김경종
173 lines
8.1 KiB
Markdown
173 lines
8.1 KiB
Markdown
---
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type: concept
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title: "Finite Element Eigenproblem Solvers"
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complexity: advanced
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domain: computational-mechanics
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aliases:
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- eigensystem solution
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- finite element eigenvalue analysis
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- modal analysis
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created: 2026-05-28
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updated: 2026-06-02
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address: c-000015
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tags:
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- concept
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- finite-element-method
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- eigenproblems
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status: current
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related:
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- "[[Finite Element Method]]"
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- "[[Direct Time Integration Methods]]"
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- "[[Static Equilibrium Equation Solvers]]"
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- "[[Dynamic Buckling Analysis]]"
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- "[[Geometric Stiffness Matrix]]"
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- "[[BLZPACK]]"
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- "[[Abaqus Analysis Procedures]]"
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- "[[Abaqus General and Linear Perturbation Steps]]"
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- "[[Midas FEA Linear Dynamics and Buckling Analyses]]"
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- "[[Midas Civil Dynamic and Seismic Analysis]]"
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- "[[Midas Civil Buckling P-Delta and Geometric Nonlinearity]]"
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- "[[Midas NFX Equation Solvers and Eigen Extraction]]"
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- "[[Midas NFX Linear Dynamics and Buckling Analyses]]"
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sources:
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- "[[Finite Element Procedures]]"
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- "[[Dynamic-Buckling-Analysis-of-Shell-Structures-using-Finite-Element-Method]]"
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- "[[Abaqus Theory Manual]]"
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- "[[Abaqus-Analysis-User-s-Guide-Volume-II|Abaqus Analysis User's Guide Volume II]]"
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- "[[Midas-FEA-Analysis-Manual|Midas FEA Analysis Manual]]"
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- "[[Midas-Civil-Analysis-Reference|Midas Civil Analysis Reference]]"
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- "[[Midas-NFX-Analysis-Manual|Midas NFX Analysis Manual]]"
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source_refs:
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- source: "[[Finite Element Procedures]]"
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raw_path: ".raw/FiniteElementProcedures/"
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raw_files:
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- "FiniteElementProcedures_086.md"
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- "FiniteElementProcedures_002.md"
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- "FiniteElementProcedures_088.md"
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- "FiniteElementProcedures_090.md"
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md_indices:
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- 86
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- 2
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- 88
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- 90
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match: "heuristic-heading-keyword"
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confidence: high
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- source: "[[Dynamic-Buckling-Analysis-of-Shell-Structures-using-Finite-Element-Method]]"
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raw_path: ".raw/유한요소해석법을이용한쉘구조물의동적좌굴해석/"
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raw_files:
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- "유한요소해석법을이용한쉘구조물의동적좌굴해석_001.md"
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md_indices:
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- 1
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match: "heuristic-heading-keyword"
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confidence: low
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- source: "[[Abaqus Theory Manual]]"
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raw_path: ".raw/AbaqusTheoriesManual/"
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raw_files:
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- "AbaqusTheoriesManual_013.md"
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- "AbaqusTheoriesManual_012.md"
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- "AbaqusTheoriesManual_008.md"
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- "AbaqusTheoriesManual_009.md"
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md_indices:
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- 13
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- 12
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- 8
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- 9
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match: "heuristic-heading-keyword"
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confidence: high
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- source: "[[Abaqus-Analysis-User-s-Guide-Volume-II|Abaqus Analysis User's Guide Volume II]]"
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raw_path: ".raw/AbaqusAnalysisUserGuide2/"
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raw_files:
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- "AbaqusAnalysisUserGuide2_015.md"
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- "AbaqusAnalysisUserGuide2_017.md"
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- "AbaqusAnalysisUserGuide2_052.md"
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- "AbaqusAnalysisUserGuide2_023.md"
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md_indices:
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- 15
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- 17
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- 52
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- 23
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match: "heuristic-heading-keyword"
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confidence: high
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- source: "[[Midas-FEA-Analysis-Manual|Midas FEA Analysis Manual]]"
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raw_path: ".raw/MidasFEAAnalysisManual/"
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raw_files:
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- "MidasFEAAnalysisManual_036.md"
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- "MidasFEAAnalysisManual_002.md"
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md_indices:
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- 36
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- 2
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match: "heuristic-heading-keyword"
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confidence: high
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- source: "[[Midas-Civil-Analysis-Reference|Midas Civil Analysis Reference]]"
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raw_path: ".raw/MidasCivilAnalysisReference/"
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raw_files:
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- "MidasCivilAnalysisReference_014.md"
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- "MidasCivilAnalysisReference_021.md"
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- "MidasCivilAnalysisReference_015.md"
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- "MidasCivilAnalysisReference_020.md"
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md_indices:
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- 14
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- 21
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- 15
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- 20
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match: "heuristic-heading-keyword"
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confidence: low
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- source: "[[Midas-NFX-Analysis-Manual|Midas NFX Analysis Manual]]"
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raw_path: ".raw/MidasNFXAnalysisManual/"
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raw_files:
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- "MidasNFXAnalysisManual_001.md"
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- "MidasNFXAnalysisManual_020.md"
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md_indices:
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- 1
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- 20
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match: "heuristic-heading-keyword"
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confidence: low
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---
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# Finite Element Eigenproblem Solvers
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## Definition
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Finite element eigenproblem solvers compute eigenvalues and eigenvectors of matrix systems such as `K phi = lambda M phi`, commonly used for free vibration, buckling, modal reduction, and stability analysis.
