forked from 1b-agent/fieldcraft-upstream
Shells and beams: MITC4/MITC3 shells, Timoshenko beams, six-DOF nodes (M4) (#47)
* fem: MITC4/MITC3 flat shells and Timoshenko beams with profile properties (ElementDofs STRUCTURAL) Co-Authored-By: Claude Opus 5.5 <[email protected]> Claude-Session: https://claude.ai/code/session_01NmAttHhUZTokgPeEJia8sm * solver: shells and beams on the DofLayout beside solids, 2D elements and trusses; analytic verification and tagged benchmarks Co-Authored-By: Claude Opus 5.5 <[email protected]> Claude-Session: https://claude.ai/code/session_01NmAttHhUZTokgPeEJia8sm * engine: probe rotations, reaction moments and shell and beam element results; docs for shells and beams Co-Authored-By: Claude Opus 5.5 <[email protected]> Claude-Session: https://claude.ai/code/session_01NmAttHhUZTokgPeEJia8sm --------- Co-authored-by: jyviko <[email protected]> Co-authored-by: Claude Opus 5.5 <[email protected]>
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//! Flat shell elements: the four-node MITC4 quadrilateral and the three-node MITC3 triangle,
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//! with six degrees of freedom per node (three translations, three rotations).
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//!
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//! Each element is flat: its nodes are projected onto the element's mean plane, where the
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//! membrane, the Reissner–Mindlin plate and a drilling stiffness are superposed in local axes and
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//! the result rotated to the global ones (the facet approach; Zienkiewicz, Taylor & Fox 2014,
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//! *The Finite Element Method for Solid and Structural Mechanics*, 7th ed., ch. 9–10). Curved
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//! shells are approximated by facets; quadrilaterals on a cylinder whose edges follow the
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//! generators and the circumference are exactly flat.
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//!
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//! - **Kinematics.** With the reference surface at the nodes and z along the element normal,
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//! u = u₀ + z θ_y, v = v₀ − z θ_x, w = w₀: membrane strains ε = (u₀,x, v₀,y, u₀,y + v₀,x),
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//! curvatures κ = (θ_y,x, −θ_x,y, θ_y,y − θ_x,x) and transverse shear strains
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//! γ = (w,x + θ_y, w,y − θ_x).
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//! - **Section.** Linear elastic and homogeneous through the thickness, so the integration
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//! through the thickness is done exactly in closed form (any number of section points gives the
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//! same stiffness): with the mid-surface at z = e from the reference surface (e = −offset · t),
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//! N = A ε + B κ, M = B ε + D κ with A = Q t, B = Q t e, D = Q (t³/12 + t e²) for the plane
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//! stress matrix Q, and Q_s = k G t γ with the shear correction k = 5/6.
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//! - **Transverse shear, quadrilateral (MITC4).** The covariant transverse shear strains are
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//! sampled at the mid-sides and interpolated linearly across the element: ẽ_rt from the points
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//! (0, ±1), ẽ_st from (±1, 0), then turned into Cartesian strains with the Jacobian
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//! (E. N. Dvorkin, K.-J. Bathe, *A continuum mechanics based four-node shell element for
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//! general nonlinear analysis*, Engineering Computations 1(1):77–88, 1984; in the flat-plate
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//! form of K.-J. Bathe, E. N. Dvorkin, *A four-node plate bending element based on
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//! Mindlin/Reissner plate theory and a mixed interpolation*, IJNME 21(2):367–383, 1985). This
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//! removes shear locking while passing the patch tests; membrane and bending use the bilinear
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//! fields, all integrated with 2 × 2 Gauss points.
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//! - **Transverse shear, triangle (MITC3).** The covariant shear strains are tied at the three
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//! mid-sides: ẽ_rt = e_rt⁽¹⁾ + c s, ẽ_st = e_st⁽²⁾ − c r with c = e_rt⁽³⁾ − e_rt⁽¹⁾ − e_st⁽³⁾ +
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//! e_st⁽²⁾, tying points (½, 0), (0, ½) and (½, ½), which keeps the tangential shear strain of
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//! each edge constant along it (P.-S. Lee, K.-J. Bathe, *Development of MITC isotropic
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//! triangular shell finite elements*, Computers & Structures 82(11–13):945–962, 2004). The
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//! membrane is the constant-strain triangle.
