Hybrid (mixed u–p) solids: hex8 and tet10 with a constant pressure, hex20 with a linear one (#49)

The isotropic elasticity splits into D_dev + K m mᵀ; an element-wise
discontinuous pressure is condensed out (p̂ = K M⁻¹ Gᵀ u), leaving
K_dev + K G M⁻¹ Gᵀ (Hughes §4.3–4.5). Q2–P1disc and P2–P0 are inf-sup
stable; Q1–P0 does not lock; tet4 and wedge6 still lock and are refused, as
are orthotropic materials (named at the section). ElementDofs counts the
condensed pressures. Verified: element patch, rigid modes and totals for
every hybrid kind, a block in uniaxial strain at ν = 0.4999 (exact), Lamé's
cylinder in plane strain at ν = 0.4999 (hex20 1e-5, tet10 1.3e-3, hex8
2.5e-3 vs the locking displacement hex8 at −80 %). Every non-hybrid result
of the test suites hashes identically before and after.


Claude-Session: https://claude.ai/code/session_01NmAttHhUZTokgPeEJia8sm

Co-authored-by: jyviko <[email protected]>
Co-authored-by: Claude Opus 5.5 <[email protected]>

ROADMAP.md

@@ -26,7 +26,7 @@ hex meshes today and it gives the best elements where stresses are read.
 | M5 Structured hex | 2D quad meshing of faces (`modelcraft-mesh::quad`: transfinite mapped meshes, free all-quad meshes of polygons with holes on prescribed boundary nodes, exact planar quad validity, shared `SizeField` and even edge divisions; done for planar faces); mapped hex of six-sided regions; sweep meshing; conforming interfaces and tet–hex transition; automatic decomposition into sweepable and mappable regions, proposed as SolidCraft splits | started |
 | M6 Automatic hex | Octree/grid-dual all-hex for any part; hex-dominant by tet recombination; polycube for blocky parts; smoothing, untangling, boundary padding, layers through thin walls; mesh editing and orphan-mesh import | planned |
 | M7 Assemblies and contact | Part instances from SolidCraft occurrences; tie between non-matching meshes; general and pairwise contact with automatic pair detection, friction; bolt pretension, connectors; amplitudes, multiple steps; nonlinear static steps in decks | planned |
-| M8 Materials, explicit, shells, composites | Plasticity, hyperelasticity, damage data per solver; explicit dynamics decks (Abaqus/Explicit, LS-DYNA); midsurface shells; composite layups and material orientations, continuum shells on swept hex; heat transfer (solids done: steady and θ-method transient conduction with flux, convection and sources, thermal strain in static steps (solids and 2D) and explicit dynamics (with plasticity), coupled temperature–displacement, `solve.heat`; deck output, 2D/shell/beam conduction planned) | started |
+| M8 Materials, explicit, shells, composites | Plasticity, hyperelasticity, damage data per solver; explicit dynamics decks (Abaqus/Explicit, LS-DYNA); midsurface shells; composite layups and material orientations, continuum shells on swept hex; hybrid (mixed u–p) hex8/hex20/tet10 for nearly incompressible materials (done); heat transfer (solids done: steady and θ-method transient conduction with flux, convection and sources, thermal strain in static steps (solids and 2D) and explicit dynamics (with plasticity), coupled temperature–displacement, `solve.heat`; deck output, 2D/shell/beam conduction planned) | started |
 | M9 Post-processing and deck readers | Results readers for public formats (Nastran OP2, LS-DYNA d3plot, Abaqus `.fil`); XY and path plots, cuts, display groups, free-body forces; deck readers that keep unknown keywords; mesh convergence studies | planned |
 | M10 Explicit dynamics | Shared element and material kernels with the implicit solver (`fem`); central-difference solver with lumped mass, stable time step, one-point hex8 with hourglass control and nodal-pressure tet4, bulk viscosity, energy accounting (X1 core: done — `modelcraft-explicit`, `explicit_dynamic` steps with initial velocities, prescribed motion and planar rigid walls, `solve.explicit`/`solve.history`, small-strain linear elastic; bar wave, rigid-wall impact, energy balance and cantilever-period tests); large-deformation materials (X2: done — updated Lagrangian elements, Jaumann/Hughes–Winget and Green–Naghdi/Flanagan–Taylor rates, J2 bilinear/table plasticity, Johnson–Cook strength with heating and damage, erosion; materials gain `plastic`, `damage`, `erosion` with `model.material.plasticity`/`damage`; Dienes shear, rigid rotation, J2 uniaxial and Taylor-bar tests); contact (global search, penalty with friction, rigid walls); runs on hex and tet meshes | started |
 | M11 Equations of state and shells | Polynomial, ideal-gas, Mie–Grüneisen, JWL with programmed burn, tabulated; energy coupling; shock verification (Rankine–Hugoniot, Sod, Noh, Sedov); Belytschko–Lin–Tsay shells with self-contact on quad meshes | planned |

crates/fem/src/element.rs

@@ -146,6 +146,11 @@ pub enum Formulation {
     Full,
     Reduced,
     IncompatibleModes,
+    /// Mixed displacement–pressure with `pressures` discontinuous pressure unknowns condensed
+    /// out of the element ([`crate::hybrid`]).
+    Hybrid {
+        pressures: usize,
+    },
 }
 
 impl Formulation {
@@ -153,14 +158,23 @@ impl Formulation {
     /// incompatible modes it condenses out.
     pub fn dofs(self) -> ElementDofs {
         let incompatible_modes = if self == Formulation::IncompatibleModes { INCOMPATIBLE_DOFS } else { 0 };
-        ElementDofs { node: DofSet::TRANSLATIONS, internal: Internal { incompatible_modes, pressure: 0 } }
+        let pressure = if let Formulation::Hybrid { pressures } = self { pressures } else { 0 };
+        ElementDofs { node: DofSet::TRANSLATIONS, internal: Internal { incompatible_modes, pressure } }
     }
 
