Adjoint-based sensitivity analysis of quantities of interest of complex combustion models

Figures & data

Figure 1. Characteristic time $t_r$ used for the definition of the ignition delay. The shift of the profile $T^{\mathrm{shift}}$ is increased for better visibility.

Figure 2. Comparison between adjoint and finite difference-based sensitivities for parameters A, treated time-dependent, of reaction #19 ${\mathrm{H}}_2+{\mathrm{O}}_2\to \mathrm{H}{\mathrm{O}}_2+\mathrm{H}$ (left) and #20 $\mathrm{H}{\mathrm{O}}_2+\mathrm{H}\to \mathrm{OH}+\mathrm{OH}$ (right) in scheme h2_v1b.

Figure 3. Adjoint-based, time-dependent sensitivities for parameters $A_j$ of all reactions in scheme h2_v1b in logarithmic scale (left). The reactions marked by the dashed lines correspond to Figure 2. Comparison between adjoint and finite difference-based sensitivity for the top six most sensitive reactions with respect to the time-independent factor $A_j$ (right).

Figure 4. Comparison between adjoint and finite difference-based sensitivity for the top six most sensitive reactions with respect to the time-independent factors ${\beta }_j$ (left) and $E_{a,j}$ (right) in scheme h2_v1b.

Figure 5. Comparison between adjoint and finite difference-based sensitivity for the top 10 most sensitive reactions with respect to the time-independent factor $A_j$ in scheme gri3.0.

Figure 6. Comparison between adjoint and finite difference-based sensitivity for the top six most sensitive reactions with respect to the time-independent factors $A_j$ (left) and $E_{a,j}$ (right) in scheme h2_v1b.

Figure 7. Comparison between adjoint and finite difference-based sensitivity for the top six most sensitive reactions with respect to the time-independent factor ${\beta }_j$ in scheme h2_v1b (left). Comparison between adjoint and finite difference-based sensitivity for the top 10 most sensitive reactions with respect to the time-independent factor $A_j$ in scheme gri3.0 (right).

Figure 8. Comparison of sensitivities based on linear objective function (32(32) $J=\frac{1}{2\tau }{\int }_{t_0}^{t_{\mathrm{end}}}\left(T-T^{\mathrm{shift}}\right)\mathrm{d}t.$(32) ) with respect to quadratic objective (28(28) $J=\frac{1}{2\tau }{\int }_{t_0}^{t_{\mathrm{end}}}{\left(T-T^{\mathrm{shift}}\right)}^2\mathrm{d}t.$(28) ), using the adjoint approach (left) and finite differences (right).

Figure 9. Comparison of sensitivities based on quadratic objective function (28(28) $J=\frac{1}{2\tau }{\int }_{t_0}^{t_{\mathrm{end}}}{\left(T-T^{\mathrm{shift}}\right)}^2\mathrm{d}t.$(28) ) with respect to the definition by the maximum temperature gradient (left) and with the maximum OH mass fraction (right).

Figure 10. Relative errors of the adjoint-based sensitivities with respect to the time-independent factor $A_j$ and different numbers of data points used for the interpolation and evaluation: in descending order with respect to $|s_A|$ in scheme h2_v1b (left) and for the top 10 most sensitive reactions in scheme gri3.0 (right).