Solid¶

Initialize a finite-element model.

Parameters:

• nodes (Tensor) –

  Nodal coordinates with shape [n_nod, n_dim].

• elements (Tensor) –

  Connectivity with shape [n_elem, n_nodes_per_element].

• material (Material | None) –

  Material model. If not vectorized, it is vectorized over elements during initialization. May be None for shells that use a laminate section instead.

Solve the quasi-static finite-element problem by load increments.

Parameters:

• increments (Tensor | None, default: None ) –

  Load scale factors, typically [0, 1]. They may rise and fall, so a load cycle is expressed as a sequence like [0, 1, 0]. Results are always returned at exactly these values. If a Newton solve does not converge, the increment is subdivided internally and retried, and the substep is grown again after each success.

• max_iter (int, default: 10 ) –

  Maximum Newton iterations before an increment is cut back.

• rtol (float, default: 1e-08 ) –

  Relative residual tolerance for Newton convergence.

• atol (float, default: 1e-06 ) –

  Absolute residual tolerance for Newton convergence.

• stol (float, default: 1e-10 ) –

  Tolerance used by iterative linear solvers.

• cutback_factor (float, default: 0.5 ) –

  Factor applied to the substep size after a Newton solve failed to converge.

• growth_factor (float, default: 1.1 ) –

  Factor applied to the substep size after a Newton solve converged, capped at the requested increment.

• max_cutbacks (int, default: 10 ) –

  Number of successive cutbacks accepted within an increment before the solve is given up.

• verbose (bool, default: False ) –

  If True, reports the solver configuration and a table of per-increment progress, updated in place inside notebooks.

• method (Literal['direct', 'cg', 'bicgstab'] | None, default: None ) –

  Linear solver method, chosen by size and tangent symmetry when omitted.

• preconditioner (Literal['amg', 'jacobi', 'none'] | None, default: None ) –

  Preconditioner for an iterative method, chosen by device and available backends when omitted.

• device (str | None, default: None ) –

  Optional device hint for the linear solver backend.

• return_intermediate (bool, default: False ) –

  If True, returns values for all increments.

• aggregate_integration_points (bool, default: True ) –

  If True, averages flux, gradient, and state over integration points.

• alpha (float, default: 0.0 ) –

  Damping factor for viscous stabilization. Dissipated energy is accumulated in self.stabilization_energy.

• differentiable_parameters (Tensor | Iterable[Tensor] | None, default: None ) –

  Explicit parameter(s) to differentiate through implicit Newton/sparse solves. Accepts either a single tensor or an iterable of tensors.

Returns:

• tuple[Tensor, Tensor, Tensor, Tensor, Tensor] –

  Tuple of displacement, internal force, flux, gradient, and material state. If return_intermediate is True, each tensor includes an increment dimension as the leading axis.

Plot the mesh with optional node and element properties.

Parameters:

• u (float or Tensor, default: 0.0 ) –

  Displacement field. Defaults to 0.0.

• node_property (Tensor or dict[str, Tensor], default: None ) –

  Nodal property to plot, optionally keyed by its color bar title. Defaults to None.

• element_property (Tensor or dict[str, Tensor], default: None ) –

  Element property to plot, keyed like node_property. Defaults to None.

• orientations (Tensor, default: None ) –

  Element orientations with shape [n_elem, k, 3] with k <= 3, drawn as red, green, and blue arrows. Defaults to None.

• show_edges (bool, default: True ) –

  Show edges. Defaults to True.

• show_undeformed (bool, default: False ) –

  Show undeformed mesh. Defaults to False.

• show_outline (bool, default: False ) –

  Show a box around the full mesh. Defaults to False.

• axes (bool, default: False ) –

  Show labeled coordinate axes around the mesh. Defaults to False.

• bcs (bool, default: False ) –

  If True, render boundary conditions (forces as arrows, prescribed displacements as arrows and tip markers, and constrained DOFs as cones). Defaults to False.

• clip (tuple[str, float], default: None ) –

  Property and value to cut the mesh at. Culls orientations and boundary conditions with it. Defaults to None.

• plotter (Plotter, default: None ) –

  PyVista plotter. Defaults to None.

• camera (str or list, default: None ) –

  Camera position, either a plane ("xy", "xz", "yz"), "iso", or an explicit position, focal point and view up. Defaults to None.

• **kwargs –

  Additional keyword arguments passed to pyvista.Plotter.add_mesh.

Compute the natural frequencies and mode shapes.

Solves the generalized eigenvalue problem

Parameters:

• n_modes (int) –

  Number of eigenpairs to compute.

Returns:

• tuple[Tensor, Tensor] –

  Tuple (omega_sq, modes) where omega_sq has shape [n_modes] (squared angular frequencies, differentiable) and modes has shape [n_modes, n_nod, n_dof_per_node] (detached).

Initialize a finite-element model.

Parameters:

• nodes (Tensor) –

  Nodal coordinates with shape [n_nod, n_dim].

• elements (Tensor) –

  Connectivity with shape [n_elem, n_nodes_per_element].

• material (Material | None) –

  Material model. If not vectorized, it is vectorized over elements during initialization. May be None for shells that use a laminate section instead.

Solve the quasi-static finite-element problem by load increments.

Parameters:

• increments (Tensor | None, default: None ) –

  Load scale factors, typically [0, 1]. They may rise and fall, so a load cycle is expressed as a sequence like [0, 1, 0]. Results are always returned at exactly these values. If a Newton solve does not converge, the increment is subdivided internally and retried, and the substep is grown again after each success.

• max_iter (int, default: 10 ) –

  Maximum Newton iterations before an increment is cut back.

• rtol (float, default: 1e-08 ) –

  Relative residual tolerance for Newton convergence.

• atol (float, default: 1e-06 ) –

  Absolute residual tolerance for Newton convergence.

• stol (float, default: 1e-10 ) –

  Tolerance used by iterative linear solvers.

• cutback_factor (float, default: 0.5 ) –

  Factor applied to the substep size after a Newton solve failed to converge.

• growth_factor (float, default: 1.1 ) –

  Factor applied to the substep size after a Newton solve converged, capped at the requested increment.

• max_cutbacks (int, default: 10 ) –

  Number of successive cutbacks accepted within an increment before the solve is given up.

• verbose (bool, default: False ) –

  If True, reports the solver configuration and a table of per-increment progress, updated in place inside notebooks.

• method (Literal['direct', 'cg', 'bicgstab'] | None, default: None ) –

  Linear solver method, chosen by size and tangent symmetry when omitted.

• preconditioner (Literal['amg', 'jacobi', 'none'] | None, default: None ) –

  Preconditioner for an iterative method, chosen by device and available backends when omitted.

• device (str | None, default: None ) –

  Optional device hint for the linear solver backend.

• return_intermediate (bool, default: False ) –

  If True, returns values for all increments.

• aggregate_integration_points (bool, default: True ) –

  If True, averages flux, gradient, and state over integration points.

• alpha (float, default: 0.0 ) –

  Damping factor for viscous stabilization. Dissipated energy is accumulated in self.stabilization_energy.

• differentiable_parameters (Tensor | Iterable[Tensor] | None, default: None ) –

  Explicit parameter(s) to differentiate through implicit Newton/sparse solves. Accepts either a single tensor or an iterable of tensors.

Returns:

• tuple[Tensor, Tensor, Tensor, Tensor, Tensor] –

  Tuple of displacement, internal force, flux, gradient, and material state. If return_intermediate is True, each tensor includes an increment dimension as the leading axis.