On accurate boundary conditions for a shape sensitivity equation method

Abstract

SUMMARYThis paper studies the application of the continuous sensitivity equation method (CSEM) for the NavierStokes equations in the particular case of shape parameters. Boundary conditions for shape parameters involve ow derivatives at the boundary. Thus, accurate ow gradients are critical to the success of the CSEM. A new approach is presented to extract accurate ow derivatives at the boundary. High order Taylor series expansions are used on layered patches in conjunction with a constrained least-squares procedure to evaluate accurate ÿrst and second derivatives of the ow variables at the boundary, required for Dirichlet and Neumann sensitivity boundary conditions. The ow and sensitivity ÿelds are solved using an adaptive ÿnite-element method. The proposed methodology is ÿrst veriÿed on a problem with a closed form solution obtained by the Method of Manufactured Solutions. The ability of the proposed method to provide accurate sensitivity ÿelds for realistic problems is then demonstrated. The ow and sensitivity ÿelds for a NACA 0012 airfoil are used for fast evaluation of the nearby ow over an airfoil of di erent thickness (NACA 0015).

Verified predictions of shape sensitivities in wall-bounded turbulent flows by an adaptive finite-element method

“…A methodology to produce accurate boundary flow gradients was recently proposed by Duvigneau and Pelletier [26].I t consists in a least-squares reconstruction, using Taylor-series as basis functions, that is constrained to satisfy the flow boundary conditions through the method of Lagrange multipliers. It has been shown that one must use at least 3 rd order Taylor-series defined over patches of four layers of neighboring elements to achieve sufficient accuracy for the Dirichlet sensitivity boundary conditions and Taylor-series expansion of degree 6 on a 8-layer patch when Neumann conditions are imposed (see Ref.…”

“…It has been shown that one must use at least 3 rd order Taylor-series defined over patches of four layers of neighboring elements to achieve sufficient accuracy for the Dirichlet sensitivity boundary conditions and Taylor-series expansion of degree 6 on a 8-layer patch when Neumann conditions are imposed (see Ref. [26] for details). For this work, we have extended the constrained Taylor-series least-squares procedure to wall-bounded turbulent flows.…”

Nodal velocity derivatives of finite element solutions: The FiND method for incompressible Navier–Stokes equations

“…First, we solve a Poiseuille flow for which the velocity changes only in the direction transverse to the flow. Then we consider a manufactured solution [32] behaving like a two-dimensional boundary layer. This problem is solved on both uniform and non-uniform meshes.…”

“…The next verification test case is a manufactured solution mimicking the flow along a flat plate [32]. The velocity field is given by (1), (2).…”

“…Duvigneau and Pelletier have recently proposed a Taylor-series-based constrained least-squares procedure to extract high-accuracy boundary derivatives that are compatible with the imposed Dirichlet conditions [32]. Improvements due to this new approach were demonstrated on examples of optimal design of thermal systems [33] and in the fast evaluation of flows on nearby geometries [34].…”