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//! - **Drilling rotation.** The rotation about the normal has no stiffness in shell theory; it
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//! is tied to the membrane's in-plane rotation ω = ½ (v,x − u,y) by the penalty
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//! ∫ γ t (θ_z − ω)² dA of the variational formulation of T. J. R. Hughes, F. Brezzi, *On drilling
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//! degrees of freedom*, CMAME 72(1):105–121, 1989, with γ = G/100. Hughes and Brezzi take
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//! γ = G with a drilling-enhanced (Allman-type) membrane; on the plain bilinear membrane the
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//! in-plane rotation of a parallelogram is linear, which θ_z interpolates exactly, so the penalty
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//! does not lock, and the smaller γ keeps distorted elements free of it too. Being a function of
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//! θ_z − ω it leaves rigid motions free. Scaled with G t like the membrane (not with the bending
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//! rigidity), it ties each node's rotation about the surface normal to the membrane as the mesh
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//! is refined, so curved shells modelled with facets converge steadily: the Scordelis–Lo and
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//! pinched-cylinder benchmarks change by less than 0.5 % for γ from G/1000 to G, whereas a
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//! penalty that is a fixed fraction of the bending stiffness lets that rotation go soft on fine
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//! meshes.
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//! - **Mass.** Consistent: ρ t on the translations, ρ (t³/12 + t e²) on the rotations and
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//! ρ t e coupling translations and bending rotations when the reference surface is offset. The
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//! drilling rotation gets the same rotary inertia as the bending ones (shell theory gives it
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//! none; it is tiny next to the in-plane inertia of the translations and keeps the mass matrix
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//! positive definite, as the frequency solver's Lanczos iteration needs). Lumped: the diagonal
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//! scaled to the element's mass (Hinton, Rock & Zienkiewicz 1976), the rotary inertia
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//! isotropic about the node so the lumped matrix stays diagonal in global axes.
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//! - **Stresses.** Section forces and the stresses at the bottom, middle and top of the section
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//! at the integration points, extrapolated bilinearly to the nodes (quadrilateral) or taken at
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//! the nodes (triangle, constant membrane and bending fields). Transverse shear stresses are
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//! reported at the mid-surface as the peak of the parabolic distribution, 3/2 Q/t.
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use modelcraft_model::{Elastic, ElementKind};
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use crate::dense::Matrix;
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use crate::dof::{DofSet, ElementDofs};
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use crate::frame::{Axes, cross, dot, norm, rotate_matrix, rotate_to_global, rotate_to_local, sub, tensor_to_global, to_local, translation, unit};
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use crate::quadrature::Rule;
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use crate::{Error, Result};
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/// Transverse shear correction factor of a homogeneous section.
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pub const SHEAR_CORRECTION: f64 = 5.0 / 6.0;
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/// The drilling penalty γ as a fraction of the shear modulus.
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pub const DRILLING: f64 = 1e-2;
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/// Degrees of freedom per shell node: translations along local (or global) x, y, z, then
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/// rotations about them.
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pub const NODE_DOFS: usize = 6;
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/// A shell section with an isotropic material.
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#[derive(Debug, Clone, Copy, PartialEq)]
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pub struct ShellSection {
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pub thickness: f64,
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/// The mid-surface's position along the element normal from the nodes (the reference
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/// surface): −offset · thickness.
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pub eccentricity: f64,
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pub young: f64,
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pub poisson: f64,
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}
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impl ShellSection {
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/// A section of the model's thickness and offset (a fraction of the thickness, 0.5 puts the
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/// nodes on the side the normal points to).
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pub fn new(thickness: f64, offset: f64, elastic: &Elastic) -> Result<ShellSection> {
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let (young, poisson) = crate::material::isotropic_constants(elastic, "shell")?;
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if !(thickness > 0.0 && thickness.is_finite() && offset.abs() <= 0.5) {
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return Err(Error::Invalid(format!(
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"a shell section needs a positive thickness and an offset in −0.5..=0.5, got {thickness} and {offset}"
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)));
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}
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Ok(ShellSection { thickness, eccentricity: -offset * thickness, young, poisson })
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}
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/// The plane stress matrix Q.
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fn plane_stress(&self) -> [[f64; 3]; 3] {
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let (e, nu) = (self.young, self.poisson);
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let c = e / (1.0 - nu * nu);
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[[c, c * nu, 0.0], [c * nu, c, 0.0], [0.0, 0.0, c * (1.0 - nu) / 2.0]]
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}
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/// The membrane–bending stiffness [[A, B], [B, D]], 6 × 6.
