edit docs

docs/index.rst

@@ -10,7 +10,40 @@ Thermo-Plastic Nonuniform Transformation Field Analysis
 
 TODO: abstract
 
-.. image:: ../data/ntfa_workflow.jpg
+.. image:: images/ntfa_workflow.jpg
+
+Offline phase: Training of the thermo-mechanical NTFA
+-----------------------------------------------------
+
+1. Generate data using thermo-elasto-plastic simulations on the microscale at select temperatures :math:`\theta \in \mathcal{T}`.
+For that, we used an in-house FE solver. However, any other suitable simulation software could be used.
+
+2. Compute a reduced basis consisting of plastic modes :math:`\underline{\underline{P}}_{\mathsf{p}}` via snapshot POD of the simulated plastic strain fields :math:`\{\varepsilon_\mathsf{p}(x | \theta)\}^n_{i=1}`.
+The corresponding implementation is available in our [AdaptiveThermoMechROM](https://github.com/DataAnalyticsEngineering/AdaptiveThermoMechROM) repository in the module [ntfa.py](https://github.com/DataAnalyticsEngineering/AdaptiveThermoMechROM/blob/ntfa/ntfa.py).
+
+3. Perform additional linear-elastic simulations to determined self-equilibrated fields for the plastic modes :math:`\underline{\underline{P}}_\mathsf{p}` at select temperatures :math:`\theta \in \mathcal{T}`.
+Again, we used our in-house FE solver for that.
+
+4. Based on the generated data at select temperatures :math:`\theta \in \mathcal{T}` we perform an interpolation to arbitrarily many intermediate temperatures :math:`\theta_j`.
+This method is published in our paper ["Reduced order homogenization of thermoelastic materials with strong temperature-dependence and comparison to a machine-learned model"](https://doi.org/10.1007/s00419-023-02411-6), where we show that it produces highly accurate results while the effort is almost on par with linear interpolation.
+
+5. Using the interpolated data, the NTFA system matrices :math:`\underline{\underline{A}}(\theta_j)`, :math:`\underline{\underline{D}}(\theta_j)`, :math:`\bar{\underline{\underline{C}}}(\theta_j)`, :math:`\underline{\tau}_{\mathrm{\theta}}(\theta_j)`, and :math:`\underline{\tau}_{\mathsf{p}}(\theta_j)` are computed and stored as tabular data.
+The corresponding implementation is available in our [AdaptiveThermoMechROM](https://github.com/DataAnalyticsEngineering/AdaptiveThermoMechROM) repository in the module [ntfa.py](https://github.com/DataAnalyticsEngineering/AdaptiveThermoMechROM/blob/ntfa/ntfa.py).
+
+Online phase: Usage of the thermo-mechanical NTFA in simulations on the macroscale
+----------------------------------------------------------------------------------
+
+1. Load the tabular data for the NTFA matrices :math:`\underline{\underline{A}}(\theta_j)`, :math:`\underline{\underline{D}}(\theta_j)`, :math:`\bar{\underline{\underline{C}}}(\theta_j)`, :math:`\underline{\tau}_{\mathrm{\theta}}(\theta_j)`, and :math:`\underline{\tau}_{\mathsf{p}}(\theta_j)` that are generated in the offline phase based on direct numerical simulations on the microscale.
+Optionally truncate the NTFA modes :math:`N_{\mathrm{modes}}` to be used.
+
+2. Perform a linear interpolation to determine the NTFA matrices at the current model temperature based on the tabular data.
+Given that the tabular data is available at sufficiently many temperatures, the linear interpolation provides results with decent accuracy.
+This is done using the class [`thermontfa.TabularInterpolation`](https://github.com/DataAnalyticsEngineering/ThermoNTFA/blob/main/thermontfa/tabular_interpolation.py).
+
+3. Use the tabular data to initialize the thermo-mechanical NTFA UMAT that is implemented in the class [`thermontfa.ThermoMechNTFA`](https://github.com/DataAnalyticsEngineering/ThermoNTFA/blob/main/thermontfa/thermoNTFA.py).
+This reference implementation in Python can be transcribed to an UMAT for other academic or commercial simulation softwares.
+The numerical experiments in our paper are conducted using an implementation of the thermo-mechanical NTFA UMAT in C++ for an in-house FE solver.
+
 
 .. toctree::
    :maxdepth: 2