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+# This file is a part of JuliaFEM.
+# License is MIT: see https://github.com/JuliaFEM/JuliaFEM.jl/blob/master/LICENSE.md
+
+"""
+Formulation API definitions.
+
+This file defines formulation abstractions - the mathematical discretization strategies
+for different types of FEM problems.
+
+Must be included after fields/api.jl (formulations work with fields).
+"""
+
+# ============================================================================
+# FORMULATION INTERFACE
+# ============================================================================
+
+"""
+    AbstractFormulation
+
+Abstract type for discretization formulations.
+
+**DESIGN PHILOSOPHY: Formulations are DOMAIN-AGNOSTIC dimensionality concepts.**
+
+# Key Distinction: Formulation vs Theory
+
+**Formulation** (this file):
+- Describes DIMENSIONALITY and geometric simplifications
+- Domain-agnostic: Used by multiple physics domains
+- Examples: FullThreeD, TwoDimensional{T}, Axisymmetric
+- Located in: `src/formulations/api.jl`
+
+**Theory** (domain-specific):
+- Describes PHYSICS assumptions (stress/strain, kinematics)
+- Domain-specific: Only meaningful for one physics domain
+- Examples: PlaneStress (continuum), Kirchhoff (plates), Timoshenko (beams)
+- Located in: `src/domains/*/theories.jl`
+
+# Why Separate Them?
+
+**Problem**: Heat transfer needs FullThreeD and Axisymmetric, just like continuum mechanics!
+If FullThreeD is defined in `domains/continuum/`, heat can't use it without duplication.
+
+**Solution**: Formulations are dimensionality (shared), theories are physics (domain-specific).
+
+```julia
+# Domain-agnostic formulations (this file)
+FullThreeD()              # Used by: continuum, heat, poisson, acoustics
+TwoDimensional{T}()       # Parametric! T is domain theory or Nothing
+Axisymmetric()            # Used by: continuum, heat, etc.
+
+# Domain-specific theories (in domains/*/theories.jl)
+PlaneStress               # domains/continuum/theories.jl
+PlaneStrain               # domains/continuum/theories.jl
+Kirchhoff                 # domains/plates/theories.jl
+Timoshenko                # domains/beams/theories.jl
+```
+
+# Parametric Formulations
+
+Use `TwoDimensional{Theory}` for 2D problems:
+
+```julia
+# Continuum mechanics with plane stress theory
+TwoDimensional{PlaneStress}()
+
+# Heat transfer (no special theory needed)
+TwoDimensional{Nothing}()
+
+# Plates with Kirchhoff theory
+TwoDimensional{Kirchhoff}()
+```
+
+# Examples
+
+```julia
+# 3D solid mechanics (continuum)
+physics = Physics(
+    formulation = FullThreeD(),           # Domain-agnostic!
+    field = Displacement{3}(),
+    mesh = mesh,
+    material = steel
+)
+
+# 3D heat transfer (reuses SAME formulation!)
+physics_heat = Physics(
+    formulation = FullThreeD(),           # SAME as continuum!
+    field = Temperature(),
+    mesh = mesh,
+    material = steel
+)
+
+# 2D plane stress (continuum with theory)
+physics_2d = Physics(
+    formulation = TwoDimensional{PlaneStress}(),  # Formulation + theory
+    field = Displacement{2}(),
+    mesh = mesh_2d,
+    material = aluminum
+)
+
+# Axisymmetric heat (reuses SAME formulation as continuum!)
+physics_axisym = Physics(
+    formulation = Axisymmetric(),         # Domain-agnostic!
+    field = Temperature(),
+    mesh = mesh_2d,
+    material = steel
+)
+```
+
+# Assembly Dispatch
+
+Assembly methods dispatch on formulation × field × domain:
+
+```julia
+# 3D continuum mechanics
+function assemble!(physics::Physics{FullThreeD, Displacement{3}, M, Mat}, ...)
+    # Standard 3D displacement-based assembly
+    # Implementation in src/assembly/continuum_3d.jl
+end
+
+# 3D heat transfer (SAME formulation, different field!)
+function assemble!(physics::Physics{FullThreeD, Temperature, M, Mat}, ...)
+    # Thermal assembly (scalar field)
+    # Implementation in src/assembly/heat.jl
+end
+
+# 2D plane stress
+function assemble!(physics::Physics{TwoDimensional{PlaneStress}, Displacement{2}, M, Mat}, ...)
