Adjoint fluid

In neko, and by extension neko-top, we solve the Navier–Stokes equations

\[{\frac {\partial \mathbf {u} }{\partial t}} + (\mathbf {u} \cdot \nabla)\mathbf {u} = -\nabla p +{\frac {1}{Re}}\nabla ^{2}\mathbf {u} + \mathbf{f}, \]

$$ \nabla \cdot \mathbf {u} = 0, $$ where $\mathbf {u}(\mathbf{x},t)$ denotes the velocity field, $p(\mathbf{x},t)$ the pressure field, $\mathbf{f}$ a forcing term and where $Re$ denotes the Reynolds number.

A common formulation of the adjoint Navier–Stokes equations reads

\[{-\frac {\partial \mathbf {u}^\dagger }{\partial t}} + (\nabla \mathbf {u})^T \mathbf {u}^\dagger - (\mathbf {u} \cdot \nabla) \mathbf {u}^\dagger = -\nabla p^\dagger +{\frac {1}{Re}}\nabla ^{2}\mathbf {u}^\dagger + \mathbf{f}^\dagger, \]

$$ \nabla \cdot \mathbf {u} ^\dagger= 0, $$ where where $\mathbf {u}^\dagger(\mathbf{x},t)$ denotes the adjoint velocity field, $p^\dagger(\mathbf{x},t)$ the adjoint pressure field and $\mathbf{f}^\dagger$ denoting a forcing term applied to the adjoint system which generally arises as a consequence of objective functions being evaluated.

The spectral element method which underpins neko solves the above system of equations using the weak formulation, which has important implications when solving the adjoint system. Primarily, when deriving the adjoint system in strong form, one introduces additional boundary terms on Neumann boundaries, and more importantly, one applies the divergence free condition strongly.

An alternative to the above adjoint system is to remain in weak form and not integrate the convective term by parts, resulting in no additional boundary terms, and more importantly, no pointwise application of the divergence free condition.

This alternative system is what is implemented in neko-top, which reads (in weak form),

\[-\int_\Omega \mathbf{v}\cdot {\frac {\partial \mathbf {u}^\dagger } {\partial t}} + \int_\Omega \mathbf{v}\cdot (\nabla \mathbf {u})^T \mathbf {u}^\dagger + \int_\Omega \nabla \mathbf{v}\cdot (\mathbf {u} \otimes \mathbf {u}^\dagger ) = -\int_\Omega \mathbf{v}\cdot \nabla p^\dagger +{\frac {1}{Re}}\int_\Omega \nabla \mathbf{v}\cdot \nabla \mathbf {u}^\dagger + \int_\Omega \mathbf{v}\cdot \mathbf{f}^\dagger, \]

$$ \int_\Omega q \nabla \cdot \mathbf {u} ^\dagger= 0. $$

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Adjoint fluid