     /// The formulation a model section asks for, if this element type has it.
     pub fn from_section(kind: ElementKind, f: &modelcraft_model::Formulation) -> Result<Formulation> {
         let ty = element_type(kind)?;
         if f.hybrid {
-            return Err(Error::NotSupported("hybrid (mixed displacement–pressure) elements".into()));
+            if f.incompatible_modes || f.integration == modelcraft_model::Integration::Reduced {
+                return Err(Error::NotSupported("hybrid elements with incompatible modes or reduced integration".into()));
+            }
+            return crate::hybrid::pressures(kind).map(|pressures| Formulation::Hybrid { pressures }).ok_or_else(|| {
+                Error::NotSupported(format!(
+                    "hybrid {} elements (with one pressure per element the linear tetrahedron and wedge still lock; use hex8, hex20 or tet10)",
+                    kind.name()
+                ))
+            });
         }
         if f.incompatible_modes {
             return if ty.incompatible_modes {
@@ -189,10 +203,10 @@ pub struct PointStress {
 }
 
 /// Shape function gradients in physical coordinates at one point.
-struct Gradients {
-    shape: Shape,
-    dx: [[f64; 3]; MAX_NODES],
-    det: f64,
+pub(crate) struct Gradients {
+    pub(crate) shape: Shape,
+    pub(crate) dx: [[f64; 3]; MAX_NODES],
+    pub(crate) det: f64,
 }
 
 /// A thermal strain field over an element: the expansion coefficients along X, Y, Z and the
@@ -219,8 +233,8 @@ impl ThermalStrain {
 #[derive(Debug, Clone, Copy)]
 pub struct Solid<'a> {
     pub ty: &'static ElementType,
-    xyz: &'a [[f64; 3]],
-    formulation: Formulation,
+    pub(crate) xyz: &'a [[f64; 3]],
+    pub(crate) formulation: Formulation,
 }
 
 /// The stiffness blocks of the incompatible-modes element.
@@ -245,6 +259,7 @@ impl<'a> Solid<'a> {
             Formulation::Full => true,
             Formulation::Reduced => ty.reduced.is_some(),
             Formulation::IncompatibleModes => ty.incompatible_modes,
+            Formulation::Hybrid { pressures } => crate::hybrid::pressures(kind) == Some(pressures),
         };
         if !ok {
             return Err(Error::NotSupported(format!("{formulation:?} formulation of {} elements", kind.name())));
@@ -283,7 +298,7 @@ impl<'a> Solid<'a> {
         j
     }
 
-    fn gradients(&self, xi: [f64; 3]) -> Result<Gradients> {
+    pub(crate) fn gradients(&self, xi: [f64; 3]) -> Result<Gradients> {
         let shape = (self.ty.shape)(xi);
         let j = self.jacobian(&shape);
         let (inv, det) = positive_inverse(&j).ok_or_else(|| self.degenerate())?;
@@ -302,11 +317,15 @@ impl<'a> Solid<'a> {
 
     /// Position of a natural point.
     pub fn position(&self, xi: [f64; 3]) -> [f64; 3] {
-        let s = (self.ty.shape)(xi);
+        self.position_of((self.ty.shape)(xi).values())
+    }
+
+    /// The position Σ Nᵢ xᵢ for shape values `n`.
+    pub(crate) fn position_of(&self, n: &[f64]) -> [f64; 3] {
         let mut p = [0.0; 3];
-        for (n, x) in s.values().iter().zip(self.xyz) {
+        for (ni, x) in n.iter().zip(self.xyz) {
             for k in 0..3 {
-                p[k] += n * x[k];
+                p[k] += ni * x[k];
             }
         }
         p
@@ -364,6 +383,9 @@ impl<'a> Solid<'a> {
 
     /// The stiffness matrix, 3n × 3n, degrees of freedom ordered node by node (x, y, z).
     pub fn stiffness(&self, d: &Elasticity) -> Result<Matrix> {
+        if let Formulation::Hybrid { .. } = self.formulation {
+            return self.hybrid_stiffness(d);
+        }
         let n = 3 * self.ty.nodes;
         if self.formulation == Formulation::IncompatibleModes {
             let IncompatibleParts { kuu: mut k, kua, kaa, .. } = self.incompatible_parts(d, true)?;
@@ -444,7 +466,7 @@ impl<'a> Solid<'a> {
     }
 
     /// The thermal strain α ΔT at a point with shape values `n` (xx, yy, zz; no shear).
-    fn thermal_strain(thermal: &ThermalStrain, n: &[f64]) -> [f64; 6] {
+    pub(crate) fn thermal_strain(thermal: &ThermalStrain, n: &[f64]) -> [f64; 6] {
         let dt: f64 = n.iter().zip(&thermal.delta).map(|(a, b)| a * b).sum();
         [thermal.alpha[0] * dt, thermal.alpha[1] * dt, thermal.alpha[2] * dt, 0.0, 0.0, 0.0]
     }
@@ -454,6 +476,9 @@ impl<'a> Solid<'a> {
     /// are condensed out as in the stiffness, f = f_u − K_ua K_aa⁻¹ f_a.
     pub fn thermal_load(&self, d: &Elasticity, thermal: &ThermalStrain) -> Result<Vec<[f64; 3]>> {
         thermal.check(self.ty.nodes)?;
+        if let Formulation::Hybrid { .. } = self.formulation {
+            return self.hybrid_thermal_load(d, thermal);
+        }
         if self.formulation == Formulation::IncompatibleModes {
             let IncompatibleParts { kua, kaa, j0, .. } = self.incompatible_parts(d, false)?;
             let (mut f, mut fa) = self.thermal_forces(d, thermal, Some(&j0))?;
@@ -511,6 +536,8 @@ impl<'a> Solid<'a> {
         if let Some(t) = thermal {
             t.check(self.ty.nodes)?;
         }
+        // Hybrid: the element's pressure field, and σ = D_dev (ε − ε_th) + p m.
+        let hybrid = if let Formulation::Hybrid { .. } = self.formulation { Some(self.pressure_field(d, u, thermal)?) } else { None };
         // Incompatible modes: recover the internal amplitudes from K_aa α + K_au u = f_a (the
         // thermal load on the modes; zero without one).
         let alpha = if self.formulation == Formulation::IncompatibleModes {
@@ -552,18 +579,18 @@ impl<'a> Solid<'a> {
                 }
             }
             let strain = [grad[0][0], grad[1][1], grad[2][2], grad[0][1] + grad[1][0], grad[0][2] + grad[2][0], grad[1][2] + grad[2][1]];
-            let mut xyz = [0.0; 3];
-            for (n, x) in g.shape.values().iter().zip(self.xyz) {
-                for k in 0..3 {
-                    xyz[k] += n * x[k];
-                }
-            }
-            let sigma = match thermal {
+            let xyz = self.position_of(g.shape.values());
+            // The mechanical strain, less the thermal strain when there is one.
+            let mechanical = match thermal {
                 Some(t) => {
                     let eth = Self::thermal_strain(t, g.shape.values());
-                    stress(d, &[0, 1, 2, 3, 4, 5].map(|k| strain[k] - eth[k]))
+                    [0, 1, 2, 3, 4, 5].map(|k| strain[k] - eth[k])
                 }
-                None => stress(d, &strain),
+                None => strain,
+            };
+            let sigma = match &hybrid {
+                Some(h) => h.stress(&mechanical, xyz),
+                None => stress(d, &mechanical),
             };
             out.push(PointStress { xyz, strain, stress: sigma });
         }
@@ -606,7 +633,7 @@ pub(crate) fn symmetrize(k: &mut Matrix) {
 }
 
