Past events

The flow in a cylinder with a rotating endwall has continued to

attract much attention since Vogel (1968) first observed the vortex

breakdown of the central core vortex that forms. Recent experiments

have observed a multiplicity of unsteady states that coexist over a

range of the governing parameters. In spite of numerous numerical and

experimental studies, there continues to be considerable controversy

with fundamental aspects of this flow, particularly with regards to

symmetry breaking. Also, it is not well understood where these

oscillatory states originate from, how they are interrelated, nor how

they are related to the steady, axisymmetric basic state.

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In the aspect ratio (height/radius) range 1.6 2.8. An efficient and

accurate numerical scheme is presented for the three-dimensional

Navier-Stokes equations in primitive variables in a cylinder. Using

these code, primary and secondary bifurcations breaking the SO(2)

symmetry are analyzed.

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We have located a double Hopf bifurcation, where an axisymmetric limit

cycle and a rotating wave bifurcate simultaneously. This codimension-2

bifurcation is very rich, allowing for several different scenarios. By

a comprehensive two-parameter exploration about this point we have

identified precisely to which scenario this case corresponds. The mode

interaction generates an unstable two-torus modulate rotating wave

solution and gives a wedge-shaped region in parameter space where the

two periodic solutions are both stable.

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For aspect ratios around three, experimental observations suggest that

the first mode of instability is a precession of the central vortex

core, whereas recent linear stability analysis suggest a Hopf

bifurcation to a rotating wave at lower rotation rates. This apparent

discrepancy is resolved with the aid of the 3D Navier-Stokes

solver. The primary bifurcation to an m=4 traveling wave, detected by

the linear stability analysis, is located away from the axis, and a

secondary bifurcation to a modulated rotating wave with dominant modes

m=1 and 4, is seen mainly on the axis as a precessing vortex breakdown

bubble. Experiments and the linear stability analysis detected

different aspects of the same flow, that take place in different

spatial locations.

We combine linear algebra techniques with finite element techniques to obtain a reliable stopping criterion for the Conjugate Gradient algorithm. The finite element method approximates the weak form of an elliptic partial differential equation defined within a Hilbert space by a linear system of equations A x = b, where A is a real N by N symmetric and positive definite matrix. The conjugate gradient method is a very effective iterative algorithm for solving such systems. Nevertheless, our experiments provide very good evidence that the usual stopping criterion based on the Euclidean norm of the residual b - Ax can be totally unsatisfactory and frequently misleading. Owing to the close relationship between the conjugate gradient behaviour and the variational properties of finite element methods, we shall first summarize the principal properties of the latter. Then, we will use the recent results of [1,2,3,4]. In particular, using the conjugate gradient, we will compute the information which is necessary to evaluate the energy norm of the difference between the solution of the continuous problem, and the approximate solution obtained when we stop the iterations by our criterion.

Finally, we will present the numerical experiments we performed on a selected ill-conditioned problem.

References

• [1] M. Arioli, E. Noulard, and A. Russo, Vector Stopping Criteria for Iterative Methods: Applications to PDE's, IAN Tech. Rep. N.967, 1995.
• [2] G.H. Golub and G. Meurant, Matrices, moments and quadrature II; how to compute the norm of the error in iterative methods, BIT., 37 (1997), pp.687-705.
• [3] G.H. Golub and Z. Strakos, Estimates in quadratic formulas, Numerical Algorithms, 8, (1994), pp.~241--268.
• [4] G. Meurant, The computation of bounds for the norm of the error in the conjugate gradient algorithm, Numerical Algorithms, 16, (1997), pp.~77--87.

Boundary layers are often studied with no pressure gradient

or with an imposed pressure gradient. Either of these assumptions

can lead to difficulty in obtaining solutions. A major advance in fluid

dynamics last century (1969) was the development of a triple deck

formulation for boundary layers where the pressure is not

specified but emerges through an interaction between

boundary layer and the inviscid outer flow. This has given rise to

new computational problems and computations have in turn

fed ideas back into theoretical developments. In this survey talk

based on my new book, I will look at three problems:

flow past a plate, flow separation and flow in channels

and discuss the interaction between theory and computation

in advancing boundary layer theory.

This talk reviews some theoretical and practical aspects

of incompressible flow modelling using finite element approximations

of the (Navier-) Stokes equations.

