University of Leicester

We use the term shape optimization when we want to find a minimizer of an objective function that assigns real values to shapes of domains. Solving shape optimization problems can be quite challenging, especially when the objective function is constrained to a PDE, in the sense that evaluating the objective function for a given domain shape requires first solving a boundary value problem stated on that domain. The main challenge here is that shape optimization methods must employ numerical methods capable of solving a boundary value problem on a domain that changes after each iteration of the optimization algorithm.

The first part of this talk will provide a gentle introduction to shape optimization. The second part of this talk will highlight how the finite element framework leads to automated numerical shape optimization methods, as realized in the open-source library fireshape. The talk will conclude with a brief overview of some academic and industrial applications of shape optimization.

This talk is about the mathematics behind an artistic project focusing on the vibrations of soap films.

It has been known for some while now that every radial basis function

in common usage for multi-dimensional interpolation has associated with

it a uniquely defined Hilbert space, in which the radial basis function

is the `minimal norm interpolant'. This space is usually constructed by

utilising the positive definite nature of the radial function, but such

constructions turn out to be difficult to handle. We will present a

direct way of constructing the spaces, and show how to prove extension

theorems in such spaces. These extension theorems are at the heart of

improved error estimates in the $L_p$-norm.

Non-selfadjoint singular differential equation eigenproblems arise in a number of contexts, including scattering theory, the study of quantum-mechanical resonances, and hydrodynamic and magnetohydrodynamic stability theory.

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It is well known that the spectra of non-selfadjoint operators can be pathologically sensitive to perturbation of the

operator. Wilkinson provides matrix examples in his classical text, while Trefethen has studied the phenomenon extensively through pseudospectra, which he argues are often of more physical relevance than the spectrum itself. E.B. Davies has studied the phenomenon particularly in the context of Sturm-Liouville operators and has shown that the eigenfunctions and associated functions of non-selfadjoint singular Sturm-Liouville operators may not even form a complete set in $L^2$.

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In this work we ask the question: under what conditions can one expect the regularization process used for selfadjoint singular Sturm-Liouville operators to be successful for non-selfadjoint operators? The answer turns out to depend in part on the so-called Sims Classification of the problem. For Sims Case I the process is not guaranteed to work, and indeed Davies has very recently described the way in which spurious eigenvalues may be generated and converge to certain curves in the complex plane.

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Using the Titchmarsh-Weyl theory we develop a very simple numerical procedure which can be used a-posteriori to distinguish genuine eigenvalues from spurious ones. Numerical results indicate that it is able to detect not only the spurious eigenvalues due to the regularization process, but also spurious eigenvalues due to the numerics on an already-regular problem. We present applications to quantum mechanical resonance calculations and to the Orr-Sommerfeld equation.

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This work, in collaboration with B.M. Brown in Cardiff, has recently been generalized to Hamiltonian systems.

In recent years, there has been considerable interest, especially in the context of

fluid-dynamics, in nonconforming finite element methods that are based on discontinuous

piecewise polynomial approximation spaces; such approaches are referred to as discontinuous

Galerkin (DG) methods. The main advantages of these methods lie in their conservation properties, their ability to treat a wide range of problems within the same unified framework, and their great flexibility in the mesh-design. Indeed, DG methods can easily handle non-matching grids and non-uniform, even anisotropic, polynomial approximation degrees. Moreover, orthogonal bases can easily be constructed which lead to diagonal mass matrices; this is particularly advantageous in unsteady problems. Finally, in combination with block-type preconditioners, DG methods can easily be parallelized.

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In this talk, we introduce DG discretizations of mixed field and potential-based formulations of

eddy current problems in the time-harmonic regime. For the electric field formulation, the

divergence-free constraint within non-conductive regions is imposed by means of a Lagrange

multiplier. This allows for the correct capturing of edge and corner singularities in polyhedral domains; in contrast, additional Sobolev regularity must be assumed in the DG formulation, and their conforming counterparts, when regularization techniques are employed. In particular, we present a mixed method involving discontinuous $P^\ell-P^\ell$ elements, which includes a normal jump stabilization term, and a non-stabilized variant employing discontinuous $P^\ell-P^{\ell+1}$ elements.The first formulation delivers optimal convergence rates for the vector-valued unknowns in a suitable energy norm, while the second (non-stabilized) formulation is designed to yield optimal convergence rates in both the $L^2$--norm, as well as in a suitable energy norm. For this latter method, we also develop the {\em a posteriori} error estimation of the mixed DG approximation of the Maxwell operator. Indeed, by employing suitable Helmholtz decompositions of the error, together with the conservation properties of the underlying method, computable upper bounds on the error, measured in terms of the energy norm, are derived.

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Numerical examples illustrating the performance of the proposed methods will be presented; here,

both conforming and non-conforming (irregular) meshes will be employed. Our theoretical and

numerical results indicate that the proposed DG methods provide promising alternatives to standard conforming schemes based on edge finite elements.

I will describe some application areas for radial basis function, and discuss how the computational problems can be overcome by the use of preconditioning methods and fast evaluation techniques.

We develop a theory of impact of viscoelastic spheres with adhesive

interactions. We assume that the collision velocities are not large to

avoid the fracture and plastic deformation of particles material and

microscopic relaxation time is much smaller than the collision duration.

The adhesive interactions are described with the use of Johnson, Kendall

and Roberts (JKR) theory, while dissipation is attributed to the

viscoelastic behavior of the material. For small impact velocities we

apply the condition of a quasi-static collision and obtain the

inter-particle force. We show that this force is a sum of four

components, having in addition to common elastic, viscous and adhesive

force, the visco-adhesive cross term. Using the derived force we compute

the coefficient of normal restitution and consider the application of our

theory to the collisions of macro and nano-particles.

Multiscale ensembles of reaction networks with well separated constants are introduced and typical properties of such systems are studied. For any given ordering of reaction rate constants the explicit approximation of steady state, relaxation spectrum and related eigenvectors (``modes") is presented. The obtained multiscale approximations are computationally cheap and robust. Some of results obtained are rather surprising and unexpected. First of all is the zero-one asymptotic of eigenvectors (asymptotically exact lumping; but these asymptotic lumps could intersect). Our main mathematical tools are auxiliary discrete dynamical systems on finite sets and specially developed algorithms of ``cycles surgery"

for reaction graphs. Roughly speaking, the dynamics of linear multiscale networks transforms into the dynamics on finite sets of reagent names.

A.N. Gorban and O. Radulescu, Dynamic and static limitation in multiscale reaction networks, Advances in Chemical Engineering, 2008 (in press), arXiv e-print: http://arxiv.org/abs/physics/0703278

A.N. Gorban and O. Radulescu, Dynamical robustness of biological networks with hierarchical distribution of time scales, IET Syst.

Biol., 2007, 1, (4), pp. 238-246.