김경종
54 lines
2.0 KiB
Markdown
54 lines
2.0 KiB
Markdown
---
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type: concept
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title: "Nonlinear Newmark-Beta Integration"
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complexity: advanced
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domain: computational-mechanics
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created: 2026-05-28
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updated: 2026-05-28
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address: c-000031
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aliases:
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- nonlinear Newmark method
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- Newmark-beta Newton iteration
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- implicit Newmark nonlinear dynamics
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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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- nonlinear-analysis
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status: current
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related:
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- "[[MITC Study Notes]]"
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- "[[Direct Time Integration Methods]]"
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- "[[Nonlinear Finite Element Analysis]]"
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- "[[Static Equilibrium Equation Solvers]]"
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sources:
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- "[[MITC Study Notes]]"
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---
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# Nonlinear Newmark-Beta Integration
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## Definition
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Nonlinear Newmark-beta integration combines Newmark time-discretization kinematics with Newton-Raphson iteration to solve nonlinear finite element dynamic equilibrium at each time step.
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## How It Works
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The study notes start from dynamic equilibrium with mass, stiffness, and external load terms. At the new time step, the residual depends on displacement, velocity, and acceleration. Newmark-beta relations express velocity and acceleration increments in terms of the unknown displacement increment, so the Newton system can be written as an effective tangent equation for that displacement increment.
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## Why It Matters
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For nonlinear structural dynamics, a time step is not just a matrix update. Internal force and tangent stiffness depend on the current trial displacement, so each step requires repeated residual evaluation, tangent assembly, displacement correction, and velocity/acceleration update until convergence.
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## Iteration Skeleton
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- Predict or initialize the new-step displacement, velocity, and acceleration.
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- Assemble residual from external load, inertia, and internal force.
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- Form the effective tangent with mass and nonlinear tangent contributions.
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- Solve for the displacement correction.
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- Update displacement, velocity, and acceleration using Newmark-beta formulas.
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- Repeat until the residual and/or correction satisfies convergence criteria.
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## Sources
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- [[MITC Study Notes]]
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