A mixed adjoint formulation for incompressible turbulent problems

Abstract

A design methodology based on a mixed adjoint a p p r o a c h for ow problems governed by the Incompressible Turbulent N a vier Stokes equations is deduced and tested. The main feature of the algorithm is that instead of solving an exact discrete adjoint equation, it solves a fastconverging low-order adjoint formulation, saving an important amount of CPU time, and giving a smoothed approximation to the real gradient. It has been shown that this type of smoothed gradients is very convenient to avoid possible diverging cycles in the whole design process, and to reduce the total optimization cost. The boundary conditions for the discrete adjoint f o r m ulation are inferred at the continuous level. In this way, the formulation is mixed. Furthermore, the methodology is general in the sense that it does not depend on the geometry representation, and all the gridpoints on the surface to be optimized can be chosen as design parameters. The partial derivatives of the ow equations with respect to the mesh movements are computed by nite di erences. Hence, this computation is independent of the numerical scheme employed to obtain the ow solution and of the mesh type. Once the sensitivities and the direction of movement h a ve been computed, the new solid surface is obtained with an improved pseudo-shell approach i n s u c h a way that local singularities, which can degrade or inhibit the convergence to the optimal solution, are avoided. Moreover, this surface parametrization allows to impose geometrical restrictions in a very easy manner. The volume mesh is updated to x the new boundary using an innovative l e v el approach for the highly stretched elements close to the solid boundary (boundary layer mesh), and a quasi-incompressible elastic movement scheme for the rest of them. Such t ype of combined mesh movement a lgorithm allows to compute the sensitivity c o n tribution of the interior mesh points by using nite di erences in a very fast manner, and avoids expensive remeshing procedures during the whole design process. The method-ologies can deal with multi-objective function problems. Some numerical examples are presented to demonstrate the methodology behaviour.

Aerodynamic Shape Optimization Using First and Second Order Adjoint and Direct Approaches

“…In this paper, the partial discrete adjoint solution is also introduced, which allowed to take into account, in an incomplete manner, all the adjoint contribution to the sensitivities. The extension of this approach to turbulent flow is found in [95]. Similarly, several discrete adjoint algorithms have been presented for handling mesh sensitivities.…”

“…There is another similar category of methods, which are named as incomplete gradient methods, [89][90][91][92][93][94][95], in which the gradient values are not exact but less expensive to compute. The simplest way to compute an approximate gradient is to omit the second term on the r.h.s.…”

An adjoint‐based design methodology for CFD problems

“…The main steps required for the evaluation of gradients based on adjoint variables are well known (Elliott and Peraire, 1998; Jameson, 1988, 1995; Mohammadi and Pironneau, 2001; Soto and Löhner, 2001a, b, 2002; Soto et al , 2002), and therefore only a summary is given here. A variation in the objective function I and the PDE constraint R result in: Equation 8 Equation 9 We can now introduce a Lagrange multiplier Ψ to merge these two expressions: Equation 10 After rearrangement of terms this results in: Equation 11 This implies that if we can solve: Equation 12 the variation of I is given by: Equation 13 The consequences of this rearrangement are profound:the variation of I exhibits only derivatives with respect to β , i.e.…”

“…While the experiment (or stand alone CFD run) only measures the performance of the product “as is”, numerical methods can also predict the effect of changes in the shape of the product. This has led, over the last decade, to a large body of literature on optimal shape design (Anderson and Venkatakrishnan, 1997; Drela, 1998; Dreyer and Matinelli, 2001; Elliott and Peraire, 1997, 1998; Gumbert et al , 2001; Jameson, 1988, 1995; Korte et al , 1997; Kuruvila et al , 1995; Li et al , 2001; Medic et al , 1998; Mohammadi, 1997; Mohammadi and Pironneau, 2001; Nielsen and Anderson, 1998; Reuther et al , 1999; Soto and Löhner, 2001a, b, 2002; Soto et al , 2002). In order to compare the merit of different designs, a function I is defined.…”

Using an Adjoint Approach to Eliminate Mesh Sensitivities in Computational Design

“…[17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35] Adjoint methods may be implemented in either a continuous or discrete context, depending on the order in which the differentiation and discretization operations are performed. The relative merits of the two approaches are the subject of much debate in the literature; one such difference is the focus of the current work.…”