The infamous Q1-P0 velocity/pressure mixed finite element approximation

method is discussed. Two practical ramifications of the inherent

instability are focused on, namely; the convergence of the approximation

with and without regularisation, and the behaviour of fast iterative

solvers (of multigrid type) applied to the pressure Poisson system

that arises when solving time-dependent Navier-Stokes equations

using classical projection methods.

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This is joint work with David Griffiths from the University of Dundee.

We examine the convergence characteristics of iterative methods based

on a new preconditioning operator for solving the linear systems

arising from discretization and linearization of the Navier-Stokes

equations. With a combination of analytic and empirical results, we

study the effects of fundamental parameters on convergence. We

demonstrate that the preconditioned problem has an eigenvalue

distribution consisting of a tightly clustered set together with a

small number of outliers. The structure of these distributions is

independent of the discretization mesh size, but the cardinality of

the set of outliers increases slowly as the viscosity becomes smaller.

These characteristics are directly correlated with the convergence

properties of iterative solvers.

The talk will focus on solution methods for augmented linear systems of

the form

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$[ A B ][x] = [b] [ B' 0 ][y] [0]$.

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Augmented linear systems of this type arise in several areas of

numerical applied mathematics including mixed finite element / finite

difference discretisations of flow equations (Darcy flow and Stokes

flow), electrical network simulation and optimisation. The general

properties of such systems are that they are large, sparse and

symmetric, and efficient solution techniques should make use of the

block structure inherent in the system as well as of these properties.

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Iterative linear solution methods will be described that

attempt to take advantage of the structure of the system, and

observations on augmented

systems, in particular the distribution of their eigenvalues, will be

presented which lead to further iterative methods and also to

preconditioners for existing solution methods. For the particular case

of Darcy flow, comments on properties of domain decomposition methods

of additive Schwarz type and similarities to incomplete factorisation

preconditioners will be made.

I am going to show that all self-scaled barriers for the

cone of symmetric positive semidefinite matrices are of the form

$X\mapsto -c_1\ln\det X +c_0$ for some constants $c_1$ > $0,c_0 \in$ \RN.

Equivalently one could state say that all such functions may be

obtained via a homothetic transformation of the universal barrier

functional for this cone. The result shows that there is a certain

degree of redundancy in the axiomatic theory of self-scaled barriers,

and hence that certain aspects of this theory can be simplified. All

relevant concepts will be defined. In particular I am going to give

a short introduction to the notion of self-concordance and the

intuitive ideas that motivate its definition.

Interval arithmetic is a way to produce guaranteed enclosures of the

results of numerical calculations. Suppose $f(x)$ is a real

expression in real variables $x= (x_1, \ldots, x_n)$, built up from

the 4 basic arithmetic operations and other 'standard functions'. Let

$X_1, \ldots, X_n$ be (compact) real intervals. The process of {\em

interval evaluation} of $f(X_1, ..., X_n)$ replaces each real

operation by the corresponding interval operation wherever it occurs

in $f$, e.g. $A \times B$ is the smallest interval containing $\{a

\times b \mid a \in A, b \in B\}$.

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As is well known, it yields a guaranteed enclosure for the true range

$\{f(x_1, \ldots, x_n) \mid x_1 \in X_1, \ldots, x_n \in X_n\}$,

provided no exceptions such as division by (an interval containing)

zero occur during evaluation.

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Interval arithmetic takes set inputs and produces set outputs. Noting

this, we show there is a consistent way to extend arithmetic to $R^* =

R \cup \{-\infty, +\infty\}$, such that interval evaluation continues

to give enclosures, and there are {\em no exceptions}. The basic

ideas are: the usual set-theory meaning of evaluating a relation at a

set; and taking topological closure of the graph of a function in a

suitable $(R^{*})^n$. It gives rigorous meaning to intuitively

sensible statements like $1/0 = \{-\infty, +\infty\}$, $0/0 = R^*$

(but $(x/x)_{|x=0} = 1$), $\sin(+\infty) = [-1,1]$, etc.

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A practical consequence is that an exception-free floating-point

interval arithmetic system is possible. Such a system is implemented

at hardware level in the new Sun Fortran compiler, currently on

beta-release.

The Cholesky factorization approach to solving the symmetric definite generalized eigenvalue problem

$Ax = \lambda Bx$, where $A$ is symmetric and $B$ is symmetric positive definite, computes a Cholesky factorization $B = LL^T$ and solves the equivalent standard symmetric eigenvalue problem $C y = \l y$ where $C = L^{-1} A L^{-T}$. Provided that a stable eigensolver is used, standard error analysis says that the computed eigenvalues are exact for $A+\dA$ and $B+\dB$ with $\max( \normt{\dA}/\normt{A}, \normt{\dB}/\normt{B} )$

bounded by a multiple of $\kappa_2(B)u$, where $u$ is the unit roundoff. We take the Jacobi method as the eigensolver and explain how backward error bounds potentially much smaller than $\kappa_2(B)u$ can be obtained.

