An accurate calculation of aerodynamic force coe cients for a given geometry is of fundamental importance for aircraft design. High-order spectral/hp element methods, which use a discontinuous Galerkin discretisation of the compressible Navier-Stokes equations, are now increasingly being used to improve the accuracy of flow simulations and thus the force coe cients. To reduce error in the calculated force coe cients whilst keeping computational cost minimal, I propose a p-adaptation method where the degree of the approximating polynomial is locally increased in the regions of the flow where low resolution is identified using a goal-based error estimator. We initially calculate a steady-state solution to the governing equations using a low polynomial order and use a goal-based error indicator to identify parts of the computational domain that require improved solution accuracy and increase the approximation order there. We demonstrate the cost-effectiveness of our method across a range of polynomial orders by considering a number of examples in two- and three-dimensions and in subsonic and transonic flow regimes. Reductions in both the number of degrees of freedom required to resolve the force coe cients to a given error, as well as the computational cost, are both observed in using the p-adaptive technique.
In addition to the adjoint-based p-adaptation strategy, I propose a mesh deformation strategy that relies on a thermo-elastic formulation. The thermal-elastic formulation is initially used to control mesh validity. Two mesh quality indicators are proposed and used to illustrate that by heating up (expanding) or cooling down (contracting) the appropriate elements, an improved robustness of the classical mesh deformation strategy is obtained. The idea is extended to perform shock wave r-adaptation (adaptation through redistribution) for high Mach number flows. The mesh deformation strategy keeps the mesh topology unchanged, contracts the elements that cover the shock wave, keeps the number of elements constant and the computation as e cient as the unrefined case. The suitability of r-adaptation for shock waves is illustrated using internal and external compressible flow problems.