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fn abd(&self) -> [[f64; 6]; 6] {
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let q = self.plane_stress();
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let (t, e) = (self.thickness, self.eccentricity);
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let (a, b, d) = (t, t * e, t * t * t / 12.0 + t * e * e);
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let mut out = [[0.0; 6]; 6];
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for i in 0..3 {
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for j in 0..3 {
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out[i][j] = a * q[i][j];
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out[i][j + 3] = b * q[i][j];
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out[i + 3][j] = b * q[i][j];
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out[i + 3][j + 3] = d * q[i][j];
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}
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}
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out
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}
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pub fn shear_modulus(&self) -> f64 {
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self.young / (2.0 * (1.0 + self.poisson))
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}
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/// The transverse shear stiffness k G t.
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fn shear(&self) -> f64 {
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SHEAR_CORRECTION * self.shear_modulus() * self.thickness
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}
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}
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/// Section forces and stresses of a shell element, per node in the element's node order.
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#[derive(Debug, Clone, PartialEq)]
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pub struct ShellStress {
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/// The element's local axes: e₁, e₂ in its plane, e₃ its normal (the right-hand normal of
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/// the node order).
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pub axes: Axes,
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/// N_x, N_y, N_xy, M_x, M_y, M_xy, Q_x, Q_y in the local axes, per unit length, about the
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/// reference surface.
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pub forces: Vec<[f64; 8]>,
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/// Stress (global xx, yy, zz, xy, xz, yz) at the bottom (−e₃ side), the middle and the top
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/// (+e₃ side) of the section.
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pub bottom: Vec<[f64; 6]>,
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pub middle: Vec<[f64; 6]>,
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pub top: Vec<[f64; 6]>,
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}
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/// The shape functions of the element's corners at (r, s) and their derivatives along r and s.
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fn shape(kind: ElementKind, r: f64, s: f64) -> ([f64; 4], [[f64; 2]; 4]) {
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if kind == ElementKind::Tri3 {
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([1.0 - r - s, r, s, 0.0], [[-1.0, -1.0], [1.0, 0.0], [0.0, 1.0], [0.0, 0.0]])
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} else {
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let c = [[-1.0, -1.0], [1.0, -1.0], [1.0, 1.0], [-1.0, 1.0]];
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let mut n = [0.0; 4];
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let mut d = [[0.0; 2]; 4];
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for (i, ci) in c.iter().enumerate() {
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n[i] = 0.25 * (1.0 + ci[0] * r) * (1.0 + ci[1] * s);
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d[i] = [0.25 * ci[0] * (1.0 + ci[1] * s), 0.25 * ci[1] * (1.0 + ci[0] * r)];
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}
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(n, d)
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}
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}
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/// The map's Jacobian at a point: J = [[x,r, y,r], [x,s, y,s]], its inverse and determinant.
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struct Map {
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n: [f64; 4],
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/// ∂Nᵢ/∂r, ∂Nᵢ/∂s.
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dn: [[f64; 2]; 4],
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/// ∂Nᵢ/∂x, ∂Nᵢ/∂y.
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dx: [[f64; 2]; 4],
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j: [[f64; 2]; 2],
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inv: [[f64; 2]; 2],
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det: f64,
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}
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/// Most degrees of freedom of a shell element (quad4).
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const MAX_DOFS: usize = 4 * NODE_DOFS;
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/// A row of a strain operator over the element's local degrees of freedom (zero past them).
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type Row = [f64; MAX_DOFS];
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/// The covariant shear rows at the element's tying points, which depend on its geometry only:
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/// quad4 e_rt at (0, 1), (0, −1) and e_st at (1, 0), (−1, 0); tri3 e_rt⁽¹⁾, e_st⁽²⁾ and the
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/// combination c.
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struct Tying([Row; 4]);
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/// A flat shell element.
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#[derive(Debug, Clone)]
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pub struct Shell {
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kind: ElementKind,
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nodes: usize,
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axes: Axes,
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/// The nodes in the element plane (local x, y).
