fem: hybrid plane strain and axisymmetric elements

One constant pressure per element (mean dilatation, Nagtegaal, Parks &
Rice 1974; the hoop strain in the divergence) for quad4, quad8 and tri6:
the linear stiffness, stresses and thermal load, and the finite- and
small-strain forms of nonlinear steps with their consistent tangent. The
linear triangle and plane stress are refused by name.

Co-Authored-By: Claude Opus 5.5 <[email protected]>
Claude-Session: https://claude.ai/code/session_01UhhZMBTviDQxZvwnyP89fo

crates/fem/src/finite.rs

@@ -515,6 +515,21 @@ impl Plane<'_> {
     }
 }
 
+/// The kinematics of one point of a 2D element in a nonlinear step: F, det F_m, the volume share, the strain columns (current at finite strain), the small strain,
+/// the temperature and the reference position.
+struct PlaneKinematics {
+    f: Mat3,
+    /// det F_m (small strain: 1), and the dilatation the mixed element averages (small strain:
+    /// 1 + tr ε).
+    j: f64,
+    dilatation: f64,
+    dv: f64,
+    current: Vec<[f64; 3]>,
+    strain: [f64; 6],
+    heat: Heat,
+    x: [f64; 3],
+}
+
 /// One integration point of a 2D element in a nonlinear step: the total deformation gradient,
 /// det F_m, the reference position and the response.
 struct PlanePoint {
@@ -525,8 +540,14 @@ struct PlanePoint {
 }
 
 impl Plane<'_> {
-    fn check_nonlinear(&self, u: &[[f64; 2]], kin: Kinematics) -> Result<()> {
+    fn check_nonlinear(&self, m: &MaterialModel, u: &[[f64; 2]], kin: Kinematics) -> Result<()> {
         check_displacements(u, self.ty.nodes)?;
+        if matches!(self.formulation, Formulation::Hybrid { .. }) && !m.is_decoupled() {
+            return Err(Error::NotSupported(format!(
+                "hybrid elements of a {} material in nonlinear steps (the mixed formulation needs a volumetric–isochoric split of the strain energy; give the material the neo-Hookean option)",
+                m.name()
+            )));
+        }
         if let Theory::PlaneStress { .. } = self.theory {
             let why = match kin {
                 Kinematics::Finite => "the thickness change needs a local iteration",
@@ -558,7 +579,7 @@ impl Plane<'_> {
         updated: &mut [f64],
         tangent: bool,
     ) -> Result<(Finite, Vec<PlanePoint>)> {
-        self.check_nonlinear(u, kin)?;
+        self.check_nonlinear(m, u, kin)?;
         let n = self.ty.nodes;
         if let Some(t) = thermal {
             t.check(n)?;
@@ -570,8 +591,9 @@ impl Plane<'_> {
         let mut k = tangent.then(|| Matrix::zeros(ndof));
         let axisymmetric = self.theory == Theory::Axisymmetric;
         let small = kin == Kinematics::Small;
-        let mut out = Vec::with_capacity(rule.len());
-        for (i, p) in rule.iter().enumerate() {
+        // The kinematics of every point first: the mixed element's pressure needs them all.
+        let mut kinematics = Vec::with_capacity(rule.len());
+        for p in &rule {
             let at = self.at(p)?;
             let cols = at.g.get(..n).unwrap_or_default();
             let mut f = IDENTITY;
@@ -585,12 +607,11 @@ impl Plane<'_> {
                     f[2][2] += ua[0] * g[2];
                 }
             }
-            let updated_point = updated.get_mut(i * len..(i + 1) * len).unwrap_or_default();
             let heat = Heat::at(thermal, at.shape.values())?;
-            let (j, dv, current, r) = if small {
+            let point = if small {
                 let strain = heat.mechanical_strain(&small_strain(&f));
-                let r = m.small_strain(&strain, heat.temperature, point_history(committed, len, i), updated_point, true)?;
-                (1.0, at.dv, cols.to_vec(), r)
+                let volume = 1.0 + strain[0] + strain[1] + strain[2];
+                PlaneKinematics { f, j: 1.0, dilatation: volume, dv: at.dv, current: cols.to_vec(), strain, heat, x: at.x }
             } else {
                 let det2 = f[0][0] * f[1][1] - f[0][1] * f[1][0];
                 let j = det2 * f[2][2];
@@ -600,9 +621,29 @@ impl Plane<'_> {
                 let inv = [[f[1][1] / det2, -f[0][1] / det2], [-f[1][0] / det2, f[0][0] / det2]];
                 let current: Vec<[f64; 3]> =
                     cols.iter().map(|g| [g[0] * inv[0][0] + g[1] * inv[1][0], g[0] * inv[0][1] + g[1] * inv[1][1], g[2] / f[2][2]]).collect();
-                let r = m.response(&heat.mechanical(&f), heat.temperature, point_history(committed, len, i), updated_point)?;
-                (j / heat.volume(), at.dv * heat.volume(), current, r)
+                let j = j / heat.volume();
+                PlaneKinematics { f, j, dilatation: j, dv: at.dv * heat.volume(), current, strain: [0.0; 6], heat, x: at.x }
             };
+            kinematics.push(point);
+        }
+        let responses: Vec<Response> = if let Formulation::Hybrid { .. } = self.formulation {
+            self.mean_dilatation(m, &kinematics, small, committed, updated, k.as_mut())?
+        } else {
+            let mut out = Vec::with_capacity(kinematics.len());
+            for (i, p) in kinematics.iter().enumerate() {
+                let updated_point = updated.get_mut(i * len..(i + 1) * len).unwrap_or_default();
+                let history = point_history(committed, len, i);
+                out.push(if small {
+                    m.small_strain(&p.strain, p.heat.temperature, history, updated_point, true)?
+                } else {
+                    m.response(&p.heat.mechanical(&p.f), p.heat.temperature, history, updated_point)?
+                });
+            }
+            out
+        };
+        let mut out = Vec::with_capacity(rule.len());
+        for (p, r) in kinematics.into_iter().zip(responses) {
+            let (dv, current) = (p.dv, &p.current);
             let tau = [r.stress[0], r.stress[1], r.stress[2], r.stress[3]];
             for (a, g) in current.iter().enumerate() {
                 if let Some(x) = force.get_mut(2 * a) {
@@ -614,7 +655,7 @@ impl Plane<'_> {
             }
             if let Some(k) = k.as_mut() {
                 let c4: [[f64; 4]; 4] = std::array::from_fn(|a| std::array::from_fn(|b| r.tangent[a][b]));
-                crate::plane::accumulate(&mut k.data, &current, &c4, dv);
+                crate::plane::accumulate(&mut k.data, current, &c4, dv);
                 if !small {
                     for (a, ga) in current.iter().enumerate() {
                         for (b, gb) in current.iter().enumerate() {
@@ -625,11 +666,56 @@ impl Plane<'_> {
                     }
                 }
             }
-            out.push(PlanePoint { f, j, x: at.x, response: r });
+            out.push(PlanePoint { f: p.f, j: p.j, x: p.x, response: r });
         }
         Ok((Finite { force, tangent: k }, out))
     }
 