 /// The strain-displacement block of one gradient g: ε = B u for a single node's (x, y, z).
-fn b_block(g: &[f64; 3]) -> [[f64; 3]; 6] {
+pub(crate) fn b_block(g: &[f64; 3]) -> [[f64; 3]; 6] {
     [[g[0], 0.0, 0.0], [0.0, g[1], 0.0], [0.0, 0.0, g[2]], [g[1], g[0], 0.0], [g[2], 0.0, g[0]], [0.0, g[2], g[1]]]
 }
 
@@ -638,7 +665,7 @@ fn accumulate(out: &mut [f64], cols: usize, ga: &[[f64; 3]], gb: &[[f64; 3]], d:
 
 /// The symmetric case of [`accumulate`] (rows and columns from the same gradients, `out` square):
 /// computes the blocks with j ≥ i and mirrors them, so the result is exactly symmetric.
-fn accumulate_symmetric(out: &mut [f64], g: &[[f64; 3]], d: &Elasticity, w: f64) {
+pub(crate) fn accumulate_symmetric(out: &mut [f64], g: &[[f64; 3]], d: &Elasticity, w: f64) {
     let cols = 3 * g.len();
     for (j, gj) in g.iter().enumerate() {
         let bj = b_block(gj);

crates/fem/src/hybrid.rs

@@ -0,0 +1,277 @@
+//! Hybrid (mixed displacement–pressure) solids for nearly incompressible materials.
+//!
+//! The isotropic elasticity splits into a deviatoric part and the bulk modulus,
+//! D = D_dev + K m mᵀ with m = (1, 1, 1, 0, 0, 0) and K = mᵀ D m / 9. The pressure (here the
+//! mean stress, positive in tension) is an independent field p = N_p · p̂, discontinuous between
+//! elements, and the volumetric strain is enforced weakly (the perturbed Lagrangian of
+//! T. J. R. Hughes, *The Finite Element Method*, Dover 2000, §4.3–4.4):
+//!
+//! K_dev = ∫ Bᵀ D_dev B dV, G = ∫ Bᵀ m N_pᵀ dV, M = ∫ N_p N_pᵀ dV,
+//! [K_dev G; Gᵀ −M/K] [u; p̂] = [f; 0],
+//!
+//! so p̂ = K M⁻¹ Gᵀ u is condensed out element by element and the element stiffness is
+//! K_dev + K G M⁻¹ Gᵀ: symmetric positive definite for any ν < ½, and bounded volumetric
+//! stiffness per element however close ν is to ½ — the constraint count no longer grows with
+//! the nodes, which is what locks the displacement elements. With a constant pressure this is
+//! the B-bar / mean dilatation element (Hughes 2000, §4.5; J. C. Nagtegaal, D. M. Parks,
+//! J. R. Rice, *On numerically accurate finite element solutions in the fully plastic range*,
+//! CMAME 4(2):153–177, 1974).
+//!
+//! Pressure fields and stability (the inf-sup or LBB condition, F. Brezzi, M. Fortin, *Mixed
+//! and Hybrid Finite Element Methods*, Springer 1991, ch. VI; K.-J. Bathe, *Finite Element
+//! Procedures*, 2nd ed., 2014, §4.5):
+//! - **hex20**: linear pressure 1, x, y, z (Q2–P1disc, inf-sup stable);
+//! - **tet10**: constant pressure (P2–P0, inf-sup stable; P2–P1disc is not on general meshes);
+//! - **tet10**: a single element has three more zero-energy modes (dilatations with a linear
+//!   volumetric strain, which its deviatoric stiffness does not see and one pressure controls
+//!   only in the mean); neighbouring elements restrain them, as they do the reduced hex20's;
+//! - **hex8**: constant pressure (Q1–P0). It passes the patch test and does not lock, but is
+//!   not inf-sup stable: on some meshes and constraints the element pressures show a
+//!   checkerboard mode while displacements stay good; nodal stresses are averaged anyway;
+//! - **tet4, wedge6**: refused — their cross-section is the linear triangle, and one pressure
+//!   per element still leaves too few displacement unknowns per constraint: they lock (verified
+//!   on the thick cylinder; a nodal-pressure formulation would be needed).
+//!
+//! A continuous nodal pressure (Taylor–Hood, P on the corner nodes) makes the global system a
+//! symmetric indefinite saddle point that the Cholesky factorization cannot solve; it waits for
+//! an indefinite (LDLᵀ) factorization.
+
+use modelcraft_model::ElementKind;
+
+use crate::dense::{Cholesky, Matrix};
+use crate::element::{Formulation, Solid, ThermalStrain, accumulate_symmetric, b_block};
+use crate::material::Elasticity;
+use crate::{Error, Result};
+
+/// The pressure unknowns of the hybrid formulation of an element kind, if it has one.
+pub fn pressures(kind: ElementKind) -> Option<usize> {
+    match kind {
+        ElementKind::Hex20 => Some(4),
+        ElementKind::Hex8 | ElementKind::Tet10 => Some(1),
+        _ => None,
+    }
+}
+
+/// The deviatoric elasticity and the bulk modulus of an isotropic elasticity; orthotropic
+/// materials have no such split and are refused.
+pub fn split(d: &Elasticity) -> Result<(Elasticity, f64)> {
+    let scale = d.iter().flatten().fold(0.0f64, |m, v| m.max(v.abs()));
