The independent set perturbation adjoint method: A new method of differentiating meshbased fluids models

A new scheme for differentiating complex mesh-based numerical models (e.g. finite element models), the Independent Set Perturbation Adjoint method (ISP-Adjoint), is presented. Differentiation of the matrices and source terms making up the discrete forward model is realized by a graph colouring approach (forming independent sets of variables) combined with a perturbation method to obtain gradients in numerical discretisations. This information is then convolved with the ‘mathematical adjoint’ which uses the transpose matrix of the discrete forward model. The adjoint code is simple to implement even with complex non-linear parameterizations, discretisation methods and governing equations. Importantly, the adjoint code is independent of the implementation of the forward code.