OneCAD-Simulation/docs/fem.md

FEM foundation: degrees of freedom and shape functions

What every element in crates/fem and every solver builds on. Code: crates/fem/src/dof.rs, crates/fem/src/basis.rs, crates/fem/src/jet.rs.

Degrees of freedom

Every unknown is a pair (node, DofKind):

DofKind	Meaning	Model / deck number
Ux, Uy, Uz	translations along X, Y, Z	1, 2, 3
Rx, Ry, Rz	rotations about X, Y, Z	4, 5, 6
P	pressure (mixed u–p elements)	8
T	temperature	11

The enum order (Ux, Uy, Uz, Rx, Ry, Rz, T, P) is the numbering order within a node. DofKind::from_number / number convert to the 1-based numbers boundary conditions use; from_direction / direction to the structural directions 0–5.

Elements declare their unknowns with ElementDofs { node: DofSet, internal: Internal }: the kinds every node of the element carries (DofSet::TRANSLATIONS for solids, PLANE for 2D elements, STRUCTURAL for shells and beams, TEMPERATURE for heat transfer, translations plus PRESSURE for continuous-pressure mixed elements) and the internal unknowns the element condenses out itself (incompatible_modes, discontinuous pressure), which never reach the global system. An element's matrices are ordered node by node, its node kinds in DofKind order within a node (a shell's 24 × 24 stiffness: node 1 Ux Uy Uz Rx Ry Rz, node 2 …). Solid elements: Formulation::dofs() / Solid::dofs().

Numbering (DofLayout): collect what every node needs with DofLayout::builder() (element(labels, dofs), node(label, kinds) for constraints and boundary conditions that need a kind no element gives, point(label, kinds) for reference points), then build(). A node carries the union of what was asked for. Unknowns are numbered mesh nodes in label order, each node's kinds in DofKind order, then the points in the order added; a slot's unknowns are contiguous, so a node pair is one dense block in the assembled matrix. The same model always gives the same numbering.

Lookups: node(label) / slot(label) (a slot is a position: mesh nodes 0..nodes(), points point_slot(k)), dof(slot, kind), range(slot), kinds(slot), locate(dof) -> (slot, kind), element_dofs(slots, kinds), describe(dof), stride() (uniform layouts).

Solvers: modelcraft_solver::system::Mesh::layout numbers the implicit system (mesh nodes, then six per coupled reference point); assembly (sparse::add_block(k, layout, slots, kinds, data), add_scalar, add_diagonal) works for any mix of kinds per node, so a solid sharing nodes with a shell adds its translation block into the shell nodes' six-unknown blocks. Boundary conditions and loads address (node, kind): a boundary condition holds the kinds a node carries, and a moment needs rotations. The explicit solver has solid nodes only and checks that its layout is the uniform node-major one (NODE_DOFS = 3).

Shape functions

Basis is every family's interpolation:

Family	Reference element	Nodes
Line2, Line3	ξ ∈ [−1, 1]	ends −1, 1; line3 adds the middle
Tri3, Tri6	ξ, η ≥ 0, ξ + η ≤ 1	corners (0,0), (1,0), (0,1); mid-sides of 1-2, 2-3, 3-1
Quad4, Quad8, Quad9	[−1, 1]²	corners counter-clockwise from (−1,−1); mid-sides of 1-2, 2-3, 3-4, 4-1; centre
Tet4, Tet10	unit tetrahedron	corners (0,0,0), (1,0,0), (0,1,0), (0,0,1); mid-sides of 1-2, 2-3, 3-1, 1-4, 2-4, 3-4
Wedge6, Wedge15	triangle × ζ ∈ [−1, 1]	bottom 1-3, top 4-6; mid-sides of the bottom edges, top edges, vertical edges
Pyramid5	base [−1, 1]² at ζ = 0, apex (0, 0, 1)	base corners as quad4, then the apex
Hex8, Hex20, Hex27	[−1, 1]³	corners 1-4 at ζ = −1, 5-8 above; hex20 mid-sides (bottom, top, vertical edges); hex27 face centres in ElementKind::Hex8.faces() order, then the centre

Node orders are the model's (ElementKind::edges() / faces() are the single source): corners, mid-side nodes in edge order, face nodes, body nodes. Basis::of(ElementKind) maps a model element; nodes() gives natural node coordinates; dim(), count(), corners(), degree() (the complete polynomial degree reproduced: 1 for linear families including the rational pyramid, 2 for the quadratic ones).

Evaluation at a natural point xi: [f64; 3] (unused coordinates are zero):

• eval(xi) -> Shape: values n and first derivatives dn[i] = ∂Nᵢ/∂(ξ, η, ζ). tet4, tet10, wedge6, hex8, hex20 use the hand-written functions of shape.rs (the solid kernels' path).
• eval2(xi) -> Shape2: adds d2n[i][a][b] = ∂²Nᵢ/∂ξₐ∂ξ_b, from second-order forward-mode automatic differentiation of the same formulas (jet::Jet): exact to round-off.
• gradients(&shape, xyz) -> Gradients: physical dx[i] = ∇ₓNᵢ = J⁻¹ ∇_ξNᵢ and det J, with J_ab = ∂x_b/∂ξₐ; an inverted or collapsed element is Error::Degenerate.
• hessians(&shape2, xyz) -> Hessians: physical d2x[i] = J⁻¹ (H_ξ − Σ_b (∂N/∂x_b) X_b″) J⁻ᵀ with X_b″ = Σᵢ x_{i,b} H_ξ(Nᵢ), the map's second derivative (T. J. R. Hughes, The Finite Element Method, 2000, §3.9, §4.2), and laplacian(i). Without the X″ term a field linear in x would have a nonzero Hessian on a curved element.

A family of dimension d maps onto the first d physical coordinates (plane elements in x–y, lines along x). Elements embedded in a higher dimension (shells, beams, trusses in 3D) pass node coordinates in their own local frame.

Verification (basis::tests, dof::tests, jet::tests)

For every family: Kronecker delta at the nodes; partition of unity (Σ N = 1, Σ ∇N = 0, Σ H = 0); exact reproduction of every monomial up to degree() with its gradient and Hessian in natural coordinates; first and second derivatives against central differences; on an affine map, physical gradients, Hessians and Laplacians exact for a complete polynomial of the family's degree; on a curved map, linear fields exact with zero Hessian, and the physical Hessian of a quadratic interpolant consistent with finite differences of its physical gradient. The solver and explicit migration to DofLayout was checked bit for bit: every result of the solver, explicit, engine and io test suites hashes identically before and after.