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local: [[f64; 2]; 4],
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section: ShellSection,
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area: f64,
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}
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impl Shell {
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pub fn new(kind: ElementKind, xyz: &[[f64; 3]], section: ShellSection) -> Result<Shell> {
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let nodes = match kind {
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ElementKind::Tri3 => 3,
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ElementKind::Quad4 => 4,
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ElementKind::Tri6 | ElementKind::Quad8 => {
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return Err(Error::NotSupported(format!("{} shells (the solver has quad4 and tri3 shells so far)", kind.name())));
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}
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_ => return Err(Error::Invalid(format!("{} is not a shell element", kind.name()))),
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};
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if xyz.len() != nodes {
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return Err(Error::Invalid(format!("a {} element has {nodes} nodes, got {}", kind.name(), xyz.len())));
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}
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modelcraft_model::check_coordinates(xyz.as_flattened()).map_err(|e| Error::Invalid(e.to_string()))?;
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let degenerate = || Error::Degenerate(format!("a {} shell element is degenerate (zero area, inverted or not convex)", kind.name()));
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let p = |i: usize| xyz.get(i).copied().unwrap_or_default();
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let (normal, along) = if nodes == 3 {
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(cross(sub(p(1), p(0)), sub(p(2), p(0))), sub(p(1), p(0)))
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} else {
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let along = sub([0, 1, 2].map(|k| p(1)[k] + p(2)[k]), [0, 1, 2].map(|k| p(0)[k] + p(3)[k]));
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(cross(sub(p(2), p(0)), sub(p(3), p(1))), along)
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};
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let e3 = unit(normal).ok_or_else(degenerate)?;
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let e1 = unit(sub(along, e3.map(|c| c * dot(along, e3)))).ok_or_else(degenerate)?;
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let e2 = cross(e3, e1);
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let axes = [e1, e2, e3];
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let mut centre = [0.0; 3];
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for x in xyz {
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for k in 0..3 {
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centre[k] += x[k] / nodes as f64;
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}
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}
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let mut local = [[0.0; 2]; 4];
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let mut size = 0.0f64;
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for (l, x) in local.iter_mut().zip(xyz) {
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let d = sub(*x, centre);
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*l = [dot(d, e1), dot(d, e2)];
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size = size.max(norm(d));
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}
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// Corners turn left (counter-clockwise about e₃) by a margin: positive area, convex.
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for i in 0..nodes {
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let (a, b, c) = (local[(i + nodes - 1) % nodes], local[i], local[(i + 1) % nodes]);
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let turn = (b[0] - a[0]) * (c[1] - b[1]) - (b[1] - a[1]) * (c[0] - b[0]);
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if !(turn > 1e-10 * size * size) {
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return Err(degenerate());
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}
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}
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// A strongly warped quadrilateral is not a facet.
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let warp = xyz.iter().map(|x| dot(sub(*x, centre), e3).abs()).fold(0.0, f64::max);
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if warp > 0.1 * size {
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return Err(Error::Degenerate(format!(
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"a quad4 shell element is warped by {warp:.3e} out of its plane (more than a tenth of its size); split it into triangles"
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)));
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}
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let mut shell = Shell { kind, nodes, axes, local, section, area: 0.0 };
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shell.area = shell.rule().points().iter().map(|q| shell.map(q.xi[0], q.xi[1]).map_or(0.0, |m| m.det * q.weight)).sum();
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if !(shell.area > 0.0) {
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return Err(degenerate());
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}
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Ok(shell)
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}
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pub fn axes(&self) -> Axes {
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self.axes
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}
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pub fn area(&self) -> f64 {
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self.area
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}
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/// The element's unknowns: three translations and three rotations per node.
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pub fn dofs(&self) -> ElementDofs {
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ElementDofs::new(DofSet::STRUCTURAL)
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}
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/// The size of the element's matrices.
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fn size(&self) -> usize {
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NODE_DOFS * self.nodes
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}
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fn rule(&self) -> Rule {
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if self.kind == ElementKind::Tri3 { Rule::TriConical(2) } else { Rule::Quad(2) }
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}
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fn map(&self, r: f64, s: f64) -> Result<Map> {
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let (n, d) = shape(self.kind, r, s);
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let mut j = [[0.0; 2]; 2];
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for (di, x) in d.iter().zip(&self.local).take(self.nodes) {
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for a in 0..2 {
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for b in 0..2 {
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j[a][b] += di[a] * x[b];
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}
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}
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}
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let det = j[0][0] * j[1][1] - j[0][1] * j[1][0];
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if !(det > 0.0 && det.is_finite()) {
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return Err(Error::Degenerate(format!("a {} shell element is inverted or degenerate", self.kind.name())));
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}
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let inv = [[j[1][1] / det, -j[0][1] / det], [-j[1][0] / det, j[0][0] / det]];
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let mut dx = [[0.0; 2]; 4];
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for (g, di) in dx.iter_mut().zip(&d) {
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g[0] = inv[0][0] * di[0] + inv[0][1] * di[1];
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g[1] = inv[1][0] * di[0] + inv[1][1] * di[1];
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}
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Ok(Map { n, dn: d, dx, j, inv, det })
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}
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/// The covariant transverse shear strain e_rt (`along` 0) or e_st (`along` 1) of the
|
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/// displacement interpolation at (r, s): w,r + θ_y x,r − θ_x y,r (and with s).
