Minimal planar example

This is a minimal example with two quadrilateral elements and six nodes.

Problem setup

import torch

from torchfem import Planar
from torchfem.materials import IsotropicElasticityPlaneStress

torch.set_default_dtype(torch.float64)

# Material
material = IsotropicElasticityPlaneStress(E=1000.0, nu=0.3)

# Nodes and elements
nodes = torch.tensor(
    [[0.0, 0.0], [1.0, 0.0], [2.0, 0.0], [0.0, 1.0], [1.0, 1.0], [2.0, 1.0]]
)
elements = torch.tensor([[0, 1, 4, 3], [1, 2, 5, 4]])

# Create model
cantilever = Planar(nodes, elements, material)

# Load at tip [Node_ID, DOF]
cantilever.forces[5, 1] = -1.0

# Constrained displacement at left end [Node_IDs, DOFs]
cantilever.constraints[[0, 3], :] = True

# Show model
cantilever.plot(node_markers=True, node_labels=True)

Solving and post-processing

We solve the problem with the solve() function, which returns nodal displacements u, nodal forces f, element-wise stresses σ, element-wise deformation gradients F, and element-wise state variables α. These results may be provided to the plot function to plot the deformed configuration (argument u), a colormap of a node property (argument node_property), or a colormap of an element property (argument element_property):

u, f, σ, F, α = cantilever.solve()

# Plot displacement magnitude on deformed state
cantilever.plot(u, node_property=torch.norm(u, dim=1))

Automatic differentiation

If we turn on gradient tracking of specific tensors, we can use them for computation of sensitivities through any part of the torch-fem package via automatic differentiation:

# Enable automatic differentiation
cantilever.thickness.requires_grad = True
u, f, _, _, _ = cantilever.solve(differentiable_parameters=cantilever.thickness)

# Compute sensitivity of compliance w.r.t. element thicknesses
compliance = torch.inner(f.ravel(), u.ravel())
torch.autograd.grad(compliance, cantilever.thickness)[0]

tensor([-0.0208, -0.0053])