+    /// The mean dilatation responses of a hybrid 2D element (one constant pressure): θ = ∫ J dV₀
+    /// / V₀ (hoop stretch included), p = U′(θ), τ = τ_iso + J p I (small strain: s + p I), and
+    /// the dilatation stiffness U″(θ) g gᵀ / V₀ added to `k`, g = ∫ J ∇ₓ·N dV₀ (small strain
+    /// ∫ ∇₀·N dV₀), the divergence taking the hoop column N/r too (the constant-pressure case of
+    /// the solids' three-field form, Simo, Taylor & Pister 1985).
+    fn mean_dilatation(
+        &self,
+        m: &MaterialModel,
+        points: &[PlaneKinematics],
+        small: bool,
+        committed: &[f64],
+        updated: &mut [f64],
+        k: Option<&mut Matrix>,
+    ) -> Result<Vec<Response>> {
+        let len = m.history_len();
+        let volume: f64 = points.iter().map(|p| p.dv).sum();
+        if !(volume > 0.0 && volume.is_finite()) {
+            return Err(inverted(&format!("2D {}", self.ty.kind.name()), volume));
+        }
+        let theta = points.iter().map(|p| p.dv * p.dilatation).sum::<f64>() / volume;
+        if !(theta > 0.0 && theta.is_finite()) {
+            return Err(inverted(&format!("2D {}", self.ty.kind.name()), theta));
+        }
+        let (pressure, stiffness) = m.volumetric(theta, small).ok_or_else(|| Error::Invalid("volumetric response".into()))?;
+        let mut out = Vec::with_capacity(points.len());
+        for (i, p) in points.iter().enumerate() {
+            let (c, u) = (point_history(committed, len, i), updated.get_mut(i * len..(i + 1) * len).unwrap_or_default());
+            out.push(if small {
+                let mut r = m.small_strain(&p.strain, p.heat.temperature, c, u, false)?;
+                r.stress.iter_mut().take(3).for_each(|s| *s += pressure);
+                r
+            } else {
+                add_pressure(&m.isochoric(&p.heat.mechanical(&p.f), p.heat.temperature, c, u)?, p.j, pressure, 0.0)
+            });
+        }
+        if let Some(k) = k {
+            let mut g = vec![0.0; k.n];
+            for p in points {
+                crate::plane::divergence(&mut g, &p.current, p.dv * p.j);
+            }
+            crate::plane::add_outer(k, &g, stiffness / volume);
+        }
+        Ok(out)
+    }
+
     /// Internal forces (u, v per node) and (with `tangent`) the tangent stiffness at nodal
     /// displacements `u`, as for [`Solid::finite_strain`].
     pub fn finite_strain(&self, m: &MaterialModel, u: &[[f64; 2]], committed: &[f64], updated: &mut [f64], tangent: bool) -> Result<Finite> {
@@ -1016,6 +1102,63 @@ mod tests {
         assert!(matches!(stress.finite_strain(&svk, &[[0.0; 2]; 4], &[], &mut [], true), Err(Error::NotSupported(_))));
     }
 
