forked from 1b-agent/fieldcraft-upstream
Explicit X2: large deformation, J2 and Johnson-Cook plasticity, damage and erosion (M10)
- model: materials gain solver-neutral plastic (bilinear isotropic/kinematic, hardening table, Johnson-Cook with adiabatic heating), damage (Johnson-Cook D1-D5) and erosion (plastic strain); explicit steps gain nonlinear_geometry (default on) and stress_rate (jaumann | green_naghdi). - fem: rotation (Hughes-Winget, Flanagan-Taylor), plasticity (radial return, capped bracketed Newton for Johnson-Cook, damage; the update takes the mechanical strain increment so thermal strains can be subtracted), closed-form hex8 gradients for deformed configurations. - explicit: updated Lagrangian hex8/tet4, per-element material state, erosion with reported energy, plastic work in the energy account. - solver: warns that the linear solver uses the elastic constants of plastic materials (results hash-identical). - decks: Abaqus *PLASTIC (+ Johnson-Cook keywords), Nastran MATS1/TABLES1, LS-DYNA *MAT_PLASTIC_KINEMATIC / *MAT_PIECEWISE_LINEAR_PLASTICITY; what a format cannot express is reported as Unsupported. - engine: model.material.plasticity, model.material.damage (catalogue ready), explicit element probe, eroded elements in the solve summary. Step commits before the rebase onto the DOF framework: branch explicit-x2-steps. Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01NmAttHhUZTokgPeEJia8sm
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jyvikoandClaude Opus 5.5 authored and Irakli Kourtis committed
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@@ -1,24 +1,41 @@
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//! Element internal forces and critical time increments.
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//! Element internal forces, the material update and critical time increments.
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//!
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//! **Kinematics.** With nonlinear geometry the elements are updated Lagrangian (T. Belytschko,
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//! W. K. Liu, B. Moran, K. Elkhodary, *Nonlinear Finite Elements for Continua and Structures*,
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//! 2nd ed., Wiley 2014, ch. 4 and 6): the strain and spin increments of an increment come from
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//! the displacement increment Δu and the gradients of the **midpoint** configuration
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//! x_{n−½} = X + u_n − ½Δu, ΔG = (1/V) Σ_I Δu_I ⊗ b_I(x_{n−½}), which makes the update
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//! incrementally objective (T. J. R. Hughes, J. Winget, IJNME 15(12):1862–1867, 1980); the nodal
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//! forces of the Cauchy stress use the gradients of the current configuration,
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//! f_iI = σ_ij b_jI(x_n). Without it, both are the gradients of the undeformed mesh and the
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//! update is small-strain elasticity (and plasticity).
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//!
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//! **Stress update** in rate form: the material state (stress, back stress, the accumulated
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//! strain and the stiffness-form hourglass forces) is rotated with the increment's rotation —
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//! the Cayley transform of the spin increment (Jaumann rate, Hughes & Winget 1980) or the
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//! increment of the polar rotation (Green–Naghdi rate, D. P. Flanagan, L. M. Taylor, CMAME
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//! 62(3):305–320, 1987), see [`modelcraft_fem::rotation`] — then the constitutive increment is
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//! added: hypoelastic, or J2 radial return with Johnson–Cook damage
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//! ([`modelcraft_fem::plasticity`]). Orthotropic elasticity is updated in the polar frame (its
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//! axes turn with the material) whichever rate the step names.
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//!
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//! **One-point hexahedron** (D. P. Flanagan, T. Belytschko, IJNME 17(5):679–706, 1981): the
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//! stress of the mean strain gives the forces f_iI = σ_ij b_jI ([`modelcraft_fem::uniform`]).
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//! The hourglass modes are resisted through the shape vectors γ_α: with the hourglass rates
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//! q̇_iα = Σ_I v_iI γ_αI (or displacements q_iα = Σ_I u_iI γ_αI for the stiffness form), the
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//! generalized forces Q_iα = C q̇_iα (viscous) or k q_iα (stiffness) give the nodal forces
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//! f_iI = Σ_α Q_iα γ_αI. Because the γ_α are orthogonal to linear fields, these forces do no work
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//! in a constant strain state. The coefficients are scaled from the dimensionless hourglass
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//! coefficient κ with the element's size h = V / √(2 λ_max(Σ b bᵀ)) (the edge, for a cube),
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//! density ρ and wave speed c: C = κ ρ c h² / 16 damps a cube's hourglass modes at a fraction κ
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//! of critical at the frequency 2c/h, and k = κ ρ c² h / 16 puts their frequency at √κ · 2c/h.