+    # 2D assembly with plane stress assumptions
+    # Implementation in src/assembly/continuum_2d.jl
+end
+```
+
+# See Also
+- Field types: src/fields/api.jl (Displacement, Temperature)
+- Physics coupling: src/physics/api.jl (AbstractPhysics)
+- Domain theories: src/domains/continuum/theories.jl, src/domains/plates/theories.jl
+- Assembly implementations: src/assembly/continuum_3d.jl, src/assembly/heat.jl
+- Architectural rationale: docs/src/developer/FORMULATIONS_AND_SOLVERS.md
+"""
+abstract type AbstractFormulation end
+
+# ============================================================================
+# CONTINUUM FORMULATION (Standard FEM)
+# ============================================================================
+
+"""
+    AbstractContinuumTheory
+
+**LEGACY**: Theory types currently in formulations/ for backward compatibility.
+
+**FUTURE ARCHITECTURE**: These should move to `src/domains/continuum/theories.jl`
+to properly separate domain-agnostic formulations from domain-specific theories.
+
+# Current Theories (Will Move to domains/continuum/theories.jl)
+- `PlaneStress` - 2D plane stress (σ_zz = 0, thin plates)
+- `PlaneStrain` - 2D plane strain (ε_zz = 0, thick plates)
+
+# Domain-Agnostic Formulations (Stay Here)
+- `FullThreeD` - Full 3D analysis (used by continuum AND heat!)
+- `Axisymmetric` - Axisymmetric analysis (used by continuum AND heat!)
+
+# Theory Selection Guidelines
+
+**PlaneStress (σ_xx, σ_yy, σ_xy, σ_zz = 0):**
+- Thin plates and membranes (thickness << length/width)
+- Out-of-plane stress σ_zz = 0
+- Examples: Sheet metal, aircraft skin, thin-walled structures
+- **Domain**: Continuum mechanics only
+
+**PlaneStrain (ε_xx, ε_yy, ε_xy, ε_zz = 0):**
+- Thick sections with no variation in z-direction
+- Out-of-plane strain ε_zz = 0
+- Examples: Dams, tunnels, retaining walls, long cylinders
+- **Domain**: Continuum mechanics only
+
+# Mathematical Details
+
+**Plane Stress (thin plate):**
+- Stress state: σ_zz = σ_xz = σ_yz = 0
+- Strain: ε_zz ≠ 0 (computed from σ_zz = 0 condition)
+- Constitutive: 3×3 reduced stiffness matrix
+
+**Plane Strain (thick section):**
+- Strain state: ε_zz = γ_xz = γ_yz = 0
+- Stress: σ_zz ≠ 0 (computed from ε_zz = 0 condition)
+- Constitutive: 3×3 reduced stiffness matrix (different from plane stress!)
+
+# See Also
+- Future location: `src/domains/continuum/theories.jl`
+- Architectural rationale: `docs/src/developer/FORMULATIONS_AND_SOLVERS.md`
+"""
+abstract type AbstractContinuumTheory end
+
+"""
+    FullThreeD <: AbstractContinuumTheory
+
+Full 3D analysis with no simplifications.
+
+**DOMAIN-AGNOSTIC**: Can be used by ANY physics domain!
+
+# Usage Across Domains
+
+```julia
+# Continuum mechanics (solid mechanics)
+physics_solid = Physics(
+    formulation = FullThreeD(),
+    field = Displacement{3}(),
+    ...
+)
+
+# Heat transfer (SAME formulation!)
+physics_heat = Physics(
+    formulation = FullThreeD(),
+    field = Temperature(),
+    ...
+)
+
+# Poisson equation (SAME formulation!)
+physics_poisson = Physics(
+    formulation = FullThreeD(),
+    field = Potential(),
+    ...
+)
+```
+
+# Details
+- All six stress/flux components in continuum context
+- No geometric simplifications
+- Most accurate but most expensive
+"""
+struct FullThreeD <: AbstractContinuumTheory end
+
+"""
+    PlaneStress <: AbstractContinuumTheory
+
+2D plane stress assumption (σ_zz = 0).
+
+Applicable to thin plates and membranes where thickness << in-plane dimensions.