+    #[test]
+    fn hybrid_2d_tangents_are_the_derivative_of_the_forces() {
+        let quad8 = [
+            [1.0, 0.0, 0.0],
+            [2.0, 0.1, 0.0],
+            [2.2, 1.0, 0.0],
+            [0.9, 1.1, 0.0],
+            [1.5, 0.0, 0.0],
+            [2.1, 0.55, 0.0],
+            [1.5, 1.1, 0.0],
+            [0.95, 0.5, 0.0],
+        ];
+        let (shear, bulk) = (300.0, 30000.0);
+        let neo = MaterialModel::NeoHookean { shear, bulk };
+        let mr = MaterialModel::Polynomial { name: "Mooney–Rivlin", c: [100.0, 50.0, 0.0, 0.0], bulk };
+        let hybrid = Formulation::Hybrid { pressures: 1 };
+        for theory in [Theory::PlaneStrain { thickness: 2.0 }, Theory::Axisymmetric] {
+            for (kind, x) in [(ElementKind::Quad8, &quad8[..]), (ElementKind::Quad4, &quad8[..4])] {
+                let plane = Plane::new(kind, x, hybrid, theory).unwrap();
+                for m in [&neo, &mr] {
+                    let u: Vec<[f64; 2]> = displacement(x.len(), 0.06).iter().map(|v| [v[0], v[1]]).collect();
+                    let k = plane.finite_strain(m, &u, &[], &mut [], true).unwrap().tangent.unwrap();
+                    let scale = k.data.iter().fold(0.0f64, |a, v| a.max(v.abs()));
+                    let h = 1e-6;
+                    for j in 0..2 * x.len() {
+                        let shifted = |s: f64| {
+                            let mut v = u.clone();
+                            v[j / 2][j % 2] += s;
+                            plane.finite_strain(m, &v, &[], &mut [], false).unwrap().force
+                        };
+                        let (p, q) = (shifted(h), shifted(-h));
+                        for i in 0..2 * x.len() {
+                            let numeric = (p[i] - q[i]) / (2.0 * h);
+                            assert!((numeric - k.get(i, j)).abs() < 1e-6 * scale, "{theory:?} {kind:?} {i} {j}: {numeric} vs {}", k.get(i, j));
+                        }
+                    }
+                }
+                // At rest the tangent is the linear hybrid stiffness of the same moduli.
+                let young = 9.0 * bulk * shear / (3.0 * bulk + shear);
+                let poisson = (3.0 * bulk - 2.0 * shear) / (2.0 * (3.0 * bulk + shear));
+                let d = isotropic(young, poisson).unwrap();
+                let k = plane.finite_strain(&neo, &vec![[0.0; 2]; x.len()], &[], &mut [], true).unwrap().tangent.unwrap();
+                let linear = plane.stiffness(&d).unwrap();
+                let scale = linear.data.iter().fold(0.0f64, |a, v| a.max(v.abs()));
+                for (a, b) in k.data.iter().zip(&linear.data) {
+                    assert!((a - b).abs() < 1e-9 * scale, "{a} {b}");
+                }
+            }
+        }
+        // The linear triangle and plane stress have none; St Venant–Kirchhoff has no split.
+        assert!(Plane::new(ElementKind::Tri3, &quad8[..3], hybrid, Theory::PlaneStrain { thickness: 1.0 }).is_err());
+        assert!(Plane::new(ElementKind::Quad4, &quad8[..4], hybrid, Theory::PlaneStress { thickness: 1.0 }).is_err());
+        let svk = MaterialModel::StVenantKirchhoff(Box::new(isotropic(1000.0, 0.3).unwrap()));
+        let plane = Plane::new(ElementKind::Quad4, &quad8[..4], hybrid, Theory::Axisymmetric).unwrap();
+        assert!(matches!(plane.finite_strain(&svk, &[[0.0; 2]; 4], &[], &mut [], true), Err(Error::NotSupported(_))));
+    }
+
     /// uᵀ K_σ u = ∫ σ_xx (∂u_y/∂x)² dV = σ_xx V for u_y = x under a uniform σ_xx, in a distorted
     /// hex20 and a plane strain quad8, and K_σ annihilates rigid translations.
     #[test]

crates/fem/src/plane.rs

@@ -35,6 +35,15 @@
 //! leaves the plane case unchanged (its mean is zero) and makes the axisymmetric element pass the
 //! patch test.
 //!
+//! **Hybrid** (mixed displacement–pressure, plane strain and axisymmetric quad4, quad8 and tri6):
+//! one constant pressure per element, the mean dilatation method (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; T. J. R. Hughes, *The Finite Element Method*, Dover 2000, §4.5 for
+//! the axisymmetric B-bar): the volumetric strain, hoop part included, is replaced by its volume
+//! mean, so K = ∫ Bᵀ D_dev B dV + κ g gᵀ / V with g = ∫ Bᵀ m dV, m = (1, 1, 1, 0). The linear
+//! triangle is refused (one pressure per element still locks it), plane stress too (σ_zz = 0
+//! leaves no volume constraint to relax). The finite-strain form is [`crate::finite`]'s.
+//!
 //! On the axis (r = 0) the hoop strain u/r is evaluated only at integration points, which lie
 //! inside the element (r > 0) for every element with some node off the axis; the solver holds
 //! the radial displacement of nodes on the axis at zero, as symmetry requires.
@@ -165,11 +174,24 @@ pub fn plane_type(kind: ElementKind) -> Result<&'static ElementType> {
         .ok_or_else(|| Error::NotSupported(format!("{} elements in a 2D section (2D sections take tri3, tri6, quad4 and quad8)", kind.name())))
 }
 
