A Continuous Lagrangian Sensitivity Equation Method for Incompressible Flow

A continuous Lagrangian sensitivity equation method (CLSEM) is presented as a cost effective alternative to the continuous (Eulerian) sensitivity equation method (CESEM) in the case of shape parameters. Boundary conditions for the CLSEM are simpler than those of the CESEM. However a mapping must be introduced to relate the undeformed and deformed configurations thus making the PDEs more complicated. We propose the use of pseudo-elasticity equations to provide a general framework to generate this mapping for unstructured meshes on complex geometries. The methodology is presented in details for the incompressible Navier-Stokes and sensitivity equations in variational form. The PDEs are solved with an adaptive FEM. Sensitivity data obtained with both approaches for a flow around a NACA 4512 are used to obtain estimates of flows around nearby geometries. Results indicate that the CLSEM produces significant improvements in terms of both accuracy and CPU time.