+    let same = |a: f64, b: f64| (a - b).abs() <= 1e-12 * scale;
+    let (c11, c12, c44) = (d[0][0], d[0][1], d[3][3]);
+    let isotropic = (0..3).all(|i| same(d[i][i], c11) && same(d[i + 3][i + 3], c44))
+        && (0..3).all(|i| (0..3).all(|j| i == j || same(d[i][j], c12)))
+        && same(c44, 0.5 * (c11 - c12))
+        && (0..6).all(|i| (0..6).all(|j| (i < 3 && j < 3) || i == j || d[i][j] == 0.0));
+    if !isotropic {
+        return Err(Error::NotSupported("hybrid elements of orthotropic materials".into()));
+    }
+    let bulk = (c11 + 2.0 * c12) / 3.0;
+    let mut dev = *d;
+    for row in dev.iter_mut().take(3) {
+        for v in row.iter_mut().take(3) {
+            *v -= bulk;
+        }
+    }
+    Ok((dev, bulk))
+}
+
+/// The parts of a hybrid element: G, 3n × np row-major; the factor of M; and K_dev when
+/// asked for.
+struct Mixed {
+    kdev: Option<Matrix>,
+    g: Vec<f64>,
+    m: Cholesky,
+    /// With a thermal strain: ∫ N_p tr ε_th dV (np) and ∫ Bᵀ D_dev ε_th dV (3n).
+    h: Vec<f64>,
+    f_dev: Vec<f64>,
+}
+
+/// The pressure field of a hybrid element under nodal displacements, for its stresses.
+pub(crate) struct PressureField {
+    dev: Elasticity,
+    bulk: f64,
+    /// M⁻¹ Gᵀ u, the coefficients of the volumetric strain's projection.
+    coefficients: [f64; 4],
+    np: usize,
+    frame: ([f64; 3], f64),
+}
+
+impl PressureField {
+    /// D_dev ε + p m at `x`.
+    pub(crate) fn stress(&self, strain: &[f64; 6], x: [f64; 3]) -> [f64; 6] {
+        let basis = pressure_basis(self.np, x, &self.frame);
+        let p = self.bulk * self.coefficients.iter().zip(basis).map(|(a, b)| a * b).sum::<f64>();
+        let mut s = crate::material::stress(&self.dev, strain);
+        for v in s.iter_mut().take(3) {
+            *v += p;
+        }
+        s
+    }
+}
+
+/// The pressure functions at a point: 1, then (x − c)/h, (y − c)/h, (z − c)/h for the linear
+/// field (`np` = 4).
+fn pressure_basis(np: usize, x: [f64; 3], frame: &([f64; 3], f64)) -> [f64; 4] {
+    let (c, h) = frame;
+    let mut out = [1.0, 0.0, 0.0, 0.0];
+    if np == 4 && *h > 0.0 {
+        for k in 0..3 {
+            out[k + 1] = (x[k] - c[k]) / h;
+        }
+    }
+    out
+}
+
+impl Solid<'_> {
+    /// The pressure unknowns of this (hybrid) element.
+    fn np(&self) -> usize {
+        match self.formulation {
+            Formulation::Hybrid { pressures } => pressures.min(4),
+            _ => 0,
+        }
+    }
+
+    /// The element's centre (mean of its corners) and size (largest corner distance from it),
+    /// which scale the linear pressure functions.
+    fn frame(&self) -> ([f64; 3], f64) {
+        let corners = self.xyz.get(..self.ty.corners).unwrap_or_default();
+        let n = corners.len().max(1) as f64;
+        let c = corners.iter().fold([0.0; 3], |a, x| [a[0] + x[0] / n, a[1] + x[1] / n, a[2] + x[2] / n]);
+        let size = corners.iter().map(|x| ((x[0] - c[0]).powi(2) + (x[1] - c[1]).powi(2) + (x[2] - c[2]).powi(2)).sqrt()).fold(0.0, f64::max);
+        (c, size)
+    }
+
+    /// G and M (and K_dev with `kdev`, and the thermal terms with `thermal`) in one pass over
+    /// the stiffness rule.
+    fn mixed(&self, dev: &Elasticity, kdev: bool, thermal: Option<&ThermalStrain>) -> Result<Mixed> {
+        let (np, nodes, frame) = (self.np(), self.ty.nodes, self.frame());
+        let mut kdev = kdev.then(|| Matrix::zeros(3 * nodes));
+        let mut g = vec![0.0; 3 * nodes * np];
+        let mut m = Matrix::zeros(np);
+        let mut h = vec![0.0; np];
+        let mut f_dev = vec![0.0; 3 * nodes];
+        for p in self.rule().points() {
+            let gr = self.gradients(p.xi)?;
+            let w = gr.det * p.weight;
+            let gu = gr.dx.get(..nodes).unwrap_or_default();
+            if let Some(k) = kdev.as_mut() {
+                accumulate_symmetric(&mut k.data, gu, dev, w);
+            }
+            let x = if np == 4 { self.position_of(gr.shape.values()) } else { [0.0; 3] };
+            let np_at = pressure_basis(np, x, &frame);
+            if let Some(t) = thermal {
+                let eth = Self::thermal_strain(t, gr.shape.values());
+                let volumetric = eth[0] + eth[1] + eth[2];
+                for (hk, nk) in h.iter_mut().zip(np_at) {
+                    *hk += w * nk * volumetric;
+                }
+                let sig = crate::material::stress(dev, &eth);
