mesh.cells: tet cells meet hex cells through pyramids

With "transition": "pyramid" (the default; "tie" keeps the old join) a tet
cell is meshed after the hex cells, with a pyramid on every hex face of its
cuts and its tetrahedra on the pyramids' sides: one conforming mesh, no tie.
Where it cannot be built the cell says why (no_pyramids) and is tied. The
apex height is 0.4 of the quad's edge, so pyramids on faces meeting at a
right angle do not cross.

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

crates/engine/src/catalogue.rs

@@ -359,7 +359,7 @@ pub static CATALOGUE: &[Capability] = &[
         "Partitioned Mesh",
         Mesh,
         "Structured Hex",
-        "One mesh of a partitioned part, a method per cell (hex, tet, or all hex as conforming blocks); cells joined across partitions",
+        "One mesh of a partitioned part, a method per cell (hex, tet, or all hex as conforming blocks); cells joined across partitions, tet cells to hex cells through conforming pyramids",
         &["mesh.hex.blocks"],
     ),
     ready(
@@ -393,7 +393,6 @@ pub static CATALOGUE: &[Capability] = &[
         "M5",
         "Find sweepable and mappable regions of 3D parts (extruded cross-sections are already cut into blocks by Hex Mesh)",
     ),
-    soon("mesh.transition", "Transition", Mesh, "Structured Hex", "M5", "Conforming tet–hex interfaces with pyramids"),
     ready(
         "mesh.hex.octree",
         "Octree Hex",

crates/engine/src/cmd/cells.rs

@@ -35,14 +35,14 @@ use fieldcraft_model::{ElementFace, GeometryRef, Model, Set, Tie};
 use serde_json::{Value, json};
 
 use super::hex::{Input, add_cells, add_hex, hex_body};
-use super::mesh::{BodyTets, add_tets, mesh_body, quality_options};
+use super::mesh::{add_tets, mesh_body, quality_options};
 use crate::requirement::Requirement;
 use crate::{CommandKind, CommandSpec, EngineError, Result, Session};
 
