Principal stress net
Definition
A mesh is in equilibrium if the sum of the forces at each vertex is zero, which means no bending forces appear in physical realization. If an orthogonal quad mesh is in equilibrium, it is a discrete version of the net of principal stress lines in the surface, i.e. Principal stress net [1].
Constraint
If a vertical load \(p_i\) is applied in an unsupported vertex \(v_i\), the equilibrium condition in \(v_i\) reads
where the sum is over 4 neighbouring connected vertices \(v_{ij}\) of \(v_i\), and \(w_{ij}\) denotes the force density in the edge from \(v_i\) to \(v_j\).
The force densities \(w_{ij}\) are introduced as auxiliary variables and meet \(w_{ij} = -w_{ji}\). The vertical load \(p_i\) can depend on the mesh in which case we update it after every iteration.
The equilibrium function refers to DOS/archgeolab/constraints/constraints_equilibrium.py/equilibrium_constraints().
[1] Martin Kilian, Davide Pellis, Johannes Wallner, Helmut Pottmann. Material-minimizing forms and structures. ACM Trans. Graphics 36, 6 (2017): 1-12.