docs(api): reorganize API documentation structure

Reorganize API documentation with organized sections for better
navigation and discoverability.

- Add structured sections: Elements, Physics, Fields, Formulations,
  Materials, Mesh, Topology, Basis Functions, DOF System, Solvers
- Include key types and functions in each section
- Change title from "API documentation" to "API Reference"
- Keep index at the end for complete reference

docs/src/api.md

@@ -1,9 +1,6 @@
-# API documentation
+# API Reference
 
-## Index
-
-```@index
-```
+Complete API documentation for JuliaFEM.jl.
 
 ```@meta
 DocTestSetup = quote
@@ -11,17 +8,143 @@ DocTestSetup = quote
 end
 ```
 
-## Types
+## Core Types
+
+### Elements
 
 ```@docs
-JuliaFEM.Element
-JuliaFEM.Problem
-JuliaFEM.Elasticity
+Element
+topology_type
+basis_type
+dof_type
+element_id
+element_dofs
+n_element_dofs
 ```
 
-# Functions
+### Physics
 
 ```@docs
-JuliaFEM.update!
+Physics
+AbstractPhysics
+DirichletBC
+NeumannBC
+Constraint
+assemble!
+solve!
+add_dirichlet!
+add_neumann!
 ```
 
+### Fields
+
+```@docs
+AbstractField
+Displacement
+Temperature
+DisplacementRotation
+```
+
+### Formulations
+
+```@docs
+AbstractFormulation
+ContinuumFormulation
+AbstractContinuumTheory
+FullThreeD
+PlaneStress
+PlaneStrain
+Axisymmetric
+```
+
+### Materials
+
+```@docs
+AbstractMaterial
+AbstractElasticMaterial
+AbstractPlasticMaterial
+LinearElastic
+NeoHookean
+compute_stress
+elasticity_tensor
+```
+
+### Mesh
+
+```@docs
+AbstractMesh
+Mesh
+nnodes_total
+nelements
+get_node
+connectivity_matrix
+get_element_set
+get_node_set
+```
+
+### Topology
+
+```@docs
+AbstractTopology
+Segment
+Triangle
+Quadrilateral
+Tetrahedron
+Hexahedron
+Pyramid
+Wedge
+Seg2
+Seg3
+Tri3
+Tri6
+Tri7
+Quad4
+Quad8
+Quad9
+Tet4
+Tet10
+Hex8
+Hex20
+Hex27
+Pyr5
+Wedge6
+Wedge15
+```
+
+### Basis Functions
+
+```@docs
+AbstractBasis
+Lagrange
+Serendipity
+get_basis_functions
+get_basis_derivatives
+eval_basis!
+eval_dbasis!
+```
+
+### DOF System
+
+```@docs
+AbstractDOF
+DOFSet
+@DOFSet
+DOFManager
+register_fields!
+create_elements!
+```
+
+### Solvers
+
+```@docs
+Solver
+Linear
+Analysis
+add_problems!
+run!
+```
+
+## Index
+
+```@index
+```