 pub const CELLS: CommandSpec = CommandSpec {
     id: "mesh.cells",
     label: "Partitioned Mesh",
-    params: r#"{"methods": {"Body1/a": "tet", "Body1/b": "swept"} (per cell; default `default`), "default": "auto", "bodies": ["Body1/a", "Body1/b"] (default: every body of the open design), "size": 2.0 (element size, mm; default 1/25 of the bounding-box diagonal of all of them), "order": 1 | 2 (hex8 and tet4, or hex20 and tet10), "tolerance": 0.05 (chordal tolerance, mm)} — replaces the cells' meshes (as mesh.tet.generate; other bodies keep theirs) with one mesh of the cells, each with its method: "auto" (structured hex where a method fits, else tet), "hex" (any structured hex method), "swept", "extrude", "sweep", "revolve", "mapped" or "tet"; or "default": "blocks" (with no `methods`): every cell a mapped block, division counts assigned over all cells together, exactly mesh.hex.blocks with the same parameters. Cells on either side of a partition (faces `partition:<name>:a` and `:b`) are joined: where their element faces match node for node the nodes are merged (conforming); otherwise, as between a hex and a tet cell, they are tied: each cell's tied cut faces make one tie, `tie_<cell>` (`/` as `_`; sets `<tie>_main`, the facing cells' cut faces, and `<tie>_secondary`, the cell's own), hex cells main to tet cells and otherwise by name, so a node is a secondary node of one such tie only; the ties an earlier run made on these cells are replaced. The cells of one design body that may sweep (auto, hex, swept, extrude, revolve) are swept together first: one cross-section and one set of layers for all of them, so they share their nodes across every cut, T-junctions included, with no edge seeds (`swept_together` names the body; `not_swept_together` says why not, and those cells are meshed one by one). Reports each cell's method (and why auto fell back to tet), its `quality` (volume, smallest and mean scaled Jacobian, largest edge ratio), and each interface: `merged` with the nodes merged or shared, or `tied` (with `why` when both cells are of one design body: the tet cell and why auto fell back, or why the body's cells were not swept together)"#,
+    params: r#"{"methods": {"Body1/a": "tet", "Body1/b": "swept"} (per cell; default `default`), "default": "auto", "bodies": ["Body1/a", "Body1/b"] (default: every body of the open design), "size": 2.0 (element size, mm; default 1/25 of the bounding-box diagonal of all of them), "order": 1 | 2 (hex8 and tet4, or hex20 and tet10), "tolerance": 0.05 (chordal tolerance, mm), "transition": "pyramid" | "tie" (how a tet cell meets hex cells; default pyramid)} — replaces the cells' meshes (as mesh.tet.generate; other bodies keep theirs) with one mesh of the cells, each with its method: "auto" (structured hex where a method fits, else tet), "hex" (any structured hex method), "swept", "extrude", "sweep", "revolve", "mapped" or "tet"; or "default": "blocks" (with no `methods`): every cell a mapped block, division counts assigned over all cells together, exactly mesh.hex.blocks with the same parameters. Cells on either side of a partition (faces `partition:<name>:a` and `:b`) are joined: where their element faces match node for node the nodes are merged (conforming); between a tet cell and the hex cells across its partitions, with "transition": "pyramid" (linear meshes), the tet cell is meshed after them with a pyramid (pyramid5) on every hex element face of the cut, its tetrahedra sharing the pyramids' sides, so the mesh is conforming there (`joined`: `pyramids`, with their count); where that cannot be built (a cut the hex faces do not cover exactly, quadratic elements, a pyramid that would reach through a thin cell) the cell says why in `no_pyramids` and is tied; otherwise, as between a hex and a tet cell with "transition": "tie", they are tied: each cell's tied cut faces make one tie, `tie_<cell>` (`/` as `_`; sets `<tie>_main`, the facing cells' cut faces, and `<tie>_secondary`, the cell's own), hex cells main to tet cells and otherwise by name, so a node is a secondary node of one such tie only; the ties an earlier run made on these cells are replaced. The cells of one design body that may sweep (auto, hex, swept, extrude, revolve) are swept together first: one cross-section and one set of layers for all of them, so they share their nodes across every cut, T-junctions included, with no edge seeds (`swept_together` names the body; `not_swept_together` says why not, and those cells are meshed one by one). Reports each cell's method (and why auto fell back to tet), its `quality` (volume, smallest and mean scaled Jacobian, largest edge ratio), and each interface: `merged` with the nodes merged or shared, or `tied` (with `why` when both cells are of one design body: the tet cell and why auto fell back, or why the body's cells were not swept together)"#,
     form: Some(super::forms::CELLS),
     kind: CommandKind::Model,
     requires: &[Requirement::Design],
@@ -80,10 +80,22 @@ impl Method {
     }
 }
 
-/// A meshed cell.
+mod transition;
+
+/// A meshed cell; tet cells are meshed after the hex cells are in the model, so they can meet
+/// them through pyramids.
 enum Meshed<'a> {
-    Hex(Input<'a>, HexMesh),
-    Tet(BodyTets),
+    Hex(Box<(Input<'a>, HexMesh)>),
+    Tet,
+}
+
+/// How a tet cell joins the hex cells across its partitions.
+#[derive(Clone, Copy, PartialEq, Eq)]
+enum Transition {
+    /// Pyramids on the hex faces, conforming (falls back to a tie with the reason).
+    Pyramid,
+    /// A tie (dual mortar), each mesh as its method made it.
+    Tie,
 }
 
