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fixes links and latex error and images
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Unsteady_NACA0012/images/wndFcts.png

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Unsteady_NACA0012/unsteady_naca0012.cfg

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@@ -179,7 +179,7 @@ HISTORY_WRT_FREQ_INNER=0
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SCREEN_WRT_FREQ_INNER =1
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%
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% Mesh input file
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MESH_FILENAME= unsteady_NACA0012_mesh.su2
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MESH_FILENAME= unsteady_naca0012_mesh.su2
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%
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% Mesh input file format (SU2, CGNS, NETCDF_ASCII)
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MESH_FORMAT= SU2
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_tutorials/compressible_flow/Unsteady_NACA0012.md

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@@ -75,7 +75,7 @@ for the physical time iterator by `TIME_ITER`. The option `TIME_DOMAIN=YES` acti
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The numerical method to solve the inner (fictional time) ODE is given by the option `TIME_DISCRE_FLOW`.
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```
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% ------------------------- UNSTEADY SIMULATION -------------------------------%
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% ----------UNSTEADY SIMULATION ---------------%
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%
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TIME_DOMAIN = YES
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%

_tutorials/design_features/Unsteady_Shape_Opt_NACA0012.md

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- Unsteady Optimization
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- Code parallelism (optional)
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This tutorial uses the windowing techniques explained in [here](../compressible_flow/Unsteady_NACA0012.md), to compute meaningful optimization objectives. Hence it is recommended to read this tutorial first.
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This tutorial uses the windowing techniques explained in [here](../Unsteady_NACA0012.md), to compute meaningful optimization objectives. Hence it is recommended to read this tutorial first.
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## Resources ##
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The resources for this tutorial can be found in the [Unsteady_NACA0012](https://github.com/su2code/su2code.github.io/tree/master/Unsteady_Shape_Opt_NACA0012) directory in the [project website repository](https://github.com/su2code/su2code.github.io).
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You will need the configuration file ([unsteady_naca0012_opt.cfg](../../Unsteady_NACA0012/unsteady_naca0012_opt.cfg)) and the mesh file ([unsteady_naca0012_FFD.su2](../../Unsteady_NACA0012/unsteady_naca0012_FFD.su2)).
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You will need the configuration file ([unsteady_naca0012_opt.cfg](../../Unsteady_Shape_Opt_NACA0012/unsteady_naca0012_opt.cfg)) and
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the mesh file ([unsteady_naca0012_FFD.su2](../../Unsteady_Shape_Opt_NACA0012/unsteady_naca0012_FFD.su2)).
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## Tutorial ##
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Using the midpoint rule for above integral, we arrive at the following constrained optimization problem
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$$ \min_{\sigma} \frac{1}{M} \sum_n_{tr}^{N} w\left(\frac{n-n_{tr}}{N-n_{tr}}\right)C_D(\sigma,n) $$
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$$ \min_{\sigma} \frac{1}{M} \sum_{n_{tr}}^{N} w\left(\frac{n-n_{tr}}{N-n_{tr}}\right)C_D(\sigma,n) $$
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$$ s.t. R(u^n) = 0 \qquad \forall n=1,\dots,N $$
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$$ s.t. \sum_n_{tr}^{N} w\left(\frac{n-n_{tr}}{N-n_{tr}}\right)C_L(\sigma,n) \geq c$$
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$$ s.t. \sum_{n_{tr}}^{N} w\left(\frac{n-n_{tr}}{N-n_{tr}}\right)C_L(\sigma,n) \geq c$$
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The optimization constraint is given by the windowed time-averaged lift, that should be greater than a specific value $$c$$. We choose arbitrarily as $$c=0.96$$, which is the windowed
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time-averaged lift of the baseline geometry.
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To compute the unsteady shape-optimization, we set up the unsteady simulation according to our test case above. More information about unsteady simulations can be found [here](../Unsteady_NACA0012.md)
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```
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% ------------------------- UNSTEADY SIMULATION -------------------------------%
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% -------- UNSTEADY SIMULATION -----------------%
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%
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TIME_DOMAIN = YES
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%

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