1.1 Mathematical Models and the Object Design Problem


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The methodology of a turbine optimal design as a complex multi-level engineering system should support the operation with diverse mathematical models, providing for each design problem communication between the neighboring subsystems levels.

One approach to turbine design with using of block-hierarchical representation consists in the transition from the original mathematical models for the subsystems and numerical methods of optimization to “all-purpose” mathematical model and general method of parameters optimization.

We will specify as original the mathematical model (OMM), which is a closed system of equations that describe the phenomena occurring in the designed object.

Regardless of the mathematical apparatus (algebraic, ordinary differential, integral, partial differential equations, etc.), OMM can be represented symbolically as follows:

Formula 1.1

where  X ⃗={x ⃗,u ⃗ };L(B ⃗,X ⃗) – the operator defining the model’s system of equations.

The parameters Y ⃗  characterize the quality indicators; B ⃗ – entering the subsystems model from adjacent levels, the operational environment of the object. Parameters X ⃗  can be either dependent, calculated by the OMM equation ( x ⃗  ) or independent, the choice of which provides the designer ( u ⃗  ). It is understood that the number of internal parameters of the object includes all internal parameters of the elements of underlying layers.

Significant simplification and unification of the subsystems description achieved by OMM approximation with a model, which we call a formal macromodel (FMM). We represent the FMM as a complete polynomial of the 2-nd degree, by which in many cases it is possible to approximate the output parameters with sufficient accuracy:

Formula 1.2

FMM parameters vector is expressed through the IMM parameters as

Formula 1.3

hence FMM may be represented symbolically as follows:

Formula 1.4

Comparing (1.4) and (1.1), we see that the FMM have no phase variables. The transformation of one model to another with a fewer number of variables or constraints, giving an approximated description of the investigated object or process compared to the initial, will be called aggregation. Thus, the FMM is aggregated with respect to (1.1).

The problems of the object’s optimal design using models (1.1) and (1.4) will be called following:

Formula 1.5
Formula 1.6

Suppose, that the problem (1.6) is solved for all possible values of the vector B ⃗  that allows you to build approximation dependencies

Formula 1.7

containing information on all kinds of optimal designs. The model (1.7) is aggregated with respect to (1.4) and (1.5). The same could be made with the OMM: by virtue of solving the equations of the model would have disappeared phase, and by optimizing – control variables. Usually, however, this task is too complex for the numerical solution.

An approximate solution can be achieved with the help of disaggregation, i.e. mapping of aggregated variables in the model space of OMM. Substituting (1.7) to (1.1), we obtain:

Formula finalwhere our   and Y  – solution of OMM.

For example, in the optimal design of turbine cascade the quality criterion is the energy loss ratio, OMM – ideal gas motion and the boundary layer on the profile equations, phase variables – flow parameters, control – profile shape, cascade spacing and others.

In practice, instead of the loss calculation OMM various empirical loss calculation methods used, which, in fact, are FMMs of form (1.7), because does not take into account information about any and just about the currently best (“optimum”) profile cascades. In this way, at higher design levels use only the information on the improved aerodynamic profiles loss ratio.

The approach described can be applied to multi-level design of complex systems.

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