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fn covariant(&self, r: f64, s: f64, along: usize) -> Result<Row> {
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let m = self.map(r, s)?;
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let (xr, yr) = (m.j[along][0], m.j[along][1]);
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let mut row = [0.0; MAX_DOFS];
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for i in 0..self.nodes {
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row[6 * i + 2] = m.dn[i][along];
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row[6 * i + 4] = m.n[i] * xr;
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row[6 * i + 3] = -m.n[i] * yr;
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}
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Ok(row)
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}
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fn tying(&self) -> Result<Tying> {
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if self.kind == ElementKind::Tri3 {
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let e1 = self.covariant(0.5, 0.0, 0)?;
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let e2 = self.covariant(0.0, 0.5, 1)?;
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let e3r = self.covariant(0.5, 0.5, 0)?;
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let e3s = self.covariant(0.5, 0.5, 1)?;
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let c: Row = std::array::from_fn(|k| e3r[k] - e1[k] - e3s[k] + e2[k]);
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Ok(Tying([e1, e2, c, [0.0; MAX_DOFS]]))
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} else {
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Ok(Tying([self.covariant(0.0, 1.0, 0)?, self.covariant(0.0, -1.0, 0)?, self.covariant(1.0, 0.0, 1)?, self.covariant(-1.0, 0.0, 1)?]))
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||||
}
|
||||
}
|
||||
|
||||
/// The assumed covariant shear strains (ẽ_rt, ẽ_st) at (r, s).
|
||||
fn assumed_shear(&self, tying: &Tying, r: f64, s: f64) -> [Row; 2] {
|
||||
let lerp = |a: &Row, wa: f64, b: &Row, wb: f64| -> Row { std::array::from_fn(|k| wa * a[k] + wb * b[k]) };
|
||||
let [t0, t1, t2, t3] = &tying.0;
|
||||
if self.kind == ElementKind::Tri3 {
|
||||
[lerp(t0, 1.0, t2, s), lerp(t1, 1.0, t2, -r)]
|
||||
} else {
|
||||
[lerp(t0, 0.5 * (1.0 + s), t1, 0.5 * (1.0 - s)), lerp(t2, 0.5 * (1.0 + r), t3, 0.5 * (1.0 - r))]
|
||||
}
|
||||
}
|
||||
|
||||
/// The generalized strain operator at (r, s): membrane strains, curvatures, transverse shear
|
||||
/// strains (8 rows) and the drilling row θ_z − ω, with the map.
|
||||
fn operator(&self, tying: &Tying, r: f64, s: f64) -> Result<([Row; 9], Map)> {
|
||||
let m = self.map(r, s)?;
|
||||
let mut b = [[0.0; MAX_DOFS]; 9];
|
||||
for i in 0..self.nodes {
|
||||
let [gx, gy] = m.dx[i];
|
||||
let (u, v, tx, ty, tz) = (6 * i, 6 * i + 1, 6 * i + 3, 6 * i + 4, 6 * i + 5);
|
||||
b[0][u] = gx;
|
||||
b[1][v] = gy;
|
||||
b[2][u] = gy;
|
||||
b[2][v] = gx;
|
||||
b[3][ty] = gx;
|
||||
b[4][tx] = -gy;
|
||||
b[5][ty] = gy;
|
||||
b[5][tx] = -gx;
|
||||
b[8][tz] = m.n[i];
|
||||
b[8][u] = 0.5 * gy;
|
||||
b[8][v] = -0.5 * gx;
|
||||
}
|
||||
let [ert, est] = self.assumed_shear(tying, r, s);
|
||||
for k in 0..MAX_DOFS {
|
||||
b[6][k] = m.inv[0][0] * ert[k] + m.inv[0][1] * est[k];
|
||||
b[7][k] = m.inv[1][0] * ert[k] + m.inv[1][1] * est[k];
|
||||
}
|
||||
Ok((b, m))
|
||||
}
|
||||
|
||||
/// The stiffness matrix in global axes, 6n × 6n, degrees of freedom node by node.
