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Adjoint-based error estimation for adaptive Petrov-Galerkin finite element methods: Application to the Euler equations for inviscid compressible flows

par D'Angelo, Stefano
Président du jury Degrez, Gérard
Promoteur Deconinck, Herman
Co-Promoteur Ricchiuto, Mario
Publication Non publié, 2015-03-24

Thèse de doctorat

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Résumé :

The current work concerns the study and the implementation of a modern algorithm for a posteriori error estimation in Computational Fluid Dynamics (CFD) simulations based on partial differential equations (PDEs). The estimate involves the use of duality argument and proper consistent discretisation of primal and dual problem.A key element is the construction of the adjoint form of the primal differential operators where the data term is a quantity of interest depending on the application. In engineering, this is typically a physical functional of the solution. So, by solving this adjoint problem, it is possible to obtain important information about local sensitivity of the error with respect to the current target quantity and thereby, we are able to perform an a posteriori error representation based on adjoint data. Through this, we provide local error indicators which can drive an adaptive meshing algorithm in order to optimally reduce the target error. Therefore, we first derive and solve the discrete primal problem in agreementwith the chosen numerical method. According to consistency and compatibility conditions, we can use the same discretisation for solving the adjoint problem, simply by swapping the position of the unknowns and the test functions in the linearised variational operator. Remembering that the corresponding adjoint problem always remains linear, the computational cost for obtaining these data is limited compared to the effort needed to solve the primal nonlinear problem.This procedure, fully developed for Discontinuous Galerkin (DG) and Finite Volume (FV) methods, is here for the first time applied in a fully consistent way for Petrov-Galerkin (PG) discretisations. Differently from the latter, the biggest issue for the PG method becomes the need to handle two different functional spaces in the discretisation, one of which is often not even continuous. Stabilized finite element schemes such as Streamline Upwind (SUPG), bubble stabilized (BUBBLE) Petrov-Galerkin and stabilized Residual Distribution (RD) have been selected for implementation and testing. Indeed, based on local advection information, these schemes are naturally more suitable for solving hyperbolic problems and therefore, interesting alternatives for fluid dynamics applications.A scalar linear advection equation is used as a model problem for convergence rate of both primal and adjoint solutions and target quantity. In addition, it is also applied in order to verify the accuracy of the adjoint-based a posteriori error estimate. Next, we apply the methods to a complete collection of numerical examples, starting from scalar Burgers’ problem till 2D compressible Euler equations. Through suited quantities of interest, we illustrate aspects of the adjoint mesh refinement by comparing its efficiency with respect to the standard a posteriori error estimation.