## Applied Mathematics Seminars

### 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.

##### 15 October 2013

**Speaker:** Thomas Lessinnes (University of Oxford).

**Title:** Birods: theory and applications.

**Abstract:** Nature abounds with filamentary, rod-like structures. Be it trunks, stems, rhubarb stalks, pollen tubes, arteries, neurons, microtubules, or linear macromolecules, filaments are known to exhibit similar instabilities and mechanical behaviours. In this seminar I will give a rapid introduction to Kirchhoff's rod theory and show how growth and remodeling is usually incorporated in it. Often in practical applications (of which I will consider a few) growth is spatially limited by walls or attachment to different mechanical bodies. This incompatibility leads to various buckling and makes for a rich landscape of shape formation. A framework will be presented that allows to practically treat "morphoelastic rods problems" and a couple of examples will be developed to illustrate how to apply it.

##### 29 October 2013

**Speaker:** Andrew Wathen (University of Oxford).

**Title:** Iterative linear solvers for PDE-constrained optimization involving fluid flow.

**Abstract:** Many control problems for PDEs can be expressed as Optimization problems with the relevant PDEs acting as constraints. As is being discovered in other areas such as multi-physics, there seem to be distinct advantages to tackling such constrained Optimization problems all-at-once' or with a `one-shot' method. That is, decoupling of the overall problem in some loosely coupled iterative fashion appears to be a rather poorer approach than to compute on the fully coupled problem.

The use of iterative methods for the relavant linear algebra is crucial here since the overall dimensions (including the Optimization and PDE) are usually very large, but matrix vector products as required in Krylov subspace methods such as MINRES are still readily computed. The work to ensure rapid convergence is in preconditioning and it is this topic that we will mostly focus on in this talk.

We will describe our general approach via block preconditioning and demonstrate its use for the control of Poisson and Stokes problems and also for the fully time-dependent heat equations.

##### 5 November 2013

**Speaker:** Tomasz Koziara (University of Durham).

**Title:** Some basic of the Discrete Element Method and how it can be improved (does anybody know, please?).

**Abstract:** I am going to share some thoughts on a "large brushstrokes" program of the refinement of DEM, idealistically aimed at more realistic/efficient simulations driven by less material parameters. I will try to sketch the avenue from the rigid particle - deformable interface law paradigm towards the deformable particle - rigid interface law one. I will show some old and, hopefully, some new preliminary results.

##### 12 November 2013

**Speaker:** Alison Ramage (University of Strathclyde, Glasgow).

**Title:** Efficient iterative solvers for director-based models of liquid crystal devices.

**Abstract:**Whenever Lagrange multipliers are used for the pointwise unit-vector constraints in director modelling of liquid crystals, or in both director and order tensor models when an electric field that stems from a constant voltage is present, the resulting discretised equations take the form of saddle-point problems. For the construction of bifurcation diagrams illustrating the various types of behaviour a particular device exhibits, these equations must be solved many times, so it is important that this is done as efficiently as possible. In this talk we discuss a preconditioned iterative solver particularly suited to director models and illustrate its performance on a model of a Twisted Nematic device. This is joint work with Eugene C. Gartland, Jr., Department of Mathematical Sciences, Kent State University.

##### 19 November 2013

**Speaker:** Hamid Ghasemi (Bauhaus University Weimar, Germany).

**Title:** Optimum fiber content and distribution in fiber-reinforced solids using a reliability and NURBS based sequential optimization approach.

**Abstract:** A comprehensive stochastic optimization algorithm for finding the optimum fiber content and its distribution in solid composites characterized by uncertain design parameters is presented. Firstly, the optimum amount of fiber in a FRC structure with uniformly distributed fibers is conducted in the framework of Reliability Based Design Optimization (RBDO) problem. Secondly, the fiber distribution optimization having the aim to increase the structure reliability more is performed by defining a fiber dispersion function through a Non-Uniform Rational B-Spline (NURBS) surface. These two stages iterate sequentially and interactively till satisfying convergence criterion. The target parameter provided by the proposed computational procedure is the reliability index (a measure for probability of structural failure) of the final structure with optimum fiber content and optimum fiber distribution. First order reliability method has been implemented in order to approximate the limit state function while a homogenization approach, based on the assumption of random orientation of fibers in the matrix, has been adopted. A number of static loading test examples are conducted to demonstrate the capability and reliability of the present model.