|
||||
pub fn stiffness(&self) -> Result<Matrix> {
|
||||
let n = self.size();
|
||||
let abd = self.section.abd();
|
||||
let ks = self.section.shear();
|
||||
let drill = DRILLING * self.section.shear_modulus() * self.section.thickness;
|
||||
let tying = self.tying()?;
|
||||
let mut k = Matrix::zeros(n);
|
||||
for p in self.rule().points() {
|
||||
let (b, m) = self.operator(&tying, p.xi[0], p.xi[1])?;
|
||||
let w = m.det * p.weight;
|
||||
// C B for the membrane–bending rows.
|
||||
let cb: [Row; 6] = std::array::from_fn(|a| std::array::from_fn(|c| (0..6).map(|d| abd[a][d] * b[d][c]).sum()));
|
||||
for i in 0..n {
|
||||
for j in i..n {
|
||||
let mut v: f64 = (0..6).map(|a| b[a][i] * cb[a][j]).sum();
|
||||
v += ks * (b[6][i] * b[6][j] + b[7][i] * b[7][j]);
|
||||
v += drill * b[8][i] * b[8][j];
|
||||
if v != 0.0 {
|
||||
k.add(i, j, w * v);
|
||||
if i != j {
|
||||
k.add(j, i, w * v);
|
||||
}
|
||||
}
|
||||
}
|
||||
}
|
||||
}
|
||||
Ok(rotate_matrix(&k, &self.axes))
|
||||
}
|
||||
|
||||
/// The consistent mass matrix in global axes, 6n × 6n.
|
||||
pub fn mass(&self, density: f64) -> Result<Matrix> {
|
||||
Ok(rotate_matrix(&self.local_mass(density)?, &self.axes))
|
||||
}
|
||||
|
||||
/// The consistent mass matrix in the element's axes.
|
||||
fn local_mass(&self, density: f64) -> Result<Matrix> {
|
||||
let (t, e) = (self.section.thickness, self.section.eccentricity);
|
||||
let (m0, m1, m2) = (density * t, density * t * e, density * (t * t * t / 12.0 + t * e * e));
|
||||
let mut mm = Matrix::zeros(self.size());
|
||||
for p in self.rule().points() {
|
||||
let m = self.map(p.xi[0], p.xi[1])?;
|
||||
let w = m.det * p.weight;
|
||||
for i in 0..self.nodes {
|
||||
for j in 0..self.nodes {
|
||||
let nn = w * m.n[i] * m.n[j];
|
||||
for c in 0..3 {
|
||||
mm.add(6 * i + c, 6 * j + c, m0 * nn);
|
||||
}
|
||||
for c in 3..6 {
|
||||
mm.add(6 * i + c, 6 * j + c, m2 * nn);
|
||||
}
|
||||
if m1 != 0.0 {
|
||||
// u θ_y and −v θ_x, both ways.
|
||||
mm.add(6 * i, 6 * j + 4, m1 * nn);
|
||||
mm.add(6 * i + 4, 6 * j, m1 * nn);
|
||||
mm.add(6 * i + 1, 6 * j + 3, -m1 * nn);
|
||||
mm.add(6 * i + 3, 6 * j + 1, -m1 * nn);
|
||||
}
|
||||
}
|
||||
}
|
||||
}
|
||||
Ok(mm)
|
||||
}
|
||||
|
||||
/// The lumped mass per degree of freedom (the same in local and global axes).
|
||||
pub fn lumped_mass(&self, density: f64) -> Result<Vec<f64>> {
|
||||
let (t, e) = (self.section.thickness, self.section.eccentricity);
|
||||
let mut diag = vec![0.0; self.size()];
|
||||
let mut sum = 0.0;
|
||||
for p in self.rule().points() {
|
||||
let m = self.map(p.xi[0], p.xi[1])?;
|
||||
for i in 0..self.nodes {
|
||||
let v = m.det * p.weight * m.n[i] * m.n[i];
|
||||
diag[6 * i] += v;
|
||||
sum += v;
|
||||
}
|
||||
}
|
||||
if !(sum > 0.0) {
|
||||
return Err(Error::Degenerate("a shell element has no area".into()));
|
||||
}
|
||||
let scale = self.area / sum;
|
||||
let (m0, m2) = (density * t, density * (t * t * t / 12.0 + t * e * e));
|
||||
for i in 0..self.nodes {
|
||||
let a = diag[6 * i] * scale;
|
||||
for c in 0..3 {
|
||||
diag[6 * i + c] = m0 * a;
|
||||
diag[6 * i + 3 + c] = m2 * a;
|
||||
}
|
||||
}
|
||||
Ok(diag)
|
||||
}
|
||||
|
||||
/// Consistent nodal forces and moments (global) of a uniform force per volume b: the
|
||||
/// consistent mass of unit density times the uniform nodal translation b.