+/// The 2D element kinds with a hybrid (constant pressure) formulation.
+const HYBRID_PRESSURES: [ElementKind; 3] = [ElementKind::Quad4, ElementKind::Quad8, ElementKind::Tri6];
+
 /// The formulation a 2D section asks for, if this element type has it.
 pub fn formulation(kind: ElementKind, f: &fieldcraft_model::Formulation) -> Result<Formulation> {
     let ty = plane_type(kind)?;
     if f.hybrid {
-        return Err(Error::NotSupported("hybrid (mixed displacement–pressure) 2D elements".into()));
+        if f.incompatible_modes || f.integration == fieldcraft_model::Integration::Reduced {
+            return Err(Error::NotSupported("hybrid 2D elements with incompatible modes or reduced integration".into()));
+        }
+        return if HYBRID_PRESSURES.contains(&kind) {
+            Ok(Formulation::Hybrid { pressures: 1 })
+        } else {
+            Err(Error::NotSupported(format!(
+                "hybrid 2D {} elements (one pressure per element still locks the linear triangle; use quad4, quad8 or tri6)",
+                kind.name()
+            )))
+        };
     }
     if f.incompatible_modes {
         return if ty.incompatible_modes {
@@ -270,11 +292,14 @@ impl<'a> Plane<'a> {
             Formulation::Full => true,
             Formulation::Reduced => ty.reduced.is_some(),
             Formulation::IncompatibleModes => ty.incompatible_modes,
-            Formulation::Hybrid { .. } => false,
+            Formulation::Hybrid { pressures } => pressures == 1 && HYBRID_PRESSURES.contains(&kind),
         };
         if !ok {
             return Err(Error::NotSupported(format!("{formulation:?} formulation of 2D {} elements", kind.name())));
         }
+        if matches!(formulation, Formulation::Hybrid { .. }) && matches!(theory, Theory::PlaneStress { .. }) {
+            return Err(Error::NotSupported("hybrid plane stress elements (σ_zz = 0 leaves no volume constraint; use full integration)".into()));
+        }
         Ok(Plane { ty, xyz, formulation, theory })
     }
 
@@ -282,7 +307,8 @@ impl<'a> Plane<'a> {
     /// modes it condenses out.
     pub fn dofs(&self) -> ElementDofs {
         let incompatible_modes = if self.formulation == Formulation::IncompatibleModes { INCOMPATIBLE_DOFS } else { 0 };
-        ElementDofs { node: DofSet::PLANE, internal: Internal { incompatible_modes, pressure: 0 } }
+        let pressure = if let Formulation::Hybrid { pressures } = self.formulation { pressures } else { 0 };
+        ElementDofs { node: DofSet::PLANE, internal: Internal { incompatible_modes, pressure } }
     }
 
     /// The stiffness integration rule.
@@ -411,6 +437,13 @@ impl<'a> Plane<'a> {
 
     /// The stiffness matrix, 2n × 2n, degrees of freedom ordered node by node (u, v).
     pub fn stiffness(&self, d: &Elasticity) -> Result<Matrix> {
+        if let Some(dev) = self.deviatoric(d)? {
+            // K_dev + κ g gᵀ / V.
+            let (mut k, points, _) = self.full_stiffness(&dev)?;
+            let mixed = self.mixed(d, &points, None)?;
+            add_outer(&mut k, &mixed.g, mixed.bulk / mixed.volume);
+            return Ok(k);
+        }
         let d2 = self.theory.elasticity(d)?;
         let (full, _, modes) = self.full_stiffness(&d2)?;
         if modes.is_none() {
@@ -433,6 +466,34 @@ impl<'a> Plane<'a> {
         Ok(k)
     }
 
