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Events
Workshop on Continuous Diversity, Complex Mixtures and Applications

MPI für Mathematik in den Naturwissenschaften Leipzig (Leipzig)

Inselstr. 22
04103 Leipzig
Germany

G3 10 (Lecture hall)

conference

05.05.05 06.05.05

Workshop on Continuous Diversity, Complex Mixtures and Applications

A medium with continuous diversity can be interpreted as an assemblage of various interacting open systems with smoothly differing qualities. Such a concept is clearly intrinsic to many natural systems. In fact, whenever the populations in a large aggregate can be distinguished by a few continuously varying characteristics, like size, color, personality, orientation, age, shape, resilience, physiology, rugosity, etc., one faces a mixture which, in the limit of a very large number of individuals, is endowed with a continuous diversity of species.
The main objective of the workshop is to bring together leading specialists in several fields of science, giving them a unique opportunity to learn and exchange new mathematical methods, models and ideas concerning continuous diversity, which have recently been developed in completely different contexts, yet being applied to problems which have a common mathematical structure. Topics to be addressed by the workshop include: granular media, structured populations and biodiversity, mechanics and thermodynamics of polycrystals, iceberg and sea ice dynamics, polymeric solutions, liquid crystals and electromagnetic media, chemical reactions in complex multicomponent (continuous) mixtures.

Program

14:20 - 14:40	Opening
14:40 - 15:40	Heiko Briesen (RWTH Aachen) Continuous composition representation and multiscale solution method for refinery simulation and optimization Because of the high economic importance of oil-related products, there is a continuing need to improve the efficiency and profitability of refinery processes. In the recent past model-based methods have gained large popularity in the refining industry and have already contributed largely to ecologically and economically more beneficial processes. When modeling refinery processes, one faces the complex composition of the processed materials (e.g. crude oil). The number of components present in crude oils quickly reaches several hundreds. Standard modeling and simulation approaches are not optimized for this type of mixtures. Currently, the complex composition is reduced to a set of fictitious so-called pseudocomponents, which are treated as real components. The loss of information and model accuracy associated with this model reduction step is hard to quantify. Thus, the results obtained from these reduced models may not be reliable. An alternative approach to characterize oil mixtures is to represent its species by a continuous density function which reflects its full complexity. This leads to sets of continuously formulated equations. By formulating the models in this continuous way, it is possible to exploit modern discretization techniques for its solution. Using error estimation and adaptation concepts, the reliability of the simulation results can therefore be improved. A tailored multilevel approach using an adaptive wavelet- Galerkin discretization scheme is presented. The presentation will address the continuous modeling as well as the solution approach. The applicability of the techniques presented will be illustrated with several examples.
15:40 - 16:40	Kolumban Hutter (Technische Universität Darmstadt) Thermodynamics of polycrystalline materials within the theory of mixtures with continuous diversity We review the key steps in the formulation of a homogenisation procedure of a polycrystalline material in which crystal orientation and crystal size are two major ingredients of the microstructure. This is done within the context of rational thermodynamics of the Müller-Liu form. Only the basic concepts are given, leaving details to a talk by L. Placidi.
16:40 - 17:00	Coffee break
17:00 - 18:00	Bob Svendsen (University of Dortmund) Continuum thermodynamic formulation of models for materials characterized by an evolving microstructural distribution The current contribution is concerned with the formulation of continuum thermodynamic field models for materials with microstructure for which the distribution of (micro)structure at each material point and its evolution relative to the continuum as a whole influences the macroscopic behaviour. Examples of such materials include metallic polycrystals (grain size, grain orientation, dislocation orientation, dislocation curvature), granular materials (grain size, grain orientation, contact), polymers (molecular chain length, chain orientation), uniaxial and biaxial nematic liquid crystals (crystallite orientation distribution). To account for the motion and evolution of such microstructure with respect to the material as a whole, the corresponding extended or structured continuum is endowed with additional kinematic degrees of freedom. Once these are established, one may proceed by analogy with the case of the standard continuum to formulate kinematics, balance relations, and constitutive relations, for the extended one. For example, the formulation of extended balance relations follows in this way in the context of the total energy balance and and invariance arguments. Indeed, analogous to the standard case, the assumed Euclidean frame-indifference of this balance, together with the transformation properties of the fields appearing in it, determine the forms of the remaining balance relations for the structured continuum. With these general results in hand, account is next taken of the fact that the degrees of freedom of the standard continuum constitute a subset of those of the structured continuum. As already established in previous work, this fact can be represented in a mathematically-precise fashion as a fibre bundle, with base space the standard kinematic space, i.e., three-dimensional Euclidean point space, and total space the kinematic space for the structured continuum. In this context, the kinematic space for the structure itself is represented by the typical fibre of the fibre bundle. Among other results, one obtains on this basis a split of the the momentum balance for the structured continuum into momentum balances for the standard continuum and for the structure. Further, the fibre bundle representation induces naturally forms of all fields and balance relations with respect to standard kinematic space, i.e., forms averaged over the degrees of freedom of the structure. In this way arises naturally in the current approach a distribution function and its role as mediator of the constitutive influence of microstructural evolution on macroscopic continuum material behaviour fields. In the last part of the work, a number of applications of the current approach are discussed and compared with previous work.
18:00 - 19:00	Mats Gyllenberg (University of Helsinki) On the formulation and analysis of structured population models A population is a collection of individuals, which typically can be characterized by properties such as age, spatial location and physiological features like size. Basic modelling consists of describing individual behaviour such as survival and reproduction as well as individual development (growth, movement, etc.) and especially how these processes depend on the environmental condition. A basic question is then to study how these processes at the individual level are reflected at the population level. In this talk I shall discuss a general mathematical framework for formulating and analysing structured population models. In particular, I shall discuss steady state and stability analysis.
19:00 - 20:00	Last remarks for the first day
20:00 -	Dinner at Gasthaus Barthels Hof

