Shape Optimization In Viscous Incompressible Flows Based On Ajoint Methods

Optimal shape design for ?uids is a combination of di?erential geometry, shape opti-mization and computational ?uid dynamics (CFD). With the rapid development of CFDand enhanced performance of the computers, these problems are of great importance inCFD. The adjoint methods for two kinds of shape optimization problems controlled bythe Navier–Stokes equations have been systematically studied in this thesis. The firstorder optimality conditions of the shape optimization problems are established in virtueof the velocity method. The numerical algorithms and the corresponding simulations aregiven with applications in wing design, cannula optimization and optimization of a bodylocated in ?uid ?ow. The main work in this thesis may be summarized as follows:(1) The continuous adjoint method for the shape inverse problem governed by the Navier–Stokes equations with nonhomogeneous Dirichlet boundary conditions has been studied.A velocity-tracking type functional is taken as the cost functional. (i) The Piola materialderivative approach is utilized to prove the weak di?erentiability of the ?uid velocity withrespect to the variable domain in the case of the weak regularity of the body force. Thenthe first order optimality condition is given by the adjoint method. (ii) The first orderoptimality condition is obtained using the function space parameterization and functionspace embedding approach which can avoid the study of the state di?erentiability. Theadvantages and disadvantages of the above three approaches are discussed.(2) The continuous adjoint method for the dissipated energy minimization problem gov-erned by the Navier–Stokes equations with mixed boundary conditions has been consid-ered. The first order optimality conditions of the steady and unsteady shape optimization problems are derived by the function space parameterization approach. The two equiv-alent and explicit formulations of Eulerian derivatives of the cost functional are obtained.(3) The discrete adjoint method for the dissipated energy minimization problem governedby the Navier–Stokes equations with mixed boundary conditions has been explored. Astabilized finite element method based on two local Gauss integrations for the Navier–Stokes equations is utilized. The variation of the finite element mesh is performed bythe discrete velocity method. The discrete adjoint equations and Eulerian derivativesof the cost functional are given by the function space parameterization approach. Thisstabilized method doesn't contain any stabilization parameters and can be simply im-plemented. The workload of the derivation procedure in the continuous adjoint methodis almost the same as that in the continuous adjoint method.(4) The numerical algorithms for the shape optimization problems are discussed withits applications in cannula optimization and shape optimization of a body located inviscous ?ows. (i) The gradient type algorithms in Banach spaces, mesh adaptation andmesh moving strategy are combined to solve the ?ow optimization problems. (ii) For thesteady and unsteady shape problems with various Reynolds numbers, the large numbersof the numerical results are given in three numerical models: a simple inverse problem,cannula optimization in Haemodynamics and shape optimization of a body located in?uid ?ow. The wing design model in the steady dissipated energy minimization problemis numerically implemented. Moreover, the standard Galerkin method and stabilizedmethod are utilized in the numerical computation of shape optimization of a body lo-cated in the steady ?uid ?ow. Comparing the numerical results for various Reynoldsnumbers, it shows that the resulted optimal shapes are almost the same, but the stabi-lized method can save lots of the computational costs.