solver, decks: symmetry about an inclined plane (#242)

The symmetry condition takes any plane normal ({"normal": [nx, ny, nz]}).
An inclined one holds n·u = 0 and, at shell, beam and reference point
nodes, the rotations about the two in-plane axes: constraint rows the
implicit elimination already handles, its support forces reported as
reactions. Decks: Abaqus *EQUATION, Nastran MPC sets per subcase, LS-DYNA
*BOUNDARY_SPC in a *DEFINE_COORDINATE_SYSTEM along the normal.

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01UhhZMBTviDQxZvwnyP89fo

crates/decks/src/abaqus.rs

@@ -148,7 +148,7 @@ use std::fmt::Write as _;
 
 use fieldcraft_model::{
     BeamProfile, BoundaryCondition, ContactSurfaces, CouplingKind, Elastic, Element, ElementKind, Hardening, Hyperelastic, InitialPenetration,
-    Integration, Load, Model, OutputVariable, Plastic, Procedure, Resolved, Section, SectionKind, Step, Symmetry, Target,
+    Integration, Load, Model, OutputVariable, Plastic, Procedure, Resolved, Section, SectionKind, Step, Target,
 };
 
 use crate::{Error, Result, unsupported};
@@ -193,6 +193,7 @@ pub fn write_resolved(model: &Model, r: &Resolved) -> Result<String> {
     d.couplings(model, r);
     d.contacts(model, r)?;
     d.ties(model)?;
+    d.equations(model, r)?;
     d.steps(model, r)?;
     Ok(d.out)
 }
@@ -370,9 +371,36 @@ fn check_expressible(model: &Model) -> Result<()> {
             bad.insert(format!("coupling `{name}`: an Abaqus distributing coupling always couples translations 1-3"));
         }
     }
+    for name in inclined_symmetries(model) {
+        if !model.steps().iter().all(|s| s.boundary_conditions.contains(name)) {
+            bad.insert(format!(
+                "boundary condition `{name}`: a symmetry plane not normal to X, Y or Z active in some steps only (it is written as `*EQUATION`, which is model data and holds in every step)"
+            ));
+        }
+    }
     unsupported(FORMAT, bad)
 }
 
+/// The symmetry conditions on inclined planes some step uses: linear equations rather than held
+/// degrees of freedom.
+fn inclined_symmetries(model: &Model) -> BTreeSet<&String> {
+    model
+        .boundary_conditions()
+        .iter()
+        .filter(|(n, bc)| {
+            matches!(bc, BoundaryCondition::Symmetry { plane, .. } if plane.axis().is_none()) && !model.boundary_condition_users(n).is_empty()
+        })
+        .map(|(n, _)| n)
+        .collect()
+}
+
+/// The boundary conditions of `names` that `*BOUNDARY` writes (inclined symmetry planes are
+/// `*EQUATION`s).
+fn held_conditions(model: &Model, names: &BTreeSet<String>) -> BTreeSet<String> {
+    let inclined = inclined_symmetries(model);
+    names.iter().filter(|n| !inclined.contains(n)).cloned().collect()
+}
+
 /// The property of a group of discrete elements.
 enum DiscreteProperty {
     Mass(f64),
@@ -1076,10 +1104,12 @@ impl Deck {
             match bc {
                 BoundaryCondition::Fixed { .. } => self.line(&format!("{target}, ENCASTRE")),
                 BoundaryCondition::Symmetry { plane, .. } => {
-                    let s = match plane {
-                        Symmetry::X => "XSYMM",
-                        Symmetry::Y => "YSYMM",
-                        Symmetry::Z => "ZSYMM",
+                    let s = match plane.axis() {
+                        Some(0) => "XSYMM",
+                        Some(1) => "YSYMM",
+                        Some(2) => "ZSYMM",
+                        // An inclined plane is an `*EQUATION` (model data), never here.
+                        _ => continue,
                     };
                     self.line(&format!("{target}, {s}"));
                 }
@@ -1097,8 +1127,74 @@ impl Deck {
         Ok(())
     }
 
+    /// The inclined symmetry planes: one `*EQUATION` per node and held direction (n · u = 0, and
+    /// the rotations about the plane's axes at shell and beam nodes), whose first term Abaqus
+    /// eliminates; it is chosen among the degrees of freedom no boundary condition, coupling
+    /// or tie constrains ([`crate::symmetry_equations`]).
+    fn equations(&mut self, model: &Model, r: &Resolved) -> Result<()> {
+        let inclined = inclined_symmetries(model);
+        if inclined.is_empty() {
+            return Ok(());
+        }
+        let steps = model.steps();
+        // Per node and degree of freedom: the value it is held at in each step that holds it.
+        let mut held: BTreeMap<(u32, u8), Vec<f64>> = BTreeMap::new();
+        for step in steps {
+            for name in held_conditions(model, &step.boundary_conditions) {
+                let Some(bc) = model.boundary_condition(&name) else { continue };
+                for &node in r.target_nodes(bc.target()) {
+                    for dof in bc.dofs().list() {
+                        let v = if matches!(step.procedure, Procedure::Frequency { .. } | Procedure::Buckle { .. }) {
+                            0.0
+                        } else {
+                            bc.value(dof).unwrap_or(0.0)
+                        };
+                        held.entry((node, dof)).or_default().push(v);
+                    }
+                }
+            }
+        }
+        let mut dependent: BTreeSet<(u32, u8)> = BTreeSet::new();
+        for c in model.couplings().values().filter(|c| c.kind == CouplingKind::Kinematic) {
+            for &node in r.sets.get(&c.set).map(|s| s.nodes.as_slice()).unwrap_or_default() {
+                dependent.extend(c.dofs.list().into_iter().map(|d| (node, d)));
+            }
+        }
+        for t in model.ties().values() {
+            for &node in r.sets.get(&t.secondary).map(|s| s.nodes.as_slice()).unwrap_or_default() {
+                dependent.extend((1..=6).map(|d| (node, d)));
+            }
+        }
+        let state = |node: u32, dof: u8| {
+            if dependent.contains(&(node, dof)) {
+                return crate::Held::Sometimes;
+            }
+            match held.get(&(node, dof)) {
+                None => crate::Held::Free,
+                Some(v) if v.len() == steps.len() && v.iter().all(|x| *x == v[0]) => crate::Held::Always(v[0]),
+                Some(_) => crate::Held::Sometimes,
+            }
+        };
+        let mut problems = BTreeSet::new();
+        let equations = crate::symmetry_equations(model, r, inclined, &model.rotating_nodes(), state, &mut problems);
+        unsupported(FORMAT, problems)?;
+        self.comment("Symmetry planes not normal to X, Y or Z: per node, no motion along the normal (and no rotation about the plane's axes)");
+        for e in &equations {
+            self.line("*EQUATION");
+            self.line(&e.terms.len().to_string());
+            let mut fields = Vec::new();
+            for (dof, c) in &e.terms {
+                fields.push(format!("{}, {dof}", e.node));
+                fields.push(number(*c)?);
+            }
+            self.line(&fields.join(", "));
+        }
+        Ok(())
+    }
+
     /// Boundary conditions of a step against the state it starts from.
     fn step_boundaries(&mut self, model: &Model, r: &Resolved, base: &BTreeSet<String>, active: &BTreeSet<String>) -> Result<()> {
+        let (base, active) = (&held_conditions(model, base), &held_conditions(model, active));
         if active == base {
             self.comment("boundary conditions: as in the base state");
             Ok(())
@@ -1188,7 +1284,8 @@ impl Deck {
             BTreeSet::new()
         } else {
             let in_every_step = |n: &String| steps.iter().all(|s| s.boundary_conditions.contains(n));
-            model.boundary_conditions().iter().filter(|(n, bc)| holds_at_zero(bc) && in_every_step(n)).map(|(n, _)| n.clone()).collect()
+            let all = model.boundary_conditions().iter().filter(|(n, bc)| holds_at_zero(bc) && in_every_step(n)).map(|(n, _)| n.clone()).collect();
+            held_conditions(model, &all)
         };
         if !initial.is_empty() {
             self.comment("Initial state: boundary conditions active in every step");

crates/decks/src/lib.rs

@@ -16,7 +16,7 @@ pub mod nastran;
 use std::collections::{BTreeMap, BTreeSet};
 
 use fieldcraft_model::nodal::edge_pressure;
-use fieldcraft_model::{Diagnostic, Load, Model, Resolved, ResolvedDirection};
+use fieldcraft_model::{BoundaryCondition, Diagnostic, Load, Model, Resolved, ResolvedDirection};
 
 #[derive(Debug, Clone, PartialEq, thiserror::Error)]
 pub enum Error {
@@ -176,6 +176,120 @@ pub fn edge_pressure_loads<'a>(model: &Model, r: &Resolved, active: impl IntoIte
     Ok(out)
 }
 