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//! (These normalizations are ours; the paper leaves the scale of the coefficient to the user.)
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//! hourglass modes are resisted through the shape vectors γ_α of the current configuration:
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//! with the hourglass rates q̇_iα = Σ_I v_iI γ_αI, the generalized forces are Q_iα = C q̇_iα
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//! (viscous) or accumulate as Q_iα += k Σ_I Δu_iI γ_αI (stiffness, rotated with the element),
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//! and give the nodal forces f_iI = Σ_α Q_iα γ_αI. Because the γ_α are orthogonal to linear
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//! fields, these forces do no work in a constant strain state. The coefficients are scaled from
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//! the dimensionless hourglass coefficient κ with the element's current size
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//! h = V / √(2 λ_max(Σ b bᵀ)) (the edge, for a cube), density ρ and wave speed c:
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//! C = κ ρ c h² / 16 damps a cube's hourglass modes at a fraction κ of critical at the frequency
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//! 2c/h, and k = κ ρ c² h / 16 puts their frequency at √κ · 2c/h. (These normalizations are ours;
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//! the paper leaves the scale of the coefficient to the user.)
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//!
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//! **Average nodal pressure tetrahedron** (J. Bonet, A. J. Burton, *A simple average nodal
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//! pressure tetrahedral element for incompressible and nearly incompressible dynamic explicit
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//! applications*, Commun. Numer. Methods Eng. 14(5):437–449, 1998): the volumetric strain of each
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//! node is the volume-weighted mean of its tetrahedra's, and each tetrahedron takes the mean of
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//! its nodes' as its own: σ = 2μ dev ε + K θ̄ I. The forces are those of the strain energy
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//! Σ_e V_e μ ‖dev ε_e‖² + ½ K Σ_a V_a θ_a², so the element conserves energy, and it does not lock
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//! when the material is nearly incompressible.
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//! **Average nodal pressure tetrahedron** (J. Bonet, A. J. Burton, Commun. Numer. Methods Eng.
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//! 14(5):437–449, 1998), in rate form: the volumetric strain increment of each node is the
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//! volume-weighted mean of its tetrahedra's, and each tetrahedron takes the mean of its nodes' as
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//! its own, so its pressure increment is K times that mean while the deviator is its own. It does
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//! not lock when the material is nearly incompressible or flows plastically.
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//!
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//! **Bulk viscosity** (J. von Neumann, R. D. Richtmyer, J. Appl. Phys. 21(3):232–237, 1950; M. L.
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//! Wilkins, J. Comput. Phys. 36(3):281–303, 1980): in compression (volumetric strain rate
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@@ -27,60 +44,118 @@
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//! Δt = L / (Q + √(Q² + c²)), Q = C_l c + C_q L |ε̇ᵥ| (Wilkins 1980).
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//!
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//! **Critical increment.** Each element's stable length L = 2c/ω_max comes from the bound on its
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//! highest frequency ([`modelcraft_fem::uniform::UniformStrain::frequency_bound`], plus the
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//! hourglass stiffness when it is used), so without bulk viscosity its critical increment is
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//! 2/ω_max (T. Belytschko, W. K. Liu, B. Moran, K. Elkhodary, 2014, §6.3; the element eigenvalue
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//! inequality makes the smallest of them stable for the mesh). For the nodal-pressure
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//! tetrahedron the averaging only lowers the volumetric energy (the mean of squares bounds the
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//! square of the mean), so the plain tetrahedron's bound holds.
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//! highest frequency in its current shape ([`modelcraft_fem::uniform`], with the element's lumped
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//! masses, plus the hourglass stiffness when it is used), so without bulk viscosity its critical
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//! increment is 2/ω_max (Belytschko et al. 2014, §6.3). The elastic moduli bound the tangent
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//! moduli of plastic flow, so the bound holds for the inelastic materials too.
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//!
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//! **Erosion** (G. R. Johnson, R. A. Stryk, Int. J. Impact Eng. 5:411–421, 1987; T. Belytschko,
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//! J. I. Lin, ibid. 5:111–127, 1987): an element whose material point fails (damage 1) or
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//! reaches its material's erosion plastic strain is removed: it carries no stress from then on,
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//! its nodes keep their mass and momentum, and its internal energy at that moment is reported.
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use modelcraft_fem::material::stress;
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use modelcraft_fem::uniform::UniformStrain;
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use modelcraft_model::{BulkViscosity, HourglassKind};
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use modelcraft_fem::plasticity::{Increment, PointState};
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use modelcraft_fem::rotation::{self, IDENTITY, Mat3, Polar};
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use modelcraft_fem::uniform::{Hex8, Tet4, UniformStrain};
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use modelcraft_model::{BulkViscosity, HourglassKind, StressRate};
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use crate::system::{Material, NODE_DOFS, System};
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use crate::{Error, Result};
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/// Scratch vectors and the element state of the latest evaluation.
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/// Scratch vectors of one evaluation.
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#[derive(Debug, Clone)]
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pub struct Workspace {
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/// Internal forces per degree of freedom, by origin.
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pub stress_force: Vec<f64>,
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pub hourglass_force: Vec<f64>,
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pub viscous_force: Vec<f64>,
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/// Per element (in [`System::elements`] order): the material stress and the strain.
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pub stress: Vec<[f64; 6]>,
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pub strain: Vec<[f64; 6]>,
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/// Per node: the mean volumetric strain of its tetrahedra (a volume-weighted sum until it is
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/// divided by `tet_volume`).
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/// Per node: the volume-weighted volumetric strain increments of its tetrahedra, and their
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/// volume.
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theta: Vec<f64>,
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tet_volume: Vec<f64>,
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/// Per tetrahedron (in [`System::tets`] order): its increment between the two passes.
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tet_increment: Vec<Option<TetIncrement>>,
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}
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impl Workspace {
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pub fn new(sys: &System) -> Workspace {
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let n = sys.labels.len();
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let mut tet_volume = vec![0.0; n];
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for t in &sys.tets {
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for node in t.nodes {
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if let Some(v) = tet_volume.get_mut(node) {
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*v += t.kinematics.volume;
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}
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}
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}
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let dofs = sys.dofs();
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let elements = sys.elements.len();
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Workspace {
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stress_force: vec![0.0; dofs],
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hourglass_force: vec![0.0; dofs],
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viscous_force: vec![0.0; dofs],
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stress: vec![[0.0; 6]; elements],
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strain: vec![[0.0; 6]; elements],
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theta: vec![0.0; n],
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tet_volume,
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tet_volume: vec![0.0; n],
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tet_increment: vec![None; sys.tets.len()],
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}
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}
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}
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/// Why an element was eroded.
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#[derive(Debug, Clone, Copy, PartialEq, Eq, serde::Serialize)]
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#[serde(rename_all = "snake_case")]
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pub enum Failure {
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/// Its damage reached 1.
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Damage,
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/// Its equivalent plastic strain reached the material's erosion strain.
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PlasticStrain,
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}
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/// The state of one element (its single material point).
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#[derive(Debug, Clone)]
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pub struct ElementState {
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pub point: PointState,
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/// The accumulated (rotated) strain increments: engineering shears.
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pub strain: [f64; 6],
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/// The polar rotation and stretch (Green–Naghdi rate and orthotropic materials).
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pub polar: Polar,
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/// Stiffness-form hourglass generalized forces Q_α (one-point hexahedra).
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pub hourglass: [[f64; 3]; 4],
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/// Work of the stress on the element (its internal energy, hourglass apart).
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pub energy: f64,
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/// Plastic work on the element.
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pub plastic_work: f64,
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pub failed: Option<Failure>,
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}
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/// The element states and the erosions of a run.
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#[derive(Debug, Clone)]
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pub struct State {
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/// Per element, in [`System::elements`] order.
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pub elements: Vec<ElementState>,
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/// Elements eroded during the latest evaluation: (index, failure).
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pub eroded: Vec<(usize, Failure)>,
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}
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impl State {
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pub fn new(sys: &System) -> State {
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let mut material = vec![usize::MAX; sys.elements.len()];
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for (index, m) in sys.hexes.iter().map(|h| (h.index, h.material)).chain(sys.tets.iter().map(|t| (t.index, t.material))) {
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if let Some(slot) = material.get_mut(index) {
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*slot = m;
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}
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}
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let elements = material
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.into_iter()
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.map(|m| {
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let temperature = sys.materials.get(m).and_then(|m| m.j2.as_ref()).map_or(0.0, |j| j.initial_temperature());
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ElementState {
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point: PointState::new(temperature),
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strain: [0.0; 6],
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polar: Polar::default(),
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hourglass: [[0.0; 3]; 4],
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energy: 0.0,
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plastic_work: 0.0,
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failed: None,
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}
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})
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.collect();
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State { elements, eroded: Vec::new() }
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}
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}
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pub(crate) fn gather<const N: usize>(x: &[f64], nodes: &[usize; N]) -> [[f64; 3]; N] {
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nodes.map(|n| [0, 1, 2].map(|k| x.get(NODE_DOFS * n + k).copied().unwrap_or(0.0)))
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}
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@@ -112,102 +187,328 @@ pub fn critical_increment(rate: f64, length: f64, wave_speed: f64, b: &BulkVisco
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length / (q + (q * q + wave_speed * wave_speed).sqrt())
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}
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/// What the hexahedron and the tetrahedron share once their stress is known: the stress and bulk
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/// viscosity forces, the element state and the critical increment.
