Past Industrial and Applied Mathematics Seminar

3 March 2016
16:00
Francesco dell'Isola
Abstract
There are relatively few results in the literature of non-linear beam theory: we recall here the very first classical results by Euler–Bernoulli and the researches stemming from von Kármán for moderately large rotations but small strains. In this paper, we consider a discretized springs model for extensible beams and propose a heuristic homogenization technique of the kind first used by Piola. The homogenized energy obtained has some peculiar features which we start to describe by solving numerically some exemplary deformation problems. Furthermore we consider pantographic structures constituted by the introduced nonlinear beams and study some planar deformation problems. Numerical solutions for these 2D problems are obtained via minimization of energy and are compared via some experimental measurements, in which the importance of elongation phenomena are clearly pointed out. In the conclusions we indicate a list of some mathematical problems which seems worth of consideration. 
 
Indeed Already Piola in 1848 introduces for microscopically discrete systems to be described via a continuum model: i) the micro-macro kinematical map, ii) the identification of micro- macro work functional and iii) the consequent determination of macro-constitutive equations in terms of the micro properties of considered mechanical system.
 
Piola uses, following the standards of his time, a rigorous mathematical deduction process and considers separately one dimensional, two dimensional and three dimensional continua as continua whose reference configuration is a curve, a surface or a regular connected subset of Euclidean three dimensional space. This subdivision of the presented matter is also followed by Cosserat Brothers: how to detect the influence on their works exerted by Piola’s pioneering ones is a historical problem which deserves further in-depth studies.
 
In the present paper we follow the spirit of Piola while looking for Lagrange density functions for a class of non-linear one dimensional continua in planar motion: we focus on modeling phenomena in which both extensional and bending deformations are of relevance.
 
Usually in literature the simultaneous extension and bending deformation of a beam is not considered: however when considering two dimensional continua embedding families of fibers as a model of some specific microstructured mechanical systems (as fiber fabrics or pantographic sheets ) the assumption that the fibers cannot extend while bending is not phenomenologically well-grounded. Therefore, we are led in the second part of the present paper to present some two dimensional continua in which the second gradient of in plane displacement (involving so called geodesic bending) appears in the expression of deformation energy.
 
The modeling assumptions are, in both cases, based on a physically reasonable discrete microstructure of used beams: in engineering literature these microstructures, constituted by extensional and rotational springs and possibly rigid bars, were introduced in order to get discrete Lagrangian approximation of continuum models in linearized regimes.
 
A natural development, involving the study of spatial placements of one dimensional or two dimensional continua or the introduction of three dimensional continua embedding reinforcement fibers will be subject of further investigations.
 
The study of pantographic sheets by means of a micro model based on Cauchy first gradient continuum models involves the choice of relatively small length scale, implying the introduction of numerical models involving finite elements with several millions of degrees of freedom: the computational burden of such models makes their use, at least in the mid term horizon, absolutely inappropriate. The higher gradient reduced order model presented in this paper involves a rather more effective numerical modeling whose performances (as will be shown in a forthcoming paper Giorgio et al. in preparation) are however absolutely comparable.
 
However the problem of formulating intermediate meso modeling, involving a class of Generalised Beam Theories, will be necessarily to be confronted: for instance the deformation of beam sections involving warping, Poisson effects, elastic necking or large shear or twist deformation can definitively be studied via reduced order models not resorting to the most detailed micro Cauchy first gradient models.
 
One should also remark that higher gradient continuum models may require novel integration schemes, more suitable to their intrinsic structure: we expect that isogeometric methods may further increase the effectiveness of the reduced models we present here, especially when completely spatial models will be considered .
  • Industrial and Applied Mathematics Seminar
25 February 2016
16:00
Abstract

Honey poured from a sufficient height onto toast undergoes the well-known `liquid rope coiling’ instability.

We have studied this instability using a combination of laboratory experiments, theory, and numerics, with the aim of determining phase diagrams and scaling laws for the different coiling modes. Finite-amplitude coiling has four distinct modes - viscous, gravitational, inertio-gravitational, and inertial - depending on how the viscous forces that resist deformation of the rope are balanced. The inertio-gravitational mode is particularly interesting as it involves resonance between the coiling portion of the rope and its long trailing `tail’. Further experiments using less viscous fluids reveal that the rope can exhibit five different morphologies, of which steady coiling is only one. We determine the detailed phase diagram of these morphologies, which includes a novel `liquid supercoiling’

state in which the coiled cylinder formed by the primary coiling instability undergoes in turn its own complex buckling instability.  We show that the onset of these different patterns is determined by a non-penetrability condition which takes different forms in the viscous, gravitational and inertial limits. To close, we will briefly evoke two additional related phenomena: spiral waves of bubbles generated by coiling, and the `fluid mechanical sewing machine’ in which the fluid falls onto a moving belt.