|
||||
pub fn body_force(&self, b: [f64; 3]) -> Result<Vec<f64>> {
|
||||
let r = translation(to_local(&self.axes, b), self.nodes);
|
||||
Ok(rotate_to_global(&self.local_mass(1.0)?.mul_vec(&r), &self.axes))
|
||||
}
|
||||
|
||||
/// Section forces and surface stresses for the element's nodal displacements and rotations
|
||||
/// (global, 6 per node).
|
||||
pub fn stresses(&self, u: &[f64]) -> Result<ShellStress> {
|
||||
if u.len() != self.size() {
|
||||
return Err(Error::Invalid(format!("{} values for a shell element with {} degrees of freedom", u.len(), self.size())));
|
||||
}
|
||||
let ul = rotate_to_local(u, &self.axes);
|
||||
let tying = self.tying()?;
|
||||
let strains = |r: f64, s: f64| -> Result<[f64; 8]> {
|
||||
let (b, _) = self.operator(&tying, r, s)?;
|
||||
Ok(std::array::from_fn(|a| b[a].iter().zip(&ul).map(|(x, y)| x * y).sum()))
|
||||
};
|
||||
let nodal: Vec<[f64; 8]> = if self.kind == ElementKind::Tri3 {
|
||||
[[0.0, 0.0], [1.0, 0.0], [0.0, 1.0]].iter().map(|c| strains(c[0], c[1])).collect::<Result<_>>()?
|
||||
} else {
|
||||
// 2 × 2 Gauss values, extrapolated with the bilinear functions of the points.
|
||||
let g = 1.0 / 3f64.sqrt();
|
||||
let corners = [[-1.0, -1.0], [1.0, -1.0], [1.0, 1.0], [-1.0, 1.0]];
|
||||
let at: Vec<[f64; 8]> = corners.iter().map(|c| strains(c[0] * g, c[1] * g)).collect::<Result<_>>()?;
|
||||
corners
|
||||
.iter()
|
||||
.map(|c| {
|
||||
let (n, _) = shape(ElementKind::Quad4, c[0] / g, c[1] / g);
|
||||
std::array::from_fn(|k| n.iter().zip(&at).map(|(w, v)| w * v[k]).sum())
|
||||
})
|
||||
.collect()
|
||||
};
|
||||
let abd = self.section.abd();
|
||||
let q = self.section.plane_stress();
|
||||
let ks = self.section.shear();
|
||||
let (t, e) = (self.section.thickness, self.section.eccentricity);
|
||||
let mut out = ShellStress { axes: self.axes, forces: Vec::new(), bottom: Vec::new(), middle: Vec::new(), top: Vec::new() };
|
||||
for g in &nodal {
|
||||
let mut f = [0.0; 8];
|
||||
for a in 0..6 {
|
||||
f[a] = (0..6).map(|b| abd[a][b] * g[b]).sum();
|
||||
}
|
||||
f[6] = ks * g[6];
|
||||
f[7] = ks * g[7];
|
||||
out.forces.push(f);
|
||||
let at = |z: f64, shear: f64| {
|
||||
let strain = [g[0] + z * g[3], g[1] + z * g[4], g[2] + z * g[5]];
|
||||
let s: [f64; 3] = std::array::from_fn(|a| (0..3).map(|b| q[a][b] * strain[b]).sum());
|
||||
let (txz, tyz) = (shear * f[6] / t, shear * f[7] / t);
|
||||
tensor_to_global(&self.axes, [[s[0], s[2], txz], [s[2], s[1], tyz], [txz, tyz, 0.0]])
|
||||
};
|
||||
out.bottom.push(at(e - t / 2.0, 0.0));
|
||||
out.middle.push(at(e, 1.5));
|
||||
out.top.push(at(e + t / 2.0, 0.0));
|
||||
}
|
||||
Ok(out)
|
||||
}
|
||||
}
|
||||
|
||||
#[cfg(test)]
|
||||
mod tests {
|
||||
use super::*;
|
||||
|
||||
const STEEL: Elastic = Elastic::Isotropic { young: 210e3, poisson: 0.3 };
|
||||
|
||||
fn rigid(xyz: &[[f64; 3]], m: usize) -> Vec<f64> {
|
||||
xyz.iter()
|
||||
.flat_map(|x| {
|
||||
let mut d = [0.0; 6];
|
||||
if m < 3 {
|
||||
d[m] = 1.0;
|
||||
} else {
|
||||
let mut w = [0.0; 3];
|
||||
w[m - 3] = 1.0;
|
||||
d[..3].copy_from_slice(&cross(w, *x));
|
||||
d[3..].copy_from_slice(&w);
|
||||
}
|
||||
d
|
||||
})
|
||||
.collect()
|
||||
}
|
||||
|
||||
#[test]
|
||||
fn shells_are_free_of_rigid_motion_and_symmetric() {
|
||||
let section = ShellSection::new(0.5, 0.25, &STEEL).unwrap();
|
||||
// A skew quadrilateral and a triangle in a tilted plane.
