Theory¶

This page summarizes the continuum and finite-element formulations implemented in torch-fem. The implementation supports solving mechanical problems with differentiable solution operators built on PyTorch autograd.

Mechanics¶

Kinematics¶

Let \(\Omega_0 \subset \mathbb{R}^d\) denote the reference domain with material point \(\mathbf{X}\) and displacement field \(\mathbf{u}(\mathbf{X})\). The deformed configuration is

and the deformation gradient is

with the displacement gradient \(\mathbf{H} = \nabla \mathbf{u}\), where the operator \(\nabla \mathbf{u} = \frac{\partial u_i}{\partial X_j}\) denotes the gradient w.r.t. the reference configuration.

Momentum balance¶

In quasi-static settings (neglecting inertia) and without body forces (neglecting gravity), the local balance of linear momentum in the reference configuration reads

where \(\mathbf{P}\) is the first Piola-Kirchhoff stress.

Boundary conditions are applied as:

• Dirichlet: prescribed displacements on \(\Gamma_u\)
• Neumann: prescribed tractions on \(\Gamma_t\)

with \(\partial\Omega_0 = \Gamma_u \cup \Gamma_t\) and \(\Gamma_u \cap \Gamma_t = \varnothing\).

Material models¶

Material models describe how the stress in a material depends on its deformation. In a general sense, we can formulate this for an isothermal material abstractly as

with a function \(\mathcal{F}\) that maps a deformation gradient \(\mathbf{F}\) and material state vector \(\pmb{\alpha}\) to a stress.

Finite element method¶

Weak form¶

Let \(\delta\mathbf{u}\) be an admissible virtual displacement. The weak form of quasi-static mechanics (without body forces) is

Discretization¶

Shape functions¶

Assembly¶

Solution procedure¶