##### 3 December 2013

**Speaker:** Peter Stewart (University of Glasgow).

**Title:** Coalescence and fracture of gas-liquid foams.

**Abstract:** We construct a large-scale network model for the dynamics and stability of gas-liquid foams which explicitly incorporates hydrodynamic effects in the liquid films between adjacent gas bubbles. We use two example problems to demonstrate how these microscale flows can play a significant role in determining the macroscale behaviour of the foam. Firstly, we consider batch processing of high-porosity metallic solids from molten metal foams (produced without surfactant) where the liquid bridges separating adjacent bubbles drain rapidly and break due to inter-molecular attractions, leading to bubble coalescence. We characterise this coalescence process over both short and long timescales, and demonstrate that the evolution of the foam is remarkably self-similar over a wide range of parameter space. Secondly, we consider the fracture of (aqueous) gas-liquid foams under an applied driving pressure, elucidating the two distinct fracture modes observed in experiment which are analogous to the brittle and ductile failure mechanisms of crystalline solids.

**Co-Host:** Dr. Maurice Blount.

##### 4 February 2014

**Speaker:** Azadeh Jafari (Université de Montréal, Canada).

**Title:** A geometrical multiscale approach to blood flow simulation: Coupling 2D Navier-Stokes and 0D lumped parameter models.

**Abstract:** Blood is a complex suspension that possesses several non-Newtonian rheological characteristics such as thixotropy, viscoelasticity and shear thinning. In this study, the time evolution of the non-Newtonian characteristics of human blood will be described by the Oldroyd-B model. Although the Oldroyd-B model is not an accurate haemorheological model in this context, it is a simplified version of a homogeneous blood model originally proposed by Owens (J. Non-Newtonian Fluid. Mech. 140, 2006, 57-70). The Navier-Stokes system of continuity and momentum equations coupled with a constitutive equation defined in some section of the cardiovascular system must be closed with initial and various boundary conditions on velocity, pressure and viscoelastic stress.

Although it is possible to prescribe, for example, Neumann conditions on the velocity and Dirichlet conditions on the pressure on outflow boundaries as part of a mathematically well-posed problem, these outflow conditions may be far from physical. The main issue is that such conditions do not take proper account of the remaining part of the circulatory system of which the flow section in question is possibly only a small part. An alternative is a geometrical multiscale approach which is a strategy for modelling the circulatory system, including the reciprocal interactions between local and systemic haemodynamics by exploiting complementary features of different possible models. Indeed, these features suggest a natural way of coupling detailed local models (in the flow domain) with coarser models able to describe the dynamics over a large part or even the whole cardiovascular system at acceptable computational cost.

In this study, we directly couple the 2D Navier-Stokes equations, the constitutive equations and a 0D lumped parameter model to simulate a cerebral aneurysm and discuss the reliability of multiscale models for computing correct boundary conditions at the outflow boundaries of the aneurysm.

**Co-Host:** Prof. Tim Phillips.

##### 18 February 2014 at 15:00-15:50 in M/2.06

**Speaker:** Dr Denis Sidorov (Energy Systems Institute, Russian Academy of Sciences).

**Title:** Machine learning for time series forecasting and defects classification: two case studies.

**Abstract:**The objective of this talk is to present the brief overview of the results concerning
the machine learning techniques employment for solution of the industrial mathematics
problems. In particular the power systems parameters forecasting and
automatic defects classification problems will be discussed.