+    /// A hybrid element's deviatoric elasticity in the theory's form; `None` for the other
+    /// formulations.
+    fn deviatoric(&self, d: &Elasticity) -> Result<Option<Elasticity2>> {
+        if !matches!(self.formulation, Formulation::Hybrid { .. }) {
+            return Ok(None);
+        }
+        Ok(Some(self.theory.elasticity(&crate::hybrid::split(d)?.0)?))
+    }
+
+    /// A hybrid element's parts over its stiffness `points`: the bulk modulus, g = ∫ Bᵀ m dV
+    /// (2n), the volume and, with a thermal strain, ∫ tr ε_th dV.
+    pub(crate) fn mixed(&self, d: &Elasticity, points: &[(Point, At)], thermal: Option<&ThermalStrain>) -> Result<Mixed2> {
+        let bulk = crate::hybrid::split(d)?.1;
+        let (mut g, mut volume, mut thermal_volume) = (vec![0.0; 2 * self.ty.nodes], 0.0, 0.0);
+        for (_, a) in points {
+            volume += a.dv;
+            divergence(&mut g, a.g.get(..self.ty.nodes).unwrap_or_default(), a.dv);
+            if let Some(t) = thermal {
+                let e = Self::thermal_strain(t, a.shape.values());
+                thermal_volume += a.dv * (e[0] + e[1] + e[2]);
+            }
+        }
+        if !(volume > 0.0) {
+            return Err(self.degenerate());
+        }
+        Ok(Mixed2 { bulk, g, volume, thermal_volume })
+    }
+
     /// The consistent mass matrix per displacement component, n × n: m_ij = ∫ ρ Nᵢ Nⱼ dV.
     pub fn mass(&self, density: f64) -> Result<Matrix> {
         let n = self.ty.nodes;
@@ -518,8 +579,18 @@ impl<'a> Plane<'a> {
     /// strains the hoop direction too); incompatible modes are condensed as in the stiffness.
     pub fn thermal_load(&self, d: &Elasticity, thermal: &ThermalStrain) -> Result<Vec<[f64; 2]>> {
         thermal.check(self.ty.nodes)?;
-        let d2 = self.theory.elasticity(d)?;
         let n = 2 * self.ty.nodes;
+        if let Some(dev) = self.deviatoric(d)? {
+            // ∫ Bᵀ D_dev ε_th dV + κ g (∫ tr ε_th dV) / V.
+            let points = self.points()?;
+            let mixed = self.mixed(d, &points, Some(thermal))?;
+            let (mut fu, _) = Self::thermal_forces(&dev, thermal, &points, None, self.ty.nodes);
+            for (f, g) in fu.iter_mut().zip(&mixed.g) {
+                *f += mixed.bulk * g * mixed.thermal_volume / mixed.volume;
+            }
+            return Ok(fu.as_chunks::<2>().0.to_vec());
+        }
+        let d2 = self.theory.elasticity(d)?;
         let mut fu = if self.formulation == Formulation::IncompatibleModes {
             let (full, points, modes) = self.full_stiffness(&d2)?;
             let (mut fu, mut fa) = Self::thermal_forces(&d2, thermal, &points, modes.as_ref(), self.ty.nodes);
@@ -549,9 +620,21 @@ impl<'a> Plane<'a> {
         if let Some(t) = thermal {
             t.check(self.ty.nodes)?;
         }
-        let d2 = self.theory.elasticity(d)?;
+        let deviatoric = self.deviatoric(d)?;
+        let d2 = match deviatoric {
+            Some(dev) => dev,
+            None => self.theory.elasticity(d)?,
+        };
         let (full, points, modes) =
             if self.formulation == Formulation::IncompatibleModes { self.full_stiffness(&d2)? } else { (Matrix::zeros(0), self.points()?, None) };
+        // Hybrid: the element's pressure κ (gᵀ u − ∫ tr ε_th dV) / V.
+        let pressure = if deviatoric.is_some() {
+            let m = self.mixed(d, &points, thermal)?;
+            let gu: f64 = m.g.iter().zip(u.iter().flatten()).map(|(g, x)| g * x).sum();
+            m.bulk * (gu - m.thermal_volume) / m.volume
+        } else {
+            0.0
+        };
         // Incompatible modes: K_aa α = f_a − K_au u (f_a the thermal load on the modes).
         let alpha = if modes.is_some() {
             let n = 2 * self.ty.nodes;
@@ -595,10 +678,11 @@ impl<'a> Plane<'a> {
                 };
             }
             // Plane stress: D̃ has no zz row or column, so the εzz just recovered adds nothing.
-            let s: [f64; 4] = match eth {
+            let mut s: [f64; 4] = match eth {
                 Some(t) => std::array::from_fn(|r| (0..4).map(|c| d2[r][c] * (e[c] - t[c])).sum()),
                 None => std::array::from_fn(|r| (0..4).map(|c| d2[r][c] * e[c]).sum()),
             };
+            s.iter_mut().take(3).for_each(|v| *v += pressure);
             out.push(PointStress { xyz: a.x, strain: [e[0], e[1], e[2], e[3], 0.0, 0.0], stress: [s[0], s[1], s[2], s[3], 0.0, 0.0] });
         }
         Ok(out)
@@ -614,6 +698,32 @@ impl<'a> Plane<'a> {
     }
 }
 