+                for (i, gi) in gu.iter().enumerate() {
+                    let b = b_block(gi);
+                    for c in 0..3 {
+                        if let Some(f) = f_dev.get_mut(3 * i + c) {
+                            *f += w * (0..6).map(|s| b[s][c] * sig[s]).sum::<f64>();
+                        }
+                    }
+                }
+            }
+            // Bᵢᵀ m is node i's gradient.
+            for (i, gi) in gu.iter().enumerate() {
+                for c in 0..3 {
+                    for k in 0..np {
+                        if let Some(v) = g.get_mut((3 * i + c) * np + k) {
+                            *v += w * gi[c] * np_at[k];
+                        }
+                    }
+                }
+            }
+            for a in 0..np {
+                for b in 0..np {
+                    m.add(a, b, w * np_at[a] * np_at[b]);
+                }
+            }
+        }
+        Ok(Mixed { kdev, g, m: Cholesky::new(&m)?, h, f_dev })
+    }
+
+    /// K_dev + K G M⁻¹ Gᵀ, symmetric by construction (H = M⁻¹ Gᵀ once, upper triangle mirrored).
+    pub(crate) fn hybrid_stiffness(&self, d: &Elasticity) -> Result<Matrix> {
+        let (dev, bulk) = split(d)?;
+        let Mixed { kdev, g, m, .. } = self.mixed(&dev, true, None)?;
+        let mut k = kdev.ok_or_else(|| Error::Invalid("hybrid stiffness".into()))?;
+        let (n, np) = (k.n, self.np());
+        let row = |i: usize| g.get(i * np..(i + 1) * np).unwrap_or_default();
+        let h: Vec<Vec<f64>> = (0..n)
+            .map(|j| {
+                let mut y = row(j).to_vec();
+                m.solve(&mut y);
+                y
+            })
+            .collect();
+        for i in 0..n {
+            for (j, hj) in h.iter().enumerate().skip(i) {
+                let v = bulk * row(i).iter().zip(hj).map(|(a, b)| a * b).sum::<f64>();
+                k.add(i, j, v);
+                if i != j {
+                    k.add(j, i, v);
+                }
+            }
+        }
+        Ok(k)
+    }
+
+    /// The equivalent nodal forces of a thermal strain: ∫ Bᵀ D_dev ε_th dV for the deviatoric
+    /// part, and K G M⁻¹ ∫ N_p tr ε_th dV for the volumetric one through the pressure.
+    pub(crate) fn hybrid_thermal_load(&self, d: &Elasticity, thermal: &ThermalStrain) -> Result<Vec<[f64; 3]>> {
+        let (dev, bulk) = split(d)?;
+        let np = self.np();
+        let Mixed { g, m, mut h, f_dev, .. } = self.mixed(&dev, false, Some(thermal))?;
+        m.solve(&mut h);
+        let f: Vec<f64> = f_dev
+            .iter()
+            .enumerate()
+            .map(|(i, fd)| fd + bulk * (0..np).map(|k| g.get(i * np + k).copied().unwrap_or(0.0) * h.get(k).copied().unwrap_or(0.0)).sum::<f64>())
+            .collect();
+        Ok(f.as_chunks::<3>().0.to_vec())
+    }
+
+    /// The element's pressure field under the nodal displacements `u` and a thermal strain:
+    /// p̂ = K M⁻¹ (Gᵀ u − ∫ N_p tr ε_th dV).
+    pub(crate) fn pressure_field(&self, d: &Elasticity, u: &[[f64; 3]], thermal: Option<&ThermalStrain>) -> Result<PressureField> {
+        let (dev, bulk) = split(d)?;
+        let np = self.np();
+        let Mixed { g, m, h, .. } = self.mixed(&dev, false, thermal)?;
+        let mut pc: Vec<f64> = h.iter().map(|x| -x).collect();
+        for (i, ui) in u.iter().flatten().enumerate() {
+            for (k, x) in pc.iter_mut().enumerate() {
+                *x += g.get(i * np + k).copied().unwrap_or(0.0) * ui;
+            }
+        }
+        m.solve(&mut pc);
+        let mut coefficients = [0.0; 4];
+        for (c, v) in coefficients.iter_mut().zip(pc) {
+            *c = v;
+        }
+        Ok(PressureField { dev, bulk, coefficients, np, frame: self.frame() })
+    }
+}
+
+#[cfg(test)]
+mod tests {
+    use super::*;
+    use crate::material::{isotropic, orthotropic};
+
+    #[test]
+    fn the_split_is_deviatoric_plus_bulk() {
+        let d = isotropic(1000.0, 0.3).unwrap();
+        let (dev, k) = split(&d).unwrap();
+        assert!((k - 1000.0 / (3.0 * (1.0 - 0.6))).abs() < 1e-9);
+        // D_dev m = 0.
+        for r in 0..6 {
+            assert!((0..3).map(|c| dev[r][c]).sum::<f64>().abs() < 1e-9);
+        }
+        assert!(split(&orthotropic([1000.0, 800.0, 600.0], [0.3, 0.25, 0.2], [400.0, 300.0, 350.0]).unwrap()).is_err());
+        assert_eq!(pressures(ElementKind::Tet4), None);
+        assert_eq!(pressures(ElementKind::Hex20), Some(4));
+    }
+}

crates/fem/src/lib.rs

@@ -50,6 +50,7 @@ pub mod element;
 pub mod face;
 pub mod frame;
 pub mod heat;
+pub mod hybrid;
 pub mod jet;
 pub mod material;
 pub mod plane;

crates/fem/src/plane.rs

@@ -270,6 +270,7 @@ impl<'a> Plane<'a> {
             Formulation::Full => true,
             Formulation::Reduced => ty.reduced.is_some(),
             Formulation::IncompatibleModes => ty.incompatible_modes,
+            Formulation::Hybrid { .. } => false,
         };
         if !ok {
             return Err(Error::NotSupported(format!("{formulation:?} formulation of 2D {} elements", kind.name())));

crates/fem/src/tests.rs

@@ -34,7 +34,7 @@ fn distorted(kind: ElementKind, curved: bool) -> Vec<[f64; 3]> {
     xyz
 }
 
-const CASES: [(ElementKind, Formulation); 7] = [
+const CASES: [(ElementKind, Formulation); 10] = [
     (ElementKind::Tet4, Formulation::Full),
     (ElementKind::Tet10, Formulation::Full),
     (ElementKind::Wedge6, Formulation::Full),
@@ -42,6 +42,9 @@ const CASES: [(ElementKind, Formulation); 7] = [
     (ElementKind::Hex8, Formulation::IncompatibleModes),
     (ElementKind::Hex20, Formulation::Full),
     (ElementKind::Hex20, Formulation::Reduced),
+    (ElementKind::Tet10, Formulation::Hybrid { pressures: 1 }),
+    (ElementKind::Hex8, Formulation::Hybrid { pressures: 1 }),
+    (ElementKind::Hex20, Formulation::Hybrid { pressures: 4 }),
 ];
 
 fn rigid_modes(xyz: &[[f64; 3]]) -> Vec<Vec<f64>> {
@@ -75,8 +78,11 @@ fn stiffness_is_symmetric_with_exactly_six_rigid_body_modes() {
         }
         // Holding six degrees of freedom (node 1 fully, node 2 in y and z, node 3 in z) leaves a
         // positive definite matrix: no other zero-energy mode. Reduced hex20 has spurious modes
-        // on a single element, so it is skipped.
-        if f != Formulation::Reduced {
+        // on a single element, and so has hybrid tet10: its deviatoric stiffness is blind to the
+        // four dilatations with a linear volumetric strain, and its constant pressure controls
+        // one; neighbouring elements restrain the other three (P2–P0, crate::hybrid). Both are
+        // skipped.
+        if f != Formulation::Reduced && !(kind == ElementKind::Tet10 && matches!(f, Formulation::Hybrid { .. })) {
             let mut held = k.clone();
             for dof in [0, 1, 2, 4, 5, 8] {
                 held.add(dof, dof, 1e3 * scale);
@@ -95,7 +101,9 @@ fn constant_strain_is_reproduced_exactly() {
     let strain = [a[0][0], a[1][1], a[2][2], a[0][1] + a[1][0], a[0][2] + a[2][0], a[1][2] + a[2][1]];
     for d in [isotropic(210e3, 0.3).unwrap(), orthotropic([140e3, 10e3, 12e3], [0.3, 0.25, 0.4], [5e3, 4e3, 3.8e3]).unwrap()] {
         let want = stress(&d, &strain);
-        for (kind, f) in CASES {
+        // Hybrid elements need an isotropic material.
+        let hybrid_ok = crate::hybrid::split(&d).is_ok();
+        for (kind, f) in CASES.into_iter().filter(|(_, f)| hybrid_ok || !matches!(f, Formulation::Hybrid { .. })) {
             // Curved mid-sides keep the field in the isoparametric span only for straight edges.
             let xyz = distorted(kind, false);
             let u: Vec<[f64; 3]> = xyz.iter().map(|x| [0, 1, 2].map(|i| (0..3).map(|j| a[i][j] * x[j]).sum())).collect();
@@ -213,9 +221,12 @@ fn section_formulations() {
 fn uniform_thermal_strain_is_free_expansion_and_its_load_balances_it() {
     // A uniform ΔT with orthotropic α: the free expansion u = diag(α ΔT) x is stress-free, and
     // the thermal load equals K u for it (the element is in equilibrium with no other load).
-    let d = orthotropic([1000.0, 800.0, 600.0], [0.3, 0.25, 0.2], [400.0, 300.0, 350.0]).unwrap();
+    let ortho = orthotropic([1000.0, 800.0, 600.0], [0.3, 0.25, 0.2], [400.0, 300.0, 350.0]).unwrap();
+    let iso = isotropic(1000.0, 0.4999).unwrap();
     let alpha = [1.2e-5, 2.0e-5, 0.7e-5];
     for (kind, f) in CASES {
+        // Hybrid elements take isotropic (here nearly incompressible) materials.
+        let d = if matches!(f, Formulation::Hybrid { .. }) { iso } else { ortho };
         let xyz = distorted(kind, true);
         let solid = Solid::new(kind, &xyz, f).unwrap();
         let th = crate::ThermalStrain { alpha, delta: vec![80.0; xyz.len()] };

crates/solver/src/system.rs

@@ -17,7 +17,7 @@ use modelcraft_fem::{DofKind, DofLayout, DofSet, Elasticity, ElementDofs, PointS
 use modelcraft_model::nodal::edge_pressure;
 use std::collections::BTreeSet;
 
-use modelcraft_model::{ElementKind, Integration, Load, Model, Resolved, SectionKind, Step, Target, centrifugal_acceleration};
+use modelcraft_model::{Elastic, ElementKind, Integration, Load, Model, Resolved, SectionKind, Step, Target, centrifugal_acceleration};
 
 use crate::analysis::MassKind;
 use crate::{Error, Result};
@@ -398,6 +398,12 @@ impl Mesh {
                 .ok_or_else(|| Error::Invalid(format!("section `{name}`: material `{}` is missing", section.material)))?;
             let elastic =
                 material.elastic.as_ref().ok_or_else(|| Error::Invalid(format!("material `{}` has no elastic constants", section.material)))?;
+            if section.formulation.hybrid && !matches!(elastic, Elastic::Isotropic { .. }) {
+                return Err(Error::NotSupported(format!(
+                    "hybrid elements of the orthotropic material `{}` (section `{name}`): the pressure split needs isotropic elasticity",
+                    section.material
+                )));
+            }
             let set = resolved.sets.get(&section.set).ok_or_else(|| Error::Invalid(format!("set `{}` is not resolved", section.set)))?;
             let bad = |e: modelcraft_fem::Error| match e {
                 modelcraft_fem::Error::NotSupported(m) => Error::NotSupported(format!("section `{name}`: {m}")),

