김경종
108 lines
7.5 KiB
Markdown
108 lines
7.5 KiB
Markdown
---
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type: concept
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title: "Direct Time Integration Methods"
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complexity: advanced
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domain: computational-mechanics
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aliases:
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- finite element dynamics
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- direct integration
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- Newmark method
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created: 2026-05-28
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updated: 2026-06-02
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address: c-000014
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tags:
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- concept
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- finite-element-method
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- dynamics
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status: current
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related:
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- "[[Finite Element Method]]"
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- "[[Nonlinear Finite Element Analysis]]"
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- "[[Nonlinear Newmark-Beta Integration]]"
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- "[[Dynamic Buckling Analysis]]"
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- "[[Finite Element Eigenproblem Solvers]]"
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- "[[Abaqus Analysis Procedures]]"
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- "[[Abaqus Explicit Analysis Efficiency Techniques]]"
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- "[[Abaqus Eulerian and Particle Methods]]"
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- "[[Beam and Frame Finite Elements]]"
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- "[[Bar and Truss Finite Elements]]"
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- "[[Elasto-Viscoplastic Finite Element Analysis]]"
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- "[[Transient Dynamic Elasto-Plastic Analysis]]"
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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 Nonlinear Time History and Hysteresis Models]]"
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- "[[Midas NFX Linear Dynamics and Buckling Analyses]]"
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- "[[Midas NFX Nonlinear Static and Dynamic Algorithms]]"
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sources:
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- "[[Finite Element Procedures]]"
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- "[[MITC Study Notes]]"
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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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- "[[A-First-Course-in-the-Finite-Element-Method|A First Course in the Finite Element Method]]"
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- "[[Abaqus-Analysis-User-s-Guide-Volume-II|Abaqus Analysis User's Guide Volume II]]"
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- "[[Finite-Elements-in-Plasticity-Theory-and-Practice|Finite Elements in Plasticity: Theory and Practice]]"
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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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---
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# Direct Time Integration Methods
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## Definition
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Direct time integration methods advance finite element equilibrium equations through time without necessarily transforming the problem into modal coordinates.
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## How It Works
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Dynamic finite element systems include mass, damping, stiffness, and time-dependent loading. The source covers central difference, Houbolt, Newmark, and Bathe methods, then analyzes approximation, load operators, stability, accuracy, numerical damping, and coupling of different integration operators.
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The MITC study notes add a focused nonlinear Newmark-beta derivation: Newton iteration is used at each time step, and Newmark relations express acceleration and velocity increments through the displacement increment.
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The dynamic buckling thesis uses time-dependent axial compression as the loading context. It connects dynamic response, natural frequency, and buckling instability boundaries rather than treating time integration as a standalone transient solve.
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[[Abaqus Analysis Procedures]] places direct integration inside the broader procedure choice: implicit dynamics, explicit dynamics, modal dynamics, and coupled transient field analyses each carry different stability, increment, and convergence requirements.
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[[A-First-Course-in-the-Finite-Element-Method|A First Course in the Finite Element Method]] adds the elementary matrix-dynamics path: spring-mass equations, lumped and consistent mass matrices for bars, beams, trusses, frames, plane elements, axisymmetric elements, and solids, plus central difference, Newmark, Wilson, and transient heat-transfer examples.
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[[Abaqus-Analysis-User-s-Guide-Volume-II|Abaqus Analysis User's Guide Volume II]] expands the production procedure choices: implicit direct integration, explicit dynamic analysis, direct-solution steady-state dynamics, modal dynamics, subspace steady-state dynamics, response spectrum, and random response analysis.
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[[Finite-Elements-in-Plasticity-Theory-and-Practice|Finite Elements in Plasticity: Theory and Practice]] adds the material-nonlinearity view of time integration: elasto-viscoplastic updates depend directly on time-step size, and transient dynamic elasto-plastic analysis couples inertia terms with evolving plastic zones.
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[[Midas-FEA-Analysis-Manual|Midas FEA Analysis Manual]] adds production time-history context: mode superposition and direct integration are treated alongside Rayleigh or modal damping, load-time interpolation, and practical time-step selection relative to modal periods and load intervals.
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[[Midas-Civil-Analysis-Reference|Midas Civil Analysis Reference]] adds civil seismic context: direct integration appears beside modal/Ritz analysis, damping choices, response spectrum procedures, multi-support excitation, nonlinear time history, and hysteretic member/link models.
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[[Midas-NFX-Analysis-Manual|Midas NFX Analysis Manual]] adds both linear and nonlinear transient details: HHT implicit integration, central-difference explicit integration, critical time-step control, artificial bulk viscosity, damping, mass scaling, residual-vector mode augmentation, and enforced-motion partitioning.
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## Why It Matters
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Time integration choices control stability, phase accuracy, numerical damping, and computational cost. Explicit methods can be efficient for very small stable time steps; implicit methods are more expensive per step but can support larger steps and nonlinear equilibrium iterations.
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## Connections
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- [[Finite Element Eigenproblem Solvers]] supports modal superposition and vibration analysis.
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- [[Nonlinear Finite Element Analysis]] couples time integration with nonlinear iteration.
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- [[Nonlinear Newmark-Beta Integration]] is the specific implicit nonlinear dynamics workflow extracted from the MITC notes.
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- [[Finite Element Heat Transfer and Field Problems]] uses related transient integration ideas for first-order field equations.
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- [[Abaqus Analysis Procedures]] connects transient integration to Standard/Explicit procedure selection.
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- [[Abaqus Explicit Analysis Efficiency Techniques]] covers mass scaling, subcycling, and steady-state detection around explicit integration.
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- [[Abaqus Eulerian and Particle Methods]] uses explicit time integration for Eulerian, DEM, and SPH workflows.
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- [[Beam and Frame Finite Elements]] and [[Bar and Truss Finite Elements]] provide simple structural examples for mass-matrix construction.
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- [[Elasto-Viscoplastic Finite Element Analysis]] and [[Transient Dynamic Elasto-Plastic Analysis]] show how time stepping interacts with rate-dependent and plastic material state variables.
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- [[Midas FEA Linear Dynamics and Buckling Analyses]] connects time history analysis to Midas modal, response spectrum, and buckling procedure choices.
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- [[Midas Civil Dynamic and Seismic Analysis]] and [[Midas Civil Nonlinear Time History and Hysteresis Models]] connect time integration to seismic damping, multi-support excitation, and hysteretic bridge components.
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- [[Midas NFX Linear Dynamics and Buckling Analyses]] and [[Midas NFX Nonlinear Static and Dynamic Algorithms]] connect time integration to NFX modal superposition, explicit stability, HHT residual iteration, damping, and mass scaling.
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## Sources
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- [[Finite Element Procedures]]
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- [[MITC Study Notes]]
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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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- [[A-First-Course-in-the-Finite-Element-Method|A First Course in the Finite Element Method]]
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- [[Abaqus-Analysis-User-s-Guide-Volume-II|Abaqus Analysis User's Guide Volume II]]
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- [[Finite-Elements-in-Plasticity-Theory-and-Practice|Finite Elements in Plasticity: Theory and Practice]]
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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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