Project : tropics
Section: New Results
Keywords : optimization, mesh adaptation, adjoint.
The team continued his reflexion on the use of smart mesh adaptation to increase the convergence order of a series of adapted computation . The innovative derivation of the adjoint and the resolution of the related optimum problem can be used in a slightly different context than shape design namely, mesh adaptation. This will be possible if we can map the mesh adaptation problem into a differentiable optimal control problem. To this end, we have introduced a new methodology that consists in setting the mesh adaptation problem under the form of a purely functional one: the mesh is reduced to a continuous property of the computational domain, the continuous metric, and we minimize a continuous model of the error resulting from that continuous property. Then the problem of searching an adapted mesh is transformed in the research of an optimal metric.
In the case of mesh interpolation minimization, the optimum is given by a close formula and gives access to a rather complete theory demonstrating that second order accuracy can be obtained on discontinuous field approximation .
In the case of adaptation for Partial Differential Equations, an Optimal Control is obtained. It involves a state equation and the optimality is expressed in terms of an adjoint state that can be derived by AD. In our first prototypes, the one-shot-SQP algorithm has been applied successfully, . This will be the focus of a cooperation with project-team GAMMA at INRIA-Rocquencourt (Paul-Louis George, Frédéric Alauzet) for the INRIA contribution to the HISAC IP European project in assocoation with 30 other partners in aeronautics. A thesis will start next year in cooperation with the SMASH project-team on 3D anisotropic mesh adaption.