김경종
62 lines
2.6 KiB
Markdown
62 lines
2.6 KiB
Markdown
---
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type: concept
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title: "Isoparametric Finite Elements"
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complexity: advanced
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domain: computational-mechanics
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aliases:
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- isoparametric elements
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- isoparametric formulation
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created: 2026-05-28
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updated: 2026-05-28
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address: c-000009
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tags:
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- concept
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- finite-element-method
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- numerical-integration
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status: current
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related:
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- "[[Finite Element Method]]"
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- "[[Displacement-Based Finite Element Formulation]]"
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- "[[Mixed Finite Element Formulations]]"
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- "[[Solid Element Notes]]"
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- "[[Isoparametric Linear Solid Elements]]"
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- "[[Solid Element Shape Functions]]"
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- "[[Continuum Mechanics Based Four-Node Shell Element]]"
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- "[[Assumed Transverse Shear Strain Interpolation]]"
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sources:
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- "[[Finite Element Procedures]]"
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- "[[A Continuum Mechanics Based Four-Node Shell]]"
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- "[[Solid Element Notes]]"
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---
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# Isoparametric Finite Elements
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## Definition
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Isoparametric finite elements use the same interpolation framework to represent both element geometry and field variables, mapping a simple natural-coordinate element to the physical element through shape functions and a Jacobian.
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## How It Works
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The element is defined in natural coordinates, shape functions interpolate geometry and unknown fields, the Jacobian maps derivatives and integration measures into physical coordinates, and numerical quadrature evaluates stiffness, mass, load, and other element matrices.
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## Why It Matters
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The isoparametric framework is the practical bridge from finite element theory to general-purpose element routines. It supports quadrilateral, triangular, solid, beam, plate, and shell elements, while exposing key numerical choices such as quadrature order, reduced integration, selective integration, and element distortion sensitivity.
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The four-node shell paper is an example of this bridge: a general quadrilateral shell can be economical and nonlinear-capable, but the interpolation must be modified to avoid shear locking in thin shell limits.
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[[Solid Element Notes]] provides the direct 3D continuum example: 4-node tetrahedral, 5-node pyramid, 6-node wedge, and 8-node hexahedral elements interpolate both position and displacement with the same natural-coordinate functions before mapping derivatives through the Jacobian.
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## Failure Modes
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- Distorted elements can degrade accuracy or convergence.
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- Under-integration can introduce spurious mechanisms.
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- Overly constrained interpolation can cause locking.
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- Incompressible behavior often requires mixed displacement/pressure treatment.
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## Sources
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- [[Finite Element Procedures]]
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- [[A Continuum Mechanics Based Four-Node Shell]]
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- [[Solid Element Notes]]
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