 fn cells(s: &mut Session, p: &Value) -> Result<Value> {
@@ -105,6 +117,11 @@ fn cells(s: &mut Session, p: &Value) -> Result<Value> {
             _ => return Err(EngineError::invalid("order", "must be 1 or 2")),
         },
     };
+    let transition = match p.get("transition").map(|v| v.as_str()) {
+        None | Some(Some("pyramid")) => Transition::Pyramid,
+        Some(Some("tie")) => Transition::Tie,
+        Some(_) => return Err(EngineError::invalid("transition", "must be \"pyramid\" or \"tie\"")),
+    };
     let default = p.get("default").map(Method::parse).transpose()?.unwrap_or(Method::Auto);
     let methods: BTreeMap<String, Method> = match p.get("methods") {
         None => BTreeMap::new(),
@@ -215,20 +232,19 @@ fn cells(s: &mut Session, p: &Value) -> Result<Value> {
             let mesh = hex_body(&hex_params, &b, m, quadratic, size)?;
             Ok((b, mesh))
         };
-        let tet = |s: &Session| mesh_body(s, &tet_params, input()?, &cdt, &tet_options, quality, true);
         let (cell, mut info) = match method {
             Method::Hex(m) => {
                 let (b, mesh) = hex(m)?;
                 let info = json!({"method": mesh.report.method.name()});
-                (Meshed::Hex(b, mesh), info)
+                (Meshed::Hex(Box::new((b, mesh))), info)
             }
-            Method::Tet => (Meshed::Tet(tet(s)?), json!({"method": "tet"})),
+            Method::Tet => (Meshed::Tet, json!({"method": "tet"})),
             Method::Auto => match hex(HexMethod::Auto) {
                 Ok((b, mesh)) => {
                     let info = json!({"method": mesh.report.method.name()});
-                    (Meshed::Hex(b, mesh), info)
+                    (Meshed::Hex(Box::new((b, mesh))), info)
                 }
-                Err(e) => (Meshed::Tet(tet(s)?), json!({"method": "tet", "why": e.to_string()})),
+                Err(e) => (Meshed::Tet, json!({"method": "tet", "why": e.to_string()})),
             },
         };
         if let Some(o) = info.as_object_mut() {
@@ -259,19 +275,50 @@ fn cells(s: &mut Session, p: &Value) -> Result<Value> {
         }
         interfaces.extend(swept_interfaces(&design, inputs, cm));
     }
-    let mut k = 0;
-    for (i, _) in shapes.iter().enumerate() {
-        if joint.get(i).copied().flatten().is_some() {
-            continue;
-        }
-        let Some((cell, mut info)) = meshed.get_mut(k).map(|(c, v)| (c, std::mem::take(v))) else { break };
-        k += 1;
-        let (kind, elements, nodes) = match &*cell {
-            Meshed::Hex(b, m) => {
+    // The cells meshed alone, in the order of `shapes`: the hex ones first, then the tet ones,
+    // which may meet the hex cells through pyramids.
+    let alone: Vec<usize> = (0..shapes.len()).filter(|&i| joint.get(i).copied().flatten().is_none()).collect();
+    let mut hex_cells: Vec<String> = swept.iter().flat_map(|(_, _, inputs, _)| inputs.iter().map(|b| b.name.clone())).collect();
+    let mut tet_cells = Vec::new();
+    for (&i, (cell, info)) in alone.iter().zip(meshed.iter_mut()) {
+        let mut info = std::mem::take(info);
+        match &*cell {
+            Meshed::Hex(hex) => {
