FerriteShells

Introduction

This package provides helper functions to assemble the different terms in the weak form of most classical shell formulations — C⁰ Kirchhoff–Love linear, C⁰ Koiter (non-linear Kirchhoff–Love), Reissner–Mindlin, and Naghi (non-linear Reissner–Mindlin) shells. Specifically, the classical membrane, bending, and shear contributions to the residuals and the consistent tangent stiffness matrix can be integrated and used with Ferrite.jl.

Some formulation that can be assembled with this package:

We refer the reader to the documentation for the specific weak form, numerical implementation and limitation of the different shell models.

ShellCellValues

Shells specialize the classical weak form obtained in continuum mechanics to a curvilinear coordinate system located on the shell's midsurface. As a result, classical continuum mechanics quantities, such as the Green–Lagrange strain tensor $\bf{E}$ of the elasticity tensor $\mathbb{C}$, change.

To help assemble these specific surface metrics, this package uses a new ShellCellValues<:AbstractCellValues, which behaves identically to Ferrite's CellValues, but additionally holds covariant basis vectors, metric tensors, and surface Jacobian at the integration points, which are used in the assembly of the different terms of the different formulations.

struct ShellCellValues{QR, IPG, IPS, T<:AbstractFloat, M} <: AbstractCellValues
    # quadrature and interpolation spaces
    qr       :: QR
    ip_geo   :: IPG
    ip_shape :: IPS
    # same as CellValues
    N, dNdξ, d2Ndξ2, detJdV :: Various{T}
    # additional fields for shells
    A₁, A₂, A₁₁, A₁₂, A₂₂ :: Vector{Vec{3, T}}
    G₃, T₁, T₂            :: Vector{Vec{3, T}}
    # shell measures
    A_metric :: Vector{SymmetricTensor{2, 2, T, 3}}
    B        :: Vector{SymmetricTensor{2, 2, T, 3}}
    # shear-locking treatment
    mitc     :: AbstractMITC
end

Calling reinit!(scv::ShellCellValues) computes the fixed covariant basis vectors $\bf{A}_1$ and $\bf{A}_2$ from the geometry of the shell's midsurface, while the current covariant basis vectors $\bf{a}_1$ and $\bf{a}_2$ are computed from the current configuration of the shell. The metric tensors $\bf{A}_{\alpha\beta}$ and $\bf{a}_{\alpha\beta}$ are then obtained as the inner product of the corresponding covariant basis vectors.

From these surface measures and the contravariant elasticity tensor $\mathbb{C}^{\alpha\beta\gamma\delta}$, the membrane, bending and shear strains can be computed, which are used in the assembly of the different terms in the different formulations.

Quickstart

Assembling the element contributions into the global system is identical to Ferrite, but instead of calling CellValues, the user needs to call ShellCellValues and use the corresponding assembly functions for the different terms in the different formulations. For example, for a non-linear Reissner–Mindlin shell, the assembly of the global consistent stiffness matrix and residual vector can be done as follows:

function assemble_shell!(K_int, r_int, dh, scv, u, mat)
    n_e = ndofs_per_cell(dh)
    ke_i = zeros(n_e, n_e); re_i = zeros(n_e)
    asm_i = start_assemble(K_int, r_int)
    for cell in CellIterator(dh)
        fill!(ke_i, 0.0); fill!(re_i, 0.0)
        reinit!(scv, cell)
        sd  = shelldofs(cell)
        u_e = u[sd]
        membrane_residuals_RM!(re_i, scv, u_e, mat)
        bending_residuals_RM!(re_i, scv, u_e, mat)
        membrane_tangent_RM!(ke_i, scv, u_e, mat)
        bending_tangent_RM!(ke_i, scv, u_e, mat)
        assemble!(asm_i, sd, ke_i, re_i)
    end
end

where shelldofs is a helper function (similar to celldofs) to get the degrees of freedom of the shell element, which are ordered as follows: first the in-plane displacements, then the out-of-plane displacements, and finally the rotations.