e., a solution in the sense of
distributions) or by considering the equations of motion as
differential inclusions rather than differential equations. In our
talk we present a discrete model that has a computable solution under
general conditions and which is very well suited for simulating
multi-body systems with friction. A simulation package based on this
model is being now implemented at the University of Iowa and some
numerical results obtained with a preliminary version of that package
(February 10) Dr. Shari Moskow :
A Finite Difference Scheme for Elliptic Equations with Rough Coefficients
Using a Cartesian Grid Non-Conforming to Interfaces -
A problem of great interest to Schlumberger, an oil services company, is to
calculate a potential function in an inhomogeneous medium quickly, easily,
and accurately. When the medium varies locally in one direction, we propose
a staggered finite difference scheme on a regular Cartesian grid with a
special cell averaging. This averaging allows for the change in conductivity
to be in any direction with respect to the grid and does not require the
grid to be small compared to the layering. We have convergence results and
numerical experiments which suggest that the method models thin, nonconformal
conductive and resistive sheets well.
(February 12) Dr. Oliver Ernst:
On Some Recurrent Theorems Concerning Krylov Subspace Methods -
The recent development of Krylov subspace methods for solving
linear systems of equations has shown that two fundamental
approaches underlie the most commonly used algorithms: the Minimal
Residual (MR) and Orthogonal Residual (OR) approaches.
We show that these two approaches can be formulated as techniques for
solving an approximation problem on a nested sequence of subspaces of
a Hilbert space. Many well-known relations among the iterates and
residuals of OR/MR pairs are shown to hold even in this abstract
setting. When applied to the solution of linear systems of equations and
when these subspaces are specialized to be a sequence of Krylov spaces,
familiar OR/MR pairs are recovered, among these CR/CG, GMRES/FOM and
their specializations MINRES/CG to the Hermitian case, QMR/BCG as well
as TFQMR/CGS. Further, we show that a common error analysis for these
methods involving the canonical angles between subspaces allows many of the
recently developed error bounds to be derived in a simple and uniform
manner. An application of this analysis to operator equations
involving compact perturbations of the identity shows that OR/MR
pairs of Krylov subspace methods converge q-superlinearly for this
class of problems.
(February 19) Professor Jian-Guo Liu:
Efficient FEM for Unsteady Viscous Incompressible Flow in Vorticity Formulation -
We will introduce an efficient time stepping procedure for
FEM approximating unsteady incompressible Navier-Stokes equations.
For large Reynolds number flows, the convection and viscous terms
are both treated explicitly in the vorticity transport equation.
The stream function (vector potential in 3D), and hence the velocity,
is then evaluated from the vorticity via the kinematic equation.
In this new time stepping procedure, the value of the vorticity
on the boundary can be obtained explicitly from the steam function
(vector potential) without any iteration, thus eliminating some
traditional difficulties associated with the vorticity formulation.
The resulting schemes are extremely simple and efficient. At each
time step, or Runge-Kutta stage, the main computation involves
solving a standard Poisson equation and inverting a standard mass
matrix. Some theoretical aspects of the method will also been
discussed.
(February 26) Professor Peter Monk:
Finite Element Methods for Maxwell's Equations
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The time harmonic Maxwell equations present a number of challenges to numerical
analysts. For example
- Scattering problems are posed on an infinite domain so that the problem
of truncating the solution domain must be faced.
- The underlying bilinear form is complex, but not Hermitian and
non-coercive.
- The Maxwell system is a vector system with unusual boundary conditions.
This talk will present a biased overview of some of the techniques used to
overcome the above problems, and some of the challenges that still must be
overcome.
(March 5) Professor Zhiming Chen:
Numerical Methods for Dynamical Ginzburg-Landau Vortices in Superconductivity -
The phenomenological Ginzburg-Landau complex superconductivitymodel is
designed to describe the phenomenon of vortex structure in the
superconducting/normal phase transitions. In this talk we report some
new development in solving the nonlinear time-dependent Ginzburg-Landau
model. The focus will be on a mixed finite element method which calculates
the magnetic field accurately, and some results on the a posteriori error
estimate which is the basis of developing adaptive finite element methods. (March 12) Dr. Barbara Wohlmuth:
Adaptive Multilevel Techniques for Mixed Finite Element Discretization -
During the last couples of years, mixed finite element methods have been
increasingly used in applications. In particular, we will consider the
two basic ingredients of an efficient adaptive algorithm in detail:
Since the arising linear algebraic systems typically become large and
ill-conditioned for discretized partial differential equations, there is
a need for fast iterative solvers. The construction of the solver is
based on an adequate splitting related with the multilevel structure. A
further important aspect is the use of adaptively generated
triangulations. Local refinement will be performed based on reliable and
efficient a posteriori error estimators. A hierarchical error estimator
obtained by defect correction in higher order mixed ansatz spaces and a
localization by suitable two-level splittings is considered in detail.
The performance of the presented algorithm is illustrated by several
numerical examples, including a benchmark problem for the neutron fluxes
in a light water reactor based on the two-group neutron diffusion
equations.
(March 18) Dr. Zhouping Xin:
On The Hyperbolic Approximations to First Order Partial
Differential Equations and Relaxation Schemes -
In this talk, I will discuss some results on a
relaxation approximation to some class of nonlinear first order
partial differential equations which includes systems of nonlinear
hyperbolic conservation laws, Hamilton-Jacobi equations, and
mean-curvature flows. The basic idea is to regard the given first order
nonlinear systems as the equilibrium for some relaxation process
as the rate of relaxation tends to zero. Such an enlarged relaxation
system is designed to be linear in convection with a localized
stiff source term which are symmetrizable. The numerical advantages
of such approximation are amusing. The convergence of such
approximation is proved in many interesting cases. The stability
condition will be discussed, and some stability results on the
approximations in the presence of strong waves such shock waves
and boundary layers will be presented. The focus will be the
advantages of the numerical methods designed by such a relaxation
procedure. Some numerical simulations will be shown.
