7. Adjoint Flux Formalism

  1. duality and inner product

  2. adjoint in Sn

The multigroup, adjoint transport equation is given by

\[\begin{split}\begin{gathered} -\frac{1}{v_g}\frac{\partial \psi^{\dagger,g}(\vec{r},\vec{\Omega},t) }{\partial t} - \vec{\Omega} \cdot \vec{\nabla} \psi^{\dagger,g}(\vec{r},\vec{\Omega},t) + \sigma_t^g(\vec{r},t)\psi^{\dagger,g}(\vec{r},\vec{\Omega},t) \\= \sum_{g'=1}^{g'=G} %\sum_{\ell=0}^{L_{\text{max}}} \sum_{m=-\ell}^{m=\ell} \sum_{\ell,m} \, \frac{2\ell+1}{4\pi}\sigma^{g\to g'}_{s,\ell}(\vec{r}) Y_{\ell,m}(\vec{\Omega},t) \phi^{\dagger,g}_{\ell,m}(\vec{r},t) + Q^{\dagger,g}_{\text{ext}}(\vec{r},\vec{\Omega},t) \\ \qquad 1\le g \le G \,. \end{gathered}\end{split}\]

Note that:

  1. streaming is reversed, the streaming term now has a \(-\) sign,

  2. time is reversed, the temporal derivative term now has a \(-\) sign,

  3. the energy transfer in the scattering term has been reserved (now they are from \(g\) to \(g'\)),

  4. a similar reversal of the energy transfer in the fission term is required when fission is included,

  5. an external adjoint source is present.

Adjoint boundary conditions of Dirichlet type are now supplied either as a known outgoing adjoint flux

\[\psi^{\dagger,g}(\vec{r},\vec{\Omega},t) = \psi^{\dagger,g}_{\text{out}}(\vec{r},\vec{\Omega},t) \qquad \forall \vec{r} \in \Gamma^+\]

where

\[\Gamma^+ = \big\{ \vec{r} \in \Gamma \text{ such that } \vec{\Omega}\cdot\vec{n}(\vec{r}) > 0 \big\} \,.\]
Adjoint final conditions are supplied in time:
\[\psi^{\dagger,g}(\vec{r},\vec{\Omega},t=T) = h^g_T(\vec{r},\vec{\Omega},g) \qquad \forall \vec{r}\in \mathcal{D},\ \forall g \in [1,G], \ \forall\vec{\Omega}\in \mathcal{S}^2\]
Multigroup \(S_n\) codes can be used to perform adjoint calculations. One need only adjust the calculation as follows:

  1. Transpose the multigroup transfer cross sections.

  2. Interpret the \(S_n\) flux solution in direction \(\vec{\Omega}\) as the adjoint flux in direction \(-\vec{\Omega}\).

  3. Interpret the S\(_n\) source in direction \(\vec{\Omega}\) as the adjoint source evaluated in direction \(-\vec{\Omega}\). The user is responsible for doing this.

Finally, it is important for the user to recognize that because the multigroup inner product is the dot product, the adjoint multigroup source for group \(g\) represents the analytic adjoint flux averaged over group \(g\) rather than integrated over group \(g\). As a consequence, the multigroup adjoint flux for group \(g\) represents analytic adjoint flux averaged over group \(g\) rather than integrated over group \(g\).
The adjoint flux is useful for computing:

  • quantities of interest,

  • first-order sensitivity in quantities of interest,

  • an importance map.

Duality statement:
We introduce the following inner products in the volume \(\mathcal{D}\) and the boundary \(\Gamma=\partial\mathcal{D}\) of the spatial domain. \(f\) and \(g\) are multigroup-valued functions.
\[(f,h) = \sum_g \int_0^T dt \int_{\mathcal{D}} d^3r \int_{\mathcal{S}^2} d\Omega \, f^g(\vec{r},\vec{\Omega},t) h^g(\vec{r},\vec{\Omega},t)\]
\[\langle f,h\rangle_\pm = \sum_g \int_0^T dt \int_{\Gamma} d^2r \int_{\vec{\Omega}\cdot \vec{n}(\vec{r}) \gtrless 0} d\Omega \, f^g(\vec{r},\vec{\Omega},t) h^g(\vec{r},\vec{\Omega},t)\]
\[\left\{ f,h\right\}_\tau = \sum_g \int_{\mathcal{D}} d^3r \int_{\mathcal{S}^2} d\Omega \, f^g(\vec{r},\vec{\Omega},\tau) h^g(\vec{r},\vec{\Omega},\tau)\]

Note that in the notation \((\Psi,\Psi^{\dagger})\), the entries in \(\Psi\) are the standard, multigroup forward fluxes (energy-dependent flux integrated over a group bin), while the entries in \(\Psi^{\dagger}\) are the standard multigroup adjoint fluxes that, as previously noted, are actually group-averaged values of the energy-dependent adjoint flux.

Given a forward transport problem with volumetric source \(Q^{g}_{\text{ext}}\), a boundary source \(\psi^{g}_{\text{inc}}\), and an initial condition \(f_0\), there is an adjoint problem with volumetric source \(Q^{\dagger,g}_{\text{ext}}\), boundary source \(\psi^{\dagger,g}_{\text{out}}\), and final condition \(h_T\) such that the following duality principle or duality conservation statement holds
\[\left( \Psi, Q^{\dagger}_{\text{ext}} \right) + \langle \Psi, \Psi^{\dagger}_{\text{out}} \rangle + \left\{ \Psi, h \right\}_T = \left( \Psi^{\dagger}, Q_{\text{ext}} \right) + \langle \Psi^{\dagger}, \Psi_{\text{inc}} \rangle + \left\{ \Psi^{\dagger}, f \right\}_0\]
Example of a quantity of interest (QoI).
Suppose one wants to compute the reaction rate in a detector (detector cross section \(\sigma_{\text{det}}\)) due to a source \(Q_{\text{ext}}\). Suppose the boundary of the problem is a vacuum. The problem is steady state. The QoI is given by:
\[\text{QoI} = \left( \Psi, \sigma_{\text{det}} \right) \,.\]

Using the duality principle, the QoI can also be computed as

\[\text{QoI} = \left( \Psi^{\dagger}, Q_{\text{ext}} \right) \,.\]

Hence, it is clear that the adjoint volumetric source should be \(Q^{\dagger}_{\text{ext}}=\sigma_{\text{det}}\) and the adjoint boundary source should be \(\Psi^{\dagger}_{\text{out}} =0\) (vacuum).