Pseudo-timestepping and verification for automatic differentiation derived CFD codes

Time-averaged steady vs. unsteady adjoint: a comparison for cases with mild unsteadiness

“…The adjoint is partly hand-written, but makes extensive use of subroutines generated from the primal solver using the source-transformation AD tool Tapenade, see [11].…”

Preparation and assembly of discrete adjoint CFD codes

“…Instead, the differentiated subroutines it calls are assembled in the driver routine in an efficient way to create the sensitivity algorithm [3].…”

“…In Section 2 we summarise our experience in how to use them in the context of preparing codes for AD, with some comments how well these features are supported by the AD tool we have used [10]. There have been a number of papers that discuss for the fully coupled compressible flow equations how to assemble the differentiated flux routines in pseudo-timestepping algorithms [3], however this methodology does not apply straightforwardly to the more complex block-iterative scheme of the incompressible equations. In Section 3 we demonstrate how to assemble an incompressible adjoint for the SIMPLE scheme [18,6] and in particular how to avoid differentiating the linear solvers used in the inner iteration.…”

CAD‐based shape optimisation with CFD using a discrete adjoint

“…In discrete adjoint methods, the sensitivity of the cost function J with respect to a perturbation in surface node coordinates X s is written as with This product can be evaluated efficiently in a consistent way by first solving the adjoint equations A T v = g and then computing the product v T f by reverse differentiating the metrics calculation (face/edge normals) to obtain and then chaining that with reverse differentiation of the same mesh smoothing algorithm of to right multiply with .…”

“…This product can be evaluated efficiently in a consistent way by first solving the adjoint equations A T v D g and then computing the product v T f by reverse differentiating the metrics calculation (face/edge normals) to obtain @R @X and then chaining that with reverse differentiation of the same mesh smoothing algorithm of (13) to right multiply with @X @X s [33]. The metrics and mesh deformation subroutines are differentiated using the AD tool Tapenade in reverse mode for use in the adjoint code.…”