GITBOOK-75: relax pattern

Author: juney-lee

gitbook/manual/2.-define-boundary-conditions/boundary-conditions-1.md

@@ -1,9 +1,13 @@
 # 2b. Relax Pattern
 
-<table><thead><tr><th></th><th width="237"></th><th></th></tr></thead><tbody><tr><td><img src="../../.gitbook/assets/RV_relax.svg" alt="" data-size="original"></td><td><p><strong>Rhino command name</strong></p><p><code>RV_pattern_supports</code></p></td><td><p><strong>source file</strong></p><p><a href="../../../plugin/RV_pattern_supports.py"><code>RV_pattern_supports.py</code></a></p></td></tr></tbody></table>
+<table><thead><tr><th></th><th width="237"></th><th></th></tr></thead><tbody><tr><td><img src="../../.gitbook/assets/RV_relax.svg" alt="" data-size="original"></td><td><p><strong>Rhino command name</strong></p><p><code>RV_pattern_relax</code></p></td><td><p><strong>source file</strong></p><p><a href="../../../plugin/RV_pattern_relax.py"><code>RV_pattern_relax.py</code></a></p></td></tr></tbody></table>
 
-An opening is a chain of edges at the boundary of a `Pattern`, in between two support vertices. In general, openings in TNA cannot be straight, unless there are no internal forces in the non-boundary edges at the openings (e.g. barrel vault or cross vault). In some applications where openings may already have some curvature, the relaxation will make the `Pattern` more "equilibrated" and optimal for the horizontal equilibrium solver later on.&#x20;
+An opening is a chain of edges at the boundary of a `Pattern`, in between two support vertices. In general, openings in shell structures cannot be straight in plan, unless there are no internal forces in the non-boundary edges at the openings (e.g. barrel vault or cross vault).
 
-### Force Density method
+`RV_pattern_relax` relaxes the entire `Pattern` to introduce curvature to the openings and boundaries. The relaxation is done based on the _force density method (FDM)_ ([Schek, 1974](https://www.sciencedirect.com/science/article/pii/0045782574900450)), which calculates the equilibrium shape by assigning a ratio of force to length (force density, **q**) to each element or edge in the structure, which allows for the efficient computation of form-finding solutions.&#x20;
 
-This feature relaxes the entire `Pattern` using the _force density method_ ([Schek, 1974](https://www.sciencedirect.com/science/article/pii/0045782574900450)), resulting in curved openings. Force density of 1 is assigned to every edge. Note that supports should be properly defined in step 2a to achieve good results.
+Force density, is the ratio of the force in an element to its length. Mathematically, q=F/L, where F is the force in the element, andL is its length. Instead of solving directly for lengths and forces, FDM assigns these force densities as fixed parameters, simplifying the equations. The method then solves for equilibrium by ensuring that forces at each node balance in three-dimensional space. This involves setting up equilibrium equations for each node in terms of force densities rather than individual force magnitudes and lengths.
+
+Because the force densities are constants, the equilibrium equations become linear, which makes FDM computationally efficient and stable. This linearity allows FDM to handle complex shapes and materials without iterative, nonlinear solutions required by some other methods.
+
+The default value for **q** for all edges are 1. When `RV_pattern_relax` is triggered, FDM is applied to the Pattern using the **q** values of all the edges. [`Identify supports`](../boundary-conditions.md) is a crucial step to complete before relaxing the `Pattern`, as the number and location of the supports will impact the geometry of the resulting relaxed `Pattern`.&#x20;

gitbook/manual/boundary-conditions.md

@@ -4,4 +4,18 @@
 
 A `Pattern` object is a mesh datastructure that describes the topology of the structure. Several additional layers of information regarding the boundary conditions need to be added in order to give the `Pattern` a structural meaning: identification of the supports, and treatment of the openings and open edges.
 
+
+
+***
+
+## Sub-commands
+
 In RhinoVAULT, a _support_ is defined as a vertex of the structure that is fixed, and can have external horizontal reactions. `RV_pattern_supports` allows user to _Add_ ore _Remove_ supports from the `Pattern`. 
+
+### Add
+
+Add support vertices to the `Pattern`. Support vertices are displayed as red points.
+
+### Remove
+
+Remove support vertices from the `Pattern`.