+/// How a degree of freedom is constrained apart from the inclined symmetry planes, for choosing
+/// the dependent one of a plane's equation (formats eliminate it, so it may carry no other
+/// constraint).
+#[derive(Debug, Clone, Copy, PartialEq)]
+pub(crate) enum Held {
+    Free,
+    /// Held at this value wherever the equation applies: it cannot depend on the others, and an
+    /// equation of held ones alone is redundant when the values satisfy it.
+    Always(f64),
+    /// Held in some steps only, or made dependent by a coupling or tie: it cannot be dependent,
+    /// nor can the equation be dropped.
+    Sometimes,
+}
+
+/// A linear equation Σ cⱼ u(node, dofⱼ) = 0 of an inclined symmetry plane, dependent degree of
+/// freedom first.
+#[derive(Debug, Clone, PartialEq)]
+pub(crate) struct Equation {
+    pub node: u32,
+    pub terms: Vec<(u8, f64)>,
+}
+
+/// A coefficient below this fraction of its row's largest is round-off of the elimination.
+const NEGLIGIBLE: f64 = 1e-12;
+
+/// The equations of the inclined symmetry conditions `names` (others are skipped), node by node
+/// in label order: n · u = 0 and, at a node in `rotating`, t₁ · θ = 0 and t₂ · θ = 0 for the
+/// plane's in-plane axes ([`fieldcraft_model::Symmetry`]). The rows of one node (several planes
+/// meet at an edge) are reduced against each other, so each has a dependent degree of freedom
+/// of its own: the largest coefficient among those `held` says are free. A row of held degrees
+/// of freedom only is left out when their values satisfy it; otherwise it is a problem, as is a
+/// row with none free that cannot be left out.
+pub(crate) fn symmetry_equations<'a>(
+    model: &Model,
+    r: &Resolved,
+    names: impl IntoIterator<Item = &'a String>,
+    rotating: &BTreeSet<u32>,
+    held: impl Fn(u32, u8) -> Held,
+    problems: &mut BTreeSet<String>,
+) -> Vec<Equation> {
+    let mut rows: BTreeMap<u32, Vec<(&str, [f64; 6])>> = BTreeMap::new();
+    let mut seen = BTreeSet::new();
+    for name in names {
+        if !seen.insert(name.as_str()) {
+            continue;
+        }
+        let Some(BoundaryCondition::Symmetry { target, plane }) = model.boundary_condition(name) else { continue };
+        let (Some(n), Some([t1, t2])) = (plane.inclined(), plane.in_plane()) else { continue };
+        for &node in r.target_nodes(target) {
+            let at = rows.entry(node).or_default();
+            at.push((name, [n[0], n[1], n[2], 0.0, 0.0, 0.0]));
+            if rotating.contains(&node) || target.reference_point().is_some() {
+                at.push((name, [0.0, 0.0, 0.0, t1[0], t1[1], t1[2]]));
+                at.push((name, [0.0, 0.0, 0.0, t2[0], t2[1], t2[2]]));
+            }
+        }
+    }
+    let mut out = Vec::new();
+    for (node, rows) in rows {
+        // The equations of this node so far: (dependent index, coefficients).
+        let mut done: Vec<(usize, [f64; 6])> = Vec::new();
+        for (name, mut c) in rows {
+            let scale = c.iter().fold(0.0f64, |m, v| m.max(v.abs()));
+            for (d, e) in &done {
+                let f = c[*d] / e[*d];
+                for k in 0..6 {
+                    c[k] -= f * e[k];
+                }
+                c[*d] = 0.0;
+            }
+            for v in &mut c {
+                if v.abs() <= NEGLIGIBLE * scale {
+                    *v = 0.0;
+                }
+            }
+            if c.iter().all(|v| *v == 0.0) {
+                continue;
+            }
+            let dof = |k: usize| k as u8 + 1;
+            let pivot = (0..6).filter(|&k| c[k] != 0.0 && held(node, dof(k)) == Held::Free).fold(None, |best: Option<usize>, k| match best {
+                Some(b) if c[b].abs() >= c[k].abs() => Some(b),
+                _ => Some(k),
+            });
+            let Some(pivot) = pivot else {
+                let mut residual = 0.0;
+                let mut size = 0.0;
+                let mut always = true;
+                for k in (0..6).filter(|&k| c[k] != 0.0) {
+                    match held(node, dof(k)) {
+                        Held::Always(v) => {
+                            residual += c[k] * v;
+                            size += (c[k] * v).abs();
+                        }
+                        _ => always = false,
+                    }
+                }
+                if !always {
+                    problems.insert(format!(
+                        "boundary condition `{name}`: at node {node} every degree of freedom the inclined symmetry plane relates is held in some steps only or follows a coupling or tie, so none can be eliminated by its equation"
+                    ));
+                } else if residual.abs() > 1e-9 * size {
+                    problems.insert(format!("boundary condition `{name}`: node {node} is held off the inclined symmetry plane"));
+                }
+                continue;
+            };
+            let mut terms = vec![(dof(pivot), c[pivot])];
+            terms.extend((0..6).filter(|&k| k != pivot && c[k] != 0.0).map(|k| (dof(k), c[k])));
+            out.push(Equation { node, terms });
+            done.push((pivot, c));
+        }
+    }
+    out
+}
+
 /// The global degree of freedom (1–6) a spring or dashpot direction acts along, when it is one
 /// (a vector or line along an axis counts: the sign does not matter to a spring).
 pub(crate) fn axis_dof(d: ResolvedDirection) -> Option<u8> {

crates/decks/src/lsdyna/mod.rs

@@ -40,7 +40,8 @@
 //!   LS-DYNA's defaults; a step without nonlinear geometry and hyperelastic materials in nonlinear
 //!   steps are refused); boundary conditions
 //!   as `*BOUNDARY_SPC_*_ID` (named) and `*BOUNDARY_PRESCRIBED_MOTION_*_ID` for non-zero
-//!   displacements; forces, moments and tractions as `*LOAD_NODE_SET`/`_POINT` from the
+//!   displacements, a symmetry plane not normal to X, Y or Z as `*BOUNDARY_SPC_*_ID` in a
+//!   `*DEFINE_COORDINATE_SYSTEM` whose x axis is the plane's normal; forces, moments and tractions as `*LOAD_NODE_SET`/`_POINT` from the
 //!   resolved nodal values; pressure as `*LOAD_SEGMENT_SET`; gravity as `*LOAD_BODY_X/Y/Z`
 //!   (with `*LOAD_BODY_PARTS` when it acts on a set); output as `*DATABASE_BINARY_D3PLOT`,
 //!   `*DATABASE_SPCFORC` and `*DATABASE_EXTENT_BINARY`.
@@ -1048,6 +1049,24 @@ fn boundary_condition(
             (r.reference_point_nodes.get(p).copied().ok_or_else(|| broken(format!("boundary condition `{name}`: reference point `{p}`")))?, "NODE")
         }
     };
+    // A symmetry plane not normal to an axis: held in a coordinate system whose x axis is its
+    // normal and whose xy plane holds its first in-plane axis (translation x, rotations y and z).
+    if let BoundaryCondition::Symmetry { plane, .. } = bc
+        && let (Some(n), Some([t1, _])) = (plane.inclined(), plane.in_plane())
+    {
+        *constraint_id = constraint_id.saturating_add(1);
+        let cid = *constraint_id;
+        d.line("*DEFINE_COORDINATE_SYSTEM");
+        d.header(&["cid", "xo", "yo", "zo", "xl", "yl", "zl", "cidl"]);
+        d.card([Field::from(cid), Field::from(0.0), Field::from(0.0), Field::from(0.0), Field::from(n[0]), Field::from(n[1]), Field::from(n[2])]);
+        d.header(&["xp", "yp", "zp"]);
+        d.card([Field::from(t1[0]), Field::from(t1[1]), Field::from(t1[2])]);
+        d.line(&format!("*BOUNDARY_SPC_{suffix}_ID"));
+        id_card(d, constraint_id, name);
+        d.header(&["nsid", "cid", "dofx", "dofy", "dofz", "dofrx", "dofry", "dofrz"]);
+        d.card([Field::from(entity), Field::from(cid)].into_iter().chain([1, 0, 0, 0, 1, 1].map(|v: i32| Field::from(v))));
+        return Ok(());
+    }
     let dofs = bc.dofs();
     // Held degrees of freedom (in a frequency step every constrained one is held).
     let held: Vec<u8> = dofs.list().into_iter().filter(|dof| !is_static || bc.value(*dof).is_none_or(|v| v == 0.0)).collect();