**Co-Host:** Dr Kirill Cherednichenko.

##### 18 February 2014

**Speaker:** Adrian Hill (University of Bath).

**Title:** Sharp Bounds on a Matrix Function.

**Abstract:** Sharp bounds on matrix functions: Consider C^N with an inner product. Let A be an N by N complex matrix with numerical range W(A) contained in some larger and more convenient subset D of C. If f: D --> C is holomorphic, then, f(A) may be defined in a number of equivalent ways. Such matrix functions appear in several branches of applied mathematics where it is desirable to accurately quantify |f(A)|.

Let S, a subset of D, contain the eigenvalues of A. A variety of approaches lead to bounds of the form |f(A)| =< K. sup{ |f(z)| : z in S}.

Here, we summarize work of Crouzeix, who considers S=W(A) and S=D, a disk or an intersection of disks. We also present a new result for S=D, a complex rectangle. These results typically yield small K in [2,11.08], at the price of bounding |f(z)| over a set larger than just the eigenvalues of A.

**Co-Host:** Prof. Marco Marletta.

##### 25 February 2014

**Speaker:** Paul Ledger (Swansea University).

**Title:** Characterising the shape and material properties of hidden targets from magnetic induction data.

**Abstract:** The first part of this talk will provide a brief introduction to inverse problems and describe some of the challenges they present. The second part of the talk will focus on how inclusions can be characterised by polarization tensors, which offer possibilities for the low-cost solution of electromagnetic inverse problems.

In particular, the talk will describe how recent developments confirm that the engineering prediction of H^T . ( M H^M) for the sensitivity of measurements of the perturbed magnetic field to the presence of a general conducting object placed in a low frequency background field is correct. Explicitly, H^T is the background field generated by the transmitter coil, H^M is the background field generated by the receiving coil as if it was used as a transmitter and M is rank 2 polarization that describes the shape and material properties of the object. The talk will also show how the new tensors for different objects can be accurately computed by solving a vector valued transmission problem by hp-version finite elements. The ability to compute these tensors for different objects holds great promise for applications that involve the location and characterisation of conducting objects such as landmine detection, ensuring food safety and security screening.

##### 4 March 2014

**Speaker:** Jennifer Pestana (University of Manchester).

**Title:** Solving saddle point systems: the Batman factorisation and preconditioners.

**Abstract:** Saddle points systems, and generalisations of them, arise in many applications as a result of applying constraints. In this talk we discuss two different ways of solving such systems.

In the first part of this talk we discuss the antitriangular factorisation of Mastronardi and van Dooren. Although applicable to any symmetric indefinite matrix, we show that the algorithm simplifies considerably when applied to saddle point matrices. Moreover, the resulting factorisation is a matrix representation of the well-known nullspace method. We also show that bounds on relevant eigenvalues can be easily obtained.

The antitriangular factorisation may not be so suitable when the saddle point matrix is large and sparse, or for more general block matrices, and in these cases preconditioned iterative methods are often used. We discuss relationships between the eigenvalues and eigenvectors of upper and lower block triangular preconditioners and show that typically the difference in convergence is small.

##### 11 March 2014

**Speaker:** David Kay (University of Oxford).

**Title:** A Porous-Elastic Model of the Lung.

**Abstract:**A stabilized conforming finite element method for the three-field (displacement, fluid flux and pressure) poroelasticity problem is presented. We use the lowest possible approximation order: piecewise constant approximation for the pressure, and piecewise linear continuous elements for the displacements and fluid flux. By applying a local pressure jump stabilization term to the mass conservation equation we avoid pressure oscillations and importantly, the discretization leads to a symmetric linear system. For the fully discretized problem we prove existence and uniqueness, an energy estimate and an optimal a-priori error estimate, including an error estimate for the fluid flux in its natural $H{div}$ norm. Numerical experiments in 2D and 3D illustrate the convergence of the method, show the effectiveness of the method to overcome spurious pressure oscillations, and evaluate the effect of the stabilization term.

