Adjoint sensitivity computations for an embedded-boundary Cartesian mesh method

1 Introduction Cartesian-mesh methods [I] are perhaps the most promising approach for addressing the issues of flow solution automation for aerodynamic design problems. In these methods, the discretization of the wetted surface is decoupled from that of the volume mesh. This not only enables fast &id robust mesh generation for geometry of arbitrary cQing&eziQ:> but &o facili@$es _accezs to geoae&rj modelinp m d manipulation us in^ parametric Computer-Aided Design (CAD) tools. Our goal is to combine the automation capabilities of Cartesian methods with an eficient computation of' design sensitivities. five address this issue using the adjoint method, where the computational cost of the design sensitivities, or objective function gradients, is esseutially indepeudent of the iiunber of desi@ variables. In previous work [2, 31, we presented m accurate and efficient algorithm for the solution pf the adjoint Euler equations discretized on Cartesian meshes with embedded, cut-cell boundaries. Novel aspects of the dgoritlm included the computation of surface &ape sensitivities for triangulations based on puametric-CAD models and the linearization of the coupling between the surface triangulation and the cut-cells. The objective of the present work is to extend our adjoint formulation to problems involving general shape changes. Central to this development is the computation of volume-mesh sensitivities to obtain a reliable turbation schemes com-due to non-smooth approximation to the ex method €or addressing the issues o the nwnn-ical method and results; 2 Problem Formulation The spatial discretization of the thee-dimensional Euler equations uses a second-order accurate fkite volume method with weak imposition of boundary conditions, resulting in a discrete system of equations WQ,M) = 0 (1) is t.he discrete solution vector for all N cells of a given mesh My and R is the flux residual vector. The iufluence of a shape design variablel X, on the residuals is implicit via the computational mesh M = f [ T (X) ] , where T denotes a surface triangulation of the CAD model. Hence, the gradient of a discrete objective function J (X , M , Q) is given by 2 M. Nemec and h4. J. Aftosmis where the vector .Ict represezlts adjoint variables given by the adjoint equation, for details see [3]. The focus of this work is on the emluation of the partial derivative terms A and B in Eq. 2, which represent the difffrentiation of the objective function and residual equations with respect to design variables that alter the surface …