crates/decks/src/nastran/mod.rs

@@ -22,6 +22,9 @@
 //!   the basic X axis through a `CORD2R`;
 //! - boundary conditions: per step, one constraint set of `SPC1` (held degrees of freedom) and
 //!   `SPC` (prescribed values); in a modes step every constrained degree of freedom is held;
+//!   symmetry about a plane not normal to X, Y or Z: one `MPC` set per step, an equation per node
+//!   and held direction (n · u = 0, and t · θ = 0 about the plane's two axes at shell and beam
+//!   grids), its dependent component one no `SPC` or rigid element constrains;
 //! - forces, moments and tractions: `FORCE` and `MOMENT` at the nodal values the model resolves,
 //!   so every writer and the solver share one distribution; pressure: `PLOAD4` per element face;
 //!   gravity: `GRAV`; a step with several loads combines them with `LOAD`;
@@ -241,6 +244,8 @@ type Constraints = BTreeMap<u32, [Option<f64>; 6]>;
 struct Subcase<'m> {
     step: &'m Step,
     spc: Option<u32>,
+    /// The equations of its inclined symmetry planes.
+    mpc: Option<u32>,
     load: Option<u32>,
     method: Option<u32>,
 }
@@ -269,6 +274,8 @@ struct Deck<'m> {
     dependent: BTreeMap<u32, (Dofs, &'m str)>,
     /// Constraint sets by id.
     spc_sets: Vec<(u32, Constraints)>,
+    /// Multipoint constraint sets (inclined symmetry planes) by id.
+    mpc_sets: Vec<(u32, Vec<crate::Equation>)>,
     /// Load name → load set id.
     load_sids: BTreeMap<&'m str, u32>,
     /// Combinations: (id, load set ids).
@@ -310,7 +317,7 @@ impl<'m> Deck<'m> {
             model,
             r,
             sol,
-            subcases: steps.into_iter().map(|step| Subcase { step, spc: None, load: None, method: None }).collect(),
+            subcases: steps.into_iter().map(|step| Subcase { step, spc: None, mpc: None, load: None, method: None }).collect(),
             problems,
             next_sid: 1,
             materials: BTreeMap::new(),
@@ -321,6 +328,7 @@ impl<'m> Deck<'m> {
             discrete: Vec::new(),
             dependent: BTreeMap::new(),
             spc_sets: Vec::new(),
+            mpc_sets: Vec::new(),
             load_sids: BTreeMap::new(),
             combinations: Vec::new(),
         }
@@ -587,6 +595,7 @@ impl<'m> Deck<'m> {
     /// One constraint set per distinct step constraint, checked against coupled nodes.
     fn plan_constraints(&mut self) -> Result<()> {
         let (model, r) = (self.model, self.r);
+        let rotating = model.rotating_nodes();
         // Conflicts, one problem per pair of definitions: the nodes they meet on.
         let mut clashes: BTreeMap<(&str, &str, &str), BTreeSet<u32>> = BTreeMap::new();
         let mut on_dependent: BTreeMap<(&str, &str), BTreeSet<u32>> = BTreeMap::new();
@@ -598,6 +607,10 @@ impl<'m> Deck<'m> {
             for name in &step.boundary_conditions {
                 let Some(bc) = model.boundary_condition(name) else { continue };
                 let dofs = bc.dofs();
+                // Inclined symmetry planes hold no global degree of freedom: they are equations.
+                if dofs.is_empty() {
+                    continue;
+                }
                 // In a modes step every constrained degree of freedom is held.
                 let values: Vec<(u8, f64)> = dofs.list().into_iter().filter_map(|d| Some((d, if modes { 0.0 } else { bc.value(d)? }))).collect();
                 for &node in r.target_nodes(bc.target()) {
@@ -622,6 +635,19 @@ impl<'m> Deck<'m> {
                     }
                 }
             }
+            let dependent = &self.dependent;
+            let held = |node: u32, dof: u8| match (
+                dependent.get(&node),
+                set.get(&node).and_then(|v| v.get(usize::from(dof).checked_sub(1)?).copied().flatten()),
+            ) {
+                (Some((d, _)), _) if d.contains(dof) => crate::Held::Sometimes,
+                (_, Some(v)) => crate::Held::Always(v),
+                _ => crate::Held::Free,
+            };
+            let equations = crate::symmetry_equations(model, r, &step.boundary_conditions, &rotating, held, &mut self.problems);
+            if !equations.is_empty() {
+                sub.mpc = Some(intern(&mut self.mpc_sets, &mut self.next_sid, equations)?);
+            }
             if !set.is_empty() {
                 sub.spc = Some(intern(&mut self.spc_sets, &mut self.next_sid, set)?);
             }
@@ -707,7 +733,7 @@ impl<'m> Deck<'m> {
             let _ = writeln!(out, "SUBCASE {}", i + 1);
             let title: String = format!("  SUBTITLE = {}", sub.step.name).chars().take(72).collect();
             let _ = writeln!(out, "{title}");
-            for (key, sid) in [("SPC", sub.spc), ("LOAD", sub.load), ("METHOD", sub.method)] {
+            for (key, sid) in [("SPC", sub.spc), ("MPC", sub.mpc), ("LOAD", sub.load), ("METHOD", sub.method)] {
                 if let Some(sid) = sid {
                     let _ = writeln!(out, "  {key} = {sid}");
                 }
@@ -715,6 +741,7 @@ impl<'m> Deck<'m> {
             for v in &sub.step.output {
                 let request = match v {
                     OutputVariable::Displacement => "DISPLACEMENT",
+                    OutputVariable::ReactionForce if sub.mpc.is_some() => "SPCFORCES = ALL\n  MPCFORCES",
                     OutputVariable::ReactionForce => "SPCFORCES",
                     OutputVariable::Stress => "STRESS",
                     OutputVariable::Strain => "STRAIN",
@@ -892,6 +919,23 @@ impl<'m> Deck<'m> {
                 Card::new("SPC").int(*sid).int(node).int(component(Dofs::from_bits(mask))).real(value).write(out)?;
             }
         }
+        for (sid, equations) in &self.mpc_sets {
+            for sub in self.subcases.iter().filter(|s| s.mpc == Some(*sid)) {
+                comment(out, &format!("step {}: inclined symmetry planes, MPC {sid} (the first grid component is dependent)", sub.step.name));
+            }
+            for e in equations {
+                // SID, then (G, C, A) triples: two on the first line (its last field blank), two
+                // per continuation after a blank first field.
+                let mut card = Card::new("MPC").int(*sid);
+                for (i, (dof, c)) in e.terms.iter().enumerate() {
+                    if i % 2 == 0 && i > 0 {
+                        card = card.blank().blank();
+                    }
+                    card = card.int(e.node).int(i64::from(*dof)).real(*c);
+                }
+                card.write(out)?;
+            }
+        }
         Ok(())
     }
 

