شکل تجزیه و تحلیل حساسیت برای معادلات ناویه استوکس تراکم پذیر توسط روش گالرکین ناپیوسته

This paper describes the formulation of adjoint-based sensitivity analysis and optimization techniques for high-order discontinuous Galerkin discretizations applied to viscous compressible flow. The flow is modeled by the compressible Navier–Stokes equations and the discretization of the viscous flux terms is based on an explicit symmetric interior penalty method. The discrete adjoint equation arising from the sensitivity derivative calculation is formulated consistently with the analysis problem, including the treatment of boundary conditions. In this regard, the influence on the sensitivity derivatives resulting from the deformation of curved boundary elements must be taken into account. Several numerical examples are used to examine the order of accuracy (up to p=4p=4) achieved by the current DG discretizations, to verify the derived adjoint sensitivity formulations, and to demonstrate the effectiveness of the discrete adjoint algorithm in steady and unsteady design optimization for both two- and three-dimensional viscous design problems.

The use of computational flow simulations in conjunction with numerical optimization techniques has become an indispensable tool in modern aerodynamic designs [1], [2], [3], [4] and [5]. This is not only because of the economic benefits from eliminating massive wind tunnel testing before a final design is obtained, but also the improved accuracy and efficiency of the optimization method in attaining a target-guided design process. While the majority of such design work has relied on second-order finite-volume methods, the need for a higher-order algorithm has become apparent due to the difficulties in delivering asymptotically grid converged solutions [6] and [7]. High-order discontinuous Galerkin (DG) methods [8], [9], [10], [11], [12] and [13] have emerged as a competitive alternative in solving a variety of computational fluid dynamics problems. Moreover, the robustness and efficiency of the DG methods have been improved significantly in the past decade. This motivates the investigation of high-order DG methods in applications of aerodynamic design optimization. As an extension of previous work [14] on sensitivity analysis for inviscid flows, this paper continues on the study of an adjoint sensitivity algorithm for high-order DG discretizations in compressible viscous flow, focusing on both steady and unsteady design problems in two and three space dimensions. To numerically solve viscous flow problems governed by the compressible Navier–Stokes (NS) equations, the DG discretization of the viscous flux terms must be carried out, together with the subsequent adjoint problem. The current work employs an explicit symmetric interior penalty (SIP) method, described in references [10], [15] and [16], due to the fact that the scheme is capable of preserving optimal error convergence rates for the flow solution and is also formulated to be dual consistent [10] for the adjoint solution. For unsteady viscous flow problems, a second-order backward difference formula (BDF2) [17] and [18] is used to avoid restrictions on the selection of time-step sizes. Since the primal flow and adjoint solutions are required in a typical optimization iteration, the efficiency of a design process is related closely to the solution strategy of these solvers. To make the proposed algorithm efficient and competitive, we consider a multigrid approach [9], [17] and [19], driven by a linearized element Gauss–Seidel smoother [17] or a Generalized Minimal Residual (GMRES) algorithm [20]. It is known that the use of high-order curved boundary elements is essential for high-order schemes to deliver an overall high-accuracy solution [21] and [22]. Therefore, the deformation of curved boundary elements and computation of the resulting mesh sensitivities remain a topic of considerable importance in the representation of smoothed surface geometries and the accuracy of sensitivity derivative calculations. Due to the fact that the present paper focuses on viscous flow with small and moderate Reynolds numbers, curvilinear elements are applied only on physical boundaries, while straight-sided elements are used in the interior meshes. In this context, both linear and higher-order geometric mappings [14] must be taken into account for calculating the mesh sensitivities involving high-order curved elements. These key ingredients have been automatically included in the adjoint sensitivity formulation developed in the present work and more attention is paid on the effects of the viscous discretization terms to the mesh sensitivities. The remainder of the paper is structured as follows. In Section 2 the governing equations are introduced and the spatial discontinuous Galerkin discretizations together with an implicit time-integration scheme are formulated. Section 3 describes the mesh parameterization and the formulation of the adjoint-based sensitivity derivative calculation. Several numerical examples are presented in Section 4 for demonstrating the accuracy of the current DG schemes and the performance of the adjoint techniques in steady and unsteady aerodynamic shape optimization. Finally, Section 5 concludes the current work and discusses the future work.

A discrete adjoint approach for high-order discontinuous Galerkin discretizations is developed in the present work and aerodynamic design optimization problems are investigated for steady and unsteady viscous flows in both two and three space dimensions. The evaluation of sensitivity derivatives for meshes involving curved boundary elements requires accounting for the mesh sensitivities arising from both mesh points and additional surface quadrature points. Moreover, the formulation of the discrete adjoint system must be consistent with the analysis problem since the former is based on linearization and a transpose operation to the forward linear problem. A similar deformation strategy is implemented for the additional surface quadratures as well as for standard surface grid points to ensure a smooth and accurate representation of the new surface geometry, and the current approach has shown success in design optimization for low Reynolds number viscous flow. Designed order of accuracy is achieved by the DG discretizations (up to p=4p=4) for the two- and three-dimensional compressible NS equations and the present work also shows capability of the current DG–NS solver in delivering smooth and accurate viscous flow solution. Since the current paper focuses on flow problems with low and moderate Reynolds numbers, high-order curved elements have been applied only on physical boundaries. In order to capture flow features in the boundary layer for higher Reynolds-number flow, highly stretched elements may be required, possibly along with curved interior elements in the boundary layer. Further work will incorporate these effects and implement a more sophisticated deformation method to handle design optimization problems in high Reynolds number viscous flow.