김경종
115 lines
4.9 KiB
Markdown
115 lines
4.9 KiB
Markdown
---
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type: concept
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title: "Isoparametric Linear Solid Elements"
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complexity: intermediate
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domain: computational-mechanics
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aliases:
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- linear solid elements
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- first-order solid elements
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- isoparametric solid elements
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- 3D solid elements
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created: 2026-05-28
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updated: 2026-06-01
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address: c-000049
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tags:
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- concept
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- finite-element-method
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- solid-elements
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- isoparametric-elements
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status: current
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related:
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- "[[Solid Element Notes]]"
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- "[[Isoparametric Finite Elements]]"
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- "[[Displacement-Based Finite Element Formulation]]"
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- "[[Solid Element Shape Functions]]"
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- "[[Solid Element Strain-Displacement Matrix]]"
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- "[[Solid Element Stiffness Integration]]"
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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 Continuum Element Families]]"
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- "[[Abaqus Element Selection and Formulation]]"
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sources:
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- "[[Solid Element Notes]]"
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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-IV|Abaqus Analysis User's Guide Volume IV]]"
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source_refs:
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- source: "[[Solid Element Notes]]"
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raw_path: ".raw/SolidElement/"
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raw_files:
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- "SolidElement_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: "[[A-First-Course-in-the-Finite-Element-Method|A First Course in the Finite Element Method]]"
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raw_path: ".raw/AFirstCourseInTheFiniteElementMethod/"
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raw_files:
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- "AFirstCourseInTheFiniteElementMethod_047.md"
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- "AFirstCourseInTheFiniteElementMethod_001.md"
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- "AFirstCourseInTheFiniteElementMethod_053.md"
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- "AFirstCourseInTheFiniteElementMethod_083.md"
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md_indices:
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- 47
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- 1
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- 53
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- 83
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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-IV|Abaqus Analysis User's Guide Volume IV]]"
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raw_path: ".raw/AbaqusAnalysisUserGuide4/"
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raw_files:
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- "AbaqusAnalysisUserGuide4_010.md"
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- "AbaqusAnalysisUserGuide4_011.md"
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- "AbaqusAnalysisUserGuide4_008.md"
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- "AbaqusAnalysisUserGuide4_022.md"
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md_indices:
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- 10
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- 8
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- 22
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match: "heuristic-heading-keyword"
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confidence: high
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---
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# Isoparametric Linear Solid Elements
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## Definition
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Isoparametric linear solid elements are first-order three-dimensional continuum finite elements that interpolate both geometry and displacement with the same nodal shape functions.
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## How They Work
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The source treats solid elements as volume elements with three translational displacement degrees of freedom per node: `u`, `v`, and `w`. They do not include rotational degrees of freedom, so connecting them directly to beam, plate, or shell elements can require care to avoid singular constraints.
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The physical position and displacement field are both interpolated from nodal values:
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```text
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x(xi) = sum N_i(xi) x_i
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u(xi) = sum N_i(xi) u_i
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```
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The covered topologies are 4-node tetrahedron, 5-node pyramid, 6-node wedge, and 8-node hexahedron. In each case, the element is defined in natural coordinates and mapped to physical space through the Jacobian.
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[[A-First-Course-in-the-Finite-Element-Method|A First Course in the Finite Element Method]] adds the introductory three-dimensional stress path through tetrahedral solid elements and isoparametric solid formulation after the plane and axisymmetric element chapters.
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[[Abaqus-Analysis-User-s-Guide-Volume-IV|Abaqus Analysis User's Guide Volume IV]] maps this theory to production element families: first-order and second-order tetrahedra, wedges, pyramids, and bricks, with reduced-integration, hybrid, incompatible-mode, thermal, pore-pressure, and piezoelectric variants.
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## Practical Notes
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- Solid elements are suited to three-dimensional volume response rather than beam or shell idealizations.
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- Aspect ratios close to one are preferred because distortion degrades the shape-function mapping and numerical integration quality.
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- The absence of rotational degrees of freedom is a modeling interface issue when solid elements meet structural elements.
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## Connections
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- [[Solid Element Shape Functions]] defines the natural-coordinate interpolation for each covered topology.
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- [[Solid Element Strain-Displacement Matrix]] converts the displacement interpolation into engineering strain components.
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- [[Solid Element Stiffness Integration]] assembles the stiffness matrix from `B`, `D`, and the Jacobian.
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- [[Axisymmetric Finite Elements]] are an efficient reduced-dimensional alternative when body and load symmetry permit.
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- [[Abaqus Continuum Element Families]] shows the Abaqus solid-element names and variants built on the same continuum interpolation idea.
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## Sources
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- [[Solid Element Notes]]
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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-IV|Abaqus Analysis User's Guide Volume IV]]
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