Date: 2026-04-17 (rev. 2026-06-05: side-connector material removed; rev. 2026-07-31: HEX8TB returned as the wall element, notes updated) Module: src/fem/interpolation/nedelec Purpose: Document BELFEM's reduced-dimension Nedelec elements used for thin-shell tape models: the surface elements QUAD4TS and PENTA6TS.

Historical note: earlier revisions of this file also documented a side-connector element (HEX8TS, and before it an experimental HEX8TB) used to model a wrap-around metallic connector on the tape edge. That wrap construction was removed from BELFEM in June 2026 after it was found to be unphysical (it could not represent the binormal-H field at the shell/connector fold and suppressed the side-curve current density). Side connectors returned in July 2026 as a different element: the degenerate HEX8TB wall element with four longitudinal edge dofs and no binormal/normal dofs — the free-parameter failure mode of the wrap is designed away. See src/fem/maxwell/doc/side_coating_wall_element.md for the full theory and status; references to the old HEX8TS wrap remain stale.

1. Overview

BELFEM models REBCO tapes as thin stacks of material layers whose thicknesses are far smaller than the other geometric dimensions. Resolving every layer with regular 3D volume elements would require extreme aspect ratios and a prohibitive number of DOFs. The thin-shell approach collapses each layer to a reduced element and performs the through-thickness behavior in the formulation rather than in the mesh.

BELFEM uses two reduced edge-element types in this context:

Element	Base geometry	Embedding	DOFs	Use case
QUAD4TS	LINE2	line in 2D	2	2D thin shells
PENTA6TS	TRI3	surface in 3D	6	3D thin shells

QUAD4TS and PENTA6TS are the classical thin-shell elements in the Alves sense: one reference direction is the collapsed through-thickness direction.

2. QUAD4TS and PENTA6TS

2.1 Geometry and Jacobian handling

QUAD4TS and PENTA6TS are surface elements embedded in a higher-dimensional physical space. Their geometric Jacobian is therefore rectangular, so the implementation uses the standard Gram-matrix / pseudo-inverse construction:

G = J J^T

J^+ = J^T G^-1

The columns of J^+ are the physical gradients of the in-surface reference coordinates. The through-thickness direction is handled separately with an explicit shell thickness and a shell normal.

This is the pattern used in:

2.2 DOF structure

The reduced shell elements carry only face-edge DOFs:

QUAD4TS has 2 edge DOFs, one on each shell face.
PENTA6TS has 6 edge DOFs, three on the bottom triangular face and three on the top triangular face.

There are no through-thickness edge DOFs. The through-thickness dependence is represented by the shell formulation itself.

2.3 Layer interfaces inside the tape stack

Inside a multilayer shell stack, neighboring shell layers may either:

share edge DOFs directly,
use duplicated edges and ghost-facet coupling on H-H interfaces, or
collapse to nodal phi behavior for insulating layers such as the buffer.

The mesh-side plumbing for this lives in src/fem/kernel/cl_ThinShellFactory.cpp, especially the layer creation, duplicate-edge creation, ghost-facet generation, and buffer handling.

2.4 Buffer treatment

For a phi-formulation buffer layer, the shell edge DOFs are not meant to represent a conducting layer. The buffer block is converted to nodal behavior and the shell-edge values are rerouted to the corresponding nodal sources in create_buffers(). The buffer is therefore locally insulating even if the surrounding conductor is globally connected elsewhere.

2a. QUAD4TS layer elements are left-handed by construction

The 2-D layer elements produced by ThinShellFactory have a negative scalar Lagrange determinant. This is a deliberate consequence of the construction, not a mesh defect, and it must not be "repaired" by renumbering the corners.

process_nodes_line2 builds each LINE2 facet normal as n = ( t_y, -t_x ) — the facet tangent rotated clockwise. Layer offsets ascend, so the top curve always lies at +h·n, and create_elements_on_blocks_line2 winds the quad ( bottom0, bottom1, top1, top0 ). Its signed area is h·( t × n ) = -h·|t|, negative for every facet at every orientation — n is derived from t, so reversing the facet reverses both.

That ordering is the only legal one. Two contracts pin it:

Element_QUAD4TS::get_nodes_of_edge declares edge 0 to be nodes {0,1} (the bottom curve) and edge 1 to be {3,2} (the top curve), while ThinShellFactory::link_elements_with_edges assigns the bottom layer's mesh Edge to local edge 0 positionally. So {0,1} must stay the bottom curve.
The edge signs mS are derived from the element's node order (Element::compute_edge_directions compares the mesh Edge's node ids against get_nodes_of_edge), whereas the Nédélec tangent ∇ξ is taken from the facet (EF_QUAD4TS::link). Their product is the basis direction, so the local edge order must follow the facet edge direction.

Keeping both forces ( bottom0, bottom1, top1, top0 ), which is clockwise. No permutation of the four corners is both counter-clockwise and sign-neutral. The 3-D sibling has no such conflict: process_nodes_tri3 uses the right-hand normal ( B-A ) × ( C-A ), and the PENTA6TS stacking agrees with it, so those elements are positively oriented.

What this means for consumers

The map is still a diffeomorphism, so only measures are affected, never gradients:

N is reference-space, and B = J⁻¹·dN/dξ returns true Cartesian gradients whichever orientation the map has (the inverse divides by the signed determinant, so the sign cancels).
The volume weight is the one orientation-sensitive quantity. Change of variables asks for |det J|, which is why Calculator::dV_quad4ts and Pipette::measure_quad4ts exist: they are QUAD4TS-only wrappers taking the absolute value. That is the exact weight, not a clamp.
The magnetic solve is unaffected in practice — it reaches those wrappers too, but takes the edge-function branch where the value is 0.25·thickness·length, positive for any thickness that reaches assembly, so the absolute value is the identity.

Two traps follow, and both are real:

MeshChecker's swap table lists QUAD4TS. Its swap_quad4 exchanges nodes 1 and 3, which on a thin-shell element turns the tangential pair into the through-thickness pair — the edge function would then read the tape length as its thickness. The checker never sees these elements (they are created later, after edges exist), but the table should not be taken as an endorsement.
A negative layer thickness in the deck inverts the stack and flips all of the above. Nothing rejects it at parse time; Kernel::compute_element_volumes catches it later.

3. Implementation map

File	Role
`cl_EF_QUAD4TS.cpp`	2D shell edge function
`cl_EF_PENTA6TS.cpp`	3D shell edge function
src/fem/kernel/cl_ThinShellFactory.cpp	Mesh creation for thin-shell layers

4. Summary

BELFEM uses two reduced Nedelec element families for tape-shell work:

QUAD4TS for 2D thin shells,
PENTA6TS for 3D thin shells.

Both are the literature-based thin-shell elements derived from the collapsed-shell construction (Alves). References to the old HEX8TS wrap side-connector construction are stale — only that construction was removed (June 2026); the HEX8TS element type and EF_HEX8TS themselves remain in the code as thin-shell machinery. HEX8TB is current again since July 2026 as a different element (the 4-dof side-connector wall element; see the historical note at the top and src/fem/maxwell/doc/side_coating_wall_element.md).

5. References

Alves, B. de Sousa et al. (2022b), "A thin-shell H-phi-formulation for superconducting devices." Supercond. Sci. Technol. 35, 024001.
Messe, C. et al. (2023), "BELFEM: a special-purpose finite-element code for the magnetodynamic modeling of high-temperature superconducting tapes." Supercond. Sci. Technol. 36, 114001.
General Nedelec references — see nedelec.md.