Shell Cell Values

ShellCellValues(geom_interpol::Interpolation, func_interpol::Interpolation, quad_rule::AbstractQuadratureRule)

A ShellCellValues object stores precomputed shape function data and reference geometry for a shell element. Works with Vec{3} node coordinates — no manual 2D projection required.

Shape function data (N, dNdξ, d2Ndξ2) are computed once at construction time from ip_shape and qr. The reference coordinates in the physical space then computed on the fly using the geometric interpolation ip_geo $x(\xi) = \sum N_{i}^\text{geo}(\xi) x_{i}$ and the solution field are interpolated via the shape functions ip_shape $u(\xi) = \sum N_{i}^\text{shape}(\xi) u_{i}$.

Arguments:

• geom_interpol: an instance of an Interpolation which is used to interpolate the geometry. By default linear Lagrange interpolation is used.
• func_interpol: an instance of an Interpolation used to interpolate the approximated function
• quad_rule: an instance of a AbstractQuadratureRule

Keyword arguments: The following keyword arguments are experimental and may change in future minor releases

• mitc: an instant of MITC to specify the shear treatment used in the element (default NoMITC)
• E : an instance of AbstractStrainMeasure to specify the strain measure used in the element (default GreenLagrangeStrain)

Common methods:

• reinit! computes the reference geometry ($A_1$, $A_2$, $G_3$, $B$, $\cdots$) by differentiating the coordinate map using ip_geo.

reinit!(scv::ShellCellValues, x::AbstractVector)
reinit!(scv::ShellCellValues, cc::CellCache)
reinit!(scv::ShellCellValues, cell::AbstractCell)

Update the ShellCellValues object for a cell with cell coordinates x. The derivatives of the shape functions, and the new integration weights are computed.

The reference surface measures such as the covariant basis are recomputed and stored per quadrature point.

Note: For ShellCellValues where a shear treatment has been specified, the MITC data is also reinit!.

function_value(scv, qp, u_e)

Interpolate the displacement field at quadrature point qp from a flat DOF vector u_e. Works for both KL (3 DOFs/node: $[u_1,u_2,u_3,\cdots]$) and RM (5 DOFs/node: $[u_1,u_2,u_3,\varphi_1,\varphi_2,\cdots]$). The DOF stride is inferred from length(u_e) ÷ n_nodes; only the first 3 DOFs of each node (the displacement components) are used.

kinematics(scv, qp, u_e)

Compute current kinematics at quadrature point qp given nodal displacements u_e (flat vector of length 3 * n_nodes: [$u_1$, $v_1$, $w_1$, $\cdots$]).

Reference geometry (A_1, A_2, A_metric) is read from scv, which must have been reinit!-ed with the element coordinates before calling this function.

Returns $(a_1, a_2, A_\text{metric}, a_\text{metric})$.