solver, model, engine: linear buckling steps

Procedure::Buckle {modes, shift}: the step's loads as a linear preload, K_σ from its stresses,
and (K + λ K_σ) φ = 0 by block Lanczos in the K inner product: K⁻¹(−K_σ) on Cholesky without a
shift, the buckling transformation (K + σ K_σ)⁻¹ K on LDLᵀ with one, whose inertia counts the
load factors below σ (Grimes–Lewis–Simon 1994). Lanczos gains an operand and an eigenvalue map
(defaults keep frequency steps bit for bit) and reports converged negative eigenvalues. Results
are frequency results with a `buckling` part; modes display with their load factor.
solve.buckle (in the Static drop-down); Abaqus *BUCKLE, Nastran SOL 105 and LS-DYNA refuse it by
name. Euler columns of hex20 within 0.18 % of Engesser's load.

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

crates/decks/src/abaqus.rs

@@ -1216,6 +1216,34 @@ impl Deck {
                     base_bcs.clone_from(&step.boundary_conditions);
                     base_loads.clone_from(&step.loads);
                 }
+                Procedure::Buckle { modes, shift } => {
+                    // Abaqus's buckling base state is the end of the previous general step;
+                    // FieldCraft's steps all start from the undeformed, unloaded model.
+                    if !base_loads.is_empty()
+                        || steps
+                            .iter()
+                            .take_while(|s| !std::ptr::eq(*s, step))
+                            .any(|s| !matches!(s.procedure, Procedure::Frequency { .. } | Procedure::Buckle { .. }))
+                    {
+                        return Err(Error::Unsupported {
+                            format: FORMAT,
+                            what: format!(
+                                "step `{}`: a buckling step after a general step (Abaqus would buckle from that step's loaded state)",
+                                step.name
+                            ),
+                        });
+                    }
+                    if shift.is_some() {
+                        return Err(Error::Unsupported { format: FORMAT, what: format!("step `{}`: a buckling shift", step.name) });
+                    }
+                    self.line(&format!("*STEP, NAME={}", step.name));
+                    self.line("*BUCKLE, EIGENSOLVER=LANCZOS");
+                    self.line(&format!("{modes}"));
+                    self.step_boundaries(model, r, &base_bcs, &step.boundary_conditions)?;
+                    let loads = StepLoads::new(model, r, step, &everything);
+                    self.step_loads(&loads, written)?;
+                    written = loads.written();
+                }
                 Procedure::Frequency { modes, min_hz, max_hz } => {
                     self.line(&format!("*STEP, NAME={}", step.name));
                     self.line("*FREQUENCY, EIGENSOLVER=LANCZOS");

crates/decks/src/lsdyna/mod.rs

@@ -899,7 +899,7 @@ fn write_step(d: &mut Deck, model: &Model, r: &Resolved, plan: &Plan, step: &Ste
                 d.card([Field::Blank, Field::Blank, Field::Blank, Field::Blank, Field::from(1)]);
             }
         }
-        Procedure::ExplicitDynamic(_) | Procedure::HeatTransfer { .. } | Procedure::CoupledTemperatureDisplacement {} => {
+        Procedure::ExplicitDynamic(_) | Procedure::HeatTransfer { .. } | Procedure::CoupledTemperatureDisplacement {} | Procedure::Buckle { .. } => {
             return Err(Error::Unsupported { format: FORMAT, what: format!("step `{}`: {} steps", step.name, step.procedure.name()) });
         }
     }

crates/decks/src/nastran/mod.rs

@@ -280,6 +280,9 @@ fn solution(steps: &[&Step]) -> std::result::Result<u32, String> {
     if let Some(s) = steps.iter().find(|s| matches!(s.procedure, Procedure::ExplicitDynamic(_))) {
         return Err(format!("step `{}`: explicit dynamic steps (Nastran's linear solutions have none)", s.name));
     }
+    if let Some(s) = steps.iter().find(|s| matches!(s.procedure, Procedure::Buckle { .. })) {
+        return Err(format!("step `{}`: buckling steps (Nastran's SOL 105; this writer writes SOL 101 and 103 so far)", s.name));
+    }
     if let Some(s) = steps.iter().find(|s| matches!(s.procedure, Procedure::NonlinearStatic(_))) {
         return Err(format!("step `{}`: nonlinear static steps (Nastran's SOL 400; this writer writes SOL 101 and 103 so far)", s.name));
     }