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Implements the Force Density Method (FDM) equilibrium equations for
pin-jointed bar systems.
References
----------
[1] Schek, H.J. (1974). The Force Density Method for form finding and
computation of general networks. CMAME, 3(1):115-134.
[2] Pastrana, R. et al. (2025). Neural FDM. ICLR 2025. Eq. 1, 8.
"""
import jax.numpy as jnp
from jax_fdm.equilibrium import EquilibriumParametersState as FDParametersState
from jax_fdm.equilibrium import EquilibriumState, LoadState, nodes_load_from_faces
# ===============================================================================
# Load helpers
# ===============================================================================
def calculate_area_loads(x, structure, load):
"""
Convert area loads into vertex loads.
Parameters
----------
x: `jax.Array`
The 3D coordinates of the vertices.
structure: `jax_fdm.EquilibriumStructure`
A structure with the discretization of the shape.
load: `float`
The vertical load per unit area in the `z` direction.
Returns
-------
vertices_load: `jax.Array`
The 3D vertex loads.
"""
x = jnp.reshape(x, (-1, 3))
# need to convert loads into face loads
num_faces = structure.num_faces
faces_load_xy = jnp.zeros(shape=(num_faces, 2)) # (num_faces, xy)
faces_load_z = jnp.ones(shape=(num_faces, 1)) * load # (num_faces, xy)
faces_load = jnp.hstack((faces_load_xy, faces_load_z))
vertices_load = nodes_load_from_faces(
x,
faces_load,
structure,
is_local=False
)
return vertices_load
def calculate_constant_loads(x, structure, load):
"""
Create constant vertical vertex loads.
Parameters
----------
x: `jax.Array`
The 3D coordinates of the vertices.
structure: `jax_fdm.EquilibriumStructure`
A structure with the discretization of the shape.
load: `float`
The vertical load per vertex in the `z` direction.
Returns
-------
vertices_load: `jax.Array`
The 3D vertex loads.
"""
num_vertices = structure.num_vertices
# (num_vertices, xy)
vertices_load_xy = jnp.zeros(shape=(num_vertices, 2))
# (num_vertices, xy)
vertices_load_z = jnp.ones(shape=(num_vertices, 1)) * load
return jnp.hstack((vertices_load_xy, vertices_load_z))
# ===============================================================================
# Form-finding helpers
# ===============================================================================
def edges_vectors(xyz, connectivity):
"""
Calculate the unnormalized edge directions (nodal coordinate differences).
Parameters
----------
xyz: `jax.Array`
The 3D coordinates of the vertices.
connectivity: `jax.Array`
The connectivity matrix of the structure.
Returns
-------
vectors: `jax.Array`
The edge vectors.
"""
return connectivity @ xyz
def edges_lengths(vectors):
"""
Compute the length of the edge vectors.
Parameters
----------
vectors: `jax.Array`
The edge vectors.
Returns
-------
lengths: `jax.Array`
The lengths.
"""
return jnp.linalg.norm(vectors, axis=1, keepdims=True)
def edges_forces(q, lengths):
"""
Calculate the force in the edges.
Parameters
----------
q: `jax.Array`
The force densities.
lengths: `jax.Array`
The edge lengths.
Returns
-------
forces: `jax.Array`
The forces in the edges.
"""
return jnp.reshape(q, (-1, 1)) * lengths
def vertices_residuals(q, loads, vectors, connectivity):
"""
Compute the residual forces on the vertices of the structure.
Parameters
----------
q: `jax.Array`
The force densities.
loads: `jax.Array`
The loads on the vertices.
vectors: `jax.Array`
The edge vectors.
connectivity: `jax.Array`
The connectivity matrix of the structure.
Returns
-------
residuals: `jax.Array`
The residual forces on the vertices.
"""
return loads - connectivity.T @ (q[:, None] * vectors)
def vertices_residuals_from_xyz(q, loads, xyz, structure):
"""
Compute the residual forces on the vertices of the structure.
Parameters
----------
q: `jax.Array`
The force densities.
loads: `jax.Array`
The loads on the vertices.
xyz: `jax.Array`
The 3D coordinates of the vertices.
structure: `jax_fdm.EquilibriumStructure`
A structure with the discretization of the shape.
Returns
-------
residuals: `jax.Array`
The residual forces on the vertices.