+"""
+struct PlaneStress <: AbstractContinuumTheory end
+
+"""
+    PlaneStrain <: AbstractContinuumTheory
+
+2D plane strain assumption (ε_zz = 0).
+
+Applicable to thick sections with no variation in z-direction.
+"""
+struct PlaneStrain <: AbstractContinuumTheory end
+
+"""
+    Axisymmetric <: AbstractContinuumTheory
+
+Axisymmetric analysis (rotation around z-axis).
+
+**DOMAIN-AGNOSTIC**: Can be used by ANY physics domain with axial symmetry!
+
+# Usage Across Domains
+
+```julia
+# Continuum mechanics (pressure vessel)
+physics_vessel = Physics(
+    formulation = Axisymmetric(),
+    field = Displacement{2}(),  # (r, z) displacements
+    ...
+)
+
+# Heat transfer in cylinder (SAME formulation!)
+physics_heat = Physics(
+    formulation = Axisymmetric(),
+    field = Temperature(),  # T(r, z)
+    ...
+)
+```
+
+# Details
+- Geometry and loading symmetric about z-axis
+- No circumferential variations (∂/∂θ = 0)
+- 2D mesh in (r, z) plane represents 3D geometry
+- Examples: Pressure vessels, pipes, rotating disks, cylinders
+"""
+struct Axisymmetric <: AbstractContinuumTheory end
+
+"""
+    ContinuumFormulation{Theory} <: AbstractFormulation
+
+Standard continuum mechanics formulation with theory variant.
+
+This is the fundamental FEM formulation for solid mechanics, heat transfer,
+and other continuum physics problems.
+
+# Type Parameter
+- `Theory <: AbstractContinuumTheory` - Dimensionality/simplification theory
+
+# Examples
+
+```julia
+# 3D elasticity
+physics = Physics(
+    formulation = ContinuumFormulation{FullThreeD}(),
+    field = Displacement{3}(),
+    mesh = mesh,
+    material = steel
+)
+
+# 2D plane stress (thin plate)
+physics_2d = Physics(
+    formulation = ContinuumFormulation{PlaneStress}(),
+    field = Displacement{2}(),
+    mesh = mesh_2d,
+    material = aluminum
+)
+
+# 2D plane strain (thick section)
+physics_2d = Physics(
+    formulation = ContinuumFormulation{PlaneStrain}(),
+    field = Displacement{2}(),
+    mesh = mesh_2d,
+    material = concrete
+)
+
+# Axisymmetric (cylinder)
+physics_axisym = Physics(
+    formulation = ContinuumFormulation{Axisymmetric}(),
+    field = Displacement{2}(),  # (r, z) displacements
+    mesh = mesh_2d,
+    material = steel
+)
+```
+
+# Assembly Dispatch
+
+Assembly methods specialize on theory × field combinations:
+
+```julia
+# 3D solid mechanics
+function assemble!(physics::Physics{ContinuumFormulation{FullThreeD}, Displacement{3}, M, Mat})
+    # Standard 3D displacement-based assembly
+    # Full 6×6 strain-displacement matrix (Bε)
+    # 6×6 constitutive matrix (Dε)
+end
+
+# 2D plane stress
+function assemble!(physics::Physics{ContinuumFormulation{PlaneStress}, Displacement{2}, M, Mat})
+    # 2D assembly with plane stress assumptions
+    # 3×3 reduced strain-displacement matrix
+    # 3×3 plane stress constitutive matrix
+end
+
+# Heat transfer (same formulation, different field!)
+function assemble!(physics::Physics{ContinuumFormulation{FullThreeD}, Temperature, M, Mat})
+    # Thermal assembly (scalar field)
+    # Thermal conductivity matrix
+end
+```
+
+# Implementation Location
+
+Concrete assembly implementations are in:
+- `src/assembly/continuum_3d.jl` - 3D continuum mechanics
+- `src/assembly/continuum_2d.jl` - 2D plane stress/strain
+- `src/assembly/axisymmetric.jl` - Axisymmetric problems
+
+# See Also
+- [`AbstractContinuumTheory`](@ref) - Theory variants
+- Field types: src/fields/api.jl (Displacement, Temperature)
+- Physics coupling: src/physics/api.jl (AbstractPhysics)
+- Assembly: src/assembly/continuum_*.jl
+"""
+struct ContinuumFormulation{Theory<:AbstractContinuumTheory} <: AbstractFormulation end