This paper presents a method for concurrent mesh and polynomial-order adaptation with the objective of direct minimization of output error using a selection process for choosing the optimal refinement option from a discrete set of choices that includes directional spatial resolution and approximation order increment. The scheme is geared towards compressible viscous aerodynamic flows, in which various solution features make certain refinement options more efficient compared to others. No attempt is made, however, to measure the solution anisotropy or smoothness directly or to incorporate it into the scheme. Rather, mesh anisotropy and approximation order distribution arise naturally from the optimization of a merit function that incorporates both an output sensitivity and a measure of the computational cost of solving on the new mesh. The method is applied to output-based adaptive simulations of the laminar and Reynolds-averaged compressible Navier-Stokes equations on body-fitted meshes in two and three dimensions. Two-dimensional results show significant reduction in the degrees of freedom and computational time to achieve output convergence when discrete choice optimization is used compared to uniform h or p refinement. Threedimensional results show that the presented method is an affordable way of achieving output convergence on notoriously difficult cases such as the third Drag Prediction Workshop W-1 configuration.

SUMMARYOver the last decade, the adjoint method has been consolidated as one of the most versatile and successful tools for aerodynamic design. It has become a research area on its own, spawning a large variety of applications and a prolific literature. Yet, some relevant aspects of the method remain relatively less explored in the literature. Such is the case with the adjoint boundary problem. In particular for Euler flows, both fluid dynamic and adjoint equations entail complementary Riemann problems, and these yield boundary conditions that are fully consistent with well‐posedness. In the literature, this approach has been pursued solely in terms of Riemann variables. This work formulates the adjoint boundary problem so as to correspond precisely to that imposed on the flow, as it is given in terms of primitive variables. Test results have shown to be in agreement with the traditional approach for external flow problems. Copyright © 2012 John Wiley & Sons, Ltd.

This paper presents the results obtained with the XFlow solver for the Fifth Drag Prediction Workshop. The discontinuous Galerkin finite element method is used for the spatial discretization of the Reynolds-Averaged Navier-Stokes (RANS) equations with a modified version of the Spalart-Allmaras (SA) turbulence model. Drag convergence is sought via mesh adaptation driven by an adjoint-weighted residual method. We present results for the drag polar of the NACA 0012 airfoil under subsonic flow conditions and for the Common Research Model (CRM) wing-body geometry under transonic flow conditions and fixed lift. The angle-of-attack that yields the desired lift is obtained via a Newton solve and a lift adjoint. We discuss how this lift constraint adds an additional term to the drag error estimate.

SummaryThis paper addresses the problem of finding a stationary point of a nonlinear dynamical system whose state variables are under inequality constraints. Systems of this type often arise from the discretization of PDEs that model physical phenomena (e.g., fluid dynamics) in which the state variables are under realizability constraints (e.g., positive pressure and density). We start from the popular pseudo‐transient continuation method and augment it with nonlinear inequality constraints. The constraint handling technique does not help in situations where no steady‐state solution exists, for example, because of an under‐resolved discretization of PDEs. However, an often overlooked situation is one in which the steady‐state solution exists but cannot be reached by the solver, which typically fails because of the violation of constraints, that is, a non‐physical state error during state iterations. This is the shortcoming that we address by incorporating physical realizability constraints into the solution path from the initial condition to steady state. Although we focus on the DG method applied to fluid dynamics, our technique relies only on implicit time marching and hence can be extended to other spatial discretizations and other physics problems. We analyze the sensitivity of the method to a range of input parameters and present results for compressible turbulent flows that show that the constrained method is significantly more robust than a standard unconstrained method while on par in terms of cost. Copyright © 2015 John Wiley & Sons, Ltd.

This paper presents a mesh adaptation scheme for direct minimization of output error using a selection process for choosing the optimal refinement option from a discrete set of choices. The scheme is geared for viscous aerodynamic flows, in which solution anisotropy makes certain refinement options more efficient compared to others. No attempt is made, however, to measure the solution anisotropy directly or to incorporate it into the scheme. Rather, mesh anisotropy arises naturally from the minimization of a cost function that incorporates both an output error estimate and a count of the additional degrees of freedom for each refinement option. The method is applied to output-based adaptive simulations of the laminar and Reynolds-averaged compressible Navier-Stokes equations on body-fitted meshes in two and three dimensions. Two-dimensional results for laminar flows show a factor of 2-3 reduction in the degrees of freedom on the final adapted meshes when the discrete choice optimization is used compared to pure isotropic adaptation. Preliminary results on a wing-body configuration show that these savings improve in three dimensions and for higher Reynolds-number flows.