Force density method (form-finding)

A self-contained FDM subsystem for cable/tension networks ships in the same package, built on the same three-layer architecture as the frame core: Network holds pure definition data, process! freezes the force-density pattern once per topology, and repeated solves are a pure scatter of the current q values plus a numeric-only refactorization — so form-finding sweeps are cheap by construction. solve! updates the node positions in place (the geometry is the result); forces and reactions are queried from network.results:

# a 2-cable chain: anchors at the ends, load at the middle
a = FDMnode([0.0, 0.0, 0.0], false)          # false = fully fixed (anchor)
m = FDMnode([1.0, 0.0, 0.0], true, :mid)     # true  = fully free
b = FDMnode([2.0, 0.0, 0.0], false)

cables = [FDMelement(a, m, 2.0), FDMelement(m, b, 2.0)]   # q = 2.0 [force/length]
net = Network([a, m, b], cables, [FDMload(m, [0.0, 0.0, -1.0])])

solve!(net)
m.position                            # the found geometry

3-element StaticArraysCore.SVector{3, Float64} with indices SOneTo(3):
  1.0
  0.0
 -0.25

member_force(net.results, cables[1])  # cable tension q·L

2.0615528128088303

reaction(net.results, a)              # anchor force, per fixed axis

3-element StaticArraysCore.SVector{3, Float64} with indices SOneTo(3):
  2.0
  0.0
 -0.5

Form-finding sweep: mutate q and re-solve (numeric refactorization only):

update_q!(net, 4.0)                   # doubles q — halves the sag
m.position

3-element StaticArraysCore.SVector{3, Float64} with indices SOneTo(3):
  0.9999999999999999
  0.0
 -0.12499999999999999

Per-axis fixity

Fixity is per-axis (true = free, like Node): prescribe the plan and let the height find equilibrium — natural for vault form-finding. FDM equilibrium is separable per coordinate, so each axis solves against its own free/fixed partition:

crown = FDMnode([1.0, 1.0, 0.0], Bool[false, false, true])   # plan fixed, z free

FDMnode
  position = [1.0, 1.0, 0.0]  [length]
  fixity   = x/y/z: FIXED/FIXED/free  (true = free)
  id       = :node

The solver keyword works here too (any LinearSolve algorithm once loaded). to_network(model) converts a solved truss model into an equivalent FDM network, and to_truss(network, section) converts back. Differentiable force-density optimization (QVariable, solve_network) lives in AsapOptim.