When a liquid drop is deposited over a solid surface whose temperature is sufficiently above the boiling point of the liquid, the drop does not experience nucleate boiling but rather levitates over a thin layer of its own vapor. This is known as the Leidenfrost effect. Whilst highly undesirable in certain cooling applications, because of a drastic decrease of the energy transferred between the solid and the evaporating liquid due to poor heat conductivity of the vapor, this effect can be of great interest in many other processes profiting from this absence of contact with the surface that considerably reduces the friction and confers an extreme mobility on the drop. During this presentation, I hope to provide a good vision of some of the knowledge on this subject through some recent studies that we have done. First, I will present a simple fitting-parameter-free theory of the Leidenfrost effect, successfully validated with experiments, covering the full range of stable shapes, i.e., from small quasi-spherical droplets to larger puddles floating on a pocketlike vapor film. Then, I will discuss the end of life of these drops that appear either to explode or to take-off. Finally, I will show that the Leidenfrost effect can also be observed over hot baths of non-volatile liquids. The understanding of the latter situation, compare to the classical Leidenfrost effect on solid substrate, provides new insights on the phenomenon, whether it concerns levitation or its threshold.

We discuss the key role that bespoke linear algebra plays in modelling PDEs with random coefficients using stochastic Galerkin approximation methods. As a specific example, we consider nearly incompressible linear elasticity problems with an uncertain spatially varying Young's modulus. The uncertainty is modelled with a finite set of parameters with prescribed probability distribution.  We introduce a novel three-field mixed variational formulation of the PDE model and and  assess the stability with respect to a weighted norm. The main focus will be  the efficient solution of the associated high-dimensional indefinite linear system of equations. Eigenvalue bounds for the preconditioned system can be  established and shown to be independent of the discretisation parameters and the Poisson ratio.  We also  discuss an associated a posteriori error estimation strategy and assess proxies for the error reduction associated with selected enrichments of the approximation spaces.  We will show by example that these proxies enable the design of efficient  adaptive solution algorithms that terminate when the estimated error falls below a user-prescribed tolerance.

This is joint work with Arbaz Khan and Catherine Powell

Just like optimization needs derivatives, shape optimization needs shape derivatives. Their definition and computation is a classical subject, at least concerning first order shape derivatives. Second derivatives have been studied as well, but some aspects of their theory still remains a bit mysterious for practitioners. As a result, most algorithms for shape optimization are first order methods.

To understand this situation better and in a general way, we consider first and second order shape sensitivities of integrals on smooth submanifolds using a variant of shape differentiation. Instead of computing the second derivative as the derivative of the first derivative, we choose a one-parameter family of perturbations  and compute first and second derivatives with respect to that parameter. The result is a  quadratic form in terms of a perturbation vector field that yields a second order quadratic model of the perturbed functional, which can be used as the basis of a second order shape optimization algorithm. We discuss the structure of this derivative, derive domain expressions and Hadamard forms in a general geometric framework, and give a detailed geometric interpretation of the arising terms.

Finally, we use our results to construct a second order SQP-algorithm for shape optimization that exhibits indeed local fast convergence.

Analytic continuation is ill-posed, but becomes merely ill-conditioned (though with an infinite condition number) if it is known that the function in question is bounded in a given region of the complex plane.
This classical, seemingly theoretical subject has many connections with numerical practice.  One argument indicates that if one tracks an analytic function from z=1 around a branch point at z=0 and back to z=1 again by a Weierstrass chain of disks, the number of accurate digits is divided by about exp(2 pi e) ~= 26,000,000.

Likely instabilities in stochastic hyperelastic solids

L. Angela Mihai

School of Mathematics, Cardiff University, Senghennydd Road, Cardiff, CF24 4AG, UK

E-mail: @email.uk

Nonlinear elasticity has been an active topic of fundamental and applied research for several decades. However, despite numerous developments and considerable attention it has received, there are important issues that remain unresolved, and many aspects still elude us. In particular, the quantification of uncertainties in material parameters and responses resulting from incomplete information remain largely unexplored. Nowadays, it is becoming increasingly apparent that deterministic approaches, which are based on average data values, can greatly underestimate, or overestimate, mechanical properties of many materials. Thus, stochastic representations, accounting for data dispersion, are needed to improve assessment and predictions. In this talk, I will consider stochastic hyperelastic material models described by a strain-energy density where the parameters are characterised by probability distributions. These models, which are constructed through a Bayesian identification procedure, rely on the maximum entropy principle and enable the propagation of uncertainties from input data to output quantities of interest. Similar modelling approaches can be developed for other mechanical systems. To demonstrate the effect of probabilistic model parameters on large strain elastic responses, specific case studies include the classic problem of the Rivlin cube, the radial oscillatory motion of cylindrical and spherical shells, and the cavitation and finite amplitude oscillations of spheres.

The problem of authorship attribution (AA) involves matching a text of unknown authorship with its creator, found among a pool of candidate authors. In this work, we examine in detail authorship attribution methods that rely on networks of function words to detect an “authorial fingerprint” of literary works. Previous studies interpreted these word adjacency networks (WANs) as Markov chains, giving transition rates between function words, and they compared them using information-theoretic measures. Here, we apply a variety of network flow-based tools, such as role-based similarity and community detection, to perform a direct comparison of the WANs. These tools reveal an interesting relation between communities of function words and grammatical categories. Moreover, we propose two new criteria for attribution based on the comparison of connectivity patterns and the similarity of network partitions. The results are positive, but importantly, we observe that the attribution context is an important limiting factor that is often overlooked in the field's literature. Furthermore, we give important new directions that deserve further consideration.

Coupling dynamics of the states of the nodes of a network to the dynamics of the network topology leads to generic absorbing and fragmentation transitions. The coevolving voter model is a typical system that exhibits such transitions at some critical rewiring. We study the robustness of these transitions under two distinct ways of introducing noise. Noise affecting all the nodes destroys the absorbing-fragmentation transition, giving rise in finite-size systems to two regimes: bimodal magnetization and dynamic fragmentation. Noise targeting a fraction of nodes preserves the transitions but introduces shattered fragmentation with its characteristic fraction of isolated nodes and one or two giant components. Both the lack of absorbing state for homogeneous noise and the shift in the absorbing transition to higher rewiring for targeted noise are supported by analytical approximations.

Paper Link:

https://journals.aps.org/pre/abstract/10.1103/PhysRevE.92.032803

In the 1990s Moy and Prasad revolutionized the representation theory of p-adic groups by showing how to use Bruhat-Tits theory to assign invariants to representations of p-adic groups. The tools they introduced resulted in rapid advancements in both representation theory and harmonic analysis -- areas of central importance in the Langlands program. A crucial ingredient for many results is an explicit construction of (types for) representations of p-adic groups. In this talk I will indicate why, survey what constructions are known (no knowledge about p-adic groups assumed) and present recent developments based on a refinement of Moy and Prasad's invariants.​