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## How It Works
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The source introduces eigenvector properties, shifting, zero-mass effects, standard-form transformations, Rayleigh-Ritz approximations, component mode synthesis, error bounds, and solution methods including inverse iteration, forward iteration, Rayleigh quotient iteration, Jacobi transformations, Householder-QR, polynomial iteration, Sturm sequence techniques, Lanczos iteration, and subspace iteration.
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The dynamic buckling thesis adds an implementation example: [[BLZPACK]], based on Block Lanczos, is used for vibration and buckling eigenvalue analyses in a shell dynamic buckling program.
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[[Abaqus Analysis Procedures]] adds the commercial-code procedure context: eigenvalue extraction supports vibration modes, modal dynamics, harmonic response, and buckling or postbuckling workflows.
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[[Abaqus-Analysis-User-s-Guide-Volume-II|Abaqus Analysis User's Guide Volume II]] adds the step-level user workflow: eigenvalue buckling, natural frequency extraction, complex eigenvalue extraction, modal dynamics, response spectrum, and random response are treated as Abaqus procedure choices, usually through linear perturbation or modal procedure contexts.
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[[Midas-FEA-Analysis-Manual|Midas FEA Analysis Manual]] adds a comparable production workflow: modal analysis reports participation and effective modal mass, Lanczos and subspace methods are available, Sturm sequence checks are used for missed eigenvalues, and buckling uses geometric stiffness with shift-invert Lanczos extraction.
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[[Midas-Civil-Analysis-Reference|Midas Civil Analysis Reference]] adds civil structural modal and stability workflows: eigenvectors and Ritz vectors support vibration/seismic response, while buckling analysis uses critical load factors and mode shapes for bridge/civil stability checks.
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[[Midas-NFX-Analysis-Manual|Midas NFX Analysis Manual]] adds detailed eigen-result checks: eigenvalue range/count controls, Lanczos versus matrix-direct extraction, mode normalization, modal assurance/cross-orthogonality ideas, generalized mass/stiffness, orthogonality loss, residual error measures, modal effective mass, and buckling-vector normalization.
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## Why It Matters
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Large finite element models can have many degrees of freedom, but engineering decisions often require only selected modes or eigenvalues. Solver choice determines whether the analysis can efficiently find the physically relevant part of the spectrum.
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## Connections
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- [[Direct Time Integration Methods]] can be contrasted with mode superposition.
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- [[Static Equilibrium Equation Solvers]] shares matrix factorization and conditioning concerns.
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- [[Finite Element Program Implementation]] must support sparse matrix operations and vector iteration workflows.
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- [[Abaqus Analysis Procedures]] frames eigen extraction as one procedure family among static, transient, and coupled analyses.
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- [[Abaqus General and Linear Perturbation Steps]] explains why many eigen and modal procedures are interpreted as perturbations about a base state.
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- [[Midas FEA Linear Dynamics and Buckling Analyses]] links eigen extraction to modal, response spectrum, and linear buckling workflows in Midas.
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- [[Midas Civil Dynamic and Seismic Analysis]] and [[Midas Civil Buckling P-Delta and Geometric Nonlinearity]] connect eigen extraction to Ritz-vector seismic analysis and buckling factors.
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- [[Midas NFX Equation Solvers and Eigen Extraction]] and [[Midas NFX Linear Dynamics and Buckling Analyses]] connect eigen extraction to solver selection, mode-superposition, effective-mass, and buckling workflows.
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## Sources
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- [[Finite Element Procedures]]
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- [[Dynamic-Buckling-Analysis-of-Shell-Structures-using-Finite-Element-Method]]
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- [[Abaqus Theory Manual]]
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- [[Abaqus-Analysis-User-s-Guide-Volume-II|Abaqus Analysis User's Guide Volume II]]
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- [[Midas-FEA-Analysis-Manual|Midas FEA Analysis Manual]]
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- [[Midas-Civil-Analysis-Reference|Midas Civil Analysis Reference]]
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- [[Midas-NFX-Analysis-Manual|Midas NFX Analysis Manual]]
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