I. We intend to develop particular mechanisms of collaboration between the center members, organize and carry out a vacation term on "Nonlinear theoretical, numerical and computer analysis in the continuum physics".
Date of the achievement of the partial goal: December 12, 2006
Results of the partial goal:
Final form and submission of results:
II. By a coordination of contemporary theoretical, numerical and computer approaches analyze systematically heat-conductive and deformation processes in compressible fluids, incompressible substances of the fluid type and in elastic matters.
Date of the achievement of the partial goal: December 12, 2008
Results of the partial goal:
A basic model of continuum thermodynamics is based on equilibrium of matter, momentum, impetus, energy and entropy. Within this general frame, the first period of the project will be focused on research of these models describing mechanical and heat-conductive procedures in fluids, gases and solid matters including an optimal steering of such procedures and shape optimization. As an example let us name the shape optimization of the paper making machine in which the state problem leads to the generalized Navier-Stokes system. In the problems considered this will mean a complex analysis, that is, from analysis of a continuous model to its discretization to its numerical realization.
This particular goal, representing the first scientific part of the center, will be focused on improving the collaboration between individual subjects in the center. It can be expected that the discussion between active members of the team on seminars and during lectures of the invited scientists from abroad will result in new conjectures for solutions of various particular questions. These conjectures will often lead to new observations which will deserve publication in international academic literature. The final form of such conjectures will be achieved in PhD. and diploma theses.
In the collaboration with K.R. Rajagopal and other experts in the field of process modeling of complicated matters, where the research is carried out within a general frame which allows certain selection) or with members of other centers within the project HYKE (analysis of hyperbolic conservation laws) and other partners from abroad will be developed the existing collaboration not only in the research itself but also in the exchange of postdocs.
To this topic, half-yearly courses by experts and lecturers from abroad will be organized, and stays of postdocs from abroad will be realized.
Final form and submission of results of the particular goal:
III. By a coordination of contemporary theoretical, numerical and computer approaches analyze processes of interaction of several substances. Carry out courses and research visits of experts in this field from abroad.
Date of the achievement of the partial goal: December 12, 2009
Results of the partial goal:
This second research period will focus on problems concerning interaction of one continuum substance with another one.
To this area belong in the first place appropriate choices of boundary values that represent interaction of a substance with another one which encompasses it. The research will focus on theoretical, numerical and computer analysis of the influence of boundary values on qualitative properties of deformation (stability, occurrence of singularities etc.).
Another closely connected research field will be the theoretical analysis and numerical realization of mathematical models which describe the behaviour of a system consisting of several deformable bodies in mutual connection with a possible effect of friction on the contact varieties. We will aim at the study of contact problems of elastic bodies with various models of friction.
Another research area, connected with the above-mentioned problems, is solution of partial differential equations in time-dependent areas. This field involves in particular interaction of flowing liquids and bodies with application in both outer and inner aerodynamics, verification of an influence of wind on stability of civil engineering constructions, flux of a liquid containing systems of mass particles or blowholes and problems from biology and medicine (flux of liquids in elastic pipes etc.). We expect to develop and analyze numerical methods based on application of the finite element method. A significant role will be played by algorithmization and development of fast and effective solvers for realization of a discrete problem. All these cases involve numerical realization and analysis of very complicated nonlinear dynamical systems.
Last but not least, the research will focus on models of theory of mixtures in the frames of continuum thermodynamics, which describe the mutual behaviour of one substance within another one such as leaking of fluids through a geological environment or penetration of nutrient fluids through muscles etc.
A principal attention in all fields will be paid to discretization of the considered problems and to development of effective computational techniques for their computer realization based on detailed theoretical analysis of the continuous problem. To this end, during for years, specialized lecture courses will be given, stays of postdocs in the center will be realized, diploma and PhD. theses focused on these problems will be offered, and new scientific results will be achieved. The culmination point of this research will be achieved in 2009.
Final form and submission of results of the particular goal:
IV. By a coordination of contemporary theoretical, numerical and computer approaches analyze chemical and biochemical processes in various substances.
Date of the achievement of the partial goal: December 12, 2010
Results of the partial goal:
The third period of research will follow the first two ones and will extend processes studied in substances modeled in frames of continuum physics to chemical and biochemical ones. The system will be extended on new equations describing chemical reactions. It will be also more tied up since certain coefficients of the model representing certain material properties depend on concentration of a chemical substance which occurs in the process.
Final form and submission of results of the particular goal:
V. We intend to obtain a deep understanding of passages between models based on dimensional analysis or other physical principles, to carry out lecture courses and stays of postdocs specialized in this topic.
Date of the achievement of the partial goal: December 12, 2010
Results of the partial goal:
This fourth period will penetrate the scientific activities of the center during the entire time of its existence. It will focus on study and deep understanding of connections between various levels of description in the whole hierarchy of models. We have in mind at least two entirely different hierarchies: the vertical one and the horizontal one. Vertical hierarchy of models leads from models of many particles in which the equations describe behaviour of individual atoms or molecules to kinetic models, where the description is characterized by scalar stochastic function of time, position and velocity, and from these to macroscopic models, that is, to models of continuum physics. This level is in the case of fluids and gases followed by turbulence models.
By a horizontal series of models we mean models obtained by certain physical simplifications, in which the order of such simplifications is important. For example, we can departure from the complete system for evolutionary deformations and heat-conductive processes of compressible visco-elastic substances; we can identify limits to stationary processes, to incompressible substances, to simplified constitutive relations (vanishing relaxation time), to heat-nonconductive processes, or to linear models.
The research is focused on understanding of a particular type of approximation (which phenomena it is able to describe) and mathematical properties of limiting models, which is significant not only form a purely theoretic point of view but also form the point of view of proposing of appropriate numerical structures.
Final form and submission of results of the particular goal: