김경종
69 lines
2.4 KiB
Markdown
69 lines
2.4 KiB
Markdown
---
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type: concept
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title: "Solid Element Shape Functions"
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complexity: intermediate
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domain: computational-mechanics
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aliases:
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- solid element interpolation functions
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- linear solid shape functions
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- tetrahedral wedge pyramid hexahedral shape functions
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created: 2026-05-28
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updated: 2026-05-28
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address: c-000050
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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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- interpolation
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status: current
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related:
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- "[[Solid Element Notes]]"
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- "[[Isoparametric Linear Solid Elements]]"
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- "[[Isoparametric Finite Elements]]"
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- "[[Solid Element Strain-Displacement Matrix]]"
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sources:
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- "[[Solid Element Notes]]"
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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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---
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# Solid Element Shape Functions
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## Definition
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Solid element shape functions interpolate three-dimensional element geometry and displacement from nodal values in natural coordinates.
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## Covered Topologies
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The notes give first-order interpolation for four common solid element shapes:
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- 4-node tetrahedron with barycentric-style coordinates.
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- 5-node pyramid connecting a quadrilateral base to an apex.
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- 6-node wedge, or triangular prism, using triangular interpolation through a two-node thickness direction.
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- 8-node hexahedron with trilinear interpolation in `xi`, `eta`, and `zeta`.
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## Why They Matter
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Shape functions are the starting point for every later element calculation. They define the displacement approximation, the geometry mapping, the Jacobian, the derivative transformation, and ultimately the strain-displacement matrix. Because the same functions interpolate geometry and field variables, the source is a concrete example of [[Isoparametric Finite Elements]].
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## Modeling Implications
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Low-order solid shape functions are economical but sensitive to distortion and limited in bending-dominated response. This is why element aspect ratio and topology selection matter before any solver choice is considered.
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## Connections
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- [[Isoparametric Linear Solid Elements]] gives the element-level context.
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- [[Solid Element Strain-Displacement Matrix]] differentiates these functions after Jacobian mapping.
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- [[Solid Element Stiffness Integration]] integrates the resulting `B^T D B` expression over the element volume.
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## Sources
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- [[Solid Element Notes]]
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