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struct Finish<'a, const N: usize> {
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kinematics: &'a UniformStrain<N>,
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nodes: &'a [usize; N],
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index: usize,
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length: f64,
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/// The largest absolute row sum of the hourglass vectors' Gram matrix Σ_I γ_αI γ_βI / m_I (an
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/// upper bound of its largest eigenvalue, exact for parallelepipeds where the γ_α are
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/// orthogonal).
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pub fn hourglass_gram_bound(gamma: &[[f64; 8]; 4], mass: &[f64; 8]) -> f64 {
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let mut worst: f64 = 0.0;
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for a in gamma {
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let row: f64 = gamma.iter().map(|b| a.iter().zip(b).zip(mass).map(|((x, y), m)| x * y / m).sum::<f64>().abs()).sum();
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worst = worst.max(row);
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}
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worst
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}
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impl<const N: usize> Finish<'_, N> {
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fn apply(&self, ws: &mut Workspace, mat: &Material, v: &[f64], sigma: [f64; 6], strain: [f64; 6], bulk: &BulkViscosity) -> f64 {
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let k = self.kinematics;
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let rate = k.divergence(&gather(v, self.nodes));
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let q = bulk_pressure(rate, mat.density, self.length, mat.wave_speed, bulk);
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scatter(&mut ws.stress_force, self.nodes, &k.forces(&sigma));
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if q != 0.0 {
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scatter(&mut ws.viscous_force, self.nodes, &k.pressure_forces(q));
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}
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if let Some(s) = ws.stress.get_mut(self.index) {
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*s = sigma;
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}
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if let Some(s) = ws.strain.get_mut(self.index) {
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*s = strain;
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}
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critical_increment(rate, self.length, mat.wave_speed, bulk)
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/// What one evaluation of the internal forces is given.
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pub struct Motion<'a> {
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/// Displacements u^n.
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pub u: &'a [f64],
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/// The displacement increment that led to `u`, Δt^{n−½} v^{n−½}.
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pub du: &'a [f64],
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/// The velocity v^{n−½}.
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pub v: &'a [f64],
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/// The increment's length Δt^{n−½} (0 at the start).
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pub dt: f64,
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pub time: f64,
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pub bulk: &'a BulkViscosity,
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}
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/// The element's nodal values: rest positions, displacements, displacement increments and
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/// velocities, gathered once per evaluation.
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struct Nodal<const N: usize> {
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x: [[f64; 3]; N],
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u: [[f64; 3]; N],
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du: [[f64; 3]; N],
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v: [[f64; 3]; N],
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}
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impl<const N: usize> Nodal<N> {
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fn gather(sys: &System, nodes: &[usize; N], m: &Motion) -> Nodal<N> {
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Nodal { x: nodes.map(|n| sys.xyz.get(n).copied().unwrap_or_default()), u: gather(m.u, nodes), du: gather(m.du, nodes), v: gather(m.v, nodes) }
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}
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/// The positions X + u − s·Δu.
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fn positions(&self, s: f64) -> [[f64; 3]; N] {
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std::array::from_fn(|i| [0, 1, 2].map(|k| self.x[i][k] + self.u[i][k] - s * self.du[i][k]))
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}
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}
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/// Evaluate the internal forces at displacements `u` and velocities `v` into `ws` and return the
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/// smallest critical increment of the elements.
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pub fn internal_forces(sys: &System, u: &[f64], v: &[f64], bulk: &BulkViscosity, ws: &mut Workspace) -> f64 {
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fn inverted(label: u32, time: f64) -> Error {
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Error::Unstable(format!("element {label} turned inside out at time {time:.6e} (its volume is no longer positive)"))
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}
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/// The rotation of an increment for an element's state, advancing its polar decomposition when
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/// it is tracked.