+                let (b, m) = &**hex;
                 add_hex(model, b, m)?;
-                (m.kind, m.element_count(), m.nodes.len())
+                hex_cells.push(b.name.clone());
+                if let Some(o) = info.as_object_mut() {
+                    o.insert("kind".into(), json!(m.kind.name()));
+                    o.insert("elements".into(), json!(m.element_count()));
+                    o.insert("nodes".into(), json!(m.nodes.len()));
+                }
+                report.push((i, info));
             }
-            Meshed::Tet(t) => {
+            Meshed::Tet => tet_cells.push((i, info)),
+        }
+    }
+    for (i, mut info) in tet_cells {
+        let Some(shape) = shapes.get(i) else { continue };
+        let input = Input { seeds: seeds.clone(), ..super::hex::shape_input(shape, Some(tolerance))? };
+        let pyramids = if transition == Transition::Tie {
+            Err("`transition` is \"tie\"".to_string())
+        } else if quadratic {
+            Err("pyramid transitions are linear (pyramid5); order 2 is tied".to_string())
+        } else {
+            transition::mesh_with_pyramids(&design, model, &tet_params, &input, &hex_cells, &cdt, &tet_options)?
+        };
+        let (kind, elements, nodes) = match pyramids {
+            Ok(c) => {
+                interfaces.extend(c.interfaces.iter().cloned());
+                let kind = c.tets.mesh.kind;
+                let (elements, nodes) = transition::add(model, c)?;
+                (kind, elements, nodes)
+            }
+            Err(why) => {
+                if let Some(o) = info.as_object_mut() {
+                    o.insert("no_pyramids".into(), json!(why));
+                }
+                let t = mesh_body(Some(&design), &tet_params, input, &cdt, &tet_options, quality, true)?;
                 add_tets(model, &t.name, &t.face_names, &t.mesh)?;
                 (t.mesh.kind, t.mesh.elements.len(), t.mesh.points.len())
             }
@@ -660,7 +707,7 @@ mod tests {
     fn hex_and_tet_cells_are_tied_and_matching_hex_cells_merged() {
         let (_dir, mut s) = cut_plate();
         // A tet cell next to a hex cell: tied, the hex cell main.
-        let r = s.execute("mesh.cells", &json!({"methods": {"Body1/a": "tet", "Body1/b": "mapped"}, "size": 2.5})).unwrap();
+        let r = s.execute("mesh.cells", &json!({"methods": {"Body1/a": "tet", "Body1/b": "mapped"}, "size": 2.5, "transition": "tie"})).unwrap();
         assert_eq!(r["cells"][0]["kind"], "tet4", "{r}");
         assert_eq!(r["cells"][1]["kind"], "hex8", "{r}");
         let i = &r["interfaces"][0];
@@ -679,7 +726,7 @@ mod tests {
         let separate = r["cells"].as_array().unwrap().iter().map(|c| c["nodes"].as_u64().unwrap()).sum::<u64>();
         assert_eq!(r["nodes"].as_u64().unwrap() + i["nodes"].as_u64().unwrap(), separate, "{r}");
         // Rerunning replaces the tie it made rather than adding another.
-        s.execute("mesh.cells", &json!({"methods": {"Body1/a": "tet"}, "size": 2.5})).unwrap();
+        s.execute("mesh.cells", &json!({"methods": {"Body1/a": "tet"}, "size": 2.5, "transition": "tie"})).unwrap();
         assert_eq!(s.model.ties().len(), 1);
     }
 
