Adaptive Meshfree Method And Its Application To Optimal Control Problems Constrained By N-S Equations

The optimal control problems constrained by partial differential equations are used extensively,such as curve and bridge design,arterial bypass design,aircraft airfoil flow and so on.Since the analytical solution of the problem cannot be achieved in most cases,the numerical algorithm for solving the optimal control problem has been vigorously developed.Because the meshfree method has the characteristics of no need to generate the mesh,less pre-processing workload,and avoiding some problems of the mesh-like method,it has received more and more attention in the application of the optimal control problem constrained by partial differential equations.Since the meshfree method does not depend on the mesh in the solution process,only the nodes are needed,so the node connection information is not required in the calculation process,and the calculation process will not be refreshed when the nodes are added,deleted or moved.The flexibility of this method in the calculation process is very suitable for adaptive calculation and analysis.In this paper,the optimal control problem constrained by Navier-Stokes equations is studied from the following two parts.(1)Adaptive meshfree method.Taking Poisson equation as an example,the discrete system equation is obtained by using the meshfree method,and then the error estimator based on function value and gradient is given to identify the high error region in the solution domain,and the appropriate node refinement strategy is selected.The corresponding h-type adaptive node algorithm based on gradient is obtained by combining both.The Poisson equation and Burgers equation with large gradient change are considered in the numerical examples.The results of the examples illustrate the effectiveness of the proposed method.(2)The adaptive meshfree method is applied to solve the optimal control problem constrained by partial differential equations.Firstly,the optimal control problem constrained by Navier-Stokes equations is used as the research model in this section.The sensitivity analysis results are obtained by conjugate method,which is the basis of gradient method to solve the optimal control problem;Furthermore,an adaptive meshfree method for solving optimal control problems is given based on local radial basis function and differential quadrature method;Then combined with the theoretical results in the previous part,an adaptive meshfree method is proposed;Finally,a numerical example is given to illustrate the feasibility of the proposed algorithm for solving the optimal control problem constrained by partial differential equations.