crates/solver/tests/hybrid.rs

@@ -0,0 +1,140 @@
+//! Hybrid (mixed u–p) solids for nearly incompressible materials, against closed forms: a block
+//! in uniaxial strain (exact for every element) and Lamé's thick cylinder in plane strain at
+//! ν = 0.4999, where the displacement elements lock and the hybrid ones do not. Run with
+//! `--nocapture` to see the errors.
+
+mod common;
+
+use common::*;
+use modelcraft_model::*;
+use modelcraft_solver::{Limits, solve_static};
+
+const NU: f64 = 0.4999;
+
+fn hybrid() -> Formulation {
+    Formulation { hybrid: true, ..Formulation::default() }
+}
+
+fn hybrids() -> Vec<(ElementKind, &'static str)> {
+    vec![(ElementKind::Hex8, "hex8 hybrid"), (ElementKind::Hex20, "hex20 hybrid"), (ElementKind::Tet10, "tet10 hybrid")]
+}
+
+#[test]
+fn block_in_uniaxial_strain() {
+    // Held normal to the bottom and the four sides, pressed on top: ε_zz = −p/M with the
+    // constrained modulus M = E(1 − ν)/((1 + ν)(1 − 2ν)); σ_xx = σ_yy = −p ν/(1 − ν).
+    let (e, p) = (10.0, 1.0);
+    let m = e * (1.0 - NU) / ((1.0 + NU) * (1.0 - 2.0 * NU));
+    for (kind, name) in hybrids() {
+        let mut g = grid([2, 2, 2], kind, |x| x);
+        g.define(iso(e, NU), None, hybrid(), Procedure::Static {});
+        g.symmetry("x0", Symmetry::X, |x| near(x[0], 0.0));
+        g.symmetry("x1", Symmetry::X, |x| near(x[0], 1.0));
+        g.symmetry("y0", Symmetry::Y, |x| near(x[1], 0.0));
+        g.symmetry("y1", Symmetry::Y, |x| near(x[1], 1.0));
+        g.symmetry("z0", Symmetry::Z, |x| near(x[2], 0.0));
+        g.face_set("top", |x| near(x[2], 1.0));
+        g.load("press", Load::Pressure { set: "top".into(), magnitude: p });
+        let r = solve_static(&g.model, None, &Limits::default()).unwrap();
+        for (i, l) in r.nodes.iter().enumerate() {
+            let z = g.model.node(*l).unwrap().xyz[2];
+            assert!((r.displacement[i][2] + p * z / m).abs() < 1e-9 * p / m, "{name} node {l}: {:?}", r.displacement[i]);
+        }
+        for s in &r.stress {
+            assert!((s[2] + p).abs() < 1e-6 && (s[0] + p * NU / (1.0 - NU)).abs() < 1e-6 && (s[1] - s[0]).abs() < 1e-6, "{name}: {s:?}");
+        }
+    }
+}
+
+/// The inner radial displacement of a quarter of a thick cylinder a = 1, b = 2 in plane strain
+/// under internal pressure 1, and Lamé's value.
+fn cylinder(kind: ElementKind, formulation: Formulation, n: [usize; 3]) -> (f64, f64) {
+    let (e, p, a, b) = (1000.0, 1.0, 1.0, 2.0);
+    let mut g = grid(n, kind, |x| {
+        let (r, t) = (a + (b - a) * x[0], std::f64::consts::FRAC_PI_2 * x[1]);
+        [r * t.cos(), r * t.sin(), 0.2 * x[2]]
+    });
+    g.define(iso(e, NU), None, formulation, Procedure::Static {});
+    g.symmetry("y0", Symmetry::Y, |x| near(x[1], 0.0));
+    g.symmetry("x0", Symmetry::X, |x| near(x[1], 1.0));
+    g.symmetry("z0", Symmetry::Z, |x| near(x[2], 0.0));
+    g.symmetry("z1", Symmetry::Z, |x| near(x[2], 1.0));
+    g.face_set("bore", |x| near(x[0], 0.0));
+    g.load("pressure", Load::Pressure { set: "bore".into(), magnitude: p });
+    let r = solve_static(&g.model, None, &Limits::default()).unwrap();
+    let inner = g.nodes_where(|x| near(x[0], 0.0) && near(x[1], 0.0) && near(x[2], 0.0))[0];
+    let u = r.displacement[r.nodes.binary_search(&inner).unwrap()][0];
+    let exact = (1.0 + NU) * p * a * a / (e * (b * b - a * a)) * ((1.0 - 2.0 * NU) * a + b * b / a);
+    (u, exact)
+}
+
+#[test]
+fn nearly_incompressible_thick_cylinder_does_not_lock() {
+    for (kind, name, n, tol) in [
+        (ElementKind::Hex8, "hex8 hybrid", [8, 12, 1], 0.01),
+        (ElementKind::Hex20, "hex20 hybrid", [4, 6, 1], 1e-4),
+        (ElementKind::Tet10, "tet10 hybrid", [4, 6, 1], 5e-3),
+    ] {
+        let (u, exact) = cylinder(kind, hybrid(), n);
+        eprintln!("{name}: u(a) = {u:.6e}, Lamé {exact:.6e}, error {:.2e}", (u - exact) / exact);
+        assert!(((u - exact) / exact).abs() < tol, "{name}: {u} vs {exact}");
+    }
+    // The displacement hex8 locks: far too stiff on the same mesh.
+    let (u, exact) = cylinder(ElementKind::Hex8, Formulation::default(), [8, 12, 1]);
+    eprintln!("hex8 (displacement): u(a) = {u:.6e}, Lamé {exact:.6e}, error {:.2e}", (u - exact) / exact);
+    assert!(u < 0.7 * exact, "full hex8 should lock at ν = {NU}: {u} vs {exact}");
+}
+
+/// Why a one-cell hybrid model of `kind` and `elastic` cannot be solved.
+fn refusal(kind: ElementKind, elastic: Elastic) -> String {
+    let mut g = grid([1, 1, 1], kind, |x| x);
+    g.define(elastic, None, hybrid(), Procedure::Static {});
+    g.fixed("base", |x| near(x[2], 0.0));
+    g.face_set("top", |x| near(x[2], 1.0));
+    g.load("press", Load::Pressure { set: "top".into(), magnitude: 1.0 });
+    solve_static(&g.model, None, &Limits::default()).unwrap_err().to_string()
+}
+
+#[test]
+fn hybrid_refusals() {
+    for (kind, needle) in [(ElementKind::Tet4, "hybrid tet4"), (ElementKind::Wedge6, "hybrid wedge6")] {
+        let e = refusal(kind, iso(10.0, NU));
+        assert!(e.contains(needle), "{e}");
+    }
+    let ortho = Elastic::EngineeringConstants { young: [10.0, 20.0, 30.0], poisson: [0.3, 0.2, 0.1], shear: [5.0, 6.0, 7.0] };
+    let e = refusal(ElementKind::Hex8, ortho);
+    assert!(e.contains("orthotropic material `Mat`"), "{e}");
+}
+
+#[test]
+fn heated_confined_block_carries_the_bulk_thermal_stress() {
+    // Held normal to all six faces and heated by ΔT: σ = −3 K α ΔT in every direction, the
+    // pressure the hybrid elements carry as their own unknown.
+    let (e, alpha, dt) = (10.0, 1e-4, 30.0);
+    let bulk = e / (3.0 * (1.0 - 2.0 * NU));
+    for (kind, name) in hybrids() {
+        let mut g = grid([2, 2, 2], kind, |x| x);
+        g.define(iso(e, NU), None, hybrid(), Procedure::Static {});
+        let thermal = Thermal { expansion: Some(Directional::Isotropic(alpha)), ..Thermal::default() };
+        g.model.replace_material("Mat", Material { elastic: Some(iso(e, NU)), thermal: Some(thermal), ..Material::default() }).unwrap();
+        for (set, plane, k, v) in [
+            ("x0", Symmetry::X, 0, 0.0),
+            ("x1", Symmetry::X, 0, 1.0),
+            ("y0", Symmetry::Y, 1, 0.0),
+            ("y1", Symmetry::Y, 1, 1.0),
+            ("z0", Symmetry::Z, 2, 0.0),
+            ("z1", Symmetry::Z, 2, 1.0),
+        ] {
+            g.symmetry(set, plane, |x| near(x[k], v));
+        }
+        g.node_set("everything", |_| true);
+        g.load("warm", Load::Temperature { target: Target::Set("everything".into()), value: dt });
+        let r = solve_static(&g.model, None, &Limits::default()).unwrap();
+        let want = -3.0 * bulk * alpha * dt;
+        for s in &r.stress {
+            for k in 0..3 {
+                assert!((s[k] - want).abs() < 1e-6 * want.abs(), "{name}: {s:?} vs {want}");
+            }
+        }
+    }
+}

