김경종
73 lines
3.5 KiB
Markdown
73 lines
3.5 KiB
Markdown
---
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type: concept
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title: "Mixed Finite Element Formulations"
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complexity: advanced
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domain: computational-mechanics
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aliases:
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- mixed formulation
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- displacement-pressure formulation
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- inf-sup condition
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- assumed strain formulation
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created: 2026-05-28
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updated: 2026-05-29
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address: c-000010
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tags:
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- concept
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- finite-element-method
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- incompressibility
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status: current
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related:
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- "[[Displacement-Based Finite Element Formulation]]"
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- "[[Isoparametric Finite Elements]]"
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- "[[Nonlinear Finite Element Analysis]]"
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- "[[Assumed Transverse Shear Strain Interpolation]]"
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- "[[Shell Locking Phenomenon]]"
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- "[[Uniform Optimal Convergence]]"
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- "[[Incompatible Mode Solid Elements]]"
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- "[[Hybrid Incompressible Elements]]"
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- "[[Reduced Integration and Hourglass Control]]"
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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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- "[[On-the-Finite-Element-Analysis-of-Shell-Structures]]"
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- "[[Solid Element Notes]]"
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- "[[Abaqus Theory Manual]]"
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---
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# Mixed Finite Element Formulations
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## Definition
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Mixed finite element formulations approximate more than one primary field, such as displacement and pressure, instead of relying only on displacement unknowns.
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## How It Works
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Additional field variables are introduced to represent constraints or stress-like quantities directly. For incompressible or nearly incompressible media, displacement/pressure formulations separate volumetric constraint behavior from deviatoric deformation. Stability depends on compatible interpolation choices, often summarized by the inf-sup condition.
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The four-node shell paper is not simply a displacement/pressure mixed formulation, but it uses the same reliability idea: a constrained or separately assumed field can remove locking when direct displacement interpolation is too restrictive.
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[[On-the-Finite-Element-Analysis-of-Shell-Structures]] adds the shell-specific stability view: MITC-style mixed interpolation is useful because it can reduce locking, but the chosen strain field still has to retain consistency, ellipticity, and thickness-uniform convergence.
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[[Solid Element Notes]] adds another local enrichment pattern: incompatible mode solid elements introduce internal deformation modes and statically condense them, improving element flexibility without adding global nodal unknowns.
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[[Hybrid Incompressible Elements]] adds the Abaqus-specific industrial case: hydrostatic pressure can be introduced as an additional field or constraint variable so incompressible materials do not force a displacement-only interpolation into volumetric locking.
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## Why It Matters
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Mixed formulations are needed when displacement-only elements lock, produce spurious pressure modes, or fail to represent constrained fields accurately. The source treats the inf-sup condition as a central test of whether the chosen interpolation spaces are stable.
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## Connections
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- [[Isoparametric Finite Elements]] supplies the element construction machinery.
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- [[Nonlinear Finite Element Analysis]] uses mixed formulations for large deformation incompressible behavior.
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- [[Finite Element Heat Transfer and Field Problems]] uses analogous ideas when multiple fields interact.
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- [[Reduced Integration and Hourglass Control]] is a related numerical remedy, but hybrid elements make the pressure constraint explicit.
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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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- [[On-the-Finite-Element-Analysis-of-Shell-Structures]]
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- [[Solid Element Notes]]
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- [[Abaqus Theory Manual]]
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