QTF Standing (Newman) Approximation (WEC-Sim#1461)

* Adds Newman Approx for the QTFs

* Bug Fix

* Bug Fix

* Adds user guide doc for the QTFs

* asserts that the QTFs are not enabled with the meanDrift

* Update docs/user/advanced_features.rst

Co-authored-by: Adam Keester <72414466+akeeste@users.noreply.github.com>

* Update docs/user/advanced_features.rst

Co-authored-by: Adam Keester <72414466+akeeste@users.noreply.github.com>

* Update docs/user/advanced_features.rst

Co-authored-by: Adam Keester <72414466+akeeste@users.noreply.github.com>

* Update docs/user/advanced_features.rst

Co-authored-by: Adam Keester <72414466+akeeste@users.noreply.github.com>

* Update docs/user/advanced_features.rst

Co-authored-by: Adam Keester <72414466+akeeste@users.noreply.github.com>

* Adds QTFs option 2 for the run from simulink feature

* fixes a bug with the QTF doc

* Pulled updated WECSim_Lib_Body_Elements.slx from dev branch to resolve conflicts

* Modifies the user doc

* Modifies lines in the Standing approx

* Modifies the user doc

---------

Co-authored-by: Adam Keester <72414466+akeeste@users.noreply.github.com>

Author: MShabara

docs/user/advanced_features.rst

@@ -734,6 +734,67 @@ Applications <https://github.com/WEC-Sim/WEC-Sim_Applications>`_ repository
 
     Morison Elements cannot be used with :code:`elevationImport`.
 
+.. _user-advanced-features-second-order-excitation-forces:
+
+Second-Order Excitation Forces and Quadratic Transfer Functions (QTFs)
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+For large offshore structures, second-order wave excitation forces can significantly influence system behavior—particularly in surge, pitch, and other low-frequency motions. These forces arise from interactions between pairs of wave frequencies, :math:`\omega_i` and :math:`\omega_j`, within the wave spectrum.
+
+There are two types of second-order forces:
+
+- **Sum-frequency forces** (:math:`\omega_i + \omega_j`): High-frequency components that can excite fast body motions.
+- **Difference-frequency forces** (:math:`\omega_i - \omega_j`): Low-frequency components that can induce slow-drift responses, commonly in surge or pitch.
+
+WEC-Sim supports **Quadratic Transfer Functions (QTFs)** to model second-order excitation forces. These can be imported from hydrodynamic solvers such as **WAMIT** and **NEMOH**. BEMIO functions will automatically read, process, and write QTF coefficients from WAMIT and NEMOH to the ``*.h5`` file.
+
+File Naming and Conventions
+"""""""""""""""""""""""""""""""""
+
+The QTF file naming depends on the hydrodynamics solver:
+
+**WAMIT**
+
+- QTF files should be located in the same folder as the `.out` file.
+- File naming format:
+  
+  ::
+  
+    qtfResults_<body#>.11s
+    qtfResults_<body#>.11d
+
+  Example: ``qtfWAMIT_1.11s``
+
+- Note: WAMIT uses **1-based indexing** for body numbers.
+
+**NEMOH**
+
+- QTF files should follow this naming format:
+
+  ::
+  
+    OUT_QTFP_N_<body#>.dat   (for sum-frequency QTFs)
+    OUT_QTFM_N_<body#>.dat   (for difference-frequency QTFs)
+
+- NEMOH uses **0-based indexing** for body numbers.
+
+Activating QTFs in WEC-Sim
+""""""""""""""""""""""""""""""""
+
+To enable second-order excitation forces in WEC-Sim, use the following flag in your `body(i)` definition:
+
+- ``body(i).QTFs = 1`` — Enables **full QTF-based** second-order excitation.
+- ``body(i).QTFs = 2`` — Enables **Standing approximation** for **difference-frequency** forces.
+  - In this case, sum-frequency components are still computed using full QTFs.
+
+.. note::
+
+   Do **not** use ``body[i].QTFs`` and ``body[i].meanDrift`` together.
+   This will result in **double-counting** of mean drift forces — once from the QTFs and once from the linear mean drift formulation.
+   If both ``body[i].QTFs`` and ``body[i].meanDrift`` are enabled, a warning will be issued and the standalone mean drift flag will be disabled. Only the QTFs will be used to compute the mean drift forces.
+
+   The **Standing approximation** is a modified form of **Newman's approximation**. While mathematically equivalent, Standing's formulation is expressed differently and is designed to support multidirectional wave environments. Unlike Newman's original version, it does not require high-frequency filtering after application.
+
 .. _user-advanced-features-non-hydro-body:
 
 Non-Hydrodynamic Bodies

source/functions/BEMIO/QTFs/triToFullMatrix.m

@@ -42,7 +42,7 @@
                 F_ij(dof_data(i, end-1), dof_data(i, end)) = elements(i);
                 F_ij(dof_data(i, end), dof_data(i, end-1)) = conj(F_ij(dof_data(i, end-1), dof_data(i, end)));
             else
-                elements = dof_data(:, col) - 1i .* dof_data(:, col + 1);
+                elements = dof_data(:, col) + 1i .* dof_data(:, col + 1);
                 F_ij(dof_data(i, end-1), dof_data(i, end)) = elements(i);
                 F_ij(dof_data(i, end), dof_data(i, end-1)) = F_ij(dof_data(i, end-1), dof_data(i, end));
             end