# Solenoid Stress & Strain
This package includes three complementary layers:
- a 1D finite-difference radial stress solver for winding-pack models with zero `rz` shear,
- a 2D quadrilateral FEM solver with axisymmetric and plane-strain formulations, reusable sparse load operators, cached Rust-side LU solves, and optional BiCGSTAB iterative solves,
- analytic reference formulas used for validation and convergence studies.
## 1D Finite-Difference Solver
::: cfsem.solenoid_stress.SolenoidStress1D
::: cfsem.solenoid_stress.SolenoidStress1DOperators
::: cfsem.solenoid_stress.solenoid_1d_structural_factor
::: cfsem.solenoid_stress.solenoid_1d_structural_rhs
## 2D FEM
The FEM path supports:
- axisymmetric and plane-strain structural formulations,
- `quad4`, inferred `quad9`, and explicit `quad9` elements,
- `gl3` and `gl4` quadrature,
- optional per-element in-plane material orientation angles,
- optional threaded stiffness assembly with `par=True`,
- reusable reduced-space operators for body force, pressure, traction, and nodal-temperature thermal strain,
- reduced quadrature-point recovery operators for strain and stress,
- direct sparse-LU or BiCGSTAB reduced-system solves,
- model-owned Dirichlet constraints applied during assembly.
The intended workflow is:
1. call `assemble_structural_2d(...)` once with mesh, materials, load topology, and prescribed Dirichlet values,
2. build each reduced load vector with `model.build_rhs(...)` or the exposed sparse operators,
3. solve with `model.solve(rhs)`, using the default cached LU factorization or
`model.solve(rhs, method="bicgstab", ...)` for an iterative solve,
4. recover quadrature strain and stress with `model.evaluate_quadrature(...)`.
By default, `model.solve(rhs)` uses the direct sparse-LU path and returns the full displacement
array. Passing `method="bicgstab"` selects the iterative solver; this path supports diagonal
preconditioning, Ruiz-style row/column equilibration, optional reuse of the previous converged
iterative solution as the initial guess, and optional diagnostics via `return_diagnostics=True`.
Equilibration scales and the scaled stiffness matrix are cached on the model for repeated
right-hand sides. The BiCGSTAB tolerance is passed unchanged to the system being solved. With
equilibration enabled, this means the tolerance applies to the equilibrated residual rather than
being rescaled to original reduced-RHS units.
### Formulation Notes
The axisymmetric and plane-strain solvers share the same 2D quadrilateral mesh, two displacement
unknowns per node, and four-component strain/stress storage. The difference is how that 2D mesh
represents a 3D body.
For `formulation="axisymmetric"`, the coordinates are interpreted as `(r, z)`. Each quadrature
area sample represents a full ring, so stiffness and load integrals use the volume scale
`2*pi*r*dA`. The strain vector is `[rr, zz, tt, rz]`; the out-of-plane hoop strain is not an
independent displacement derivative, but is recovered from the radial displacement as
`epsilon_tt = u_r / r`.
For `formulation="plane_strain"`, the coordinates are interpreted as `(x, y)`. Each area sample
represents a prismatic slice with user-supplied `thickness`, so integrals use the volume scale
`thickness*dA`. The strain vector is `[xx, yy, zz, xy]`; the out-of-plane strain is constrained to
`epsilon_zz = 0`, while `sigma_zz` can still be nonzero through the constitutive matrix.
Plane strain and plane stress are different 2D reductions. Plane strain models a body that is long,
periodic, or otherwise constrained in the out-of-plane direction, with zero out-of-plane strain and
generally nonzero out-of-plane stress. Plane stress models a thin sheet or plate with traction-free
faces through the thickness, with zero out-of-plane stress and generally nonzero out-of-plane
strain. This FEM path currently implements axisymmetric and plane-strain reductions; it does not
implement a plane-stress constitutive reduction.
::: cfsem.solenoid_stress.fem2d
## Analytic Reference Formulas
::: cfsem.solenoid_stress.s_long_solenoid
::: cfsem.solenoid_stress.s_radial_thick_wall_cylinder
::: cfsem.solenoid_stress.s_hoop_thick_wall_cylinder