+/// The parts of a hybrid 2D element ([`Plane::mixed`]).
+pub(crate) struct Mixed2 {
+    bulk: f64,
+    pub(crate) g: Vec<f64>,
+    volume: f64,
+    thermal_volume: f64,
+}
+
+/// g += w Bᵀ m over strain columns `cols` (u takes ∂N/∂x and the hoop N/r, v takes ∂N/∂y): the
+/// divergence of each nodal motion, which the hybrid elements' volume constraint acts on.
+pub(crate) fn divergence(g: &mut [f64], cols: &[Column], w: f64) {
+    for (pair, col) in g.as_chunks_mut::<2>().0.iter_mut().zip(cols) {
+        pair[0] += w * (col[0] + col[2]);
+        pair[1] += w * col[1];
+    }
+}
+
+/// k += c g gᵀ.
+pub(crate) fn add_outer(k: &mut Matrix, g: &[f64], c: f64) {
+    for (i, gi) in g.iter().enumerate() {
+        for (j, gj) in g.iter().enumerate() {
+            k.add(i, j, c * gi * gj);
+        }
+    }
+}
+
 /// The `size` × `size` block of `m` starting at (`row`, `col`).
 fn sub_block(m: &Matrix, row: usize, col: usize, size: usize) -> Matrix {
     let mut out = Matrix::zeros(size);
@@ -661,6 +771,53 @@ mod tests {
 
     const THEORIES: [Theory; 3] = [Theory::PlaneStress { thickness: 0.7 }, Theory::PlaneStrain { thickness: 1.0 }, Theory::Axisymmetric];
 
+    #[test]
+    fn hybrid_elements_reproduce_a_uniform_strain_and_do_not_lock() {
+        // u = a x, v = b y: a uniform strain (axisymmetric: the hoop strain a too), whose
+        // stress the mean dilatation reproduces exactly at every point.
+        let d = isotropic(1000.0, 0.4999).unwrap();
+        let (a, b) = (0.002, -0.003);
+        for theory in [Theory::PlaneStrain { thickness: 1.5 }, Theory::Axisymmetric] {
+            for kind in [ElementKind::Quad4, ElementKind::Quad8, ElementKind::Tri6] {
+                let x = element(kind);
+                let plane = Plane::new(kind, &x, Formulation::Hybrid { pressures: 1 }, theory).unwrap();
+                let u: Vec<[f64; 2]> = x.iter().map(|p| [a * p[0], b * p[1]]).collect();
+                let hoop = if theory == Theory::Axisymmetric { a } else { 0.0 };
+                let e = [a, b, hoop, 0.0, 0.0, 0.0];
+                let exact = crate::material::stress(&d, &e);
+                for p in plane.stresses(&d, &u).unwrap() {
+                    for c in 0..4 {
+                        assert!((p.stress[c] - exact[c]).abs() < 1e-9 * exact[0].abs().max(1.0), "{theory:?} {kind:?} {c}: {:?} {exact:?}", p.stress);
+                    }
+                }
+                // K u is the forces of that stress: the stiffness agrees with the stresses.
+                let k = plane.stiffness(&d).unwrap();
+                let flat: Vec<f64> = u.iter().flatten().copied().collect();
+                let full = Plane::new(kind, &x, Formulation::Full, theory).unwrap().stiffness(&d).unwrap();
+                let (ku, kf): (Vec<f64>, Vec<f64>) = (0..flat.len())
+                    .map(|i| (0..flat.len()).fold((0.0, 0.0), |(s, t), j| (s + k.get(i, j) * flat[j], t + full.get(i, j) * flat[j])))
+                    .unzip();
+                for (p, q) in ku.iter().zip(&kf) {
+                    assert!((p - q).abs() < 1e-8 * kf.iter().fold(0.0f64, |m, v| m.max(v.abs())), "{p} {q}");
+                }
+            }
+        }
+        // Nearly incompressible: the hybrid element's volume is one constraint, so a motion that
+        // keeps the element's volume (orthogonal to g) costs only shear, where the full element's
+        // four point constraints lock it.
+        let x = element(ElementKind::Quad4);
+        let hybrid = Plane::new(ElementKind::Quad4, &x, Formulation::Hybrid { pressures: 1 }, Theory::PlaneStrain { thickness: 1.0 }).unwrap();
+        let g = hybrid.mixed(&d, &hybrid.points().unwrap(), None).unwrap().g;
+        let raw: Vec<f64> = (0..8).map(|i| ((i * 7 % 5) as f64 - 2.0) * 0.01).collect();
+        let along = raw.iter().zip(&g).map(|(a, b)| a * b).sum::<f64>() / g.iter().map(|v| v * v).sum::<f64>();
+        let shear: Vec<f64> = raw.iter().zip(&g).map(|(a, b)| a - along * b).collect();
+        let energy = |f: Formulation| {
+            let k = Plane::new(ElementKind::Quad4, &x, f, Theory::PlaneStrain { thickness: 1.0 }).unwrap().stiffness(&d).unwrap();
+            (0..8).map(|i| (0..8).map(|j| shear[i] * k.get(i, j) * shear[j]).sum::<f64>()).sum::<f64>()
+        };
+        assert!(energy(Formulation::Hybrid { pressures: 1 }) < 0.01 * energy(Formulation::Full));
+    }
+
     #[test]
     fn rigid_motion_strains_nothing_and_matrices_are_symmetric() {
         let d = isotropic(1000.0, 0.3).unwrap();
@@ -783,6 +940,9 @@ mod tests {
         let on_axis = sq.map(|p| [p[0] - 1.0, p[1], 0.0]);
         let e = Plane::new(ElementKind::Quad4, &on_axis, Formulation::IncompatibleModes, Theory::Axisymmetric).unwrap();
         assert!(e.stiffness(&d).is_ok());
-        assert!(formulation(ElementKind::Quad4, &fieldcraft_model::Formulation { hybrid: true, ..Default::default() }).is_err());
+        let hybrid = fieldcraft_model::Formulation { hybrid: true, ..Default::default() };
+        assert_eq!(formulation(ElementKind::Quad4, &hybrid).unwrap(), Formulation::Hybrid { pressures: 1 });
+        assert!(formulation(ElementKind::Tri3, &hybrid).is_err());
+        assert!(formulation(ElementKind::Quad4, &fieldcraft_model::Formulation { incompatible_modes: true, ..hybrid }).is_err());
     }
 }