@@ -762,11 +809,10 @@ mod tests {
     }
 
     /// A 40 × 10 × 10 bar in uniaxial tension, cut in the middle: one half hex, the other tet,
-    /// tied. With ν = 0, symmetry conditions on three faces and the end load spread by consistent
-    /// weights, the stress is uniform, which both elements and the dual mortar tie carry exactly:
-    /// the end moves F L / (E A).
-    #[test]
-    fn a_tied_hex_tet_bar_in_tension_stretches_as_the_formula_says() {
+    /// joined by `transition` (`joined` in the reply). With ν = 0, symmetry conditions on three
+    /// faces and the end load spread by consistent weights, the stress is uniform, which the
+    /// elements, the pyramids and the dual mortar tie all carry exactly: the end moves F L / (E A).
+    fn hex_tet_bar_in_tension(transition: &str, joined: &str) -> Session {
         let commands = json!({"commands": [
             {"command": "solid.box", "params": {"length": 40, "width": 10, "height": 10, "body_name": "Bar"}},
         ]});
@@ -786,8 +832,8 @@ mod tests {
         point[axis] = mid;
         s.execute("geometry.datum", &json!({"name": "Mid", "by": {"type": "point_normal", "point": point, "normal": normal}})).unwrap();
         s.execute("geometry.partition", &json!({"name": "Cut", "datum": "Mid"})).unwrap();
-        let r = s.execute("mesh.cells", &json!({"methods": {"Bar/a": "mapped", "Bar/b": "tet"}, "size": 2.5})).unwrap();
-        assert_eq!(r["interfaces"][0]["joined"], "tied", "{r}");
+        let r = s.execute("mesh.cells", &json!({"methods": {"Bar/a": "mapped", "Bar/b": "tet"}, "size": 2.5, "transition": transition})).unwrap();
+        assert_eq!(r["interfaces"][0]["joined"], joined, "{r}");
 
         let m = &s.model;
         let on = |k: usize, v: f64| -> Vec<u32> { m.nodes().iter().filter(|(_, n)| (n.xyz[k] - v).abs() < 1e-9).map(|(l, _)| *l).collect() };
@@ -829,6 +875,27 @@ mod tests {
             let u = n["displacement"][axis].as_f64().unwrap();
             assert!((u - expected).abs() < 1e-6 * expected, "{u} vs {expected}: {n}");
         }
+        s
+    }
+
+    #[test]
+    fn a_tied_hex_tet_bar_in_tension_stretches_as_the_formula_says() {
+        let s = hex_tet_bar_in_tension("tie", "tied");
+        assert_eq!(s.model.ties().len(), 1);
+    }
+
+    /// Through pyramids the bar is one conforming mesh: no tie, no free faces inside, every
+    /// element valid.
+    #[test]
+    fn a_hex_tet_bar_joined_by_pyramids_is_conforming_and_stretches_as_the_formula_says() {
+        let mut s = hex_tet_bar_in_tension("pyramid", "pyramids");
+        assert!(s.model.ties().is_empty());
+        let kinds: BTreeSet<&str> = s.model.elements().values().map(|e| e.kind.name()).collect();
+        assert_eq!(kinds.into_iter().collect::<Vec<_>>(), ["hex8", "pyramid5", "tet4"]);
+        let d = s.execute("mesh.check.duplicates", &json!({})).unwrap();
+        assert_eq!(d["counts"], json!({"coincident": 0, "cracks": 0, "duplicate_elements": 0, "free_edges": 0}), "{d}");
+        let q = s.execute("mesh.quality", &json!({})).unwrap();
+        assert_eq!(q["invalid"]["count"], 0, "{q}");
     }
 
     /// A box cut by planes into cells, `methods` per cell, a heat source in every cell and