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fn rotation_of(sys: &System, mat: &Material, es: &mut ElementState, g: &Mat3, label: u32, time: f64) -> Result<Mat3> {
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if !sys.nonlinear {
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return Ok(IDENTITY);
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}
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if sys.stress_rate == StressRate::GreenNaghdi || mat.j2.is_none() {
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es.polar.advance(g).ok_or_else(|| inverted(label, time))
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} else {
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let (_, w) = rotation::split(g);
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rotation::cayley(&w).ok_or_else(|| inverted(label, time))
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}
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}
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/// Rotate an element's state with `q`, then add the constitutive increment for the strain
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/// increment `d` (engineering shears) with volumetric part `vol` (the tetrahedron's averaged
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/// one; the trace of `d` for the hexahedron), track its energy at the midpoint volume `volume`,
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/// and check for erosion.
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#[allow(clippy::too_many_arguments)]
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fn update(mat: &Material, es: &mut ElementState, q: &Mat3, d: &[f64; 6], vol: f64, dt: f64, density: f64, volume: f64) {
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let p = &mut es.point;
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if *q != IDENTITY {
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// Only the parts of the state that are in use (a back stress with kinematic hardening,
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// hourglass forces with stiffness-form control).
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p.stress = rotation::rotate(q, &p.stress);
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if p.back != [0.0; 6] {
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p.back = rotation::rotate(q, &p.back);
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}
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es.strain = rotation::rotate_strain(q, &es.strain);
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if es.hourglass != [[0.0; 3]; 4] {
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for h in &mut es.hourglass {
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*h = rotation::apply(q, h);
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}
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}
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}
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let before = p.stress;
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let inc = match &mat.j2 {
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Some(j2) => j2.update(p, d, vol, dt, density),
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None => {
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// Orthotropic: in the polar frame, σ̂ += D d̂.
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let r = es.polar.rotation;
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if r == IDENTITY {
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for (s, x) in p.stress.iter_mut().zip(stress(&mat.elasticity, d)) {
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*s += x;
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}
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} else {
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let local = rotation::rotate_strain(&rotation::transpose(&r), d);
|
||||
for (s, x) in p.stress.iter_mut().zip(rotation::rotate(&r, &stress(&mat.elasticity, &local))) {
|
||||
*s += x;
|
||||
}
|
||||
}
|
||||
Increment::default()
|
||||
}
|
||||
};
|
||||
// Stress work over the increment, ½(σ_n + σ_{n+1}) : Δε, with the volumetric strain the
|
||||
// stress took.
|
||||
let shift = (d[0] + d[1] + d[2] - vol) / 3.0;
|
||||
let mut work = 0.0;
|
||||
for k in 0..6 {
|
||||
let dk = if k < 3 { d[k] - shift } else { d[k] };
|
||||
work += 0.5 * (before[k] + p.stress[k]) * dk;
|
||||
es.strain[k] += dk;
|
||||
}
|
||||
es.energy += work * volume;
|
||||
es.plastic_work += inc.plastic_work * volume;
|
||||
if es.failed.is_none() {
|
||||
if p.damage >= 1.0 {
|
||||
es.failed = Some(Failure::Damage);
|
||||
} else if mat.erosion.is_some_and(|e| p.plastic_strain >= e) {
|
||||
es.failed = Some(Failure::PlasticStrain);
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
/// The density and dilatational wave speed of an element of rest volume `rest`, now of volume
|
||||
/// `volume`.
|
||||
fn current(mat: &Material, rest: f64, volume: f64) -> (f64, f64) {
|
||||
let rho = mat.density * rest / volume;
|
||||
(rho, (mat.stiffest / rho).sqrt())
|
||||
}
|
||||
|
||||
/// The bound ω² ≤ k λ_max(Σ b bᵀ) / (V m_min) on an element's highest frequency
|
||||
/// ([`modelcraft_fem::uniform`]), with the lumped masses of the undeformed element and
|
||||
/// `gradient_norm` = λ_max(Σ b bᵀ).
|
||||
pub fn frequency_bound<const N: usize>(k: &UniformStrain<N>, gradient_norm: f64, modulus: f64, mass: &[f64; N]) -> f64 {
|
||||
let lightest = mass.iter().copied().fold(f64::INFINITY, f64::min);
|
||||
modulus * gradient_norm / (k.volume * lightest)
|
||||
}
|
||||
|
||||
/// A hexahedron's size h, hourglass coefficient (damping or stiffness) and frequency bound ω²
|
||||
/// (with the hourglass stiffness), at density `density`.