crates/model/src/sections.rs

@@ -278,8 +278,9 @@ impl Formulation {
         if !kind.is_continuum() && self.incompatible_modes {
             return Err(Error::Invalid(format!("incompatible modes are for solid and 2D sections, not {}", kind.name())));
         }
-        if kind.dimension() != 3 && self.hybrid {
-            return Err(Error::Invalid(format!("hybrid formulations are for solid sections, not {}", kind.name())));
+        let volume_constrained = kind.dimension() == 3 || matches!(kind, SectionKind::PlaneStrain { .. } | SectionKind::Axisymmetric {});
+        if self.hybrid && !volume_constrained {
+            return Err(Error::Invalid(format!("hybrid formulations are for solid, plane strain and axisymmetric sections, not {}", kind.name())));
         }
         Ok(())
     }
@@ -391,7 +392,10 @@ mod tests {
         let im = Formulation { incompatible_modes: true, ..Formulation::default() };
         im.validate_for(&stress).unwrap();
         im.validate_for(&axi).unwrap();
-        assert!(Formulation { hybrid: true, ..Formulation::default() }.validate_for(&strain).is_err());
+        let hybrid = Formulation { hybrid: true, ..Formulation::default() };
+        hybrid.validate_for(&strain).unwrap();
+        hybrid.validate_for(&axi).unwrap();
+        assert!(hybrid.validate_for(&stress).is_err());
         assert!(im.validate_for(&SectionKind::Shell { thickness: 1.0, offset: 0.0, points: 5 }).is_err());
     }
 }

docs/fem.md

@@ -146,7 +146,10 @@ pressure field. The choice of pressure space follows the inf-sup analysis (docs
 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
+when confined. Plane strain and axisymmetric quad4, quad8 and tri6 sections take one constant
+pressure (the mean dilatation, the hoop strain in the divergence): K_dev + κ g gᵀ / V with
+g = ∫ Bᵀ m dV (`crates/fem/src/plane.rs`); the linear triangle locks and plane stress has no
+volume constraint, so both are refused. A continuous
 nodal pressure (`DofKind::P`, Taylor–Hood) needs a symmetric indefinite factorization and is
 planned.
 
@@ -251,6 +254,7 @@ differences of its forces.
 | Element | Nonlinear steps |
 |---|---|
 | tet4, tet10, wedge6, hex8, hex20 (full), hex20 (reduced) | total Lagrangian |
+| hybrid quad4, quad8, tri6 (plane strain, axisymmetric) | mean dilatation: θ = ∫ J dV₀ / V₀ with the hoop stretch, p = U′(θ), stiffness U″ g gᵀ / V₀ |
 | hybrid hex8, hex20, tet10 | Simo–Taylor–Pister three-field form on the linear hybrid elements' pressure fields (θ the projection of J, p the projection of U′(θ)); needs the neo-Hookean option (St Venant–Kirchhoff has no volumetric split) |
 | hex8 and quad4 incompatible modes | refused: their finite-strain form (enhanced assumed strain, Simo–Armero 1992) needs the modes as unknowns carried between iterations and hourglasses in large compression; use full integration, hybrid or quadratic elements |
 | plane strain, axisymmetric (tri3, tri6, quad4, quad8) | total Lagrangian with F₃₃ = 1 or r/R (hoop column N/r, geometric term τ_θθ N_a N_b/r²) |

tests/oracle/nl-mooney-rivlin-tube-axisymmetric/README.md

@@ -0,0 +1,8 @@
+# Inflation of a stretched Mooney–Rivlin tube, axisymmetric
+
+The tube of `nl-mooney-rivlin-tube` (A = 10, B = 20, one unit long; E = 3, ν = 0.4995,
+C₀₁ = 0.2) as an axisymmetric model of 12 × 2 hybrid quad4 (one constant pressure each, the mean
+dilatation with the hoop strain): the end y = 0 on a symmetry line, the end y = 1 stretched to
+λ_z = 1.5 while the follower pressure p = 0.540850 inflates the bore. The closed form puts the
+bore at a = 15 and the outer wall at b = 20.615528 (see `nl-mooney-rivlin-tube`); this case checks
+the nearly incompressible 2D hybrid elements at large strain against it.

