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Applied Mathematics Seminars 2012 - 2013

Programme

These seminars take place on Tuesdays, in Room M/2.06, Senghennydd Road, Cardiff from 4pm, unless otherwise stated.

When a seminar is not scheduled there is a collaborative workshop with other groups within the College of Physical Sciences & Engineering or a SIAM Chapter Meeting. Further details can be found on the School Diary.

For more information or if you wish to give a talk, please contact the programme organiser Dr Angela Mihai.

2nd October 2012

Speaker: Coralia Cartis (Edinburgh).

Title: Optimal Newton-type methods for nonconvex smooth optimization.

Abstract: This talk addresses global rates of convergence and the worst-case evaluation complexity of methods for nonconvex optimization problems. We show that the classical steepest-descent and Newton's methods for unconstrained nonconvex optimization under standard assumptions may both require a number of iterations and function evaluations arbitrarily close to the steepest-descent's global worst-case complexity bound.

This implies that the latter upper bound is essentially tight for steepest descent and that Newton's method may be as slow as the steepest-descent method in the worst case. Then the cubic regularization of Newton's method (Griewank (1981), Nesterov & Polyak (2006)) is considered and extended to large-scale problems, while preserving the same order of its improved worst-case complexity (by comparison to that of steepest-descent); this improved worst-case bound is also shown to be tight.

We further show that the cubic regularization approach is, in fact, optimal from a worst-case complexity point of view amongst a wide class of second-order methods for nonconvex optimization. The worst-case problem evaluation complexity of constrained optimization will also be discussed. This is joint work with Nick Gould (Rutherford Appleton Laboratory, UK) and Philippe Toint (University ofNamur,Belgium).

30th October 2012

Speaker: Tony Shardlow (Bath).

Title: Approximation of Gaussian random fields.

Abstract: With the growing importance of stochastic PDEs, there is much interest in random fields generators. We look at random field generators through the eyes of a numerical analyst. The most well known generators give approximate realisations to a large class of Gaussian random fields. We will analyses the errors and give rates of convergence for several well known (e.g., circulant embedding, turning bands, quadrature, KL) methods.

19th November 2012 in Room S/0.38, School of Engineering

Speaker: Feng Xiao (Tokyo).

Title: High order conservative collocation method using multi-moment constraints.

Abstract: In this talk, I will present a general formulation to devise high order conservative collocation schemes by using multi-moment constraint conditions for flux function reconstruction. Different from the existing methods, such as the nodal type discontinuous Galerkin method and the spectral collocation method, where only the point values are used to construct the numerical flux, the new method makes use of different types of quantities (moments), such as cell integrated average, point value and derivatives. This formulation can be also interpreted as a blend of the Lagrange interpolation and the Hermite interpolation, which leads to a new class of high order schemes. Some representative schemes will be presented and evaluated through Fourier analysis and numerical tests.

12th March 2013

Speaker: Dr. Mikito Furuichi (Japan Agency for Marine-Earth Science and Technology (JAMSTEC).

Title: Development of fluid-particle coupled simulation method in the Stokes flow regime: toward 3-D geodynamic magma simulation including granular media.

Abstract: A fluid-particle two-phase flow has been widely studied in geodynamics, because particle-saturated fluid layer is important for understanding the dynamics of solidifying and melting process in the magma chamber or magma ocean.

In order to deal with such particle-fluid systems as the geodynamical modeling in 3-D geometry, we develop a new coupled simulation code of Finite Difference method (FDM) for fluid flow and Discrete Element method (DEM) for solid particles. In the geodynamic modeling with highly viscous fluid, the fluid motion can be treated as the Stokes flow. Although this type of coupled simulation method has been well developed in the engineering field especially for a fluidized bed with high Reynolds number, the method for viscous granular media over long time scales has not yet been fully addressed.