09:30 - 10:30	Daniel Feltham (University College London) Continuous diversity in sea ice dynamics and thermodynamics: Some recent research After a short description of geophysically occuring sea ice, the established theory of sea ice dynamics is outlined. The discussion then focusses on recent research into sea ice dynamics and thermodynamics that may be of more general interest: (1) A new model of anisotropic sea ice dynamics that makes use of internal variables characterising the degree of anisotropy of the ice cover such as the structure tensor; (2) A homogenisation procedure for determining a geophysical-scale sea ice stress rheology from laboratory scale measurements; (3) The use of granular flow theory to describe observed phenomena at the margins of the sea ice pack; and (time permitting) (4) The simulation of melt ponds on sea ice, which have a large impact on albedo.
10:30 - 11:20	Luca Placidi (Università degli studi La Sapienza di Roma) Characterization of grain boundary migration and polygonization with the use of the postulate of global action Incompressible polycrystalline materials are treated by the theory of mixtures with continuous diversity. Each species of the mixture is characterized by grain size and lattice orientation, that is assumed to be represented solely by one unit vector. Thus, the evolution of the distributions of grain sizes and of lattice orientations are modelled by the balance of mass, in which the specific production rate of mass "Gamma" has to be assigned constitutively. In this talk, I will present a method to assign a constitutive equation for "Gamma". The problem is that such a constitutive quantity must fulfil an integral restriction due to the conservation of mass. Thus, among the classical principles of determinism and local action, a new postulate is formulated, that is an extension of these two to take the variables belonging to the species assemblage into account and that is already known, in other forms, in the literature of mixtures with continuous diversity: The postulate of global action. The formulation of this axiom is general. However, it gives a simple method to assign "Gamma" coherently with the conservation of mass. In the second part of the talk, I will give a brief introduction of the concepts of grain boundary migration and polygonization. Besides, I will discuss their modelling through the specific production rate of mass "Gamma" and finally I will use the postulate of global action and give explicit constitutive equations for "Gamma" to model these phenomena within the application of the theory of mixtures with continuous diversity that is presented in the talk of Prof. K. Hutter.
11:20 - 11:30	Coffee break
11:30 - 12:00	Closing discussion: Open issues in orthodox, less orthodox, and exotic applications of the notion of continuous diversity.

Workshop on Continuous Diversity, Complex Mixtures and Applications

Program

Participants

Heiko Briesen

Isaac Chenchiah

Nicolas Dirr

Sergio Faria

Daniel Feltham

Mats Gyllenberg

Kolumban Hutter

Faustino Maestre Caballero

Stefan Müller

Walter Pedra

Luca Placidi

Marcello Ponsiglione

Patrick Shipman

Bob Svendsen

Organizers

Sergio H. Faria

Administrative Contact

Katja Bieling