tests/oracle/nl-mooney-rivlin-tube-axisymmetric/expected.json

@@ -0,0 +1,53 @@
+{
+ "title": "Inflation of a stretched Mooney–Rivlin tube (axisymmetric hybrid quad4)",
+ "kind": "theory",
+ "source": "Finite inflation and extension of an incompressible Mooney–Rivlin tube (R. S. Rivlin, Large elastic deformations of isotropic materials VI, Phil. Trans. R. Soc. A 242:173–195, 1949; R. W. Ogden, Non-Linear Elastic Deformations, Dover 1997, §5.3): at axial stretch λ_z, r² = a² + (R² − A²)/λ_z, σ_θ − σ_r = 2 (λ_θ² − λ_r²)(C₁₀ + C₀₁ λ_z²) with λ_θ = r/R, λ_r = 1/(λ_θ λ_z), integrated through the wall: p = ∫ (σ_θ − σ_r) dr/r (Simpson, 2·10⁵ intervals)",
+ "quantities": [
+  {
+   "id": "ua",
+   "quantity": "radial displacement of the bore (10, 0, 0) at the pressure that inflates it to a = 15 at λ_z = 1.5",
+   "measure": {
+    "command": "solve.probe",
+    "params": {
+     "step": "Inflate",
+     "at": [
+      [
+       10.0,
+       0,
+       0.0
+      ]
+     ]
+    },
+    "pointer": "/nodes/0/displacement/0"
+   },
+   "value": 5.0,
+   "tolerance": {
+    "relative": 0.01
+   },
+   "reason": "16 × 6 hybrid hex20 around the quarter; ν = 0.4995 stands in for incompressibility (a volume change of order p/κ ≈ 10⁻³ shifts the radii by about 0.1 %), and the tube is stretched axially by its ends, which a nearly incompressible solid takes with larger pressures than the plane-strain neo-Hookean tube"
+  },
+  {
+   "id": "ub",
+   "quantity": "radial displacement of the outer wall (20, 0, 0): b − B, a small difference of radii",
+   "measure": {
+    "command": "solve.probe",
+    "params": {
+     "step": "Inflate",
+     "at": [
+      [
+       20.0,
+       0,
+       0.0
+      ]
+     ]
+    },
+    "pointer": "/nodes/0/displacement/0"
+   },
+   "value": 0.6155281280883038,
+   "tolerance": {
+    "absolute": 0.04
+   },
+   "reason": "0.2 % of the outer radius b = 20.616: the same volume change of order 10⁻³ moves b by about 0.03, which is 4.5 % of b − B; the incompressible kinematics r² = a² + (R² − A²)/λ_z amplify it there"
+  }
+ ]
+}

tests/oracle/nl-mooney-rivlin-tube-axisymmetric/fixture.json

@@ -0,0 +1,13 @@
+{
+  "note": "FieldCraft's accepted results for this case, written by `cargo xtask oracle --bless`. A result outside its band fails CI.",
+  "values": {
+    "ua": {
+      "band": 5.03287906011305e-9,
+      "value": 5.03287906011305
+    },
+    "ub": {
+      "band": 0.000040000644728979273,
+      "value": 0.6447289792693816
+    }
+  }
+}

tests/oracle/nl-mooney-rivlin-tube-axisymmetric/recipe.json

@@ -0,0 +1,14 @@
+[
+ {"command": "mesh.block", "params": {"body": "Pipe", "kind": "quad4", "corners": [[10.0, 0, 0], [20.0, 0, 0], [20.0, 1, 0], [10.0, 1, 0]], "divisions": [12, 2], "edges": [{"corners": [1, 4], "name": "bore"}, {"corners": [1, 2], "name": "bottom"}, {"corners": [3, 4], "name": "top"}]}},
+ {"command": "model.set.create", "params": {"name": "all", "geometry": [{"body": "Pipe", "face": "area"}]}},
+ {"command": "model.set.create", "params": {"name": "bore", "geometry": [{"body": "Pipe", "edge": "bore"}]}},
+ {"command": "model.set.create", "params": {"name": "bottom", "geometry": [{"body": "Pipe", "edge": "bottom"}]}},
+ {"command": "model.set.create", "params": {"name": "top", "geometry": [{"body": "Pipe", "edge": "top"}]}},
+ {"command": "model.material.create", "params": {"name": "Rubber", "elastic": {"type": "isotropic", "young": 3.0, "poisson": 0.4995}, "hyperelastic": {"type": "mooney_rivlin", "c01": 0.2}}},
+ {"command": "model.section.create", "params": {"name": "pipe", "set": "all", "material": "Rubber", "kind": {"type": "axisymmetric"}, "formulation": {"hybrid": true}}},
+ {"command": "model.boundary_condition.create", "params": {"name": "sb", "type": "symmetry", "target": {"set": "bottom"}, "plane": "y"}},
+ {"command": "model.boundary_condition.create", "params": {"name": "stretch", "type": "displacement", "target": {"set": "top"}, "values": [null, 0.5, null, null, null, null]}},
+ {"command": "model.load.create", "params": {"name": "p", "type": "pressure", "set": "bore", "magnitude": 0.5408503415684837}},
+ {"command": "model.step.create", "params": {"name": "Inflate", "procedure": {"type": "nonlinear_static"}, "boundary_conditions": ["sb", "stretch"], "loads": ["p"]}},
+ {"command": "solve.nonlinear", "params": {"step": "Inflate"}, "id": "solve"}
+]