Many continuum systems in physics exhibit a spatially localized response to spatially homogeneous forcing. The resulting structures are frequently called dissipative solitons. In this talk I will describe the phenomenon of homoclinic snaking that is responsible for the presence of these states, together with the associated snakesand- ladders structure of the snaking region, i.e. the parameter region containing such localized structures. I will describe different types of snaking in both one and two spatial dimensions, focusing on the manner in which localized structures grow as one follows the solution branch from small to large amplitude. I will illustrate the results using both model equations such as the Swift-Hohenberg and forced Ginzburg-Landau equations, and "real" systems such as binary fluid convection and magnetoconvection.

Accurate modeling of realistic industrial turbomachinery requires overcoming several challenges, including: (1) handling the complexity of turbomachinery models, which often include multiple stages; (2) modeling the aerodynamic loads; (3) modeling the nonlinear dynamics caused by cracks in multi-stage systems. These challenges are addressed in this talk. First, a methodology is presented to construct reduced order models (ROMs) of multi-stage systems with mistuning using single-sector calculations only. These ROMs are then used to perform a statistical characterization of structural mistuning in multi-stage systems, and to introduce a new classification method for characterizing the properties of all modes. Next, the effects of the aerodynamics on the multi-stage response are explored. The methodology consists of first creating efficient structural ROMs, and then iteratively calculating the aerodynamic stiffness matrices for each stage. Finally, the effects of cracks are discussed. A novel methodology for modeling the nonlinear vibration analysis of multistage systems with mistuning and a cracked blade is proposed. The nonlinear effects of a crack opening and closing are considered in conjunction with the effects of mistuning. A variety of results are provided for a two-stage industrial rotor, and the accuracy and efficiency of the proposed methods are discussed.

Resonances can be defined as complex eigenvalues of a self-adjoint differential operator on an unbounded domain, for which the corresponding eigenfunction satisfies a radiation condition. Such eigenfunctions are typically highly improper since they increase exponentially at infinity. Resonances appear in a variety of applications including acoustics (e.g. musical instruments or slat noise in air-planes), molecular dynamics, and optics (e.g. lasers or leaky modes in wave-guides). Excitation of a resonant system by a resonant frequency (the square root of the real part of a resonance) leads to a large system response, which can be approximately computed with the help of the corresponding resonance and eigenfunction. To compute such resonances numerically, we split the domain into a bounded computational domain and an unbounded exterior domain. For the discretization of the exterior domain we suggest so-called Hardy-space infinite elements, which preserve the eigenvalue structure of the problem and exhibit super-algebraic convergence. Their performance is illustrated by numerical simulations for acoustic and electromagnetic resonance problems.

La simulación numérica del calentamiento de piezas en hornos industriales presenta grandes beneficios potenciales, tanto en el diseño de las condiciones de operación como en el control de los hornos. Sin embargo, la complejidad del problema hace inviable el uso de simulación numérica directa, salvo quizá en situaciones muy concretas. En esta charla se presenta la aplicación de modelos de orden reducido que permiten trabajar ‘en tiempo real’ (entendiendo como tal un tiempo de cálculo compatible con la implementación en un código de simulación de procesos o que se encuentre por debajo de los tiempos característicos empleados en la estrategia de control del horno). Se considerarán dos casos concretos, relacionados con proyectos de carácter industrial que se desarrollan actualmente en la Universidad de Vigo. El primero de ellos se refiere a la predicción del calentamiento de ejes de automoción forjados en un horno de tratamiento térmico (previo al baño de temple) en condiciones estacionarias, con vistas al rediseño del horno y/o de su operación. El segundo caso corresponde a la predicción del calentamiento de grandes piezas de acero (para la alimentación de un tren de laminación en caliente) destinada a integrarse en el esquema de regulación del horno.

A relevant problem in the computational simulation of (time-harmonic and transient) waves is the devising of the correct boundary conditions that allow us to cut-off the computational domain and, therefore, to use any of the common PDE solvers. The main issue arises from the fact that waves have to be allowed to scatter away but the artificial boundary has to avoid them bouncing back. This problem is nowadays very well understood in the time-harmonic case, for the most relevant types of linear waves: acoustic, elastic, electromagnetic,... Much recent progress is occurring in the field of transient problems. Most possible approaches can be categorized in three classes (almost corresponding to tribal groups): ABC, PML and BIE. I will try to explain the advantages and disadvantages of some Absorbing Boundary Conditions and Perfectly Matched Layers for transient acoustic waves. I will then move to the realm of Boundary Integral Operators/Equations, explaining what they have to offer (exactness and arbitrary closeness to where "stuff happens") and what is the price to pay, both at the computational level and at the steep learning process they require. The talk will be informative. There will be plenty of formulas and some pictures, but no theorems or Sobolev spaces.

The invariant manifolds of a dynamical system are organizing centers, whose computation and study help to understand its dynamics. The continuation of steady solutions of parabolic systems of partial differential equations with respect to parameters is now a common tool in Science and Engineering. The computation of other invariant manifolds, such as periodic orbits and invariant tori is not so usual, although all of them can be cast into a common framework, through the calculation of fixed points of an appropriately defined map. Both the definition of the map and the calculation of its fixed points through the Newton-Krylov method requires some care, especially in the case of invariant tori. The computation of periodic orbits by parallel multiple shooting will also be considered. Codes using this method can achieve linear or close to linear speedups. The MPI library has been used in our calculations but OMP could also be used on architectures based on shared-memory multi-core processors. As a test problem, the thermal convection of a binary fluid will be considered. This is joint work with Marta Net and Carles Simò.

La teoría de los sistemas din´micos ha resultado exitosa para describir el transporte por advección en fluidos. Sin embargo su potencia se ve mermada por la falta de una teoría matemática establecida para describir los sistemas dinámicos no autónomos. En esta presentación introducir nuevos conceptos que generalizan ideas bien asentadas de los sistemas dinámicos autónomos y periódicos, al caso dependiente del tiempo de forma aperiódica. Este es un problema de gran interés para la descripción del transporte en flujos geofísicos. Se ilustrara la eficacia de estas herramientas en el análisis de datos de altimetría, tomados sobre la superficie oceánica en la región de la corriente de Kuroshio.

Se consideran ecuaciones de evolución de tipo convección-difusión, en el caso límite en que domina la convección. Es bien conocido que los métodos clásicos de elementos finitos no son adecuados para aproximar problemas de convección dominante. En la primera parte de la charla se expone un procedimiento que permite eliminar las oscilaciones espúreas de los métodos clásicos. Se propone también un algoritmo adaptativo con el cual se obtiene una malla espacial automáticamente adaptada sobre la cual la aproximación Galerkin no es oscilante. En la segunda parte de la charla se estudia la aproximación numérica de las mismas ecuaciones usando métodos de tipo ENO (essentially non oscillatory) en espacio. Este tipo de métodos se usan con mucha frecuencia en el contexto de la resolución de ecuaciones hiperbólicas. En este trabajo mostraremos que estos métodos son también adecuados para resolver problemas de convecció-nreacción- difusión en los que domina la convección. Nuevamente se propone un procedimiento adaptativo en espacio para mejorar la eficiencia de los métodos considerados.

This talk will describe the origin and properties of spatially localized structures in one and two dimensions using a model equation, the Swift-Hohenberg equation, and will use this understanding to explain the properties of similar structures recently identified in convection and shear flows using both direct numerical simulations and numerical branch-following.