The normal DEM-fluid formulation requires a solution of dumped oscillation with a small time step dt ~1/? for high fluid viscosity ?. Thus the normal formulation is not suitable for our target problems. We therefore propose to drop off the inertial term from the equation of particle motion likewise the Stokes flow. With this approach, we can employ the large dt~? for the problems with highly viscos fluid. In the talk, we introduce the details of our coupled model treatment and its implementation on the vector/many core parallel architectures.

19th March 2013

Speaker: Serafim Kalliadasis (Imperial College London).

Title: Recent progress on the moving contact line problem.

Abstract: The moving contact line problem is a long-standing and fundamental challenge in the field of fluid dynamics, occurring when one fluid replaces another as it moves along a solid surface. Moving contact lines occur in a vast range of applications, where an apparent paradox of motion of a fluid-fluid interface, yet static fluid velocity at the solid satisfying the no-slip boundary condition arises. In this talk we will review recent progress on the problem made by our group.

The motion of a contact line is examined, and comparisons drawn, for a variety of proposed models in the literature. We first scrutinise a number of models in the classic test-bed system of spreading of a thin two-dimensional droplet on a planar substrate, showing that slip, precursor film and interface formation models effectively reduce to the same spreading behaviour. This latter model, developed by Shikhmurzaev a few years ago, is a complex and somewhat controversial one, differentiating itself by accounting for a variation in surface layer quantities and having finite-time surface tension relaxation. Extensions to consider substrate heterogeneities in this prototype system for slip models are also considered, such as for surface roughness and fluctuations in wetting properties through chemical variability. Analysis of a solid-liquid-gas diffuse-interface model is then presented, with no-slip at the solid and where the fluid phase is specified by a continuous density field. We first obtain a wetting boundary condition on the solid that allows us to consider the motion without any additional physics, i.e. without density gradients at the wall away from the contact line associated with precursor films.

Careful examination of the asymptotic behaviour asthe contact line is approached is then shown to resolve the singularities associated with the moving contact line problem. Various features of the model are scrutinised alongside extensions to incorporate slip, finite-time relaxation of the chemical potential, or a precursor film at the wall. But these are not necessary to resolve the moving contact line problem.

Co-Host: Nikos Savva

8th April 2013 at 10:30 in M/0.40

Speaker: John Bush (MIT).

Title: Hydrodynamic quantum analogs: Droplets walking on the impossible pilot wave

Abstract: Yves Couder and coworkers have recently discovered that droplets walking on a vibrating fluid bath exhibit several features previously thought to be peculiar to the microscopic, quantum realm. Theoretical developments provide rationale for the complex behavior of the bouncing droplets, and yield a trajectory equation for the walking droplets. Experimental results reveal the emergence of wave-like statistics from pilot-wave dynamics for droplets walking in confined geometries, and for droplets walking on a rotating fluid bath. Theoretical results indicate the manner in which wave-like statistics emerge from pilot-wave dynamics for droplets walking on a rotating bath, or under the influence of a central force. The relation between this fluid system and de Broglie’s relativistic pilot-wave theory of quantum mechanics is discussed.

Co-Host: Nikos Savva

4th June 2013

Speaker: Christopher Davies (Cardiff University School of Mathematics)

Title: Evolution of disturbance wavepackets in an oscillatory Stokes layer

Abstract: Numerical simulation results are presented for the evolution of disturbances in a flat Stokes layer. The response to a spatially localised impulsive forcing is investigated and it is found that the spatial-temporal development of linearized disturbances displays an intriguing family tree-like structure. This involves the birth of successive generations of distinct wavepacket components. It is shown that some important features of the disturbance behaviour can be predicted using the results of a linear stability analysis based on Floquet theory. The ramifications of the simulation results are also discussed with regard to the interpretation of observations obtained from physical experiments.

11th June 2013

Speaker: Greg King (Population and Conservation Genetics Group, Instituto Gulbenkian de Ciência Oeiras, Portugal)

Title: The energy cascade in the atmospheric mesoscales at the bottom of the marine boundary layer: Is it upscale? Downscale? Or is nature not so simple?

