MITC{N,M,T}

Mixed Interpolation of Tensorial Components data for the N-node shell element (Bucalem & Bathe 1993). Eliminates transverse shear locking by evaluating the covariant shear strains $\gamma_\alpha = a_\alpha \cdot d$ at fixed tying points and interpolating back to Gauss points.

The M tying entries are component-tagged: entry k ties the covariant component α_tie[k] at ξ_tie[k], and both h_tie_1 and h_tie_2 span all entries, $\gamma_\alpha(\xi_q) = \sum_k h^\alpha_{qk}\,\gamma_k$. Quadrilateral schemes leave the off-component columns zero ($\gamma_1$ is built from $\gamma_1$ tying values only); triangular ones do not, since the hypotenuse condition couples the two components (Lee & Bathe 2004).

Every scheme is declared the same way: a tying_conditions(::typeof(MITCx)) method returning its tying conditions and the assumed-strain space they are tied against, which the shared MITC{N}(ip_shape, qr, scheme) constructor feeds to tying_weights.

Static fields (N_tie, dN_tie, h_tie_*) are precomputed once at construction. Mutable fields (A_tie, d₀_tie, *_node) are updated each reinit! call.

MITC{9,12,T}

Mixed Interpolation of Tensorial Components data for the 9-node quadrilateral shell element (Bucalem & Bathe 1993).

Tying points (reference domain $[-1,1]^2$), on the $\pm1/\sqrt3$ Gauss lines: $\gamma_1 = a_1 \cdot d$ at $(\pm1/\sqrt3,\ -1), (\pm1/\sqrt3,\ 0), (\pm1/\sqrt3,\ +1)$ $\gamma_2 = a_2 \cdot d$ at $(-1,\ \pm1/\sqrt3), (0,\ \pm1/\sqrt3), (+1,\ \pm1/\sqrt3)$ Each condition ties a single covariant component, so $M = 12$ and the off-component columns of h_tie_1/h_tie_2 vanish.

Assumed field: $\tilde\gamma_1 \in$ span$\{1,\xi_1\}\otimes\{1,\xi_2,\xi_2^2\}$ — linear in $\xi_1$, quadratic in $\xi_2$ — and $\tilde\gamma_2$ its transpose. The $\pm1/\sqrt3$ stations are the superconvergent points of the degree that is dropped.

MITC{4,4,T}

Mixed Interpolation of Tensorial Components data for the 4-node quadrilateral shell element (Dvorkin & Bathe 1984).

Tying points (reference domain $[-1,1]^2$), at the edge midpoints: $\gamma_1 = a_1 \cdot d$ at $(0, \pm1)$ $\gamma_2 = a_2 \cdot d$ at $(\pm1, 0)$ Each condition ties a single covariant component, so $M = 4$ and the off-component columns of h_tie_1/h_tie_2 vanish.

Assumed field: $\tilde\gamma_1 = a_1 + b_1\xi_2$ (constant in $\xi_1$, linear in $\xi_2$) and $\tilde\gamma_2 = a_2 + b_2\xi_1$, the spaces obtained by differentiating the bilinear displacement field $w = c_0 + c_1\xi_1 + c_2\xi_2 + c_3\xi_1\xi_2$.

MITC{6,10,T}

Mixed Interpolation of Tensorial Components data for the 6-node triangular shell element, variant MITC6-a (Lee & Bathe 2004, Eq. 35–39) — linear transverse shear along the edges plus one interior tying point carrying the quadratic part.

Tying points (reference domain $r,s \ge 0$, $r+s \le 1$): the two Gauss stations $1/2 \mp 1/(2\sqrt3)$ of each edge — $\gamma_1$ on $s=0$, $\gamma_2$ on $r=0$, and $\gamma_q = (\gamma_2-\gamma_1)/\sqrt{2}$ on the hypotenuse — plus the centroid $(1/3,1/3)$. The hypotenuse and centroid conditions need both covariant components, giving $M = 10$ entries.

Assumed field (rotated Raviart–Thomas $RT_1$, the MITC7 plate space), Lee & Bathe Eq. (39): $\tilde\gamma_1 = a_1 + b_1 r + c_1 s + s(d r + e s)$, $\tilde\gamma_2 = a_2 + b_2 r + c_2 s - r(d r + e s)$.

Dropping the two interior conditions and the last two basis fields gives the linear variant MITC6-b (Eq. 40–41), which is stiffer in bending-dominated problems.

MITC{3,4,T}

Mixed Interpolation of Tensorial Components data for the 3-node triangular shell element (Lee & Bathe 2004). The transverse shear is assumed constant along each edge, the triangular counterpart of MITC4.

Tying points (reference domain $r,s \ge 0$, $r+s \le 1$), at the edge midpoints: $\gamma_1 = a_1 \cdot d$ at $(1/2, 0)$ $\gamma_2 = a_2 \cdot d$ at $(0, 1/2)$ $\gamma_q = (\gamma_2-\gamma_1)/\sqrt{2}$ at $(1/2, 1/2)$ — the hypotenuse condition, which needs both covariant components there, giving $M = 4$ tying entries.

Assumed field (rotated Raviart–Thomas $RT_0$), Lee & Bathe Eq. (25): $\tilde\gamma_1 = a_1 + c\,s$, $\tilde\gamma_2 = a_2 - c\,r$, i.e. the span of $(1,0)$, $(0,1)$ and $(s,-r)$.

tying_shear_strains(mitc::MITC{N,M,T}, u_e)

Compute the covariant shear strain $\gamma_k = a_{\alpha_k} \cdot d$ of every tying entry k from the current DOF vector u_e (5 DOFs/node: [$u_1$,$u_2$,$u_3$,$\varphi_1$,$\varphi_2$,$\cdots$]). Returns an NTuple of length M, ForwardDiff-safe. Each value subtracts its own reference $A_{\alpha_k}\cdot G_3$, so the tying strains vanish in the reference configuration. Call once before the quadrature-point loop and pass to shear_strains.

shear_strains(a₁, a₂, d, qp, γ_k, mitc)

Return (γ₁, γ₂) at quadrature point qp. With MITC: weighted sum of the tying-entry values from tying_shear_strains. Without MITC: direct dot(a₁, d), dot(a₂, d).

tying_weights(qr, conds, basis)

Build the tying entries and interpolation weights of an assumed transverse-shear field $\tilde\gamma(\xi) = \sum_j c_j P_j(\xi)$ from its tying conditions (Lee & Bathe 2004, §3.2).

• basis[j](ξ) -> Vec{2} — the $j$-th assumed field, as covariant components $(\gamma_1,\gamma_2)$.
• conds[i] = (ξ, w) — condition $w \cdot \tilde\gamma(\xi) = w \cdot \gamma(\xi)$, with w the tied direction: Ê₁/Ê₂ on the $\xi_1$/$\xi_2$ edges, Ê_q on the hypotenuse.

A condition needs the displacement-based $\gamma_\alpha(\xi)$ for every component with $w_\alpha \neq 0$; the union of those $(\xi,\alpha)$ pairs forms the tying entries $k = 1\ldots M$. With C[i,j] = w_i ⋅ P_j(ξ_i) and W[i,k] = w_i[α_k] (entry k located at ξ_i, else 0), the coefficients are $c = C^{-1} W \gamma^\text{tie}$, hence $h_\alpha[q,k] = \sum_j P_j(\xi_q)_\alpha (C^{-1}W)_{jk}$.