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Given a PDE solution methodology for the forward (analysis) problem, adjoint techniques are used to integrate backwards in time to efficiently perform sensitivity analyses of output functions with respect to any number of input parameters.
The unique strength of adjoint-based methods in the context of formal mesh adaptation and error estimation is also noted, where the approach can be used to provide mathematically-rigorous mesh adaptation that has proven far superior to traditional feature-driven techniques based on heuristics.
Why use Adjoints in Shape Optimization?
Unlike forward-mode sensitivity analysis techniques whose cost scales linearly with the number of design variables, the cost associated with the adjoint solution is equivalent to that of the primal problem.
In this manner, adjoint-based schemes can provide discretely-consistent sensitivity derivatives for very large numbers of design variables at the same cost as the baseline analysis problem.
This ultimately enables formal gradient-based optimization for realistic large-scale computational aerodynamics problems which would otherwise be intractable.
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