crates/engine/src/cmd/cells/transition.rs

@@ -0,0 +1,172 @@
+//! Pyramid transitions between a tet cell and the hex cells across its partitions (#439).
+//!
+//! The tet cell is meshed after the hex cells: its cut faces that face hex cells are replaced by
+//! a pyramid on every hex element face there ([`fieldcraft_mesh::transition`]), the tetrahedra
+//! fill the rest and keep the pyramids' sides, and the pyramids join the hex nodes to the tet
+//! nodes. The mesh is conforming across the cut, so no tie is needed there.
+
+use std::collections::{BTreeMap, BTreeSet};
+
+use fieldcraft_mesh::cdt::{CdtOptions, tetrahedralize};
+use fieldcraft_mesh::surface::BoundarySurface;
+use fieldcraft_mesh::tet::{QualityMesh, QualityOptions, quality_mesh};
+use fieldcraft_mesh::transition::{Transition, apex_points, pyramid_surface};
+use fieldcraft_model::{ElementKind, GeometryRef, Model};
+use serde_json::{Value, json};
+
+use super::super::hex::Input;
+use super::super::mesh::{BodyTets, add_tets, analytic_surfaces};
+use crate::{EngineError, Result};
+
+/// A tet cell meshed against the pyramids on its hex neighbours' faces.
+pub(super) struct Pyramids {
+    pub(super) tets: BodyTets,
+    transition: Transition,
+    /// Each pyramid's base: the hex nodes of its face, oriented into the tet cell.
+    bases: Vec<[u32; 4]>,
+    /// The interfaces it makes (reply entries).
+    pub(super) interfaces: Vec<Value>,
+}
+
+/// The base name of a cut face (`partition:P:a` of `partition:P:a#1`): its partition and side.
+fn cut_of(name: &str) -> Option<(&str, usize)> {
+    let base = name.split_once('#').map_or(name, |(b, _)| b);
+    let (partition, side) = base.strip_prefix("partition:")?.rsplit_once(':')?;
+    Some((partition, usize::from(side == "b")))
+}
+
+/// Mesh tet cell `input` with pyramids on the faces of the hex cells `hex` (already in `model`)
+/// across its cut faces. `Ok(Err(why))` when there is no such face or the transition cannot be
+/// built (the caller meshes the cell plainly and ties it); limits and a cancel are errors.
+pub(super) fn mesh_with_pyramids(
+    design: &fieldcraft_geometry::Design,
+    model: &Model,
+    p: &Value,
+    input: &Input,
+    hex: &[String],
+    cdt: &CdtOptions,
+    options: &QualityOptions,
+) -> Result<std::result::Result<Pyramids, String>> {
+    // The hex element faces across each of the cell's cut faces.
+    let sides = super::sides(model, hex);
+    let mut cut: Vec<u32> = Vec::new();
+    let mut quads: Vec<[u32; 4]> = Vec::new();
+    let mut interfaces = Vec::new();
+    let mut seen: BTreeSet<(String, String)> = BTreeSet::new();
+    for (i, name) in input.face_names.iter().enumerate() {
+        let Some((partition, side)) = cut_of(name) else { continue };
+        let Some(across) = sides.get(partition).and_then(|s| s.get(1 - side)) else { continue };
+        if across.is_empty() {
+            continue;
+        }
+        cut.push(i as u32);
+        for other in across {
+            if !seen.insert((partition.to_string(), other.body.clone())) {
+                continue;
+            }
+            for f in &other.faces {
+                let nodes = model.face_nodes(*f).unwrap_or_default();
+                let Ok(q) = <[u32; 4]>::try_from(nodes.as_slice()) else {
+                    return Ok(Err(format!("`{}` has quadratic or non-quadrilateral faces on the cut; pyramid transitions are linear", other.body)));
+                };
+                quads.push(q);
+            }
+            interfaces.push((partition.to_string(), other.body.clone(), other.faces.len()));
+        }
+    }
+    if quads.is_empty() {
+        return Ok(Err("no hex cell faces it across a partition".into()));
+    }
+    // The hex faces must cover the cut faces exactly (a cut shared with a tet cell, or a hex
+    // cell larger than this one, is tied instead).
+    let xyz = |n: u32| model.node(n).map(|x| x.xyz).unwrap_or([f64::NAN; 3]);
+    let positions: Vec<[[f64; 3]; 4]> = quads.iter().map(|q| q.map(xyz)).collect();
+    let quad_area: f64 = positions.iter().map(|q| 0.5 * norm(cross(sub(q[2], q[0]), sub(q[3], q[1])))).sum();
+    let m = &input.mesh;
+    let cut_area: f64 = m
+        .triangles
+        .iter()
+        .zip(&m.tri_face)
+        .filter(|(_, f)| cut.contains(f))
+        .map(|(t, _)| {
+            let c = t.map(|v| m.positions.get(v as usize).copied().unwrap_or([f64::NAN; 3]));