docs/elements.md

@@ -73,7 +73,7 @@ hex8/20/27 with first and second derivatives).
 | wedge6 | Swept meshes from triangles, transitions | done |
 | hex8 one-point with hourglass control, nodal-pressure tet4 (explicit) | Explicit dynamics | done |
 | wedge15, pyramid5/13 | Quadratic transitions, tet–hex transition | planned (M5); wedge15 and pyramid5 shape functions done |
-| Hybrid (mixed u–p) solids | Nearly incompressible materials (rubber) | in progress (FEM foundation, phase C); pressure DOF kind done |
+| Hybrid (mixed u–p) solids | Nearly incompressible materials (rubber) | done for hex8 (constant pressure), hex20 (linear pressure) and tet10 (constant pressure), discontinuous pressure condensed, isotropic materials; tet4/wedge6 refused (they lock); continuous-pressure (Taylor–Hood) elements planned with an indefinite factorization |
 | Modified quadratic tet (tet10 for contact/plasticity) | Robust quadratic tets | planned |
 
 ## Special-purpose and multiphysics

docs/fem.md

@@ -133,3 +133,19 @@ or per axis, and `reference_temperature`), `BoundaryCondition::Temperature`, loa
 in a static step), procedures `HeatTransfer { transient }` and `CoupledTemperatureDisplacement`.
 Commands: `solve.heat`, and `solve.static` for coupled steps. The deck writers refuse heat
 transfer by name for now.
+## Hybrid (mixed u–p) solids
+
+A section with `"formulation": {"hybrid": true}` gives `Formulation::Hybrid { pressures }`
+(`crate::hybrid`): the isotropic elasticity splits as D = D_dev + K m mᵀ, the mean stress is an
+independent field discontinuous between elements (hex20: 1, x, y, z; hex8 and tet10: constant),
+and it is condensed out element by element, p̂ = K M⁻¹ Gᵀ u, leaving the stiffness
+K_dev + K G M⁻¹ Gᵀ. `ElementDofs.internal.pressure` counts the condensed unknowns; nothing
+global changes, and Cholesky still applies. Stresses are D_dev ε + p m with the element's
+pressure field. The choice of pressure space follows the inf-sup analysis (docs in
+`crates/fem/src/hybrid.rs`): Q2–P1disc and P2–P0 are stable; Q1–P0 does not lock but may show
+checkerboard pressures; linear tetrahedra and wedges lock even so and are refused. A thermal
+strain splits the same way: ∫ Bᵀ D_dev ε_th on the nodes and, through the pressure,
+K G M⁻¹ ∫ N_p tr ε_th, so a heated nearly incompressible solid takes its bulk stress −3KαΔT
+when confined. A continuous
+nodal pressure (`DofKind::P`, Taylor–Hood) needs a symmetric indefinite factorization and is
+planned.

docs/solvers/literature.md

@@ -1141,3 +1141,17 @@ update are written from (cited again in the module docs):
   constrained thermal expansion (σ = −EαΔT), the bimetal strip (S. Timoshenko, *Analysis of
   bi-metal thermostats*, J. Opt. Soc. Am. 11(3):233–255, 1925), and the coupled step against
   the heat step followed by the static step.
+
+## Part L: Hybrid (mixed u–p) solids, as implemented
+
+- Perturbed-Lagrangian u–p formulation with element-wise condensed discontinuous pressure:
+  T. J. R. Hughes, *The Finite Element Method*, Dover 2000, §4.3–4.5 (B-bar and mean
+  dilatation as the constant-pressure case); J. C. Nagtegaal, D. M. Parks, J. R. Rice, CMAME
+  4(2):153–177, 1974 (locking of displacement elements near incompressibility).
+- Pressure spaces and the inf-sup condition: F. Brezzi, M. Fortin, *Mixed and Hybrid Finite
+  Element Methods*, Springer 1991, ch. VI; K.-J. Bathe, *Finite Element Procedures*, 2nd ed.,
+  2014, §4.5 (Q2–P1disc and P2–P0 stable, Q1–P0 checkerboard-prone, P1–P0 locks).
+- Verification (`crates/solver/tests/hybrid.rs`, analytic): a block in uniaxial strain at
+  ν = 0.4999 (exact), Lamé's thick cylinder in plane strain at ν = 0.4999 (hex20 1e-5,
+  tet10 1e-3, hex8 3e-3 relative error in the bore displacement, where the displacement hex8
+  is 80 % too stiff). Cook's membrane is a published benchmark, not part of the analytic gate.