"""
connectivity = structure.connectivity
xyz = jnp.reshape(xyz, (-1, 3))
vectors = edges_vectors(xyz, connectivity)
return vertices_residuals(q, loads, vectors, connectivity)
def calculate_equilibrium_state(q, xyz, loads_nodes, structure):
"""
Assembles an equilibrium state object.
Parameters
----------
q: `jax.Array`
The force densities.
xyz: `jax.Array`
The 3D coordinates of the vertices.
loads_nodes: `jax.Array`
The loads on the vertices.
structure: `jax_fdm.EquilibriumStructure`
A structure with the discretization of the shape.
Returns
-------
state: `jax_fdm.EquilibriumState`
The equilibrium state.
"""
connectivity = structure.connectivity
vectors = edges_vectors(xyz, connectivity)
lengths = edges_lengths(vectors)
residuals = vertices_residuals(q, loads_nodes, vectors, connectivity)
forces = edges_forces(q, lengths)
return EquilibriumState(
xyz=xyz,
residuals=residuals,
lengths=lengths,
forces=forces,
loads=loads_nodes,
vectors=vectors
)
def calculate_fd_params_state(q, xyz_fixed, loads_nodes):
"""
Assembles an simulation parameters state.
Parameters
----------
q: `jax.Array`
The force densities.
xyz_fixed: `jax.Array`
The 3D coordinates of the fixed vertices.
loads_nodes: `jax.Array`
The loads on the vertices.
Returns
-------
state: `jax_fdm.EquilibriumParametersState`
The current state of the simulation parameters.
"""
return FDParametersState(q, xyz_fixed, LoadState(loads_nodes, 0.0, 0.0))
# =============================================================================
# Reaction forces at supports
# =============================================================================
def compute_reactions(q, loads, xyz, structure):
"""
Compute reaction forces at support (fixed) nodes.
In equilibrium, the residual at free nodes is zero. At fixed nodes,
the residual represents the reaction force the support must provide:
R_i = sum_j K_ij (x_j - x_i) * q_j - P_i
This is the negative of the residual at fixed nodes.
Parameters
----------
q : jax.Array (E,)
Force densities per edge.
loads : jax.Array (N, 3)
Applied loads at all vertices.
xyz : jax.Array (N, 3)
Vertex positions.
structure : EquilibriumStructure
The mesh structure with connectivity.
Returns
-------
reactions : jax.Array (N_fixed, 3)
Reaction force vectors at each support node.
indices_fixed : jax.Array (N_fixed,)
Indices of the fixed (support) nodes.
"""
# Compute residuals at ALL nodes (not just free)
residuals_all = vertices_residuals_from_xyz(q, loads, xyz, structure)
# Reactions are the negative residuals at fixed nodes
# At equilibrium, free node residuals are ~0
# At fixed nodes, residuals represent the imbalance that supports carry
indices_fixed = structure.indices_fixed
reactions = -residuals_all[indices_fixed]
return reactions, indices_fixed
def compute_total_reactions(reactions):
"""
Compute total reaction forces (sum over all supports).
Parameters
----------
reactions : jax.Array (N_fixed, 3)
Reaction force vectors per support.
Returns
-------
total : jax.Array (3,)
Sum of all reactions [Rx, Ry, Rz].
"""
return jnp.sum(reactions, axis=0)
def compute_l_physics(x_hat, q, loads, structure, free_indices=None):
"""Compute L_physics: Euclidean norm (L2) of residual forces at free nodes.
Matches the paper's physics loss definition (losses.py compute_error_residual):
sqrt(sum(residual_vectors_free^2)). This is the standard metric used
throughout all benchmarks for consistency.
Parameters
----------
x_hat : jax.Array
Predicted vertex positions (flat or (N,3)).
q : jax.Array
Force densities per edge.
loads : jax.Array
Applied loads per vertex (N,3).
structure : EquilibriumStructure
Mesh connectivity structure.
free_indices : array-like, optional
Indices of free (non-support) nodes. If None, uses all nodes.
Returns
-------
l_physics : float
Euclidean norm of residual forces at free nodes.
"""
xyz = jnp.reshape(x_hat, (-1, 3))
residuals = vertices_residuals_from_xyz(q, loads, xyz, structure)
if free_indices is not None:
residuals = residuals[jnp.array(free_indices), :]
return float(jnp.sqrt(jnp.sum(jnp.square(residuals))))
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