김경종
66 lines
2.9 KiB
Markdown
66 lines
2.9 KiB
Markdown
---
|
|
type: concept
|
|
title: "Mixed Finite Element Formulations"
|
|
complexity: advanced
|
|
domain: computational-mechanics
|
|
aliases:
|
|
- mixed formulation
|
|
- displacement-pressure formulation
|
|
- inf-sup condition
|
|
- assumed strain formulation
|
|
created: 2026-05-28
|
|
updated: 2026-05-28
|
|
address: c-000010
|
|
tags:
|
|
- concept
|
|
- finite-element-method
|
|
- incompressibility
|
|
status: current
|
|
related:
|
|
- "[[Displacement-Based Finite Element Formulation]]"
|
|
- "[[Isoparametric Finite Elements]]"
|
|
- "[[Nonlinear Finite Element Analysis]]"
|
|
- "[[Assumed Transverse Shear Strain Interpolation]]"
|
|
- "[[Shell Locking Phenomenon]]"
|
|
- "[[Uniform Optimal Convergence]]"
|
|
- "[[Incompatible Mode Solid Elements]]"
|
|
sources:
|
|
- "[[Finite Element Procedures]]"
|
|
- "[[A Continuum Mechanics Based Four-Node Shell]]"
|
|
- "[[On-the-Finite-Element-Analysis-of-Shell-Structures]]"
|
|
- "[[Solid Element Notes]]"
|
|
---
|
|
|
|
# Mixed Finite Element Formulations
|
|
|
|
## Definition
|
|
|
|
Mixed finite element formulations approximate more than one primary field, such as displacement and pressure, instead of relying only on displacement unknowns.
|
|
|
|
## How It Works
|
|
|
|
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.
|
|
|
|
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.
|
|
|
|
[[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.
|
|
|
|
[[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.
|
|
|
|
## Why It Matters
|
|
|
|
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.
|
|
|
|
## Connections
|
|
|
|
- [[Isoparametric Finite Elements]] supplies the element construction machinery.
|
|
- [[Nonlinear Finite Element Analysis]] uses mixed formulations for large deformation incompressible behavior.
|
|
- [[Finite Element Heat Transfer and Field Problems]] uses analogous ideas when multiple fields interact.
|
|
|
|
## Sources
|
|
|
|
- [[Finite Element Procedures]]
|
|
- [[A Continuum Mechanics Based Four-Node Shell]]
|
|
- [[On-the-Finite-Element-Analysis-of-Shell-Structures]]
|
|
- [[Solid Element Notes]]
|