  • Industrial and Applied Mathematics Seminar
18 February 2016
16:00
Abstract

While there have been recent advances for analyzing the complex deterministic
behavior of systems with discontinuous dynamics, there are many open questions about
understanding and predicting noise-driven and noise-sensitive phenomena in the
non-smooth context.  Stochastic effects can often change the picture dramatically,
particularly if multiple time scales are present.  We demonstrate novel approaches
for exploring and explaining surprising phenomena driven by the interplay of
nonlinearities, delays, randomness, in specific applications with piecewise smooth
dynamics - nonlinear models of balance,  relay control, and impacting dynamics.
Effective techniques typically depend on the combination of mathematical techniques,
multiple scales techniques, and phenomenological intuition from seemingly unrelated
canonical models of biophysics, mechanics, and chemical dynamics.  The appropriate
strategy is not always immediately obvious from the area of application or model
type. This gap may follow from the limited attention that stochastic models with
discontinuous dynamics have received in the past, or it may be the reason for this
limited attention.  Combining the geometrical perspective with asymptotic approaches
in physical and phase space appears to be a critical part of developing effective
approaches.

  • Industrial and Applied Mathematics Seminar
11 February 2016
16:00
Anand Oza
Abstract
Roughly a decade ago, Yves Couder and coworkers demonstrated that droplets walking on a vibrating fluid bath exhibit several features previously thought to be peculiar to the microscopic quantum realm, including single-particle diffraction, tunneling, quantized orbits, and wave-like statistics in a corral. We here develop an integro-differential trajectory equation for these walking droplets with a view to gaining insight into their subtle dynamics. We then rationalize the emergence of orbital quantization in a rotating frame by assessing the stability of the orbital solutions. In the limit of large vibrational forcing, the chaotic walker dynamics gives rise to a coherent statistical behavior with wave-like features.
 
I will then describe recent efforts to model the dynamics of interacting flapping swimmers. Our study is motivated by recent experiments using a one-dimensional array of wings in a water tank, in which the system adopts “schooling modes” characterized by specific spatial phase relationships between swimmers. We develop a discrete dynamical system that models the swimmers as airfoils shedding point vortices, and study the existence and stability of steady solutions. We expect that our model may be used to understand how schooling behavior is influenced by hydrodynamics in more general contexts.
 
  • Industrial and Applied Mathematics Seminar
4 February 2016
16:00
Barbara Mahler, Thomas Woolley, Julian A. Garcia Grajales
Abstract

Barbara Mahler: 15+5 min

Thomas Woolley: 15+5 min

Julian A. Garcia Grajales: 15+5 min
 

  • Industrial and Applied Mathematics Seminar
28 January 2016
16:00
Laura Nicolaou
Abstract

Respiratory illnesses, such as asthma and chronic obstructive pulmonary disease, account for one in five deaths worldwide and cost the UK over £6 billion a year. The main form of treatment is via inhaled drug delivery. Typically, however, a low fraction of the inhaled dose reaches the target areas in the lung. Predictive numerical capabilities have the potential for significant impact in the optimisation of pulmonary drug delivery. However, accurate and efficient prediction is challenging due to the complexity of the airway geometries and of the flow in the airways. In addition, geometric variation of the airways across subjects has a pronounced effect on the aerosol deposition. Therefore, an accurate model of respiratory deposition remains a challenge.

High-fidelity simulations of the flow field and prediction of the deposition patterns motivate the use of direct numerical simulations (DNS) in order to resolve the flow. Due to the high grid resolution requirements, it is desirable to adopt an efficient computational strategy. We employ a robust immersed boundary method developed for curvilinear coordinates, which allows the use of structured grids to model the complex patient-specific airways, and can accommodate the inter-subject geometric variations on the same grid. The proposed approach reduces the errors at the boundary and retains the stability guarantees of the original flow solver.

