|»|| Applied Analysis Group
The Applied Analysis Group
The field of Applied analysis brings together many mathematical topics, such differential equations, dynamical systems, variational calculus, functional analysis, geometry, and approximation theory. The Applied Analysis Group at TU/e, headed by Mark Peletier, focuses on the interaction between these mathematical disciplines and the real world around us.
A number of themes are common to much of the research of the group:
- (Nonlinear) partial differential equations: well-posedness and qualitative properties such as parameter dependence and asymptotic behaviour, rigorous convergence analysis of numerical schemes
- Multiscale problems: homogenization, upscaling, rough and moving boundaries, discrete-to-continuum transitions
- Variational methods for PDEs: minimization, critical point theory, gradient flows, Gamma-convergence
While there are many, many different applications, some of the more important application areas are
Below we list some of the projects that we are currently working on.
- Continuum mechanics: elasticity, viscous flows, reactive flows, plasticity, acoustics, and many others
- Biology: biochemistry, biophysics, but also agent-based models of groups of people and animals
Some active projects
Flow through porous media
Reactive flows in porous media involve hierarchically organized structures and processes at different scales. The mathematical modelling of such processes, as well as the mathematical and numerical analysis (including upscaling and homogenization) of resulting models is a long-term research topic within CASA. Non-standard porous media flow models (involving dynamic capillarity effects, or interfacial areas) are analyzed by Clement Cances, Catherine Choquet, Hans van Duijn, Majid Hassanizadeh, Bert Peletier, Sorin Pop, and in the doctoral theses of Yabin Fan and Xiulei Cao. Regularization methods for degenerate parabolic equations for either one- or two phase flows (including outflow boundary conditions or a-posteriori error estimates) or for reactive flows (convergent mixed/finite element discretization and linearization schemes) are considered by Sorin Pop, Florin Radu, Ben Schweizer, Mauricio Sepulveda, Martin Vohralik and in the doctoral thesis of Kundan Kumar. Part of this work is carried out within the International Research Training Group NUPUS (Non-linearities and Upscaling in Porous Media). On this topic, further collaboration exists with Willi Jäger, Andro Mikelic, Jan Nordbotten, Christian Rohde, and Barbara Wohlmuth.
Where do the Wasserstein gradient flows come from?
Since the seminal work of Jordan, Kinderlehrer, and Otto in 1997, it has become clear that many, many well-known systems are Wasserstein gradient flows. In two papers Stefan Adams, Nicolas Dirr, Mark Peletier, and Johannes Zimmer give a first explanation why Wasserstein gradient flows are so ubiquitous. The key observation is that there is a high-level connection between gradient flows on one hand and the large deviations in stochastic particle systems on the other. PhD student Michiel Renger extended some results to systems with convection and decay, and PhD student Giovanni Bonaschi is studying lattice systems.
Aircraft noise is a major
design problem for new aircraft. The spectacular reduction of
noise emitted per aircraft, achieved over the past 50 years, is entirely compensated by an
equally spectacular increase of numbers of aircraft in use. Altogether this hasn't mended the
troubled relationship between society and air traffic,
and a continuous research effort for further
reduction is required.
Models and modelling tools at all levels (from crude and fast to refined but slow) are necessary.
In the PhD work of Mireal Darau, supervised by
and Bob Mattheij, a modelling problem was studied about
the anomalous unstable interaction
between the thin mean flow boundary layer and the lined wall of a turbofan engine duct,
and related problems with
Ed Brambley (DAMTP).
The goal of the PhD project of Martien Oppeneer is to further develop and make available a
class of semi-analytical solutions for the propagation and attenuation of sound in flow
ducts with strong shear and temperature gradients, aiming in particular to the reduction
of noise from the APU, the aircraft electric power station. This is
done in cooperation with Bob Mattheij and Pieter Sijtsma (NLR)
The ITN project FlowAirS of Deepesh Singh will involve the modelling of
nonlinear Helmholtz resonators as used in acoustic lining, and the acoustics
related with hydrodynamic instabilities in flow ducts.
The design of public space requires a deep understanding of the way large numbers of people move. Of particular importance are their dynamical interactions and the resulting collective behaviour. This is a key issue not only in designing e.g. railway stations, airports, but also in ensuring the safety of crowds at large public events. However, the modelling and computation of such groups of people is still in its infancy.
In principle, there are two main approaches to modelling crowds: as a particle system (microscopic) or as a PDE for the crowd density (macroscopic).Adrian Muntean and Joep Evers develop and study models in a measure-theoretical framework, which incorporates the micro- and macroscopic views as special cases. Their emphasis is on the influence of walls and obstacles (boundary conditions); anisotropy (people do not perceive their surroundings equally well in all directions); and two-scale effects (leader-group interaction). In March 2013, Joep Evers submitted video description of this work to the NWO Bessensap competition.
Free- and moving-boundary problems arise in a wide variety of applications, and present both modelling and mathematical challenges. In these problems, the domain on which a (partial) differential equation is supposed to hold is not known a priori, and finding it or its evolution in time is a crucial part of the problem. Currently, Georg Prokert is working on two problems of this type: a cell growth model for fungal hyphae (with J. Hulshof and R. Nolet, VU Amsterdam) and a model describing diffusion through a semipermeable membrane with surface tension (with F. Lippoth, Leibniz University Hannover).
Three papers by Anozie Ebigbo, Rainer Helmig, Kundan Kumar, Tycho van Noorden, and Sorin Pop deal with models for biological and chemical processes in porous media. biofilm growth and dissolution and precipitation in porous media, or chemical vapor deposition. In all these papers free boundaries are encountered at the micro/pore scale. The aim is to derive upscaled equations for situations including a dominating flow regime (Taylor dispersion), or for domains with rough boundaries.
Adrian Muntean and his collaborators T. Aiki (Tokyo) and M. Böhm (Bremen) studied the large-time behavior of macroscopic sharp-reaction fronts traveling through cement-based materials (e.g. carbonation and sulfatation thin reaction zones and/or interfaces), viewed as unsaturated highly heterogeneous reactive porous materials. The target is to give qualitative estimates on the lifetime of concrete structures exposed to chemically aggressive environments.
Discrete-to-continuum transitions in plasticity
Dislocations are the elementary quanta of plastic deformation: bending a piece of steel involves moving large numbers of dislocations, which are defects in the metal crystal structure, from one side of the crystal to the other. Together with the Mechanics of Materials group at TU/e and the Materials innovation institute, Lucia Scardia has characterized the continuum, many-dislocation limit in a simple, stylized case, rigorously proving five different limiting behaviours corresponding to five different scalings. PhD students Patrick van Meurs and Marleen Kooiman are continuing this work in different directions.