(March 19) Dr. John C. Strikwerda:
Preconditioning for Regular Elliptic Systems of Differential Equations -
In solving linear systems of equations arising from discretizations
of elliptic systems of equations, it is common to precondition the
system so that the resulting system is easier to solve by an iterative
method. What this means is that the linear system Ax = b is replaced
by the system PA x = Pb , where P, the preconditioning matrix, is chosen
to mimic the inverse of A in some way. If A results for the discretization of an elliptic differential
operator E, then one way to choose P is as an approximation to the
inverse of a differential operator G so that G^(-1) E is a bounded
operator. In this talk I will present a general theory for preconditioning
elliptic systems, especially for the Stokes equations and the equations
of linear elasticity by using other elliptic systems as preconditioners. (April 2) Professor G. W. Stewart:
The Eigenstructure of Graded Matrices -
A matrix A is column graded if it can be written in the form A = BD,
where B is a "nice" matrix. For example, the matrix 6.8563e-01 -1.0540e-04 1.7984e-09 2.3076e-13
-6.3679e-01 -7.1539e-06 -5.4202e-09 6.7163e-13
-1.0026e+00 2.7920e-05 1.6342e-08 -5.0808e-13
-1.8562e-01 1.3733e-04 8.2522e-09 8.5635e-13 is column graded, with a grading ratio of about 1e-3 from column to
column. The eigensystem of a graded matrix generally shares the grading of the
matrix. For example, the eigenvalues of the above matrix are 6.8573e-01 -1.0504e-04 2.3476e-08 2.0918e-12 The eigenvectors 1.0000e+00 1.5370e-04 -8.1092e-09 5.3325e-13
-9.2862e-01 1.0000e+00 -3.5687e-05 6.5557e-09
-1.4621e+00 1.2011e+00 1.0000e+00 5.2613e-05
-2.7088e-01 -1.0359e+00 2.0688e-01 1.0000e+00 exhibit a more complicated grading, but one that is related to the
original. Unfortunately, this state of affairs is dependent on the "niceness'
of the base matrix B. For example, if we change the (2,2)-element of the
above matrix to 9.8895e-05, then the eigenvalues become 6.8573e-01 1.3543e-06 -3.3548e-07 6.9367e-12 and the second eigenvalue is too small. In this talk we will derive the eigenstructure of graded matrices and
give conditions under which base matrix B is nice. Along the way we
will review the perturbation theory of eigenspaces--a theory which
applies in infinite as well as finite dimensional spaces.
(April 9) Dr. Alison Ramage:
Towards Parameter-Free Streamline Upwinding For Advection-Diffusion Problems -
The subject of this talk is the design of robust and efficient finite
element approximation methods for solving advection-diffusion
equations. Specifically, we consider the stabilisation of discrete
approximations using uniform grids which do not resolve boundary
layers, as might arise using a multi-level (or multigrid) iteration
strategy. In particular, we show that when using SUPG
(streamline-upwind) finite element methodology, there is a symbiotic
relationship between `optimal' solution approximation and fast
convergence of smoothers based on the standard GMRES iteration.
(April 16) Professor Charalambos Makridakis:
Error estimates for approximations to scalar conservation laws:
Relaxation and second-order schemes -
A general framework for deriving error estimates
for numerical approximations to scalar conservation laws is presented.
Applications include finite difference and finite volume relaxation
schemes, as well as standard monotone finite volume schemes.
Error estimates are derived also for schemes satisfying "strong"
as well as "weak" entropy inequalities. (April 23) Professor Chi-Wang Shu:
A Discontinuous Galerkin Finite Element Method for Hamilton-Jacobi Equations -
We present a discontinuous Galerkin finite element method for solving the
nonlinear Hamilton-Jacobi equations. This method is based on the Runge-Kutta
discontinuous Galerkin finite element method for solving conservation laws.
The method has the flexibility of treating complicated geometry by using
arbitrary triangulation, can achieve high order accuracy with a local,
compact stencil, and are suited for efficient parallel implementation.
One and two dimensional numerical examples are given to illustrate the
capability of the method. (April 30) Dr. Z. Jane Wang:
Aerodynamics of Insect Flight -
Flapping insect flight generates high mean lift by the interaction of the
flying surfaces with the shed vorticity. Typical Reynolds numbers of
insects are around 5000. The high lift of a hovering insect is not
usually explained by conventional quasi-steady aerodynamics. In this
study, we compute numerically, for the full Navier-Stokes equations, the
unsteady viscous flow that involve vorticity shed from a
two-dimensional wing. We model an insect wing by a thin ellipse, which
undergoes various translational and rotational motions. We present
results describing the vorticity field and the time history of lift and
drag forces for various Reynolds numbers and different wing motions.
We also discuss computational aspects of the force measurements
and the role of the far-field boundary conditions. (May 7) Professor Celso Grebogi:
Obstruction to Shadowing and Modeling -
Dynamical conditions for the loss of validity of numerical chaotic solutions of
dynamical systems are already understood. However, the fundamental questions of
``how good'' and for ``how long'' the solutions are valid remained unanswered.
In this talk, I will address these questions by establishing scaling laws for
the shadowing distance and for the shadowing time in terms of meaningful
quantities that are easily computable in practice. The scaling theory is
verified against a physical model. I will also argue that a new level of
mathematical difficulty, brought from the theory of dynamical systems, limits
our ability to model nature using deterministic models.
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