|
||||
let tilt = |x: f64, y: f64| [x, 0.6 * y, 0.8 * y + 1.0];
|
||||
let quad = [tilt(0.0, 0.0), tilt(2.0, 0.2), tilt(2.3, 1.7), tilt(-0.2, 1.4)];
|
||||
let tri = [tilt(0.0, 0.0), tilt(2.0, 0.2), tilt(0.4, 1.5)];
|
||||
for (kind, xyz) in [(ElementKind::Quad4, &quad[..]), (ElementKind::Tri3, &tri[..])] {
|
||||
let shell = Shell::new(kind, xyz, section).unwrap();
|
||||
let k = shell.stiffness().unwrap();
|
||||
let scale = (0..k.n).map(|i| k.get(i, i)).fold(0.0, f64::max);
|
||||
for i in 0..k.n {
|
||||
for j in 0..k.n {
|
||||
assert_eq!(k.get(i, j), k.get(j, i));
|
||||
}
|
||||
}
|
||||
for m in 0..6 {
|
||||
let f = k.mul_vec(&rigid(xyz, m));
|
||||
assert!(f.iter().all(|v| v.abs() < 1e-10 * scale), "{kind:?} rigid {m}: {f:?}");
|
||||
}
|
||||
// Mass: translations sum to ρ t A per direction (the offset couples rotations only).
|
||||
let m = shell.mass(2.0).unwrap();
|
||||
let total = 2.0 * 0.5 * shell.area();
|
||||
let s: f64 =
|
||||
(0..k.n).filter(|i| i % 6 == 0).flat_map(|i| (0..k.n).filter(|j| j % 6 == 0).map(move |j| (i, j))).map(|(i, j)| m.get(i, j)).sum();
|
||||
assert!((s - total).abs() < 1e-12 * total, "{s} {total}");
|
||||
let lumped = shell.lumped_mass(2.0).unwrap();
|
||||
let s: f64 = lumped.iter().step_by(6).sum();
|
||||
assert!((s - total).abs() < 1e-12 * total);
|
||||
let body = shell.body_force([0.0, 0.0, -1.0]).unwrap();
|
||||
let fz: f64 = body.iter().skip(2).step_by(6).sum();
|
||||
assert!((fz + 0.5 * shell.area()).abs() < 1e-12, "{fz}");
|
||||
}
|
||||
let flat = [[0.0, 0.0, 0.0], [1.0, 0.0, 0.0], [1.0, 1.0, 0.0], [0.5, 0.2, 0.0]];
|
||||
assert!(Shell::new(ElementKind::Quad4, &flat, section).is_err(), "not convex");
|
||||
let line = [[0.0, 0.0, 0.0], [1.0, 0.0, 0.0], [2.0, 0.0, 0.0]];
|
||||
assert!(Shell::new(ElementKind::Tri3, &line, section).is_err());
|
||||
assert!(Shell::new(ElementKind::Quad8, &flat, section).is_err());
|
||||
assert!(ShellSection::new(1.0, 0.0, &Elastic::EngineeringConstants { young: [1.0; 3], poisson: [0.0; 3], shear: [1.0; 3] }).is_err());
|
||||
}
|
||||
}
|
||||
Reference in new issue
Block a user