|
||||
#[derive(Debug, Clone, Copy)]
|
||||
pub struct HexScales {
|
||||
pub size: f64,
|
||||
pub coefficient: f64,
|
||||
pub omega2: f64,
|
||||
}
|
||||
|
||||
/// The [`HexScales`] of a hexahedron in configuration `k` with shape vectors `gamma`.
|
||||
pub fn hex_scales(
|
||||
k: &Hex8,
|
||||
gamma: &[[f64; 8]; 4],
|
||||
mass: &[f64; 8],
|
||||
mat: &Material,
|
||||
density: f64,
|
||||
hourglass: modelcraft_model::Hourglass,
|
||||
) -> HexScales {
|
||||
let norm = k.gradient_norm();
|
||||
// h: the edge of the cube with the element's volume and gradient norm.
|
||||
let size = k.volume / (2.0 * norm).sqrt();
|
||||
let wave2 = mat.stiffest / density;
|
||||
let mut omega2 = frequency_bound(k, norm, mat.modulus, mass);
|
||||
let kappa = hourglass.coefficient;
|
||||
let coefficient = match hourglass.kind {
|
||||
HourglassKind::Viscous => kappa * density * wave2.sqrt() * size * size / 16.0,
|
||||
HourglassKind::Stiffness => {
|
||||
let stiffness = kappa * density * wave2 * size / 16.0;
|
||||
omega2 += stiffness * hourglass_gram_bound(gamma, mass);
|
||||
stiffness
|
||||
}
|
||||
};
|
||||
HexScales { size, coefficient, omega2 }
|
||||
}
|
||||
|
||||
/// Evaluate the internal forces at the motion `m` into `ws`, updating the element states, and
|
||||
/// return the smallest critical increment of the elements still in the analysis.
|
||||
pub fn internal_forces(sys: &System, m: &Motion, state: &mut State, ws: &mut Workspace) -> Result<f64> {
|
||||
for f in [&mut ws.stress_force, &mut ws.hourglass_force, &mut ws.viscous_force] {
|
||||
f.fill(0.0);
|
||||
}
|
||||
state.eroded.clear();
|
||||
let mut critical = f64::INFINITY;
|
||||
let hourglass = modelcraft_model::Hourglass { kind: sys.hourglass, coefficient: sys.hourglass_coefficient };
|
||||
for h in &sys.hexes {
|
||||
let Some(mat) = sys.materials.get(h.material) else { continue };
|
||||
let ue = gather(u, &h.nodes);
|
||||
let strain = h.kinematics.strain(&ue);
|
||||
let sigma = stress(&mat.elasticity, &strain);
|
||||
// Hourglass forces on the displacement (stiffness) or velocity (viscous) modes.
|
||||
let modes = match sys.hourglass {
|
||||
HourglassKind::Viscous => gather(v, &h.nodes),
|
||||
HourglassKind::Stiffness => ue,
|
||||
let (Some(mat), Some(es)) = (sys.materials.get(h.material), state.elements.get_mut(h.index)) else { continue };
|
||||
if es.failed.is_some() {
|
||||
continue;
|
||||
}
|
||||
let e = Nodal::gather(sys, &h.nodes, m);
|
||||
let x_now = e.positions(0.0);
|
||||
let (mid, now) = if sys.nonlinear {
|
||||
let at = |x: &[[f64; 3]; 8]| Hex8::hex8_deformed(x).map_err(|_| inverted(h.label, m.time));
|
||||
(at(&e.positions(0.5))?, at(&x_now)?)
|
||||
} else {
|
||||
(h.kinematics, h.kinematics)
|
||||
};
|
||||
let g = mid.gradient(&e.du);
|
||||
let q = rotation_of(sys, mat, es, &g, h.label, m.time)?;
|
||||
let (d, _) = rotation::split(&g);
|
||||
let (rho, c) = current(mat, h.kinematics.volume, now.volume);
|
||||
update(mat, es, &q, &d, d[0] + d[1] + d[2], m.dt, rho, mid.volume);
|
||||
if let Some(failure) = es.failed {
|
||||
state.eroded.push((h.index, failure));
|
||||
continue;
|
||||
}
|
||||
// Hourglass forces with the current shape vectors and size (fixed without nonlinear
|
||||
// geometry).