crates/decks/tests/inclined_symmetry.rs

@@ -0,0 +1,193 @@
+//! Symmetry about a plane not normal to X, Y or Z in the three decks (#242): Abaqus `*EQUATION`s,
+//! Nastran `MPC` sets and an LS-DYNA `*BOUNDARY_SPC` in a local coordinate system, on a small
+//! shell plate of our own whose diagonal x = y is the plane.
+
+use fieldcraft_decks::{Error, abaqus, lsdyna, nastran};
+use fieldcraft_model::{
+    BoundaryCondition, Elastic, ElementFace, ElementKind, Formulation, Load, Material, Model, Procedure, Section, SectionKind, Set, Step, Symmetry,
+    Target,
+};
+
+/// A 2 × 2 quad4 shell plate in the XY plane (nodes 1–9, x and y in 0, 1, 2), clamped on x = 0,
+/// the inclined plane (normal (1, −1, 0)) on the diagonal nodes 1, 5 and 9, under pressure;
+/// steps `Load` (static) and `Modes`, `in_modes` saying whether the plane is active in both.
+fn plate(in_modes: bool) -> Model {
+    let mut m = Model::default();
+    for j in 0..3u32 {
+        for i in 0..3u32 {
+            m.set_node(1 + i + 3 * j, [f64::from(i), f64::from(j), 0.0]).unwrap();
+        }
+    }
+    for (e, n) in [(1, [1, 2, 5, 4]), (2, [2, 3, 6, 5]), (3, [4, 5, 8, 7]), (4, [5, 6, 9, 8])] {
+        m.set_element(e, ElementKind::Quad4, n.to_vec()).unwrap();
+    }
+    m.add_material(
+        "Steel",
+        Material { elastic: Some(Elastic::Isotropic { young: 210e3, poisson: 0.3 }), density: Some(7.85e-9), ..Material::default() },
+    )
+    .unwrap();
+    m.add_set("all", Set { elements: vec![1, 2, 3, 4], ..Set::default() }).unwrap();
+    let kind = SectionKind::Shell { thickness: 0.1, offset: 0.0, points: 5 };
+    m.add_section("plate", Section { set: "all".into(), material: "Steel".into(), kind, formulation: Formulation::default() }).unwrap();
+    m.add_set("edge", Set { nodes: vec![1, 4, 7], ..Set::default() }).unwrap();
+    m.add_set("diagonal", Set { nodes: vec![1, 5, 9], ..Set::default() }).unwrap();
+    let faces = (1..=4).map(|element| ElementFace { element, face: 1 }).collect();
+    m.add_set("top", Set { faces, ..Set::default() }).unwrap();
+    m.add_boundary_condition("clamp", BoundaryCondition::Fixed { target: Target::Set("edge".into()) }).unwrap();
+    let plane = Symmetry::from_normal([1.0, -1.0, 0.0]).unwrap();
+    m.add_boundary_condition("sym", BoundaryCondition::Symmetry { target: Target::Set("diagonal".into()), plane }).unwrap();
+    m.add_load("q", Load::Pressure { set: "top".into(), magnitude: 0.01 }).unwrap();
+    let mut load = Step::new("Load", Procedure::Static {});
+    load.boundary_conditions = ["clamp", "sym"].map(String::from).into();
+    load.loads = ["q".to_string()].into();
+    m.add_step(load, None).unwrap();
+    let mut modes = Step::new("Modes", Procedure::Frequency { modes: 3, min_hz: None, max_hz: None });
+    modes.boundary_conditions = if in_modes { ["clamp", "sym"].map(String::from).into() } else { ["clamp".to_string()].into() };
+    m.add_step(modes, None).unwrap();
+    m
+}
+
+/// A Nastran real: an exponent may come without `E`.
+fn real(s: &str) -> f64 {
+    let split = s.char_indices().skip(1).find(|(i, c)| (*c == '+' || *c == '-') && !s[..*i].ends_with(['E', 'e']));
+    match split {
+        Some((i, _)) => format!("{}E{}", &s[..i], &s[i..]).parse().unwrap(),
+        None => s.parse().unwrap(),
+    }
+}
+
+fn dot(a: [f64; 3], b: [f64; 3]) -> f64 {
+    (0..3).map(|k| a[k] * b[k]).sum()
+}
+
+/// Every equation is one of the plane's at a free node: its terms on one node, its coefficients
+/// along the normal (translations) or along an in-plane axis (rotations), unit length; nodes 5
+/// and 9 get three each (n · u, t₁ · θ, t₂ · θ); node 1 is clamped, so its equations are left out;
+/// no dependent degree of freedom repeats.
+fn check(equations: &[(u32, Vec<(u8, f64)>)]) {
+    let n = [1.0 / 2f64.sqrt(), -1.0 / 2f64.sqrt(), 0.0];
+    let mut dependents = std::collections::BTreeSet::new();
+    let mut per_node = std::collections::BTreeMap::<u32, usize>::new();
+    for (node, terms) in equations {
+        assert!(dependents.insert((*node, terms[0].0)), "{node} {terms:?}");
+        *per_node.entry(*node).or_default() += 1;
+        let mut v = [0.0; 3];
+        let rotation = terms[0].0 > 3;
+        for (dof, c) in terms {
+            assert_eq!(*dof > 3, rotation, "{terms:?}");
+            v[usize::from((dof - 1) % 3)] = *c;
+        }
+        assert!((dot(v, v) - 1.0).abs() < 1e-12, "{terms:?}");
+        if rotation {
+            assert!(dot(v, n).abs() < 1e-12, "{terms:?}");
+        } else {
+            assert!((dot(v, n).abs() - 1.0).abs() < 1e-12, "{terms:?}");
+        }
+    }
+    assert_eq!(per_node, [(5, 3), (9, 3)].into());
+}
+
+#[test]
+fn abaqus_writes_equations_in_the_model_data() {
+    let deck = abaqus::write(&plate(true)).unwrap();
+    let lines: Vec<&str> = deck.lines().collect();
+    let mut equations = Vec::new();
+    for (i, l) in lines.iter().enumerate() {
+        if *l == "*EQUATION" {
+            let count: usize = lines[i + 1].trim().parse().unwrap();
+            let f: Vec<&str> = lines[i + 2].split(',').map(str::trim).collect();
+            assert_eq!(f.len(), 3 * count, "{}", lines[i + 2]);
+            let node: u32 = f[0].parse().unwrap();
+            let terms = f.chunks(3).map(|t| (t[1].parse().unwrap(), t[2].parse().unwrap())).collect();
+            assert!(f.chunks(3).all(|t| t[0] == f[0]));
+            equations.push((node, terms));
+        }
+    }
+    check(&equations);
+    // Model data, before the first step; the plane is no `*BOUNDARY` line.
+    let first_step = deck.find("*STEP").unwrap();
+    assert!(deck.rfind("*EQUATION").unwrap() < first_step);
+    assert!(!deck.contains("SYMM"), "{deck}");
+
+    let e = abaqus::write(&plate(false)).unwrap_err();
+    assert!(matches!(&e, Error::Unsupported { what, .. } if what.contains("boundary condition `sym`") && what.contains("some steps only")), "{e}");
+}
+
+#[test]
+fn nastran_writes_one_mpc_set_per_step() {
+    // A static deck and a modes deck (one solution per run).
+    for (steps, sol) in [("Load", 101), ("Modes", 103)] {
+        let m = fieldcraft_decks::with_steps(&plate(true), &[steps.to_string()]).unwrap();
+        let deck = nastran::write(&m).unwrap();
+        assert!(deck.contains(&format!("SOL {sol}")));
+        let sid = deck.lines().find_map(|l| l.trim().strip_prefix("MPC = ")).unwrap().to_string();
+        // MPC entries: SID, then (grid, component, coefficient) triples; the continuation after
+        // two triples starts with a blank field. Large-field lines are 16 columns per field.
+        let mut equations = Vec::new();
+        let mut fields: Vec<String> = Vec::new();
+        let flush = |fields: &mut Vec<String>, equations: &mut Vec<(u32, Vec<(u8, f64)>)>| {
+            if fields.is_empty() {
+                return;
+            }
+            assert_eq!(fields[0], sid);
+            let data: Vec<&String> = fields[1..].iter().enumerate().filter(|(i, _)| !matches!(i % 8, 6 | 7)).map(|(_, f)| f).collect();
+            let triples: Vec<(u32, u8, f64)> = data
+                .chunks(3)
+                .filter(|t| t.len() == 3 && !t[0].is_empty())
+                .map(|t| (t[0].parse().unwrap(), t[1].parse().unwrap(), real(t[2])))
+                .collect();
+            assert!(triples.iter().all(|t| t.0 == triples[0].0));
+            equations.push((triples[0].0, triples.iter().map(|t| (t.1, t.2)).collect()));
+            fields.clear();
+        };
+        for line in deck.lines() {
+            let (head, rest) = line.split_at(8.min(line.len()));
+            let row: Vec<String> = rest.as_bytes().chunks(16).map(|c| String::from_utf8_lossy(c).trim().to_string()).collect();
+            let mut row = row;
+            row.resize(4, String::new());
+            match head.trim() {
+                "MPC*" => {
+                    flush(&mut fields, &mut equations);
+                    fields.extend(row);
+                }
+                "*" if !fields.is_empty() => fields.extend(row),
+                _ => flush(&mut fields, &mut equations),
+            }
+        }
+        flush(&mut fields, &mut equations);
+        check(&equations);
+    }
+}
+
+#[test]
+fn lsdyna_holds_the_plane_in_a_local_system() {
+    let deck = lsdyna::write(&plate(true)).unwrap().text;
+    let block = |keyword: &str| -> Vec<Vec<f64>> {
+        let at = deck.find(keyword).unwrap();
+        deck[at..]
+            .lines()
+            .skip(1)
+            .filter(|l| !l.starts_with('$'))
+            .take_while(|l| !l.starts_with('*'))
+            .map(|l| l.as_bytes().chunks(20).filter_map(|c| String::from_utf8_lossy(c).trim().parse().ok()).collect())
+            .collect()
+    };
+    let system = block("*DEFINE_COORDINATE_SYSTEM");
+    let (cid, x, p) = (system[0][0], [system[0][4], system[0][5], system[0][6]], [system[1][0], system[1][1], system[1][2]]);
+    assert!((x[0] + x[1]).abs() < 1e-12 && x[2] == 0.0 && (dot(x, x) - 1.0).abs() < 1e-12, "{x:?}");
+    assert!(dot(x, p).abs() < 1e-12, "{p:?}");
+    // The SPC after the system: its id card, then nsid, cid and the held local directions.
+    let at = deck.find("*DEFINE_COORDINATE_SYSTEM").unwrap();
+    let spc = deck[at..].find("*BOUNDARY_SPC_SET_ID").unwrap() + at;
+    let card: Vec<f64> = deck[spc..]
+        .lines()
+        .skip(2)
+        .find(|l| !l.starts_with('$'))
+        .unwrap()
+        .as_bytes()
+        .chunks(20)
+        .map(|c| String::from_utf8_lossy(c).trim().parse().unwrap())
+        .collect();
+    assert_eq!(card[1], cid);
+    assert_eq!(&card[2..], [1.0, 0.0, 0.0, 0.0, 1.0, 1.0]);
+}

crates/model/src/bcs.rs

@@ -110,25 +110,101 @@ impl std::fmt::Display for Target {
     }
 }
 
-/// The plane of a symmetry boundary condition, named by its normal axis.
-#[derive(Debug, Clone, Copy, PartialEq, Eq, PartialOrd, Ord, Hash, Serialize, Deserialize)]
+/// The plane of a symmetry boundary condition, named by its normal: an axis (`"x"`), or any
+/// direction (`{"normal": [1, 1, 0]}`, not necessarily of unit length).
+#[derive(Debug, Clone, Copy, PartialEq, Serialize, Deserialize)]
 #[serde(rename_all = "snake_case")]
 pub enum Symmetry {
     /// The plane normal to X: no X translation, no rotation about Y or Z.
     X,
     Y,
     Z,
+    /// A plane normal to a direction: no translation along it, no rotation about the two axes
+    /// in the plane (n · u = 0, and θ ∥ n at nodes with rotations).
+    Normal([f64; 3]),
 }
 
