김경종
156 lines
6.5 KiB
Markdown
156 lines
6.5 KiB
Markdown
---
|
|
type: concept
|
|
title: "Isoparametric Finite Elements"
|
|
complexity: advanced
|
|
domain: computational-mechanics
|
|
aliases:
|
|
- isoparametric elements
|
|
- isoparametric formulation
|
|
created: 2026-05-28
|
|
updated: 2026-06-01
|
|
address: c-000009
|
|
tags:
|
|
- concept
|
|
- finite-element-method
|
|
- numerical-integration
|
|
status: current
|
|
related:
|
|
- "[[Finite Element Method]]"
|
|
- "[[Displacement-Based Finite Element Formulation]]"
|
|
- "[[Mixed Finite Element Formulations]]"
|
|
- "[[Solid Element Notes]]"
|
|
- "[[Isoparametric Linear Solid Elements]]"
|
|
- "[[Solid Element Shape Functions]]"
|
|
- "[[Continuum Mechanics Based Four-Node Shell Element]]"
|
|
- "[[Assumed Transverse Shear Strain Interpolation]]"
|
|
- "[[Abaqus Element Library]]"
|
|
- "[[Abaqus Continuum Element Families]]"
|
|
- "[[Abaqus Element Selection and Formulation]]"
|
|
- "[[Reduced Integration and Hourglass Control]]"
|
|
- "[[Hybrid Incompressible Elements]]"
|
|
- "[[Plane Stress and Plane Strain Elements]]"
|
|
- "[[Axisymmetric Finite Elements]]"
|
|
sources:
|
|
- "[[Finite Element Procedures]]"
|
|
- "[[A Continuum Mechanics Based Four-Node Shell]]"
|
|
- "[[Solid Element Notes]]"
|
|
- "[[Abaqus Theory Manual]]"
|
|
- "[[Abaqus-Analysis-User-s-Guide-Volume-IV|Abaqus Analysis User's Guide Volume IV]]"
|
|
- "[[A-First-Course-in-the-Finite-Element-Method|A First Course in the Finite Element Method]]"
|
|
source_refs:
|
|
- source: "[[Finite Element Procedures]]"
|
|
raw_path: ".raw/FiniteElementProcedures/"
|
|
raw_files:
|
|
- "FiniteElementProcedures_050.md"
|
|
- "FiniteElementProcedures_036.md"
|
|
- "FiniteElementProcedures_048.md"
|
|
- "FiniteElementProcedures_049.md"
|
|
md_indices:
|
|
- 50
|
|
- 36
|
|
- 48
|
|
- 49
|
|
match: "heuristic-heading-keyword"
|
|
confidence: high
|
|
- source: "[[A Continuum Mechanics Based Four-Node Shell]]"
|
|
raw_path: ".raw/AContinuumMechanicsBasedFourNodeShell/"
|
|
raw_files:
|
|
- "AContinuumMechanicsBasedFourNodeShell_001.md"
|
|
- "AContinuumMechanicsBasedFourNodeShell_002.md"
|
|
md_indices:
|
|
- 1
|
|
- 2
|
|
match: "heuristic-heading-keyword"
|
|
confidence: high
|
|
- source: "[[Solid Element Notes]]"
|
|
raw_path: ".raw/SolidElement/"
|
|
raw_files:
|
|
- "SolidElement_001.md"
|
|
md_indices:
|
|
- 1
|
|
match: "heuristic-heading-keyword"
|
|
confidence: low
|
|
- source: "[[Abaqus Theory Manual]]"
|
|
raw_path: ".raw/AbaqusTheoriesManual/"
|
|
raw_files:
|
|
- "AbaqusTheoriesManual_029.md"
|
|
- "AbaqusTheoriesManual_027.md"
|
|
- "AbaqusTheoriesManual_028.md"
|
|
- "AbaqusTheoriesManual_index2.md"
|
|
md_indices:
|
|
- 29
|
|
- 27
|
|
- 28
|
|
- 87
|
|
match: "heuristic-heading-keyword"
|
|
confidence: high
|
|
- source: "[[Abaqus-Analysis-User-s-Guide-Volume-IV|Abaqus Analysis User's Guide Volume IV]]"
|
|
raw_path: ".raw/AbaqusAnalysisUserGuide4/"
|
|
raw_files:
|
|
- "AbaqusAnalysisUserGuide4_036.md"
|
|
- "AbaqusAnalysisUserGuide4_037.md"
|
|
- "AbaqusAnalysisUserGuide4_008.md"
|
|
- "AbaqusAnalysisUserGuide4_045.md"
|
|
md_indices:
|
|
- 36
|
|
- 37
|
|
- 8
|
|
- 45
|
|
match: "heuristic-heading-keyword"
|
|
confidence: high
|
|
- source: "[[A-First-Course-in-the-Finite-Element-Method|A First Course in the Finite Element Method]]"
|
|
raw_path: ".raw/AFirstCourseInTheFiniteElementMethod/"
|
|
raw_files:
|
|
- "AFirstCourseInTheFiniteElementMethod_047.md"
|
|
- "AFirstCourseInTheFiniteElementMethod_083.md"
|
|
- "AFirstCourseInTheFiniteElementMethod_001.md"
|
|
- "AFirstCourseInTheFiniteElementMethod_052.md"
|
|
md_indices:
|
|
- 47
|
|
- 83
|
|
- 1
|
|
- 52
|
|
match: "heuristic-heading-keyword"
|
|
confidence: high
|
|
---
|
|
|
|
# Isoparametric Finite Elements
|
|
|
|
## Definition
|
|
|
|
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.
|
|
|
|
## How It Works
|
|
|
|
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.
|
|
|
|
## Why It Matters
|
|
|
|
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.
|
|
|
|
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.
|
|
|
|
[[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.
|
|
|
|
[[Abaqus Element Library]] shows the same framework at software-library scale: isoparametric interpolation, numerical integration points, full or reduced quadrature, multi-field interpolation, and hybrid pressure variables become selectable element families.
|
|
|
|
[[Abaqus-Analysis-User-s-Guide-Volume-IV|Abaqus Analysis User's Guide Volume IV]] turns those ideas into named continuum elements: bricks, wedges, tetrahedra, plane elements, axisymmetric elements, fluid continuum elements, and coupled-field variants whose suffixes indicate reduced integration, hybrid pressure treatment, incompatible modes, temperature, pore pressure, or piezoelectric fields.
|
|
|
|
[[A-First-Course-in-the-Finite-Element-Method|A First Course in the Finite Element Method]] adds the teaching path: it develops isoparametric bar, rectangular plane stress, general plane, higher-order, and three-dimensional stress elements before connecting them to Gaussian and Newton-Cotes quadrature.
|
|
|
|
## Failure Modes
|
|
|
|
- Distorted elements can degrade accuracy or convergence.
|
|
- Under-integration can introduce spurious mechanisms.
|
|
- Overly constrained interpolation can cause locking.
|
|
- Incompressible behavior often requires mixed displacement/pressure treatment.
|
|
|
|
## Sources
|
|
|
|
- [[Finite Element Procedures]]
|
|
- [[A Continuum Mechanics Based Four-Node Shell]]
|
|
- [[Solid Element Notes]]
|
|
- [[Abaqus Theory Manual]]
|
|
- [[Abaqus-Analysis-User-s-Guide-Volume-IV|Abaqus Analysis User's Guide Volume IV]]
|
|
- [[A-First-Course-in-the-Finite-Element-Method|A First Course in the Finite Element Method]]
|