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.