+/// A normal whose component along an axis is within this of 1 (unit length) is that axis.
+const ALONG_AXIS: f64 = 1e-12;
+
 impl Symmetry {
-    /// The fixed degrees of freedom.
-    pub fn dofs(self) -> Dofs {
+    /// The plane normal to `normal`: [`Symmetry::X`], `Y` or `Z` when it lies along an axis,
+    /// otherwise [`Symmetry::Normal`] with the unit normal; `None` for a zero or non-finite one.
+    pub fn from_normal(normal: [f64; 3]) -> Option<Symmetry> {
+        let unit = unit(normal)?;
+        Some(match (0..3).find(|&k| (unit[k].abs() - 1.0).abs() < ALONG_AXIS) {
+            Some(0) => Symmetry::X,
+            Some(1) => Symmetry::Y,
+            Some(2) => Symmetry::Z,
+            _ => Symmetry::Normal(unit),
+        })
+    }
+
+    /// The unit normal (`None` for a zero or non-finite [`Symmetry::Normal`]).
+    pub fn unit_normal(self) -> Option<[f64; 3]> {
         match self {
-            Symmetry::X => Dofs(0b110_001),
-            Symmetry::Y => Dofs(0b101_010),
-            Symmetry::Z => Dofs(0b011_100),
+            Symmetry::X => Some([1.0, 0.0, 0.0]),
+            Symmetry::Y => Some([0.0, 1.0, 0.0]),
+            Symmetry::Z => Some([0.0, 0.0, 1.0]),
+            Symmetry::Normal(n) => unit(n),
         }
     }
+
+    /// The axis (0–2) the normal lies along, so the condition holds global degrees of freedom;
+    /// `None` for an inclined plane, which is a linear constraint instead.
+    pub fn axis(self) -> Option<usize> {
+        let n = self.unit_normal()?;
+        (0..3).find(|&k| (n[k].abs() - 1.0).abs() < ALONG_AXIS)
+    }
+
+    /// The held global degrees of freedom: those of an axis-aligned plane; none for an inclined
+    /// one, which holds combinations of them ([`Symmetry::unit_normal`]).
+    pub fn dofs(self) -> Dofs {
+        match self.axis() {
+            Some(0) => Dofs(0b110_001),
+            Some(1) => Dofs(0b101_010),
+            Some(2) => Dofs(0b011_100),
+            _ => Dofs::none(),
+        }
+    }
+
+    /// The inclined plane's unit normal; `None` for one normal to an axis.
+    pub fn inclined(self) -> Option<[f64; 3]> {
+        if self.axis().is_some() { None } else { self.unit_normal() }
+    }
+
+    /// Two unit vectors in the plane, perpendicular to each other and to the unit normal n:
+    /// t₁ = n × e / |n × e| with e the axis n leans on least, and t₂ = n × t₁ (`None` for a
+    /// zero or non-finite normal).
+    pub fn in_plane(self) -> Option<[[f64; 3]; 2]> {
+        let n = self.unit_normal()?;
+        let least = (0..3).fold(0, |m, k| if n[k].abs() < n[m].abs() { k } else { m });
+        let mut e = [0.0; 3];
+        e[least] = 1.0;
+        let cross = |a: [f64; 3], b: [f64; 3]| [a[1] * b[2] - a[2] * b[1], a[2] * b[0] - a[0] * b[2], a[0] * b[1] - a[1] * b[0]];
+        let t1 = unit(cross(n, e))?;
+        Some([t1, cross(n, t1)])
+    }
+
+    fn validate(self) -> Result<()> {
+        if let Symmetry::Normal(n) = self
+            && unit(n).is_none()
+        {
+            return Err(Error::Invalid(format!("the symmetry plane's normal must be a finite, non-zero vector, got {n:?}")));
+        }
+        Ok(())
+    }
+}
+
+/// `v` scaled to unit length; `None` for a zero or non-finite vector.
+fn unit(v: [f64; 3]) -> Option<[f64; 3]> {
+    let scale = v.iter().fold(0.0f64, |m, c| m.max(c.abs()));
+    if !(scale > 0.0 && v.iter().all(|c| c.is_finite())) {
+        return None;
+    }
+    let w = v.map(|c| c / scale);
+    let length = w.iter().map(|c| c * c).sum::<f64>().sqrt();
+    Some(w.map(|c| c / length))
 }
 
 /// A boundary condition.
@@ -198,6 +274,9 @@ impl BoundaryCondition {
     }
 
     pub fn validate(&self) -> Result<()> {
+        if let BoundaryCondition::Symmetry { plane, .. } = self {
+            plane.validate()?;
+        }
         if let BoundaryCondition::Temperature { target, value } = self {
             finite("temperature", *value)?;
             if let Target::ReferencePoint(r) = target {
@@ -285,4 +364,35 @@ mod tests {
         let parsed: BoundaryCondition = serde_json::from_value(json.clone()).unwrap();
         assert_eq!(serde_json::to_value(&parsed).unwrap(), json);
     }
+
+    #[test]
+    fn inclined_symmetry_planes() {
+        let tilt = Symmetry::from_normal([1.0, 1.0, 0.0]).unwrap();
+        let h = 1.0 / 2f64.sqrt();
+        assert_eq!(tilt, Symmetry::Normal([h, h, 0.0]));
+        assert_eq!((tilt.axis(), tilt.dofs(), tilt.inclined()), (None, Dofs::none(), Some([h, h, 0.0])));
+        assert_eq!(Symmetry::from_normal([0.0, -2.0, 0.0]), Some(Symmetry::Y));
+        assert_eq!(Symmetry::Normal([0.0, 0.0, 3.0]).dofs(), Symmetry::Z.dofs());
+        assert_eq!(Symmetry::Normal([0.0, 0.0, 3.0]).inclined(), None);
+        for bad in [[0.0; 3], [f64::NAN, 1.0, 0.0], [f64::INFINITY, 0.0, 0.0]] {
+            assert_eq!(Symmetry::from_normal(bad), None);
+            let bc = BoundaryCondition::Symmetry { target: Target::Set("s".into()), plane: Symmetry::Normal(bad) };
+            assert!(bc.validate().is_err(), "{bad:?}");
+        }
+        // The tiniest and the largest finite normals still give a unit one.
+        assert_eq!(Symmetry::from_normal([1e-300, 1e-300, 0.0]), Some(Symmetry::Normal([h, h, 0.0])));
+        assert_eq!(Symmetry::from_normal([f64::MAX, f64::MAX, 0.0]), Some(Symmetry::Normal([h, h, 0.0])));
+        for plane in [tilt, Symmetry::X, Symmetry::Normal([0.3, -2.0, 7.0])] {
+            let n = plane.unit_normal().unwrap();
+            let [a, b] = plane.in_plane().unwrap();
+            let dot = |u: [f64; 3], v: [f64; 3]| (0..3).map(|k| u[k] * v[k]).sum::<f64>();
+            for (u, v, want) in [(a, a, 1.0), (b, b, 1.0), (a, b, 0.0), (a, n, 0.0), (b, n, 0.0)] {
+                assert!((dot(u, v) - want).abs() < 1e-15, "{plane:?}");
+            }
+        }
+        let json = serde_json::json!({"type": "symmetry", "target": {"set": "s"}, "plane": {"normal": [1.0, 1.0, 0.0]}});
+        let parsed: BoundaryCondition = serde_json::from_value(json.clone()).unwrap();
+        assert_eq!(parsed, BoundaryCondition::Symmetry { target: Target::Set("s".into()), plane: Symmetry::Normal([1.0, 1.0, 0.0]) });
+        assert_eq!(serde_json::to_value(&parsed).unwrap(), json);
+    }
 }

crates/model/src/check.rs

@@ -176,6 +176,19 @@ impl Model {
             .collect()
     }
 
+    /// The mesh nodes with rotational degrees of freedom under the model's own sections: those of
+    /// shells and beams, not of plane stress, plane strain or axisymmetric elements.
+    pub fn rotating_nodes(&self) -> BTreeSet<u32> {
+        let planar: BTreeSet<u32> = self
+            .sections
+            .values()
+            .filter(|s| s.kind.extent().is_some())
+            .filter_map(|s| self.resolve_set_partial(&s.set))
+            .flat_map(|r| r.elements)
+            .collect();
+        self.rotational_nodes(|l| planar.contains(&l))
+    }
+
     fn analyze(&self) -> (Resolved, Vec<Diagnostic>) {
         let mut d = Diagnostics::default();
         let mut resolved = Resolved::default();

crates/render/src/glyphs.rs

@@ -354,19 +354,6 @@ fn scan<T>(items: &[T]) -> impl Iterator<Item = &T> {
     items.iter().step_by(items.len().div_ceil(MAX_SCAN).max(1))
 }
 
-/// The nodes that can rotate: shell and beam nodes (not those of 2D sections: plane stress,
-/// plane strain, axisymmetric), as the model check counts them.
-fn rotating_nodes(model: &Model) -> BTreeSet<u32> {
-    let planar: BTreeSet<u32> = model
-        .sections()
-        .values()
-        .filter(|s| s.kind.extent().is_some())
-        .filter_map(|s| model.resolve_set_partial(&s.set))
-        .flat_map(|r| r.elements)
-        .collect();
-    model.rotational_nodes(|l| planar.contains(&l))
-}
-
 /// The corner node labels of an element face, in the face's order.
 fn face_corners(model: &Model, f: ElementFace) -> Option<Vec<u32>> {
     let kind = model.element(f.element)?.kind;
@@ -589,7 +576,7 @@ impl Builder<'_> {
         }
         let (spots, total) = self.spots(bc.target());
         if dofs & 0b111_000 != 0 && self.rotating.is_none() {
-            self.rotating = Some(rotating_nodes(self.model));
+            self.rotating = Some(self.model.rotating_nodes());
         }
         let none = BTreeSet::new();
         let rotating = self.rotating.as_ref().unwrap_or(&none);

crates/solver/src/analysis.rs

@@ -12,7 +12,7 @@ use serde::Serialize;
 use fieldcraft_parallel as parallel;
 
 use crate::cholesky::{Factor, FactorError, UpperColumns};
-use crate::constraints::{Budget, Reduction, coupling_rows};
+use crate::constraints::{Budget, Reduction, coupling_rows, symmetry_rows};
 use crate::eigen::{Operator, Wanted, lanczos};
 use crate::ldlt::{Kind, Symmetric};
 use crate::ordering::nested_dissection;
@@ -342,6 +342,13 @@ pub(crate) struct Setup<'a> {
 pub(crate) fn reduction(s: &Setup, values: bool, order: &[usize], budget: Budget, warnings: &mut Vec<String>) -> Result<Reduction> {
     let mut rows = coupling_rows(s.model, s.resolved, s.mesh)?;
     rows.extend(crate::tie::rows(s.model, s.resolved, s.mesh, warnings)?);
+    // Reference point directions exist when a coupling, tie or discrete element uses them.
+    let used: std::collections::BTreeSet<usize> = rows.iter().flat_map(|r| r.terms.iter().map(|t| t.0)).chain(s.mesh.discrete.dofs()).collect();
+    let node_dofs = s.mesh.layout.node_dofs();
+    // Boundary conditions first, as the elimination takes them.
+    let mut symmetry = symmetry_rows(s.model, s.resolved, s.mesh, s.step, |d| d < node_dofs || used.contains(&d))?;
+    symmetry.append(&mut rows);
+    let rows = symmetry;
     Reduction::new(
         &s.mesh.layout,
         s.mesh.discrete.dofs(),

crates/solver/src/constraints.rs

@@ -53,7 +53,7 @@ use std::collections::BTreeMap;
 
 use fieldcraft_fem::dense::inverse3;
 use fieldcraft_fem::{DofKind, DofLayout, DofSet};
-use fieldcraft_model::{CouplingKind, Model, Resolved};
+use fieldcraft_model::{BoundaryCondition, CouplingKind, Model, Resolved, Step};
 
 use crate::sparse::Csr;
 use crate::system::Mesh;
@@ -74,6 +74,9 @@ pub struct Row<'m> {
 #[derive(Debug, Clone, Copy, PartialEq, Eq)]
 pub enum Source {
     Coupling,
+    /// A symmetry boundary condition on an inclined plane: n · u = 0, and θ ∥ n at nodes with
+    /// rotations.
+    Symmetry,
     /// A tie: its secondary node follows a weighted average of main nodes.
     Tie,
 }
@@ -82,6 +85,7 @@ impl Source {
     fn what(self) -> &'static str {
         match self {
             Source::Coupling => "coupling",
+            Source::Symmetry => "boundary condition",
             Source::Tie => "tie",
         }
     }
@@ -224,6 +228,64 @@ pub fn coupling_rows<'m>(model: &'m Model, resolved: &Resolved, mesh: &Mesh) ->
     Ok(rows)
 }
 
