Cyclic-symmetry reduction#

A rotor built from \(N\) geometrically-identical sectors spanning \(360°\) admits a cyclic-symmetry modal reduction: the full-rotor eigenproblem decouples into \(\lfloor N/2 \rfloor + 1\) smaller eigenproblems, one per harmonic index. This page derives the reduction from the rotational-symmetry property of \(K\) and \(M\), shows how femorph-solver maps it onto the real, base-sector linear-algebra primitives the femorph_solver.solvers.cyclic backend works with, and lists the references the kernel cites.

The driving API (sector mesh → cyclic faces → per-harmonic modal results) lives in CyclicModel; see Cyclic-symmetry modal analysis for the user-facing walk-through.

Setting#

A rotor with \(N\) identical sectors has assembled matrices \(K, M \in \mathbb{R}^{Nn \times Nn}\), where \(n\) is the per-sector free-DOF count. Index the sectors \(j = 0, 1, \dots, N-1\) and partition each matrix accordingly:

\[\begin{split}K = \begin{bmatrix} K_{00} & K_{01} & 0 & \cdots & K_{0,N-1} \\ K_{10} & K_{11} & K_{12} & \cdots & 0 \\ 0 & K_{21} & K_{22} & \cdots & 0 \\ \vdots & & & \ddots & \vdots \\ K_{N-1,0} & 0 & \cdots & K_{N-1,N-2} & K_{N-1,N-1} \end{bmatrix},\end{split}\]

with \(K_{ij} = 0\) whenever \(|i - j| > 1 \pmod{N}\) — i.e. each sector only couples to its two cyclic neighbours through the shared cyclic face.

Cyclic symmetry of the geometry implies

\[K_{j,j} = K_{0,0}, \qquad K_{j,j+1} = K_{0,1}, \qquad K_{j,j-1} = K_{0,1}^\top \quad \forall j.\]

— every diagonal block is the same self-stiffness \(A \equiv K_{00}\), and every off-diagonal block is the same inter-sector coupling \(B \equiv K_{01}\). The same structure holds for \(M\); call its blocks \(A^M, B^M\).

Block-circulant decomposition#

Matrices of the form

\[\begin{split}K = \begin{bmatrix} A & B & 0 & \cdots & 0 & B^\top \\ B^\top & A & B & \cdots & 0 & 0 \\ \vdots & & & \ddots & & \vdots \\ B & 0 & \cdots & 0 & B^\top & A \end{bmatrix}\end{split}\]

are block circulant. The discrete Fourier transform on the sector index block-diagonalises them — exactly, not approximately. Define the per-sector phase shift \(\theta_k = 2\pi k / N\) for \(k \in \{0, 1, \dots, N-1\}\) and the harmonic basis vectors

\[\mathbf{e}_k = \frac{1}{\sqrt{N}} \bigl[ 1,\; e^{i \theta_k},\; e^{i 2 \theta_k},\; \dots,\; e^{i (N-1) \theta_k} \bigr]^\top.\]

The block-Fourier projection

\[\tilde{K}_k \;=\; A + e^{i \theta_k} B + e^{-i \theta_k} B^\top\]

gives \(N\) reduced \(n \times n\) complex-Hermitian problems, each independent of the others:

\[\tilde{K}_k\, \tilde{\boldsymbol{\varphi}}_k \;=\; \omega^2_k\, \tilde{M}_k\, \tilde{\boldsymbol{\varphi}}_k, \qquad k = 0, 1, \dots, N-1,\]

with \(\tilde{M}_k = A^M + e^{i \theta_k} B^M + e^{-i \theta_k} {B^M}^\top\) defined identically.

The full-rotor mode is recovered by re-expansion:

\[\boldsymbol{\varphi}_k^{(j)} \;=\; e^{i j \theta_k}\, \tilde{\boldsymbol{\varphi}}_k, \qquad j = 0, 1, \dots, N-1.\]

— the same base-sector mode shape repeated around the rotor with a per-sector phase advance \(\theta_k\).

Physical harmonic indices#

The \(N\) Fourier indices \(\{0, 1, \dots, N-1\}\) contain redundancy: the spectrum at \(k\) is the complex conjugate of the spectrum at \(N - k\). For a real rotor, this collapses to the physical harmonic indices

\[k \in \{0, 1, \dots, \lfloor N/2 \rfloor\}.\]

  • \(k = 0\) — every sector moves in phase; axisymmetric (umbrella) modes; mode shape is real.

  • \(k = N/2\) (only when \(N\) is even) — adjacent sectors move \(180°\) out of phase; mode shape is real.

  • \(0 < k < N/2\) — travelling-wave modes that come in forward / backward pairs of identical frequency, often called the standing-wave doublets. The two members of a doublet are the real and imaginary parts of a single complex eigenvector.

The “nodal diameter” terminology used in turbomachinery (forward / backward \(k\)-nodal-diameter modes) is exactly this harmonic index.

The number of distinct frequencies a cyclic modal solve returns is therefore not \(N \cdot n_\text{modes}\) but \((\lfloor N/2 \rfloor + 1) \cdot n_\text{modes}\), counting each \(0 < k < N/2\) mode once with multiplicity two.