Finally, the method is applied to a model of a lung, where an airway tree is coupled to a porous approximation of the tissue.

##### 1 April 2014

**Speaker:** David Silvester (University of Manchester).

**Title:** A posteriori error estimation for stochastic Galerkin approximation.

**Abstract:** Stochastic Galerkin finite element approximation is an increasingly popular approach for the solution of elliptic PDE problems with correlated random data. Given a parametrisation of the data in terms of a large, possibly infinite, number of random variables, this approach allows the original PDE problem to be reformulated as a parametric, deterministic PDE on a parameter space of high dimension. A typical strategy is to combine conventional (h-) finite element approximation on the spatial domain with spectral (p-) approximation on a finite-dimensional manifold in the (stochastic) parameter space.

For approximations relying on low-dimensional manifolds in the parameter space, stochastic Galerkin finite element methods have superior convergence properties to standard sampling techniques. On the other hand, the desire to incorporate more and more parameters (random variables) together with the need to use high-order polynomial approximations in these parameters inevitably generates huge-dimensional discretised systems. This in turn means that adaptive algorithms are needed to efficiently construct approximations, and fast and robust linear algebra techniques are essential for solving the discretised problems.

Both strands will be discussed in the talk. We outline the issues involved in a posteriori error analysis of computed solutions and present a practical a posteriori estimator for the approximation error. We describe a novel energy error estimator that uses a parameter-free part of the underlying differential operator: this effectively exploits the tensor-product structure of the approximation space (and simplifies the linear algebra). We prove that our error estimator is reliable and efficient. We also discuss different strategies for enriching the discrete space and establish two-sided estimates of the error reduction for the corresponding enhanced approximations.

##### 8 April 2014 at 15:10

**Speaker:** Rebecca Lingwood (KTH, The Royal Institute of Technology, Sweden, and University of Cambridge).

**Title:** The rotating-disk boundary layer: edge effects and other recent results.

**Abstract:** Recent experimental and numerical results from the ASTRID (A STudy of Rotation In Developing boundary-layer flows) programme at KTH, Stockholm, will be presented. These results focus on the interplay between convective and absolute instabilities and the onset of laminar-turbulent transition in the rotation-induced boundary-layer flow over a rotating disk. Particular attention is given to the effects that a finite disk radius have on the stability characteristics of the flow; a feature that is naturally part of any physical flow but is neglected from standard linear-stability analyses, where an infinite radius is assumed. Preliminary results from global linear direct numerical simulations (DNS) provide interesting insights into the action of a finite radius. Global nonlinear DNS have also been undertaken and show the onset of turbulence for the rotating-disk flow.

**Co-Host:** Dr. Chris Davies.

##### 13 May 2014 at 15:10

**Speaker:** Claire Heaney (Cardiff School of Engineering).

**Title:** Summation rules for the Quasi-Continuum method.

**Abstract:** The behaviour of materials that possess a discrete meso-, micro- or nano-structure can be modelled by lattice models, which take into account the discrete structure directly. Instead of using phenomenologically derived continuum models to relate stresses and strains, the small-scale lattice interactions are used in the calculation of the governing equations. The small spatial scales that are involved preclude the modelling of realistically sized domains. The Quasi-continuum (QC) method was introduced to alleviate this problem for (conservative) atomistic lattices, but has recently been shown to also be applicable to non-conservative structural lattice models.

To apply the QC method, the domain should be split into two parts, a fully resolved region and a coarse region. The lattice model is used without simplification in the fully resolved region where the solution experiences significant fluctuations in deformation, for instance, in the vicinity of a defect. In coarse regions, where the solution varies less rapidly, it is sufficient to interpolate the lattice model and select a small number of lattice interactions to estimate the governing equations, thereby reducing the computational burden. The lattice interactions which are selected for this purpose are determined by a summation rule. It is within the setting of structural lattice models that this paper investigates the efficiency and accuracy of several different summation rules for QC methods using higher-order interpolation functions.