+/// The constraint rows of a step's symmetry conditions on inclined planes (those on a plane
+/// normal to an axis hold global degrees of freedom and are prescribed values instead), in the
+/// step's order, nodes by label. At every target node the translation along the unit normal n
+/// is zero, n · u = 0, eliminating the translation of the largest |nₖ|; at a node with
+/// rotations (shells, beams, a reference point whose rotations exist) the rotation about each of
+/// two in-plane axes t₁, t₂ is zero too, tᵢ · θ = 0: a mirror image turns only about the normal.
+/// `exists` says whether a reference point direction exists (mesh node ones always do).
+pub fn symmetry_rows<'m>(model: &Model, resolved: &Resolved, mesh: &Mesh, step: &'m Step, exists: impl Fn(usize) -> bool) -> Result<Vec<Row<'m>>> {
+    let mut rows = Vec::new();
+    for name in &step.boundary_conditions {
+        let Some(BoundaryCondition::Symmetry { target, plane }) = model.boundary_condition(name) else { continue };
+        let (Some(n), Some([t1, t2])) = (plane.inclined(), plane.in_plane()) else { continue };
+        let rp = target.reference_point();
+        if rp.is_some_and(|r| mesh.point(r).is_none()) {
+            // Reported by `prescribed` as holding nothing, like any condition on such a point.
+            continue;
+        }
+        for &label in resolved.target_nodes(target) {
+            let slot = match rp {
+                Some(r) => mesh.point(r).map(|k| mesh.layout.point_slot(k)),
+                None => mesh.layout.slot(label),
+            };
+            let Some(slot) = slot else { continue };
+            let kinds = mesh.layout.kinds(slot);
+            let dof = |kind: Option<DofKind>| kind.and_then(|k| kinds.contains(k).then(|| mesh.layout.dof(slot, k)).flatten()).filter(|d| exists(*d));
+            let row = |axis: [f64; 3], rotation: bool| -> Result<Option<Row<'m>>> {
+                let mut terms = Vec::with_capacity(3);
+                let mut missing = false;
+                for (k, c) in axis.into_iter().enumerate().filter(|(_, c)| *c != 0.0) {
+                    match dof(if rotation { DofKind::rotation(k) } else { DofKind::translation(k) }) {
+                        Some(d) => terms.push((d, c)),
+                        None => missing = true,
+                    }
+                }
+                if missing {
+                    // No rotation exists at a solid node: only its translation is held.
+                    if rotation && terms.is_empty() {
+                        return Ok(None);
+                    }
+                    return Err(Error::Invalid(format!(
+                        "boundary condition `{name}`: node {label} cannot move along the symmetry plane's normal {n:?} (it has no {} along some axis the normal leans on); in a 2D model the plane must contain the Z axis",
+                        if rotation { "rotation" } else { "translation" }
+                    )));
+                }
+                let pivot = terms.iter().fold(None, |best: Option<(usize, f64)>, t| match best {
+                    Some(b) if b.1.abs() >= t.1.abs() => Some(b),
+                    _ => Some(*t),
+                });
+                Ok(pivot.map(|(pivot, _)| Row { terms, pivot, coupling: name, source: Source::Symmetry }))
+            };
+            rows.extend(row(n, false)?);
+            rows.extend(row(t1, true)?);
+            rows.extend(row(t2, true)?);
+        }
+    }
+    Ok(rows)
+}
+
 /// The degrees of freedom that exist: every mesh node's, and the reference point directions
 /// some constraint row or `extra` (the discrete elements') involves.
 fn active_dofs(layout: &DofLayout, extra: impl Iterator<Item = usize>, rows: &[Row]) -> Vec<bool> {
@@ -346,6 +408,10 @@ pub struct Reduction {
     prescribed_motion: Vec<f64>,
     /// Whether any degree of freedom depends on others (T is not a plain selection).
     has_dependents: bool,
+    /// The degrees of freedom an inclined symmetry plane supports (those of its rows that are not
+    /// prescribed): their share of K u − f is the plane's reaction on them, n λ, and is reported
+    /// where it acts rather than carried to prescribed degrees of freedom.
+    supports: Vec<bool>,
     budget: Budget,
 }
 
@@ -447,7 +513,7 @@ impl Reduction {
         for row in rows {
             let scale = row.terms.iter().fold(0.0f64, |m, t| m.max(t.1.abs()));
             if !(scale > 0.0 && scale.is_finite()) {
-                return Err(Error::Invalid(format!("coupling `{}`: a constraint with no finite coefficient", row.coupling)));
+                return Err(Error::Invalid(format!("{} `{}`: a constraint with no finite coefficient", row.source.what(), row.coupling)));
             }
             // The row in current free and prescribed degrees of freedom.
             let mut expanded = Vec::with_capacity(row.terms.len());
@@ -497,7 +563,8 @@ impl Reduction {
                 let size: f64 = terms.iter().map(|t| value(t).abs()).sum();
                 if residual.abs() > 1e-9 * size.max(f64::MIN_POSITIVE) && residual.abs() > 0.0 {
                     return Err(Error::Invalid(format!(
-                        "coupling `{}` contradicts the prescribed values (boundary conditions or other couplings hold its nodes in a way it cannot follow)",
+                        "{} `{}` contradicts the prescribed values (boundary conditions or couplings hold its nodes in a way it cannot follow)",
+                        row.source.what(),
                         row.coupling
                     )));
                 }
@@ -572,7 +639,17 @@ impl Reduction {
         }
         let prescribed_motion = (0..n).map(|d| fixed.row(d).map(|(p, c)| c * values.get(p).copied().unwrap_or(0.0)).sum()).collect();
         let columns = free.transpose(eq_dof.len());
-        Ok(Reduction { eq_dof, active, free, columns, fixed, prescribed_motion, has_dependents: !exprs.is_empty(), budget })
+        let mut supports = vec![false; n];
+        for row in rows.iter().filter(|r| r.source == Source::Symmetry) {
+            for &(dof, _) in &row.terms {
+                if state.get(dof) != Some(&State::Fixed)
+                    && let Some(x) = supports.get_mut(dof)
+                {
+                    *x = true;
+                }
+            }
+        }
+        Ok(Reduction { eq_dof, active, free, columns, fixed, prescribed_motion, has_dependents: !exprs.is_empty(), supports, budget })
     }
 
     /// Free equations.
@@ -737,10 +814,17 @@ impl Reduction {
     }
 
     /// T_pᵀ g for a vector g over every degree of freedom (the reactions of the out-of-balance
-    /// forces g = f_int − f_ext), zero where nothing is prescribed.
+    /// forces g = f_int − f_ext), zero where nothing is prescribed; at the degrees of freedom an
+    /// inclined symmetry plane supports, g itself.
     pub fn reactions_of(&self, g: &[f64]) -> Vec<f64> {
         let mut out = vec![0.0; self.prescribed_motion.len()];
         for (dof, v) in g.iter().enumerate() {
+            if self.supports.get(dof).copied().unwrap_or(false) {
+                if let Some(x) = out.get_mut(dof) {
+                    *x += v;
+                }
+                continue;
+            }
             for (p, c) in self.fixed.row(dof) {
                 if let Some(x) = out.get_mut(p) {
                     *x += c * v;
@@ -750,24 +834,23 @@ impl Reduction {
         out
     }
 