Real \(2n\) augmentation#

The reduced problem \(\tilde{K}_k\, \tilde{\boldsymbol{\varphi}}_k = \omega_k^2\, \tilde{M}_k\, \tilde{\boldsymbol{\varphi}}_k\) is complex-Hermitian. Most mature sparse eigensolvers (ARPACK, LOBPCG, PRIMME) ship a real-symmetric path optimised for the SPD case; the complex- Hermitian path is less mature and slower per iteration.

Grimes, Lewis & Simon (1994) pointed out that any complex- Hermitian generalised eigenproblem can be lifted to a real-symmetric \(2n \times 2n\) problem with the same spectrum. Write \(\tilde{K}_k = K_R + i K_I\), \(\tilde{M}_k = M_R + i M_I\) (with \(K_R, M_R\) symmetric and \(K_I, M_I\) antisymmetric since the original \(A, B\) are real) and form

\[\begin{split}K_k^{\,2n} = \begin{bmatrix} K_R & -K_I \\ K_I & K_R \end{bmatrix}, \qquad M_k^{\,2n} = \begin{bmatrix} M_R & -M_I \\ M_I & M_R \end{bmatrix}.\end{split}\]

Both lifted matrices are real symmetric (and SPD, when the original sector matrices are SPD on the constraint quotient). Their generalised eigenvalues are exactly the eigenvalues of the complex-Hermitian problem, each with double multiplicity that mirrors the real / imaginary part decomposition.

femorph-solver uses this real-\(2n\) lift on every reduced harmonic-index problem so the same shift-invert ARPACK / SPD- direct path the standard modal solve uses applies unchanged.

Practical mapping in femorph-solver#

The user-facing path is:

  1. Author the base-sector mesh as a regular Model carrying one wedge of the rotor. No need to assemble the full rotor.

  2. Identify the cyclic faces — the two θ-bounding surfaces of the wedge. Either supply them explicitly via the low_face / high_face keyword arguments, or let CyclicModel auto-detect them by matching every node on the low face to its rotated image on the high face within a \(pair\_tol\) tolerance.

  3. Call CyclicModel.solve_modal with the harmonic indices of interest (default: every physical index \(0 \le k \le N/2\)). One per-\(k\) reduced eigenproblem runs internally, on the \(2n\)-lifted real-symmetric system.

  4. Each CyclicModalResult carries the harmonic index \(k\), the per-mode frequencies, and the base-sector mode shapes. Full-rotor mode-shape recovery is handled by the result’s expansion helpers.

The compute saving over assembling the full rotor scales as \(O((\lfloor N/2 \rfloor + 1) \cdot n^p / N \cdot n^p) = O((\lfloor N/2 \rfloor + 1) / N^p)\) for an \(O(n^p)\) solver — close to the obvious “factor of \(N\)” speedup, plus an extra factor \(p-1\) from the better factorisation cost on the smaller system. For a typical \(N = 36\)-sector turbine rotor this is ~50× faster than the full-rotor solve.

Validity assumptions#

Cyclic-symmetry reduction is exact when:

  • The sector geometry repeats identically under the per-step rotation — no mistuning, no per-sector mass / stiffness variation. Real turbomachinery is mistuned at the 0.1-1 % level, which the cyclic reduction cannot capture; mistuned analysis requires a separate technique (component mode synthesis or full-rotor solve).

  • The cyclic faces of adjacent sectors mate one-to-one. Mesh generators sometimes produce slightly-misaligned faces that the femorph-solver auto-pairing can match within tolerance, but mismatch beyond that fails the pairing check explicitly.

  • The boundary conditions on the non-cyclic faces of every sector are identical. A bracket bolted to the rotor at one sector position breaks cyclic symmetry.

Violations of these assumptions don’t produce a wrong answer silently — the pairing check catches the geometry mismatch, the modal-solve sanity check catches BC asymmetry, and the mistuning case is intercepted by the user reading these constraints up front.

References#

  • D. L. Thomas, “Dynamics of rotationally periodic structures,” International Journal for Numerical Methods in Engineering 14 (1979), 81–102 — the foundational paper for cyclic-symmetry FE reduction in modal analysis.

  • A. Wildheim, “Excitation of rotationally periodic structures,” Journal of Applied Mechanics 46 (1979), 878–882 — companion paper covering the harmonic-excitation case.

  • R. Grimes, J. Lewis, H. Simon, “A shifted block Lanczos algorithm for solving sparse symmetric generalized eigenproblems,” SIAM J. Matrix Anal. Appl. 15 (1994), 228–272 — origin of the real-\(2n\) augmentation that lets a real-symmetric Lanczos drive the complex-Hermitian per-harmonic problem.

  • R. R. Craig & A. J. Kurdila, Fundamentals of Structural Dynamics, 2nd ed., 2006, §16.3 — accessible textbook treatment of cyclic / dihedral group representations applied to FE eigenproblems.

  • A. V. Srinivasan, “Flutter and resonant vibration characteristics of engine blades,” Journal of Engineering for Gas Turbines and Power 119 (1997), 742–775 — application to bladed-disc / rotor problems with mistuning.

See also#