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/ui-egui/src/results.rs

@@ -64,11 +64,18 @@ pub struct Prepared {
     pub contour: Option<Arc<Contour>>,
 }
 
+/// A mode's value as the results show it: its frequency, or a buckling mode's load factor.
+pub fn mode_value(buckling: bool, eigenvalue: f64, frequency: f64) -> String {
+    if buckling { format!("λ = {}", number(eigenvalue)) } else { format!("{} Hz", number(frequency)) }
+}
+
 impl Plot {
-    /// A frequency step's mode shown, and its frequency.
-    pub fn mode(&self) -> Option<(usize, f64)> {
+    /// A frequency or buckling step's mode shown, and its frequency (Hz) or load factor.
+    pub fn mode(&self) -> Option<(usize, String)> {
         match self.result.kind() {
-            Kind::Frequency(f) => f.modes.get(self.pick.mode.checked_sub(1)?).map(|m| (self.pick.mode, m.frequency)),
+            Kind::Frequency(f) => {
+                f.modes.get(self.pick.mode.checked_sub(1)?).map(|m| (self.pick.mode, mode_value(f.buckling.is_some(), m.eigenvalue, m.frequency)))
+            }
             Kind::Static(_) | Kind::Explicit(_) | Kind::Fluid(_) | Kind::Heat(_) => None,
         }
     }