+            0.5 * norm(cross(sub(c[1], c[0]), sub(c[2], c[0])))
+        })
+        .sum();
+    if !((quad_area - cut_area).abs() <= 1e-3 * cut_area) {
+        return Ok(Err(format!("the hex cells' faces ({quad_area:.6} mm²) do not cover its cut faces ({cut_area:.6} mm²)")));
+    }
+    let tag = u32::try_from(input.face_names.len()).map_err(|_| EngineError::Params("too many faces".into()))?;
+    let welding = super::super::mesh::surface_options(p, m)?;
+    let built = || -> fieldcraft_mesh::Result<(QualityMesh, Transition, usize, usize, f64)> {
+        let surface = BoundarySurface::new(&m.positions, &m.triangles, &m.tri_face, &welding)?;
+        let t = pyramid_surface(&surface, &cut, &positions, tag)?;
+        // Already welded: nothing more merges.
+        let exact = fieldcraft_mesh::surface::SurfaceOptions { weld_tolerance: 0.0, weld_distance: 0.0, ..welding };
+        let region = BoundarySurface::new(&t.positions, &t.triangles, &t.tri_face, &exact)?;
+        let tets = tetrahedralize(&region, cdt)?;
+        let mut analytic = design.body(&input.name).map(analytic_surfaces).unwrap_or_default();
+        analytic.resize(tag as usize, None);
+        analytic.push(None);
+        let o = QualityOptions { seeds: input.seeds.clone(), frozen_faces: vec![tag], ..options.clone() };
+        let mesh = quality_mesh(&region, &tets, analytic, &o)?;
+        Ok((mesh, t, tets.steiner.len(), tets.stats.cavities, region.volume()))
+    };
+    let (mesh, transition, steiner, cavities, volume) = match built() {
+        Ok(x) => x,
+        Err(fieldcraft_mesh::Error::Model(fieldcraft_model::Error::Cancelled)) => return Err(EngineError::Model(fieldcraft_model::Error::Cancelled)),
+        Err(e @ fieldcraft_mesh::Error::Limit(_)) => return Err(EngineError::Params(e.to_string())),
+        Err(e) => return Ok(Err(e.to_string())),
+    };
+    if let Err(e) = apex_points(&mesh, &transition) {
+        return Ok(Err(e.to_string()));
+    }
+    let tets = BodyTets::new(input.name.clone(), input.face_names.clone(), mesh, steiner, cavities, volume);
+    let interfaces = interfaces
+        .into_iter()
+        .map(|(partition, other, faces)| json!({"partition": partition, "cells": [other, input.name], "joined": "pyramids", "pyramids": faces}))
+        .collect();
+    Ok(Ok(Pyramids { tets, transition, bases: quads, interfaces }))
+}
+
+/// Add the cell's tetrahedra and pyramids to `model`: the tetrahedra's nodes on the hex faces
+/// are the hex nodes, the pyramids join them to the apexes.
+pub(super) fn add(model: &mut Model, mut c: Pyramids) -> Result<(usize, usize)> {
+    let tag = c.tets.face_names.len() as u32;
+    let apexes = apex_points(&c.tets.mesh, &c.transition).map_err(|e| EngineError::Params(e.to_string()))?;
+    // The pyramid sides are inside the part: no geometry face.
+    c.tets.mesh.boundary.retain(|b| b.face != tag);
+    let (first_node, _) = add_tets(model, &c.tets.name, &c.tets.face_names, &c.tets.mesh)?;
+    // The tet mesh keeps the hex nodes' coordinates exactly: merge its copies into them.
+    let hex: BTreeMap<[u64; 3], u32> = c.bases.iter().flatten().filter_map(|&n| model.node(n).map(|x| (x.xyz.map(f64::to_bits), n))).collect();
+    let mut merge = BTreeMap::new();
+    for (i, x) in c.tets.mesh.points.iter().enumerate() {
+        if let Some(&h) = hex.get(&x.map(f64::to_bits)) {
+            merge.insert(first_node + i as u32, h);
+        }
+    }
+    let shared = merge.len();
+    model.merge_nodes(&merge)?;
+    let (_, first) = crate::cmd::fresh_labels(model, 0, c.bases.len())?;
+    let body = model.geometry_tag(&GeometryRef::body(c.tets.name.as_str())?)?;
+    for (k, (base, apex)) in c.bases.iter().zip(apexes).enumerate() {
+        let label = first + k as u32;
+        let mut nodes = base.to_vec();
+        nodes.push(first_node + apex);
+        model.set_element(label, ElementKind::Pyramid5, nodes)?;
+        model.tag_element(label, body)?;
+    }
+    Ok((c.tets.mesh.elements.len() + c.bases.len(), c.tets.mesh.points.len() - shared))
+}
+
+fn sub(a: [f64; 3], b: [f64; 3]) -> [f64; 3] {
+    [a[0] - b[0], a[1] - b[1], a[2] - b[2]]
+}
+
+fn cross(a: [f64; 3], b: [f64; 3]) -> [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]]
+}
+
+fn norm(a: [f64; 3]) -> f64 {
+    (a[0] * a[0] + a[1] * a[1] + a[2] * a[2]).sqrt()
+}

