A Hybridized Adaptive Method for Compressible Flow Simulation

Outline

The proposed research revolves around high-order consistent, hp-adaptive finite-element methods for convection-dominated problems. More specifically, we consider hybridized schemes for the compressible Navier-Stokes equations.

As an implementation technique hybridization is a classic paradigm for dual-mixed finite-element discretization of linear elliptic equations. Hybridization of finite-element discretization has the main advantage, that the resulting set of algebraic equations has globally coupled degrees of freedom (DOFs) only on the skeleton of the numerical mesh (i.e. the edges in 2D, and faces 3D). Solving for these DOFs thus involves solution of a potentially much smaller system. This not only reduces storage requirement, but also allows faster solution with iterative solvers. Only recently has the concept been extended to other finite-element paradigms, and equations involving hyperbolic operators.

While hybridization can have a profound impact on efficiency, higher-order schemes for nonlinear convection-dominated problems face other difficulties as well. In fact, for such problems, high-order methods have not yet made a great impact on industrial applications. For example, low solution regularity, and the application to complex geometry require sophisticated adaptation techniques to exploit benefits of high-order schemes in practice. Furthermore, for compressible flow simulation, turbulence modeling must be incorporated into the hybridized finite-element paradigm. To enable the transition of hybridized schemes for compressible flow simulation to more complex engineering problems, we propose the extension of the framework to incorporate both target-based hp-adaptivity, via a discrete adjoint approach, and the hybridized discretization of suitable turbulence models for the Reynolds-Averaged Navier-Stokes (RANS) equations. Combining hybridized methods for nonlinear advection-diffusion systems with adjoint-based optimal control techniques has, to the best of my knowledge, not previously been attempted. The same is true for high-order hybridized RANS models. This is basis of the research outlined below.

The main milestones for the doctoral research phase are

  • Formulate an adjoint-consistent, hybridized discretization of the RANS equations
  • Devise, implement, and validate target-based mesh-adaptivity for the compressible Navier-Stokes equations and RANS equations via discrete adjoint approach
  • Assess the efficiency of the overall method, in comparison with a (non-hybridized) Discontinuous Galerkin method

Target-based Adaptivity

In the first phase of the doctoral research, focus is on target-based mesh-adaptive methods. Target-based adaption aims to allocate computational resources, for example local mesh resolution, and possibly polynomial degree of approximation, in such a way that certain output functionals are computed in an optimal fashion. This can be achieved by using adjoint methods for the computation of the sensitivities with respect to the target. In compressible flow applications, such target functionals are often given by lift and drag coefficients. The adjoint-based sensitivity approach has been investigated from a quite general theoretical perspective for hybridized schemes in a previous dissertation. Based on this work, the extension of the current hybridized Navier-Stokes solver to target-based mesh adaptivity is the next imminent task. Using an adjoint consistent hybridized formulation of the compressible Navier-Stokes equations, a discrete adjoint approach is used to drive the mesh adaptivity. Target-based p-adaptivity is addressed in a concurrent thesis (Aravind Balan). Ultimately the integration of both mesh adaptation and p-adaptive method into a common codebase is needed, with contributions expected from the present project.

Target-based Adaptivity

Hybridized Discretization of the RANS Equations

The adaptive scheme for the Navier-Stokes equations is augmented with turbulence modeling via the RANS approach. To this end, a suitable RANS model must be formulated in a hybridized, adjoint consistent manner, to allow incorporation into the discrerete adjoint framework. While the implementation must be general enough to allow extension to general n-equation models, initial tests will be performed with one- and two-equation models that are traditionally used for applications in transonic external aerodynamics (e.g. the Spalart-Allmaras model or the k-ω model).