To show the practical utility of our bounds we describe a vibration problem from structural engineering in which $B$ is ill conditioned yet the error bounds are small. We show how, in cases of instability, iterative refinement based on Newton's method can be used to produce eigenpairs with small backward errors.

Our analysis and experiments also give insight into the popular Cholesky--QR method used in LAPACK, in which the QR method is used as the eigensolver.

In contrast to spectra, pseudospectra of large Toeplitz matrices

behave as nicely as one could ever expect. We demonstrate some

basic phenomena of the asymptotic distribution of the spectra

and pseudospectra of Toeplitz matrices and show how by employing

a few simple $C^*$-algebra arguments one can prove rigorous

convergence results for the pseudospectra. The talk is a survey

of the development since a 1992 paper by Reichel and Trefethen

and is not addressed to specialists, but rather to a general

mathematically interested audience.

This lecture is about the extension of our CAD-Free platform to the simulation and sensitivity analysis for design and control of multi-model configurations. We present the different ingredients of the platform using simple models for the physic of the problem. This presentation enables for an easy evaluation of coupling, design and control strategies.

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Sensitivity analysis has been performed using automatic differentiation in direct or reverse modes and finite difference or complex variable methods. This former approach is interesting for cases where only one control parameter is involved as we can evaluate the state and sensitivity in real time with only one evaluation.

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We show that for some class of applications, incomplete sensitivities can be evaluated dropping the state dependency which leads to a drastic cost reduction.

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The design concerns the structural characteristic of the model and our control approach is based in perturbating the inflow incidence by a time dependent flap position described by a dynamic system encapsulating a gradient based minimization algorithm expressed as dynamic system. This approach enables for the definition of various control laws for different minimization algorithm.

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We present dynamic minimization algorithms based on the coupling of several heavy ball method. This approach enables for global minimization at a cost proportional to the number of balls times the cost of one steepest descent minimization.

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Two and three dimension flutter problem simulation and control are presented and the sensitivity of the different parameters in the model is discussed.

This seminar has been cancelled.

There is a growing interest in the study of periodic phenomena in

large-scale nonlinear dynamical systems. Often the high-dimensional

system has only low-dimensional dynamics, e.g., many reaction-diffusion

systems or Navier-Stokes flows at low Reynolds number. We present an

approach that exploits this property in order to compute branches of

periodic solutions of the large system of ordinary differential

equations (ODEs) obtained after a space discretisation of the PDE. We

call our approach the Newton-Picard method. Our method is based on the

recursive projection method (RPM) of Shroff and Keller but extends this

method in many different ways. Our technique tries to combine the

performance of straightforward time integration with the advantages of

solving a nonlinear boundary value problem using Newton's method and a

direct solver. Time integration works well for very stable limit

cycles. Solving a boundary value problem is expensive, but works also

for unstable limit cycles.

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We will present some background material on RPM. Next we will explain

the basic features of the Newton-Picard method for single shooting. The

linearised system is solved by a combination of direct and iterative

techniques. First, we isolate the low-dimensional subspace of unstable

and weakly stable modes (using orthogonal subspace iteration) and

project the linearised system on this subspace and on its

(high-dimensional) orthogonal complement. In the high-dimensional

subspace we use iterative techniques such as Picard iteration or GMRES.

In the low-dimensional (but "hard") subspace, direct methods such as

Gaussian elimination or a least-squares are used. While computing the

projectors, we also obtain good estimates for the dominant,

stability-determining Floquet multipliers. We will present a framework

that allows us to monitor and steer the convergence behaviour of the

method.

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RPM and the Newton-Picard technique have been developed for PDEs that

reduce to large systems of ODEs after space discretisation. In fact,

both methods can be applied to any large system of ODEs. We will

indicate how these methods can be applied to the discretisation of the

Navier-Stokes equations for incompressible flow (which reduce to an

index-2 system of differential-algebraic equations after space

discretisation when written in terms of velocity and pressure.)

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The Newton-Picard method has already been extended to the computation

of bifurcation points on paths of periodic solutions and to multiple

shooting. Extension to certain collocation and finite difference

techniques is also possible.