University of Cambridge > > Fluid Mechanics (DAMTP) > An optimization framework for adaptive PDE solutions applied to fluid dynamics

An optimization framework for adaptive PDE solutions applied to fluid dynamics

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Improving the autonomy, efficiency, and reliability of computational fluid dynamics algorithms has become increasingly important as powerful computers enable characterization of the input-output relationship of complex PDE -governed processes. This work is a step towards the development of a versatile PDE solver that accurately predicts output quantities of interest to user-prescribed accuracy in a fully automated manner. Given a discretization and a localizable error estimate, the framework iterates toward a mesh that minimizes the error for a given number of degrees of freedom by considering a continuous optimization problem of the Riemannian metric field. The adaptation procedure consists of three key steps: sampling of the anisotropic error behavior using element-wise local solves; synthesis of the local errors to construct a surrogate error model based on an affine-invariant metric interpolation framework; and optimization of the surrogate model to drive the mesh toward optimality. The combination of the framework with a discontinuous Galerkin discretization and an a posteriori output error estimate results in a versatile PDE solver for reliable output prediction.

The versatility and effectiveness of the adaptive framework are demonstrated in a number of applications. First, the optimality of the method is verified against anisotropic polynomial approximation theory in the context of L2 projection. Second, the behavior of the method is studied in the context of output-based adaptation using advection-diffusion problems with manufactured primal and dual solutions. Third, the framework is applied to the steady-state Euler and Reynolds-averaged Navier-Stokes equations. The results highlight the importance of adaptation for high-order discretizations and demonstrate the robustness and effectiveness of the proposed method in solving complex aerodynamic flows exhibiting a wide range of scales. Fourth, fully-unstructured space-time adaptivity is realized, and its competitiveness is assessed for wave propagation problems. Finally, the framework is applied to enable spatial error control of parametrized PDEs, producing universal optimal meshes applicable for a wide range of parameters.

This talk is part of the Fluid Mechanics (DAMTP) series.

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