Controllability Analysis and Optimization of Hydrocannon Nozzle Shape Based on Direct Extreme Approach
Year: 2025
Type: article
Abstract: Introduction. Controllability analysis is a required stage for the correct formulation and solution of any optimal control problem. This problem becomes specifically relevant in the context of optimizing systems with distributed parameters, which are described by partial difference equations. Such problems include the considered problem of optimization of the shape of the nozzle of a hydrocannon. The optimal nozzle should provide the maximum value of the functional expressed through the average force of the impulse of the jet of a hydrocannon. The relevance of this research is due to the lack of a unified approach to the analysis of controllability of systems with distributed parameters, which complicates the correct formulation and solution of optimization problems. In particular, previous attempts to solve the problem of hydrocannon nozzle optimization using classical variational calculus were unsuccessful due to ignoring aspects of controllability. The objective of this research was to apply a new approach proposed by V.K. Tolstykh to controllability analysis to solve the problem of optimal design of the shape of a hydrocannon nozzle. Materials and Methods. The research method used was controllability analysis based on the Tikhonov conditional correctness of the inverse problem. This approach allowed us to identify the conditions for the existence of the gradient of the objective functional and construct a regularization of the solution to the inverse problem using adaptive gradient methods. It was of current interest for multiextremal problems, including the problem of the optimum nozzle shape. It was solved by a direct extreme approach in the form of direct maximization of the objective functional based on its gradient. In the process of research, a nonlinear, quasi-one-dimensional mathematical model of isentropic water flow in a hydrocannon nozzle was used. The flow was considered inviscid, compressible, and subsonic. Results. As part of the research, controllability conditions were obtained that allowed us to radically simplify the problem of optimizing the shape of the hydrocannon nozzle. It was found that in order to correctly determine the gradient of the objective functional, it was required to narrow the solution area of the conjugate problem to a small rectangular area. The use of adaptive gradient methods with satisfactory step factors provided for the regularization of the solution. For the first time, two optimum shapes of the hydrocannon nozzle were found. The first shape provided a local maximum of the objective functional, the second — a global maximum of the functional with a restriction on the expansion of the nozzle. Discussion and Conclusion. The results obtained show that it is impossible to perform a directed search for an optimal solution using the Frechet derivative without taking into account controllability conditions. The first proposed approach, in combination with the desired adaptive gradient optimization methods, allowed us not only to correctly formulate the optimization problem, but also to find optimal nozzle shapes that provided the maximum average pulse force of the ultrajet. In some cases, for the stability of the solution, it was necessary to introduce expansion limitation of the nozzle beyond the barrel of the hydrocannon. This made it possible to meet the requirements of the controllability theorem and guaranteed the correctness of the results obtained. The theoretical relevance of the research is in the development of controllability analysis methods for systems with distributed parameters, which creates new opportunities for solving similar problems in other areas. The research results can be used to optimize devices operating on the basis of pulsed jets, as well as for further study of more complex models of fluid flow.
Source: Advanced Engineering Research
Authors: V. K. Tolstykh, Yu. V. Dmitruk
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: diamond
OpenAlex ID: W4408830783