In this article a new principle of geometric design for blade's surface of an impeller is provided.This is an optimal control problem for the boundary geometric shape of flow and the control variable is the surfac...In this article a new principle of geometric design for blade's surface of an impeller is provided.This is an optimal control problem for the boundary geometric shape of flow and the control variable is the surface of the blade.We give a minimal functional depending on the geometry of the blade's surface and such that the flow's loss achieves minimum.The existence of the solution of the optimal control problem is proved and the Euler-Lagrange equations for the surface of the blade are derived.In addition,under a new curvilinear coordinate system,the flow domain between the two blades becomes a fixed hexahedron,and the surface as a mapping from a bounded domain in R2 into R3,is explicitly appearing in the objective functional.The Navier-Stokes equations,which include the mapping in their coefficients,can be computed by using operator splitting algorithm.Furthermore,derivatives of the solution of Navier-Stokes equations with respect to the mapping satisfy linearized Navier-Stokes equations which can be solved by using operator splitting algorithms too.Hence,a conjugate gradient method can be used to solve the optimal control problem.展开更多
In this paper, the geometrical design for the blade's surface in an impeller or for the profile of an aircraft, is modeled from the mathematical point of view by a boundary shape control problem for the Navier-Sto...In this paper, the geometrical design for the blade's surface in an impeller or for the profile of an aircraft, is modeled from the mathematical point of view by a boundary shape control problem for the Navier-Stokes equations. The objective function is the sum of a global dissipative function and the power of the fluid. The control variables are the geometry of the boundary and the state equations are the Navier-Stokes equations. The Euler-Lagrange equations of the optimal control problem are derived, which are an elliptic boundary value system of fourth order, coupled with the Navier-Stokes equations. The authors also prove the existence of the solution of the optimal control problem, the existence of the solution of the Navier-Stokes equations with mixed boundary conditions, the weak continuity of the solution of the Navier-Stokes equations with respect to the geometry shape of the blade's surface and the existence of solutions of the equations for the Gateaux derivative of the solution of the Navier-Stokes equations with respect to the geometry of the boundary.展开更多
基金This work was supported bythe National Natural Science Foundation of China(No.50306019,40375010,10471110,10471109).
文摘In this article a new principle of geometric design for blade's surface of an impeller is provided.This is an optimal control problem for the boundary geometric shape of flow and the control variable is the surface of the blade.We give a minimal functional depending on the geometry of the blade's surface and such that the flow's loss achieves minimum.The existence of the solution of the optimal control problem is proved and the Euler-Lagrange equations for the surface of the blade are derived.In addition,under a new curvilinear coordinate system,the flow domain between the two blades becomes a fixed hexahedron,and the surface as a mapping from a bounded domain in R2 into R3,is explicitly appearing in the objective functional.The Navier-Stokes equations,which include the mapping in their coefficients,can be computed by using operator splitting algorithm.Furthermore,derivatives of the solution of Navier-Stokes equations with respect to the mapping satisfy linearized Navier-Stokes equations which can be solved by using operator splitting algorithms too.Hence,a conjugate gradient method can be used to solve the optimal control problem.
基金supported by the National High-Tech Research and Development Program of China (No.2009AA01A135)the National Natural Science Foundation of China (Nos. 10926080, 10971165, 10871156)Xian Jiaotong University (No. XJJ2008033)
文摘In this paper, the geometrical design for the blade's surface in an impeller or for the profile of an aircraft, is modeled from the mathematical point of view by a boundary shape control problem for the Navier-Stokes equations. The objective function is the sum of a global dissipative function and the power of the fluid. The control variables are the geometry of the boundary and the state equations are the Navier-Stokes equations. The Euler-Lagrange equations of the optimal control problem are derived, which are an elliptic boundary value system of fourth order, coupled with the Navier-Stokes equations. The authors also prove the existence of the solution of the optimal control problem, the existence of the solution of the Navier-Stokes equations with mixed boundary conditions, the weak continuity of the solution of the Navier-Stokes equations with respect to the geometry shape of the blade's surface and the existence of solutions of the equations for the Gateaux derivative of the solution of the Navier-Stokes equations with respect to the geometry of the boundary.