Abstract: A long-standing question in atmospheric dynamics has been: Is horizontal kinetic energy transferred to small scales through a downscale cascade as in ideal three-dimensional (3D) turbulence? Or is it transferred to large scales via a two-dimensional (2D) inverse cascade?

The classic papers by Nastrom et al (1984, 1985) and more recent papers by Lindborg (1999) and Cho and Lindborg (2001) have addressed this question through an analysis of global datasets of winds near the tropopause measured by instruments carried on commercial aircraft. Here we use winds at the bottom of the marine boundary layer inferred from radar backscatter from the ocean surface measured by the Advanced Scatterometer (ASCAT) on the MetOp-A satellite and the SeaWinds scatterometer on the QuikSCAT satellite. Our results indicate that nature is not so simple.

Co-Host: Chris Davies

18th June 2013

Speaker: Vinh Phu Nguyen (Cardiff University School of Engineering)

Title: Isogeometric cohesive elements for two and three dimensional composite delamination analysis.

Abstract: Isogeometric cohesive elements are presented for modeling two and three dimensional delaminated composite structures. We exploit the knot insertion algorithm offered by NURBS (Non Uniform Rational B-splines) to generate cohesive elements along delamination planes in an automatic fashion. A complete computational framework is presented including pre-processing, processing and post-processing. They are explained in details and implemented in MIGFEM an open source Matlab Isogemetric Analysis code developed by the authors.

The composite laminates are modeled using both NURBS solid and rotation-free shell elements. Several two and three dimensional examples ranging from standard delamination tests (the mixed mode bending test) to the L-shaped specimen with a fillet, three dimensional (3D) double cantilever beam and a 3D singly curved thick-walled laminate are provided. To the authors knowledge, it is the first time that NURBS-based isogeometric analysis for two/three dimensional delamination modeling is presented. IGA provides a bi-directional system in which one can go forward from CAD to analysis and backwards from analysis to CAD. This is believed to facilitate the design of composite structures.

9th July 2013

Speaker: Derek Moulton (University of Oxford)

Title: Morphorods: modelling growing elastic rods.

Abstract: Filamentary structures are abundant in nature, and exist across many length scales, from biopolymers and microtubules to umbilical cords to vines and branches. Here, we develop a framework to study mechanics and questions of stability in the case of growing elastic rods, what we term morphorods. To demonstrate a particular application of the theory, we turn to the fascinating world of seashells. By considering the mechanical nature of the growth process of seashells, we find an intriguing relationship between morphorods and the formation of intricate ornamentations found on certain shells, such as sharp spines.

16th July 2013

Speaker: Elena Atroshchenko )Department of Mechanical Engineering, University of Chile)

Title: Boundary element method in two-dimensional Cosserat elasticity.

Abstract: A wide range of engineering materials can be modeled using theory of linear elasticity. However, methods of classical elasticity are not efficient in predicting mechanical behavior of solids in cases when it is significantly affected by material microstructure. One of the possibilities to include the microstructural effects in consideration is by means of generalized continuum theories, such as micropolar (also known as Cosserat or asymmetric) elasticity. In micropolar elasticity material points are endowed with additional rotational degrees of freedom. This assumption leads to the more accurate description of a deformation in terms of asymmetric stress and couple stress tensors. Micropolar elasticity incorporates the intrinsic material length scale parameters associated with the size of the microstructure (grains, fibers, pores).

This allows one to replace a highly heterogeneous Cauchy media with a homogeneous micropolar continuum described by only 6 constants. Examples of Cosserat materials include fiber-reinforce composites, layered and blocky materials (sandwich structures), geomaterials and human bones. The main objective of the present work is to develop and implement the boundary element method for two-dimensional Cosserat elasticity, including plane strain and anti-plane strain states. Fundamental solutions have been derived using method of associated matrices and used to represent displacement/microrotation and stress/couple stress fields in the form of single-layer and double-layer potentials. The fundamental solutions have been shown to be weakly singular, singular and hypersingular near the boundary. The boundary integral equations have been formulated and subsequently solved by boundary element method.