### Multilevel optimization in details

##### Our multilevel procedure

The typical situation while solving a problem of optimizing a complex engineering system is that the user has several tools of various degree of fidelity to perform the analysis. These tools differ in their level of complexity of modeling the actual physical phenomena and in their level of numerical accuracy. The high-fidelity (true) tools could be represented by detailed non-linear mathematical models of the researched systems or even by the experimental samples of such systems. However, the use of such tools in optimization is associated with significant time expenditures. The low-fidelity (surrogate) models could also be employed in optimization search, but the reliability of the obtained results can be rather low. Therefore, the methods based on a combination of various fidelity analysis tools are widely practiced. Our multilevel optimization procedure is based upon the adaptive use of analysis tools of various levels of complexity. The intention is to minimize the use of complicated time-consuming tools for the analysis.

##### The basic scheme of multilevel optimization

The simplified scheme of work for the multilevel optimization procedure can be represented as follows.

1. Solving the optimization problem using a surrogate model. For this purpose, the method of indirect optimization based on the self-organization (IOSO) is used. This method allows finding the single solution for single-objective optimization or the Pareto-optimal set of solutions for multi-objective problems.
2. For the obtained solution the indicators of efficiency are updated using the high-fidelity analysis tools.
3. The adjustment of a current search region is performed.
4. The adjustment of the surrogate model is performed. Depending upon the particular features of the applied mathematical simulation, the adjustment procedure can performed using several approaches. One such approach involves constructing non-linear corrective dependencies. This includes evaluation of the results obtained with different fidelity analysis tools. The other possible approach is application of nonlinear estimation of surrogate model internal parameters.

The information stored during the search is used to improve the surrogate model. After the analysis procedure terminates, one can monitor the evaluated response functions. However, both adjusted model and response functions are correct not for the entire initial search area but only for a certain neighborhood of the obtained optimal solution. This ensures purposeful improvement of approximating properties only in the area of the optimal solution. Such a procedure noticeably reduces the computation effort of solving complex optimization problems.

#### Multilevel Optimization of the Multistage Axial Compressor Design

Problem features:
 variable parameters: inlet an exit angles of 7 blade rows in 3 sections by radius (42 variables) criteria: the efficiency at two operating modes (2 criteria) constraints: stall margins at two operating modes; the constraint by criteria computability high-fidelity tool: quasi-3D model with viscosity effects simulation low-fidelity tool: 2D axis symmetric model capable to be identified

Result:

#### The Multilevel Optimization of Temporal Control Laws of STOVL Aircraft Power Plant for Short Take-Off

 variable parameters: the temporal control laws of 10 power plant independent controllable elements and the aircraft pitch criteria: the take-off run length; the take-off fuel expenditures (2 criteria) constraints: the maximum rotors rotation rates and the turbine inlet temperature; the minimum compressors surge margins; the flight safety; the criteria computability high-fidelity tool: the quasi-steady model of the power plant in the system of the aircraft. The aircraft was modeled as the material point low-fidelity tool: the same model with simplified procedures of differential equations numerical integration and the computation of thermodynamical processes of the power plant
Result:

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