-    /// The reactions at the prescribed degrees of freedom, T_pᵀ (K u − f), over every degree of
-    /// freedom (zero where nothing is prescribed).
+    /// The reactions at the prescribed degrees of freedom, T_pᵀ (K u − f), and at those an
+    /// inclined symmetry plane supports ([`Reduction::reactions_of`]), over every degree of
+    /// freedom (zero elsewhere).
     pub fn reactions(&self, k: &Csr, u: &[f64], f: &[f64]) -> Vec<f64> {
-        let mut out = vec![0.0; self.prescribed_motion.len()];
-        for (dof, fv) in f.iter().enumerate() {
-            if self.fixed.is_empty_row(dof) {
-                continue;
-            }
-            let (cols, vals) = k.row(dof);
-            let ku: f64 = cols.iter().zip(vals).map(|(j, v)| v * u.get(*j).copied().unwrap_or(0.0)).sum();
-            let r = ku - fv;
-            for (p, c) in self.fixed.row(dof) {
-                if let Some(x) = out.get_mut(p) {
-                    *x += c * r;
+        let residual: Vec<f64> = f
+            .iter()
+            .enumerate()
+            .map(|(dof, fv)| {
+                if self.fixed.is_empty_row(dof) && !self.supports.get(dof).copied().unwrap_or(false) {
+                    return 0.0;
                 }
-            }
-        }
-        out
+                let (cols, vals) = k.row(dof);
+                let ku: f64 = cols.iter().zip(vals).map(|(j, v)| v * u.get(*j).copied().unwrap_or(0.0)).sum();
+                ku - fv
+            })
+            .collect();
+        self.reactions_of(&residual)
     }
 }
 