|
||||
let (gamma, scales) = if sys.nonlinear {
|
||||
let gamma = now.hourglass_vectors(&x_now);
|
||||
let scales = hex_scales(&now, &gamma, &h.mass, mat, rho, hourglass);
|
||||
(gamma, scales)
|
||||
} else {
|
||||
(h.gamma, h.rest)
|
||||
};
|
||||
let mut f = [[0.0; 3]; 8];
|
||||
for g in &h.gamma {
|
||||
let mut generalized = [0.0; 3];
|
||||
for (x, gk) in modes.iter().zip(g) {
|
||||
for i in 0..3 {
|
||||
generalized[i] += x[i] * gk;
|
||||
let modes = if sys.hourglass == HourglassKind::Viscous { &e.v } else { &e.du };
|
||||
for (k, g) in gamma.iter().enumerate() {
|
||||
let rate: [f64; 3] = [0, 1, 2].map(|i| modes.iter().zip(g).map(|(x, gk)| x[i] * gk).sum());
|
||||
// Viscous: Q = C q̇; stiffness: Q accumulates k Δq.
|
||||
let generalized = match (sys.hourglass, es.hourglass.get_mut(k)) {
|
||||
(HourglassKind::Stiffness, Some(qa)) => {
|
||||
for i in 0..3 {
|
||||
qa[i] += scales.coefficient * rate[i];
|
||||
}
|
||||
*qa
|
||||
}
|
||||
}
|
||||
_ => rate.map(|r| scales.coefficient * r),
|
||||
};
|
||||
for (fk, gk) in f.iter_mut().zip(g) {
|
||||
for i in 0..3 {
|
||||
fk[i] += h.hourglass * generalized[i] * gk;
|
||||
fk[i] += generalized[i] * gk;
|
||||
}
|
||||
}
|
||||
}
|
||||
scatter(&mut ws.hourglass_force, &h.nodes, &f);
|
||||
let finish = Finish { kinematics: &h.kinematics, nodes: &h.nodes, index: h.index, length: h.length };
|
||||
critical = critical.min(finish.apply(ws, mat, v, sigma, strain, bulk));
|
||||
let element = Element { now: &now, mid: &mid, nodes: &h.nodes, density: rho, wave_speed: c, length: 2.0 * c / scales.omega2.sqrt() };
|
||||
critical = critical.min(element.finish(ws, &es.point.stress, &e.v, m.bulk));
|
||||
}
|
||||
if !sys.tets.is_empty() {
|
||||
// Pass 1: each tetrahedron's strain, and each node's mean volumetric strain.
|
||||
// Pass 1: each tetrahedron's increment, and each node's mean volumetric increment.
|
||||
ws.theta.fill(0.0);
|
||||
for t in &sys.tets {
|
||||
let strain = t.kinematics.strain(&gather(u, &t.nodes));
|
||||
ws.tet_volume.fill(0.0);
|
||||
for (k, t) in sys.tets.iter().enumerate() {
|
||||
let Some(slot) = ws.tet_increment.get_mut(k) else { continue };
|
||||
*slot = None;
|
||||
if state.elements.get(t.index).is_none_or(|e| e.failed.is_some()) {
|
||||
continue;
|
||||
}
|
||||
let e = Nodal::gather(sys, &t.nodes, m);
|
||||
let (mid, now) = if sys.nonlinear {
|
||||
let at = |x: &[[f64; 3]; 4]| Tet4::tet4(x).map_err(|_| inverted(t.label, m.time));
|
||||
(at(&e.positions(0.5))?, at(&e.positions(0.0))?)
|
||||
} else {
|
||||
(t.kinematics, t.kinematics)
|
||||
};
|
||||
let g = mid.gradient(&e.du);
|
||||
for node in t.nodes {
|
||||
if let Some(x) = ws.theta.get_mut(node) {
|
||||
*x += t.kinematics.volume * (strain[0] + strain[1] + strain[2]);
|
||||
if let (Some(x), Some(w)) = (ws.theta.get_mut(node), ws.tet_volume.get_mut(node)) {
|
||||
*x += mid.volume * (g[0][0] + g[1][1] + g[2][2]);
|
||||
*w += mid.volume;
|
||||
}
|
||||
}
|
||||
if let Some(s) = ws.strain.get_mut(t.index) {
|
||||
*s = strain;
|
||||
*slot = Some(TetIncrement { g, mid, now, v: e.v });
|
||||
}
|
||||
for (x, w) in ws.theta.iter_mut().zip(&ws.tet_volume) {
|
||||
if *w > 0.0 {
|
||||
*x /= w;
|
||||
}
|
||||
}
|
||||
for (x, v) in ws.theta.iter_mut().zip(&ws.tet_volume) {
|
||||
if *v > 0.0 {
|
||||
*x /= v;
|
||||
}
|
||||
}
|
||||
// Pass 2: each tetrahedron's pressure from the mean of its nodes' volumetric strains.