crates/engine/src/cmd/forms.rs

@@ -1169,6 +1169,9 @@ pub const CELLS: &[Field] = &[
     HEX_SIZE,
     HEX_ORDER,
     TOLERANCE,
+    choice("transition", "Hex–tet joint", &[("pyramid", "Pyramids (conforming)"), ("tie", "Tie")])
+        .help("How a tet cell meets the hex cells across its partitions: a layer of pyramids on the hex faces, one conforming mesh (linear elements), or a tie between the two meshes.")
+        .def("\"pyramid\""),
 ];
 
 pub const HEX_PROJECT: &[Field] = &[

crates/engine/src/cmd/mesh.rs

@@ -255,10 +255,16 @@ pub(super) struct BodyTets {
     surface_volume: f64,
 }
 
+impl BodyTets {
+    pub(super) fn new(name: String, face_names: Vec<String>, mesh: QualityMesh, steiner: usize, cavities: usize, surface_volume: f64) -> Self {
+        BodyTets { name, face_names, mesh, steiner, cavities, surface_volume }
+    }
+}
+
 /// Mesh one body or imported surface; the model is not touched. With `several` bodies an error
 /// names the body.
 pub(super) fn mesh_body(
-    s: &Session,
+    design: Option<&fieldcraft_geometry::Design>,
     p: &Value,
     input: super::hex::Input,
     cdt: &CdtOptions,
@@ -269,7 +275,7 @@ pub(super) fn mesh_body(
     let mesh = || -> Result<BodyTets> {
         // The exact faces of a design body; an imported surface keeps to its triangles.
         let analytic = match p.get("surface") {
-            None => s.design.as_deref().and_then(|d| d.body(&input.name)).map(analytic_surfaces).unwrap_or_default(),
+            None => design.and_then(|d| d.body(&input.name)).map(analytic_surfaces).unwrap_or_default(),
             Some(_) => Vec::new(),
         };
         fieldcraft_model::progress::stage(&format!("{}: boundary", input.name), None, "");
@@ -303,7 +309,7 @@ fn tet_generate(s: &mut Session, p: &Value) -> Result<Value> {
     let several = inputs.len() > 1;
     let mut meshed = Vec::with_capacity(inputs.len());
     for input in inputs {
-        meshed.push(mesh_body(s, p, input, &cdt_options, &options, quality, several)?);
+        meshed.push(mesh_body(s.design.as_deref(), p, input, &cdt_options, &options, quality, several)?);
     }
 
     // Replace the bodies' meshes (replace.rs); other bodies and the definitions stay.