Introduction to Optimization for Problems with Distributed Parameters. Mathematical Statements Reflecting the Physical Effects

The hierarchical HSP-VAT governing equations for medium or material resulting from the VAT based analysis are mostly the parabolic partial integro-differential equations (PPDE), also they are nonlinear and have additional integral and integro-differential terms. Meanwhile, the great number of optimization problems for Heterogeneous and Scaled media as, for example, for diffusion or electrical transport can be formulated using VAT governing equations. The problems have attractive advantages having the physical as well as morphological (structural) requirements at the same time. This kind of models seems not addressed in research, because the type of HSP-VAT based models were not presented before and there were not need to study the problems.

Models that are ordinary surfaced in the control and optimization theory are usually based on the technology requests and up to now have Homogeneous physics mathematical formulation including problems that should be formulated in an area of heterogeneous medium transport as the Heterogeneous ones. We started this field in 80s, while studying the optimization problems for a variety of tasks in chemical engineering and environmental engineering involving the HSP-VAT based formulations of designs that are more complex than traditional Homogeneous models and designs which require the new optimization simulation techniques and a complete research to be done for this class of equations, including analysis of necessary conditions and existence of optimal control, as well as developing computational methods for solving various optimal control problems.

2011. Insertion at this time was required by the needed comments on some declarations which better were not done - I mean by my collaborator on the optimization of scaled thermal physics problems, see on this in - "Announcements. Announcement #2:"

I have never thought that I would write some definitions of my collaborators with such straight language, but I should, thinking about claims on the leading role of some of my collaborators in the development, of HSP-VAT sciences. To this web section particular science - Optimization and Control for Heterogeneous scaled disciplines, for example, by prof. I.Catton on the development of Optimization science for Heterogeneous scaled Heat Exchangers physics. A person to whom I conveyed in the 1990s along with the grad students the conceptual options, mathematical tools, commencing of problems' formulation, solution of some problems for HEs, now is saying about his leading role in all of this?

That does have to be answered in a proper way - who was commencing this discipline and what has been done. So, below, when I talk about any developments done firstly - that means I was the person who has done the leading role and major actions. This should be clear for the future scripts.

There were firstly developed the main definitions and understanding of optimization problems for scaled problems in heterogeneous hierarchical media. Formulated the optimization p roblem statements for - 1) Heat Exchangers; 2) Semiconductor Chip Heat Sinks. There are a number of traditional techniques applied to the optimization of heat transfer devices such as heat sink enhanced surfaces or heat exchangers. There are, however, no methods or mathematical studies devoted to optimization of hierarchical heat transfer devices. Design optimization procedures for transport in porous structures and enhanced heat transfer surfaces are formulated and developed.

In 90s we firstly formulated the definitions of Class and Absolute Optimization in the hierarchical heterogeneous scaled medium. Were developed the methods of solution of distributed optimization problems in heterogeneous media. Has been developed heterogeneous two and three scale volume averaging theory (VAT) volumetric heat dissipation device (VHDD) models and mathematical methods with closure of heterogeneous terms in optimization governing equations. Simulated few canonical and test morphology designs on the lower level of the heat sink heat transport modeling. Volume averaging theory (HSP-VAT) was used to optimize heat transport and flow resistance within a specified heterogeneous hierarchical media with multiple scales. VAT has been advanced to the extent that a statement about the absolute upper limit can be made and is being used in a practical application. Practical applications include heat generation in solid state devices and semiconductor chips. The work presents itself as a heterogeneous distributed parameters optimization problem.

Among few applications of hierarchical matter and media description the heat generation in solid state devices and semiconductor chips is addressed with the aim to reduce the rate of heat generation and optimize its dissipation.

Were developed First Elements of Statistical Theory of Design of Experiments Optimization (DOE) Method for Hierarchical Transport Problems. Ran the simulation experiments and obtained preliminary results. Obtained the results which allow to support the statement about achievable the absolute upper limit of heat dissipation effectiveness on upper scale of VHDD in outlined morphological classes as, for example:

  • a) 2D longitudinal or transverse regular rib fins; or

  • b) 3D transversely (cross flow) streamlined regularly located circular cross section arbitrary in 2D( x,z) pin fins; or

  • c) 3D arbitrary shape regularly located pin fins; or

  • d) 3D random location circular cross section arbitrary in 2D pin fins; etc.

    Mathematical formulation of a hypothetical heat transfer surface with a priori unknown heat transfer enhancing elements has been developed using a two scale description based on volume averaging theory. Second order turbulent model equation sets based on VAT are used to determine turbulent transport and two temperature diffusion in a non-isotropic porous media and inter-phase exchange at a rough wall. Though several different closure models for the source terms for spatial uniform, non-uniform, non-isotropic highly porous layers have been successfully developed, quite different situations arise when attempting to describe processes occurring in irregular, random or even unknown morphologies. After simplification by assuming regularity of the spatial morphology, this problem is still has a large number of optimization space dimensions. In a laminar heat transfer region, the problem is 6 to 8-D and in turbulent it is 8 to 9-D.

    As an illustration was elaborated the method of hierarchical optimization of two- and three scale heat transport in a heterogeneous media of a semiconductor heat sink. It is shown how traditional governing equations developed using rigorous HSP-VAT methods can be used to optimize surface transport processes in support of heat transport technology. The difficulty in treating a multiparameter (more than 3 ) problem, even linear, known to be very difficult to overcome using a parameter sorting process. The combination of VAT based equations and the theory of statistical design was used to effectively begin treating 6D or 8D optimization volumes.


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