Modal sparse/dense eigensolvers

Algorithm / executionDENSE ROUTE SIZE-LIMITED

The normal sparse eigenroute calls ARPACK/eigsh; dense conversion is explicit and limited by active DOFs.

01 Model and scope

The sparse route extracts only selected generalized eigenpairs, often around a spectral shift. Dense conversion stores all matrix entries and costs roughly O(n²) memory and O(n³) work, so the active-DOF cap is part of the method boundary.

Route assumptions and calculation checks

Mode count, spectral shift, tolerance, iteration limit, and dense DOF cap are input parameters.
ARPACK non-convergence, excessive residual, or degraded mass orthogonality is not a pass.
Rotating QEP uses a different dense complex solve and remains experimental.

Useful foundations and prerequisites

Units, signs and error measures Interpolation, mapping and assembly

02 Essential equations and symbols

K \varphi{} = \lambda{} M \varphi{}

Definitions and conventions

$K\varphi{} = \lambda{}M\varphi{}$: Generalized structural eigenproblem.
$k$: Requested number of modes.
$\sigma{}$: Spectral shift used by the sparse extraction.
$n$: Number of active degrees of freedom.

01Sparse ARPACK

02Dense DOF cap

03Eigenpair metrics

04QEP kept separate

Modal sparse/dense eigensolvers · method mapSparse extraction and explicitly bounded dense conversion use different cost and memory contracts; the rotating QEP is separate.

4 detailed steps

03 Step by step derivation

Follow the assumptions, equations and discrete operations in order. The full explanation stays visible.

1 · Choose a sparse extraction or a bounded dense route

The modal problem is generalized: Kφ=λMφ. Sparse extraction requests a limited set of eigenpairs near a target shift; it avoids constructing a full dense matrix. Dense conversion is explicit and guarded by an active-DOF cap.

\mathbf K\boldsymbol\phi=\lambda\mathbf M\boldsymbol\phi

2 · Understand why dense size changes quickly

A dense n×n matrix stores n² entries, while a dense eigensolve requires work that grows cubically with n. The cap is therefore part of the solver contract and protects memory/runtime; it is not a cosmetic configuration option.

\text{dense storage}=O(n^2),\qquad\text{dense eigensolve}=O(n^3)

3 · Check the returned spectrum

Verify convergence flags, eigenpair residuals, requested mode count, positive mass, and mass orthogonality. A sparse solver that returns a partial set or a loose residual must report that outcome. Shift selection influences which part of the spectrum is extracted.

r_i=\frac{\|\mathbf K\boldsymbol\phi_i-\lambda_i\mathbf M\boldsymbol\phi_i\|_2}{\max(\|\mathbf K\boldsymbol\phi_i\|_2,|\lambda_i|\|\mathbf M\boldsymbol\phi_i\|_2,1)}

4 · Keep the rotating QEP separate

The rotating quadratic pencil contains a gyroscopic term and generally complex eigenpairs. It uses a distinct bounded dense route; classical ARPACK generalized symmetric settings and qualification claims do not transfer to that problem.

04 Algorithm and implementation

This sheet provides explanatory references rather than an inline source excerpt. Use the labeled sources to inspect the precise boundary.

05 Verification and limitations

DENSE ROUTE SIZE-LIMITED

The formulation and its explanatory checks do not by themselves establish a qualified application domain. Consult the capability matrix and the applicable recorded case.

Recorded repository calculations

See this method in a calculation.

Each case keeps its mesh, measured quantities, source record and limitations attached to the result.

First bending mode of the TET10 cantilever with mode shape color map.

VNV-TET10-MASS-MODAL-LOADS-013-MTET10 first bending modeFirst bending frequency pair: 12.7961 / 12.7965 HzRead model, result and limits →

Browse every calculation case →

06 Sources and related sheets

Modal solver details Rotating QEP

Continue the reasoning

Sparse direct factorization CG and MINRES

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