Research On Adaptive Mesh Refinement Based On Local Pseudospectral Method

Optimal control is widely used in engineering technology,economic planning,transportation and other fields.With the advancement of aerospace technology,optimal control problems are developing towards non-linear,high-dimensional and complex constraints,which make the numerical methods and optimization of optimal control problems more and more important.In recent years,much attention has been paid to the good performance of pseudospectral methods in optimal control numerical methods.Pseudospectral method parameterizes state variables and control variables by means of orthogonal point interpolation,so that the variables strictly satisfy the constraints at the interpolation points,thus transforming the continuous problem into a finite-dimensional optimization problem.However,on the one hand,in the case of multiple variables and complex constraints,increasing the number of allocations will lead to a rapid increase in the size of the optimization problem and the solution time.On the other hand,in the case of non-linearity,too few collocations can result in poor quality fitting of control and state curves.The traditional global pseudospectral method cannot take into account both the quality of fit and the efficiency of solution.To overcome these shortcomings,this paper focuses on the optimal control numerical method and adaptive grid configuration.Firstly,the research and development status of optimal control numerical methods and mesh refinement strategies were introduced.In response to the existing problems in optimal control numerical theory,the main research content of this article has been determined to be the study of optimal control numerical methods based on local pseudospectral method.The first order necessary conditions for the optimal control problem were derived,the theory of symplectic algorithm and pseudospectral parameterization method were introduced,and the construction idea of the adaptive mesh refinement method in this paper was elaborated.Secondly,the numerical solution of optimal control problems based on local pseudospectral method was studied.To address the limitations of the global pseudospectral method in balancing computational efficiency and fitting quality,an adaptive mesh refinement based local pseudospectral method is proposed based on local error estimation and curvature function.This method adaptively adjusts the position and number of allocation points based on the curvature distribution of the state curve and control curve on the interval,which can maintain computational efficiency while improving fitting quality.Numerical simulations have verified the applicability and effectiveness of the method.By repeatedly iterating,the fitting quality can be gradually improved with a small increase in the number of matching points.After the iteration is terminated,the calculation results that meet the accuracy constraints are obtained.Finally,the study investigates the numerical solution of unconstrained optimal control problems based on the symplectic preserving pseudospectral method.To further improve the computational efficiency and applicability of the algorithm,an adaptive hp mesh refinement method based on curvature density function is proposed,utilizing the characteristics of introducing local pseudospectral method in an indirect framework.This algorithm determines the interval and strategy of mesh refinement based on necessary conditions and curvature density function,which not only increases the accuracy of point matching but also saves the use of point matching.Numerical simulation shows that the computational efficiency is maintained while the computational accuracy is improved.Compared with traditional mesh refinement methods,under the same precision constraints,there are fewer coordination points and higher computational efficiency.