crates/solver/tests/inclined_symmetry.rs

@@ -0,0 +1,289 @@
+//! Symmetry about an inclined plane (#242): n · u = 0 on the plane, and zero rotation about the
+//! two axes in it at shell nodes, eliminated as linear constraints.
+//!
+//! The checks need no reference program: a finite element model turned by a rotation R solves
+//! to R times the displacements of the unturned one (the elements are frame-indifferent), so an
+//! inclined plane is checked against the axis-aligned plane it is the image of; and half of a
+//! part whose mesh is mirror-symmetric about the plane, with the condition and half the load,
+//! solves exactly as the whole does at the nodes they share.
+
+mod common;
+
+use std::collections::BTreeMap;
+
+use common::{full, grid, iso, near, reduced};
+use fieldcraft_model::{
+    BoundaryCondition, Elastic, ElementFace, ElementKind, Formulation, Load, Material, Model, Procedure, Section, SectionKind, Set, Step, Symmetry,
+    Target,
+};
+use fieldcraft_solver::{FrequencyOptions, Limits, StaticResult, solve_frequency, solve_static};
+
+/// A rotation by `angle` about the unit axis `a` (Rodrigues).
+fn rotation(a: [f64; 3], angle: f64) -> [[f64; 3]; 3] {
+    let l = (a[0] * a[0] + a[1] * a[1] + a[2] * a[2]).sqrt();
+    let a = a.map(|c| c / l);
+    let (s, c) = angle.sin_cos();
+    let mut r = [[0.0; 3]; 3];
+    for i in 0..3 {
+        for j in 0..3 {
+            let k = [[0.0, -a[2], a[1]], [a[2], 0.0, -a[0]], [-a[1], a[0], 0.0]][i][j];
+            r[i][j] = if i == j { c } else { 0.0 } + (1.0 - c) * a[i] * a[j] + s * k;
+        }
+    }
+    r
+}
+
+fn apply(r: &[[f64; 3]; 3], v: [f64; 3]) -> [f64; 3] {
+    [0, 1, 2].map(|i| (0..3).map(|j| r[i][j] * v[j]).sum())
+}
+
+fn norm(v: [f64; 3]) -> f64 {
+    v.iter().map(|c| c * c).sum::<f64>().sqrt()
+}
+
+fn key(x: [f64; 3]) -> [i64; 3] {
+    x.map(|c| (c * 1e6).round() as i64)
+}
+
+/// The displacements of a result by node position (rounded).
+fn by_position(model: &Model, r: &StaticResult) -> BTreeMap<[i64; 3], ([f64; 3], f64)> {
+    r.nodes.iter().enumerate().map(|(i, l)| (key(model.nodes()[l].xyz), (r.displacement[i], r.von_mises[i]))).collect()
+}
+
+/// A cantilever L × 2b × h along x (y from −b to b when `whole`, 0 to b otherwise) clamped at
+/// x = 0, under a downward traction on its end; the half has the symmetry condition on y = 0.
+/// Everything is turned by `turn` (nodes, the traction and the plane's normal).
+fn cantilever(kind: ElementKind, formulation: Formulation, whole: bool, turn: &[[f64; 3]; 3]) -> (Model, StaticResult) {
+    let (l, b, h) = (10.0, 1.0, 1.0);
+    let n = if whole { [8, 4, 2] } else { [8, 2, 2] };
+    let mut g = grid(n, kind, |p| apply(turn, [l * p[0], if whole { b * (2.0 * p[1] - 1.0) } else { b * p[1] }, h * p[2]]));
+    g.define(iso(210e3, 0.3), None, formulation, Procedure::Static {});
+    g.fixed("root", |p| near(p[0], 0.0));
+    g.face_set("tip", |p| near(p[0], 1.0));
+    g.load("push", Load::Traction { set: "tip".into(), vector: apply(turn, [0.0, 0.0, -10.0]) });
+    if !whole {
+        let normal = apply(turn, [0.0, 1.0, 0.0]);
+        g.symmetry("mid", Symmetry::from_normal(normal).unwrap(), |p| near(p[1], 0.0));
+    }
+    let r = solve_static(&g.model, None, &Limits::default()).unwrap();
+    (g.model, r)
+}
+
+/// The turned half cantilever with its inclined plane solves to the turned displacements of the
+/// unturned half (and, for the mirror-symmetric hexahedral meshes, of the whole), at every node;
+/// the von Mises stress is the same; the plane's reactions balance the load.
+#[test]
+fn a_turned_half_solid_matches_the_axis_aligned_half_and_the_whole() {
+    let turn = rotation([1.0, 2.0, 3.0], 0.7);
+    let identity = rotation([0.0, 0.0, 1.0], 0.0);
+    for (kind, formulation, mirrored) in [
+        (ElementKind::Hex8, full(), true),
+        (ElementKind::Hex20, full(), true),
+        (ElementKind::Hex20, reduced(), true),
+        (ElementKind::Tet10, full(), false),
+        (ElementKind::Wedge6, full(), false),
+    ] {
+        let (m_axis, axis) = cantilever(kind, formulation, false, &identity);
+        let (m_turned, turned) = cantilever(kind, formulation, false, &turn);
+        let size = axis.displacement.iter().fold(0.0f64, |m, u| m.max(norm(*u)));
+        let stress = axis.von_mises.iter().fold(0.0f64, |m, s| m.max(*s));
+        let reference = by_position(&m_axis, &axis);
+        let mut worst = (0.0f64, 0.0f64);
+        let unturn = |x: [f64; 3]| [0, 1, 2].map(|i| (0..3).map(|j| turn[j][i] * x[j]).sum::<f64>());
+        for (i, l) in turned.nodes.iter().enumerate() {
+            let (u, s) = reference[&key(unturn(m_turned.nodes()[l].xyz))];
+            let want = apply(&turn, u);
+            let got = turned.displacement[i];
+            worst.0 = worst.0.max(norm([0, 1, 2].map(|k| got[k] - want[k])) / size);
+            worst.1 = worst.1.max((turned.von_mises[i] - s).abs() / stress);
+        }
+        eprintln!(
+            "{kind:?} {formulation:?}: tip deflection {size:.6e}, turned vs axis-aligned: displacement {:.1e}, von Mises {:.1e}",
+            worst.0, worst.1
+        );
+        assert!(worst.0 < 1e-9 && worst.1 < 1e-8, "{kind:?}: {worst:?}");
+        for r in [&axis, &turned] {
+            let imbalance = (0..3).map(|c| (r.applied_total[c] + r.reaction_total[c]).abs()).fold(0.0, f64::max);
+            assert!(imbalance < 1e-8 * norm(r.applied_total), "{kind:?}: {:?} vs {:?}", r.applied_total, r.reaction_total);
+            assert!(!r.warnings.iter().any(|w| w.contains("reactions")), "{:?}", r.warnings);
+        }
+        if mirrored {
+            let (m_whole, whole) = cantilever(kind, formulation, true, &identity);
+            let all = by_position(&m_whole, &whole);
+            let mut worst = 0.0f64;
+            for (x, (u, _)) in &reference {
+                let w = all[x].0;
+                worst = worst.max(norm([0, 1, 2].map(|k| u[k] - w[k])) / size);
+            }
+            eprintln!("{kind:?}: half with the plane vs whole: displacement {worst:.1e}");
+            assert!(worst < 1e-9, "{kind:?}: {worst}");
+        }
+    }
+}
+
+/// A shell model: one material and step; `n` × `n` cells of the quarter of a simply supported
+/// square plate (side 1), two triangles per cell split along x = y, so the mesh is mirror-
+/// symmetric about the diagonal plane x = y; `eighth` keeps the cells below the diagonal and
+/// puts the inclined symmetry plane (normal (1, −1, 0)) on it.
+fn plate(n: usize, eighth: bool, procedure: Procedure) -> (Model, BTreeMap<u32, [f64; 3]>) {
+    let (a, t) = (1.0, 0.01);
+    let mut m = Model::default();
+    m.add_material("Mat", Material { elastic: Some(Elastic::Isotropic { young: 1e7, poisson: 0.3 }), density: Some(1.0), ..Material::default() })
+        .unwrap();
+    m.add_step(Step::new("Step", procedure), None).unwrap();
+    let label = |i: usize, j: usize| (1 + i + (n + 1) * j) as u32;
+    let mut xyz = BTreeMap::new();
+    for j in 0..=n {
+        for i in 0..=n {
+            if eighth && j > i {
+                continue;
+            }
+            let p = [i as f64 / n as f64 * a / 2.0, j as f64 / n as f64 * a / 2.0, 0.0];
+            m.set_node(label(i, j), p).unwrap();
+            xyz.insert(label(i, j), p);
+        }
+    }
+    let mut elements = Vec::new();
+    for j in 0..n {
+        for i in 0..n {
+            let (p, q, r, s) = (label(i, j), label(i + 1, j), label(i + 1, j + 1), label(i, j + 1));
+            for (tri, below) in [([p, q, r], true), ([p, r, s], false)] {
+                if eighth && (j > i || (i == j && !below)) {
+                    continue;
+                }
+                let e = elements.len() as u32 + 1;
+                m.set_element(e, ElementKind::Tri3, tri.to_vec()).unwrap();
+                elements.push(e);
+            }
+        }
+    }
+    let faces = elements.iter().map(|e| ElementFace { element: *e, face: 1 }).collect();
+    m.add_set("plate", Set { elements, ..Set::default() }).unwrap();
+    m.add_section(
+        "plate",
+        Section {
+            set: "plate".into(),
+            material: "Mat".into(),
+            kind: SectionKind::Shell { thickness: t, offset: 0.0, points: 5 },
+            formulation: Formulation::default(),
+        },
+    )
+    .unwrap();
+    let on = |x: f64, c: f64| (x - c).abs() < 1e-9;
+    let free = None;
+    let hold = |m: &mut Model, name: &str, bc: BoundaryCondition, f: &dyn Fn([f64; 3]) -> bool| {
+        let nodes = xyz.iter().filter(|(_, x)| f(**x)).map(|(l, _)| *l).collect();
+        m.add_set(name, Set { nodes, ..Set::default() }).unwrap();
+        m.add_boundary_condition(name, bc).unwrap();
+        m.set_boundary_condition_active("Step", name, true).unwrap();
+    };
+    let held = |values| move |name: &str| BoundaryCondition::Displacement { target: Target::Set(name.into()), values };
+    // Hard simple support on x = 0 and y = 0; symmetry on x = a/2 and y = a/2.
+    let x0 = held([free, free, Some(0.0), Some(0.0), free, free]);
+    let y0 = held([free, free, Some(0.0), free, Some(0.0), free]);
+    let sym = |name: &str, plane| BoundaryCondition::Symmetry { target: Target::Set(name.into()), plane };
+    if !eighth {
+        hold(&mut m, "x0", x0("x0"), &|p| on(p[0], 0.0));
+        hold(&mut m, "ys", sym("ys", Symmetry::Y), &|p| on(p[1], a / 2.0));
+    }
+    hold(&mut m, "y0", y0("y0"), &|p| on(p[1], 0.0));
+    hold(&mut m, "xs", sym("xs", Symmetry::X), &|p| on(p[0], a / 2.0));
+    if eighth {
+        hold(&mut m, "diagonal", sym("diagonal", Symmetry::from_normal([1.0, -1.0, 0.0]).unwrap()), &|p| on(p[0], p[1]));
+    }
+    if matches!(m.steps()[0].procedure, Procedure::Static {}) {
+        m.add_set("top", Set { faces, ..Set::default() }).unwrap();
+        m.add_load("q", Load::Pressure { set: "top".into(), magnitude: 1.0 }).unwrap();
+        m.set_load_active("Step", "q", true).unwrap();
+    }
+    (m, xyz)
+}
+
+/// An eighth of the simply supported plate, cut on the diagonal with the inclined plane, solves
+/// as the quarter (deflections and rotations at every shared node, to round-off; the first
+/// frequency, whose mode is symmetric about the diagonal), and both match Navier's series.
+#[test]
+fn an_eighth_plate_on_the_diagonal_plane_matches_the_quarter() {
+    let n = 16;
+    let (mq, quarter) = plate(n, false, Procedure::Static {});
+    let (me, eighth) = plate(n, true, Procedure::Static {});
+    let rq = solve_static(&mq, None, &Limits::default()).unwrap();
+    let re = solve_static(&me, None, &Limits::default()).unwrap();
+    let at = |r: &StaticResult, l: u32| r.nodes.binary_search(&l).unwrap();
+    let size = rq.displacement.iter().fold(0.0f64, |m, u| m.max(norm(*u)));
+    let turn = rq.rotation.iter().fold(0.0f64, |m, u| m.max(norm(*u)));
+    let (mut du, mut dr) = (0.0f64, 0.0f64);
+    for l in eighth.keys() {
+        assert!(quarter.contains_key(l));
+        let (a, b) = (at(&rq, *l), at(&re, *l));
+        du = du.max(norm([0, 1, 2].map(|k| rq.displacement[a][k] - re.displacement[b][k])) / size);
+        dr = dr.max(norm([0, 1, 2].map(|k| rq.rotation[a][k] - re.rotation[b][k])) / turn);
+    }
+    // The diagonal turns only about its normal: θx + θy = 0 and θz = 0 there.
+    for (l, x) in &eighth {
+        if (x[0] - x[1]).abs() < 1e-9 {
+            let r = re.rotation[at(&re, *l)];
+            assert!((r[0] + r[1]).abs() < 1e-12 * turn && r[2].abs() < 1e-12 * turn, "node {l}: {r:?}");
+        }
+    }
+    let centre = *eighth.iter().find(|(_, x)| (x[0] - 0.5).abs() < 1e-9 && (x[1] - 0.5).abs() < 1e-9).unwrap().0;
+    let w = -re.displacement[at(&re, centre)][2];
+    let d = 1e7 * 0.01f64.powi(3) / (12.0 * (1.0 - 0.09));
+    let navier = navier(1.0, 1.0, d);
+    eprintln!("eighth vs quarter plate: displacement {du:.1e}, rotation {dr:.1e}; centre deflection {w:.6e} vs Navier {navier:.6e}");
+    assert!(du < 1e-9 && dr < 1e-9, "{du} {dr}");
+    assert!((w - navier).abs() < 0.01 * navier, "{w} vs {navier}");
+    let imbalance = (0..3).map(|c| (re.applied_total[c] + re.reaction_total[c]).abs()).fold(0.0, f64::max);
+    assert!(imbalance < 1e-9 * norm(re.applied_total), "{:?} vs {:?}", re.applied_total, re.reaction_total);
+    assert!(!re.warnings.iter().any(|w| w.contains("reactions")), "{:?}", re.warnings);
+    // An eighth carries an eighth of the plate's load (the quarter a quarter).
+    assert!((2.0 * re.applied_total[2] - rq.applied_total[2]).abs() < 1e-12 * rq.applied_total[2].abs());
+
+    let frequency = |eighth| {
+        let (m, _) = plate(n, eighth, Procedure::Frequency { modes: 1, min_hz: None, max_hz: None });
+        solve_frequency(&m, None, &FrequencyOptions::default(), &Limits::default()).unwrap().modes[0].frequency
+    };
+    let (fq, fe) = (frequency(false), frequency(true));
+    eprintln!("first frequency: quarter {fq:.9}, eighth {fe:.9}");
+    assert!((fq - fe).abs() < 1e-8 * fq, "{fq} vs {fe}");
+}
+
+/// Navier's series for the centre deflection of a simply supported square plate under uniform
+/// pressure (S. Timoshenko, S. Woinowsky-Krieger, *Theory of Plates and Shells*, 2nd ed.,
+/// McGraw-Hill 1959, §30).
+fn navier(q: f64, a: f64, d: f64) -> f64 {
+    let mut s = 0.0;
+    for m in (1..400).step_by(2) {
+        for n in (1..400).step_by(2) {
+            let (mf, nf) = (f64::from(m), f64::from(n));
+            let sign = if ((m + n) / 2 - 1) % 2 == 0 { 1.0 } else { -1.0 };
+            s += sign / (mf * nf * (mf * mf + nf * nf).powi(2));
+        }
+    }
+    16.0 * q * a.powi(4) / (std::f64::consts::PI.powi(6) * d) * s
+}
+
+/// A node that is also held, or listed in two planes, is no conflict; a prescribed motion off
+/// the plane is.
+#[test]
+fn conflicts() {
+    // The clamp's nodes on y = 0 are also on the plane: held there, the plane's rows are redundant.
+    let (_, r) = cantilever(ElementKind::Hex8, full(), false, &rotation([0.0, 0.0, 1.0], 0.3));
+    assert!(!r.warnings.iter().any(|w| w.contains("reactions")), "{:?}", r.warnings);
+    let mut g = grid([4, 1, 1], ElementKind::Hex8, |p| [4.0 * p[0], p[1], p[2]]);
+    g.define(iso(1e3, 0.3), None, full(), Procedure::Static {});
+    g.fixed("root", |p| near(p[0], 0.0));
+    g.symmetry("a", Symmetry::Normal([1.0, 1.0, 0.0]), |p| near(p[1], 0.0) && near(p[0], 1.0));
+    g.symmetry("b", Symmetry::Normal([2.0, 2.0, 0.0]), |p| near(p[1], 0.0) && near(p[0], 1.0));
+    g.face_set("tip", |p| near(p[0], 1.0));
+    g.load("push", Load::Traction { set: "tip".into(), vector: [0.0, 0.0, -1.0] });
+    let r = solve_static(&g.model, None, &Limits::default()).unwrap();
+    let u = r.displacement[r.nodes.binary_search(&g.nodes_where(|p| near(p[1], 0.0) && near(p[0], 1.0))[0]).unwrap()];
+    assert!((u[0] + u[1]).abs() < 1e-12 * norm(u), "{u:?}");
+    // A prescribed motion the plane cannot follow is a contradiction, reported by name.
+    let nodes = g.nodes_where(|p| near(p[1], 0.0) && near(p[0], 1.0));
+    g.model.add_set("pulled", Set { nodes, ..Set::default() }).unwrap();
+    g.bc("pull", BoundaryCondition::Displacement { target: Target::Set("pulled".into()), values: [Some(0.1), Some(0.1), None, None, None, None] });
+    let e = solve_static(&g.model, None, &Limits::default()).unwrap_err().to_string();
+    assert!(e.contains("boundary condition `") && e.contains("contradicts"), "{e}");
+}