|
||||
for t in &sys.tets {
|
||||
let Some(mat) = sys.materials.get(t.material) else { continue };
|
||||
let strain = ws.strain.get(t.index).copied().unwrap_or_default();
|
||||
// Pass 2: each tetrahedron's update with the mean of its nodes' volumetric increments.
|
||||
for (k, t) in sys.tets.iter().enumerate() {
|
||||
let (Some(mat), Some(es), Some(Some(inc))) =
|
||||
(sys.materials.get(t.material), state.elements.get_mut(t.index), ws.tet_increment.get(k).copied())
|
||||
else {
|
||||
continue;
|
||||
};
|
||||
let averaged = t.nodes.iter().map(|n| ws.theta.get(*n).copied().unwrap_or(0.0)).sum::<f64>() / 4.0;
|
||||
let own = strain[0] + strain[1] + strain[2];
|
||||
let mut sigma = stress(&mat.elasticity, &strain);
|
||||
let shift = mat.bulk.unwrap_or(0.0) * (averaged - own);
|
||||
for s in sigma.iter_mut().take(3) {
|
||||
*s += shift;
|
||||
let q = rotation_of(sys, mat, es, &inc.g, t.label, m.time)?;
|
||||
let (d, _) = rotation::split(&inc.g);
|
||||
let (rho, c) = current(mat, t.kinematics.volume, inc.now.volume);
|
||||
update(mat, es, &q, &d, averaged, m.dt, rho, inc.mid.volume);
|
||||
if let Some(failure) = es.failed {
|
||||
state.eroded.push((t.index, failure));
|
||||
continue;
|
||||
}
|
||||
let finish = Finish { kinematics: &t.kinematics, nodes: &t.nodes, index: t.index, length: t.length };
|
||||
critical = critical.min(finish.apply(ws, mat, v, sigma, strain, bulk));
|
||||
let omega2 = if sys.nonlinear { frequency_bound(&inc.now, inc.now.gradient_norm(), mat.modulus, &t.mass) } else { t.omega2 };
|
||||
let element = Element { now: &inc.now, mid: &inc.mid, nodes: &t.nodes, density: rho, wave_speed: c, length: 2.0 * c / omega2.sqrt() };
|
||||
critical = critical.min(element.finish(ws, &es.point.stress, &inc.v, m.bulk));
|
||||
}
|
||||
}
|
||||
critical
|
||||
Ok(critical)
|
||||
}
|
||||
|
||||
/// A tetrahedron's increment between the two passes: the gradient of its displacement increment,
|
||||
/// its midpoint and current kinematics, and its nodal velocities.
|
||||
#[derive(Debug, Clone, Copy)]
|
||||
struct TetIncrement {
|
||||
g: Mat3,
|
||||
mid: Tet4,
|
||||
now: Tet4,
|
||||
v: [[f64; 3]; 4],
|
||||
}
|
||||
|
||||
/// What the hexahedron and the tetrahedron share once their stress is known.
|
||||
struct Element<'a, const N: usize> {
|
||||
now: &'a UniformStrain<N>,
|
||||
mid: &'a UniformStrain<N>,
|
||||
nodes: &'a [usize; N],
|
||||
density: f64,
|
||||
wave_speed: f64,
|
||||
length: f64,
|
||||
}
|
||||
|
||||
impl<const N: usize> Element<'_, N> {
|
||||
/// Add the stress and bulk viscosity forces and return the critical increment.
|
||||
fn finish(&self, ws: &mut Workspace, sigma: &[f64; 6], v: &[[f64; 3]; N], bulk: &BulkViscosity) -> f64 {
|
||||
let rate = self.mid.divergence(v);
|
||||
let q = bulk_pressure(rate, self.density, self.length, self.wave_speed, bulk);
|
||||
scatter(&mut ws.stress_force, self.nodes, &self.now.forces(sigma));
|
||||
if q != 0.0 {
|
||||
scatter(&mut ws.viscous_force, self.nodes, &self.now.pressure_forces(q));
|
||||
}
|
||||
critical_increment(rate, self.length, self.wave_speed, bulk)
|
||||
}
|
||||
}
|
||||
Reference in new issue
Block a user