modify wiki

wiki/concepts/Mixed Finite Element Formulations.md

@@ -9,7 +9,7 @@ aliases:
   - inf-sup condition
   - assumed strain formulation
 created: 2026-05-28
-updated: 2026-05-28
+updated: 2026-05-29
 address: c-000010
 tags:
   - concept
@@ -24,11 +24,14 @@ related:
   - "[[Shell Locking Phenomenon]]"
   - "[[Uniform Optimal Convergence]]"
   - "[[Incompatible Mode Solid Elements]]"
+  - "[[Hybrid Incompressible Elements]]"
+  - "[[Reduced Integration and Hourglass Control]]"
 sources:
   - "[[Finite Element Procedures]]"
   - "[[A Continuum Mechanics Based Four-Node Shell]]"
   - "[[On-the-Finite-Element-Analysis-of-Shell-Structures]]"
   - "[[Solid Element Notes]]"
+  - "[[Abaqus Theory Manual]]"
 ---
 
 # Mixed Finite Element Formulations
@@ -47,6 +50,8 @@ The four-node shell paper is not simply a displacement/pressure mixed formulatio
 
 [[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.
 
+[[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.
+
 ## 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.
@@ -56,6 +61,7 @@ Mixed formulations are needed when displacement-only elements lock, produce spur
 - [[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.
+- [[Reduced Integration and Hourglass Control]] is a related numerical remedy, but hybrid elements make the pressure constraint explicit.
 
 ## Sources
 
@@ -63,3 +69,4 @@ Mixed formulations are needed when displacement-only elements lock, produce spur
 - [[A Continuum Mechanics Based Four-Node Shell]]
 - [[On-the-Finite-Element-Analysis-of-Shell-Structures]]
 - [[Solid Element Notes]]
+- [[Abaqus Theory Manual]]