A Lagrangian particle tracking scheme is adopted to model the transport of aerosol particles. In order to characterise deposition, we propose the use of an instantaneous Stokes number based on the local properties of the flow field. The effective Stokes number is then defined as the time-average of the instantaneous value. This effective Stokes number thus encapsulates the flow history and geometric variability. Our results demonstrate that the effective Stokes number can deviate significantly from the reference value based solely on a characteristic flow velocity and length scale. In addition, the effective Stokes number shows a clear correlation with deposition efficiency.

  • Industrial and Applied Mathematics Seminar
21 January 2016
16:00
Tmoslav Plesa, John Ockendon, Hilary Ockendon
Abstract

Tmoslav Plesa: Chemical Reaction Systems with a Homoclinic Bifurcation: An Inverse Problem, 25+5 min;

John Ockendon: Wave Homogenisation, 10 min + questions; 

Hilary Ockendon: Sloshing, 10 min + questions
 

 

  • Industrial and Applied Mathematics Seminar
3 December 2015
16:00
Leonid v Berlyand
Abstract

We study the motion of a eukaryotic cell on a substrate and investigate the dependence of this motion on key physical parameters such as strength of protrusion by actin filaments and adhesion. This motion is modeled by a system of two PDEs consisting of the Allen-Cahn equation for the scalar phase field function coupled with a vectorial parabolic equation for the orientation of the actin filament network. The two key properties of this system are (i) presence of gradients in the coupling terms and (ii) mass (volume) preservation constraints. We pass to the sharp interface limit to derive the equation of the motion of the cell boundary, which is mean curvature motion perturbed by a novel nonlinear term. We establish the existence of two distinct regimes of the physical parameters. In the subcritical regime, the well-posedness of the problem is proved (M. Mizuhara et al., 2015). Our main focus is the supercritical regime where we established surprising features of the motion of the interface such as discontinuities of velocities and hysteresis in the 1D model, and instability of the circular shape and rise of asymmetry in the 2D model. Because of properties (i)-(ii), classical comparison principle techniques do not apply to this system. Furthermore, the system can not be written in a form of gradient flow, which is why Γ-convergence techniques also can not be used. This is joint work with V. Rybalko and M. Potomkin.

  • Industrial and Applied Mathematics Seminar
26 November 2015
16:00
Adilson E Motter
Abstract

Much of the recent interest in complex networks has been driven by the prospect that network optimization will help us understand the workings of evolutionary pressure in natural systems and the design of efficient engineered systems.  In this talk, I will reflect on unanticipated attributes and artifacts in three classes of network optimization problems. First, I will discuss implications of optimization for the metabolic activity of living cells and its role in giving rise to the recently discovered phenomenon of synthetic rescues. Then I will comment on the problem of controlling network dynamics and show that theoretical results on optimizing the number of driver nodes/variables often only offer a conservative lower bound to the number actually needed in practice. Finally, I will discuss the sensitive dependence of network dynamics on network structure that emerges in the optimization of network topology for dynamical processes governed by eigenvalue spectra, such as synchronization and consensus processes.  Optimization is a double-edged sword for which desired and adverse effects can be exacerbated in complex network systems due to the high dimensionality of their dynamics.

  • Industrial and Applied Mathematics Seminar
19 November 2015
16:00
Robert Style, Samuel Crew and Phil Trinh
Abstract
New singularities for Stokes waves
Samuel Crew (Lincoln College) and Philippe Trinh
 
In 1880, Stokes famously demonstrated that the singularity that occurs at the crest of the steepest possible water wave in infinite depth must correspond to a corner of 120°. Here, the complex velocity scales like the one-third power of the complex potential. Later in 1973, Grant showed that for any wave away from the steepest configuration, the singularity moves into the complex plane, and is instead of order one-half. Grant conjectured that as the highest wave is approached, other singularities must coalesce at the crest so as to cancel the square-root behaviour. Even today, it is not well understood how this process occurs, nor is it known what other singularities may exist. 
 
In this talk, we shall explain how we have been able to construct the Riemann surface that represents the extension of the water wave into the complex plane. We shall also demonstrate the existence of a countably infinite number of singularities, never before noted, which coalesce as Stokes' highest wave is approached. Our results demonstrate that the singularity structure of a finite amplitude wave is much more complicated than previously anticipated, 
 
  • Industrial and Applied Mathematics Seminar

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