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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
+
+#=
+Classical Hellinger–Reissner mixed formulation for small-strain linear
+elasticity (3D): displacement `u` at vertices and a piecewise-constant
+symmetric stress tensor `σ` on each cell (`DOF{SymmetricTensor{2,3}, Cell}`).
+
+DOF layout (element template `S`):
+
+    `@DOFSet{u::DOF{Displacement{3}, Vertex}, σ::DOF{SymmetricTensor{2,3}, Cell}}`
+
+Symmetric HR bilinear form (steady, no body-force contribution in `K`):
+
+    a((u,σ), (v,τ)) = −∫ C⁻¹σ : τ dΩ + ∫ σ : ε(v) dΩ + ∫ τ : ε(u) dΩ
+
+with `C` the fourth-order elasticity from `LinearElastic` at each
+quadrature point (constant here). The discrete σ–σ block uses the
+Voigt-unit tensors implied by `SymmetricTensor{2,3}(ntuple(...))` —
+the same ordering as `local_dof_layout` components 1…6 — so the
+6×6 matrix `G M⁻¹ G` with `Mᵢⱼ = Eᵢ : C : Eⱼ` and `Gᵢⱼ = Eᵢ : Eⱼ`
+matches the double-contraction inner product on `span{E₁…E₆} = Sym(3)`.
+
+The displacement–displacement block vanishes (`K_uu = 0`). The global
+system is symmetric indefinite; `MatrixFreeOperator` declares
+`isposdef == false` like `MixedUPKernel`.
+
+Currently `material` must be `LinearElastic` (constant `C`); the kernel
+precomputes the discrete compliance `G M⁻¹ G` once at construction.
+=#
+
+using LinearAlgebra
+using StaticArrays: SMatrix, MMatrix
+using Tensors
+using Tensors: basevec
+
+using ..JuliaFEM: AbstractKernel
+using ..JuliaFEM: ContinuumFormulation, AbstractContinuumTheory
+using ..JuliaFEM: LinearElastic, AssemblyMaterialWorkspace, compute_stress
+import ..JuliaFEM: qpoint_buffer_eltype, update_qpoint_buffer!, evaluate_entry,
+                   evaluate_mass_entry,
+                   reference_fields, get_field, dofs_per_node
+using ..JuliaFEM: DOFLayoutEntry, field_idx, entity_local, component, extract_tangent!
+
+# Unit one-hot symmetric tensors in Tensors.jl Voigt layout (matches `local_dof_layout` components).
+const _HR_VOIGT_UNITS = ntuple(
+    c -> SymmetricTensor{2,3}(ntuple(i -> Float64(i == c), 6)),
+    6,
+)::NTuple{6,SymmetricTensor{2,3,Float64,6}}
+
+@inline _hr_unit_stress(c::Int) = @inbounds _HR_VOIGT_UNITS[c]
+
+function _hr_discrete_compliance(C::SymmetricTensor{4,3,Float64,36})
+    Es = _HR_VOIGT_UNITS
+    Mm = MMatrix{6,6,Float64}(undef)
+    Gm = MMatrix{6,6,Float64}(undef)
+    @inbounds for j in 1:6, i in 1:6
+        Mm[i, j] = dcontract(Es[i], dcontract(C, Es[j]))
+        Gm[i, j] = dcontract(Es[i], Es[j])
+    end
+    Ms = Symmetric(SMatrix(Mm))
+    Gs = SMatrix(Gm)
+    return Gs * (Ms \ Gs)
+end
+
+"""
+    HellingerReissnerKernel{Theory}
+
+Hellinger–Reissner mixed kernel: 3D vertex displacement + cell-wise
+`SymmetricTensor{2,3}` stress (6 scalars per element). See the file-level
+docstring for the weak form.
+
+Only [`LinearElastic`](@ref) is supported; `C` is taken at zero strain and
+the discrete `σ`–`σ` block uses `G M⁻¹ G` in the Voigt component basis.
+
+# Example
+
+```julia
+S = @DOFSet{u::DOF{Displacement{3}, Vertex}, σ::DOF{SymmetricTensor{2,3}, Cell}}
+kernel = HellingerReissnerKernel(
+    ContinuumFormulation{FullThreeD}(),
+    LinearElastic(E = 210e9, ν = 0.3),
+)
+```
+"""
+struct HellingerReissnerKernel{Theory<:AbstractContinuumTheory} <: AbstractKernel
+    formulation::ContinuumFormulation{Theory}
+    material::LinearElastic
+    """Discrete compliance in Voigt σ-components: `K_σσ[i,j] = -vol * σσ[i,j]`."""
+    σσ::SMatrix{6,6,Float64,36}
+end
+
+function HellingerReissnerKernel(
+    formulation::ContinuumFormulation{Theory},
+    material::LinearElastic,
+) where {Theory<:AbstractContinuumTheory}
+    ε_ref = zero(SymmetricTensor{2,3,Float64,6})
+    _, C, _ = compute_stress(material, ε_ref, NamedTuple(), 0.0)
+    σσ = _hr_discrete_compliance(C)
+    return HellingerReissnerKernel{Theory}(formulation, material, σσ)
+end
+
+# Mixed u–σ system; matrix-free K is symmetric indefinite.
+@inline operator_is_posdef(::HellingerReissnerKernel) = false
+
+function get_field(::K) where {K<:HellingerReissnerKernel}
+    error("$(K) is mixed u–σ — use `local_dof_layout(E)` and `elem.dof_indices`; " *
+          "do not call `get_field(kernel)`.")
+end
+
+# 3 displacement + 6 stress "per cell slot" is not uniform; use an upper
+# bound on (total element DOFs / nnodes) for `ElementCache.dofs` sizing.
+# `ceil((24+6)/8) = 4` for Hex8; linear Tet4 needs `ceil(18/4) = 5`.
+@inline dofs_per_node(::HellingerReissnerKernel) = 5
+
+@inline qpoint_buffer_eltype(::HellingerReissnerKernel) = SymmetricTensor{4,3,Float64,36}
+
+@inline function reference_fields(kernel::HellingerReissnerKernel)
+    ε_ref = zero(SymmetricTensor{2,3,Float64,6})
+    σ_ref, 𝔻_ref, _ = compute_stress(kernel.material, ε_ref, NamedTuple(), 0.0)
+    return ((σ = σ_ref, 𝔻 = 𝔻_ref), NamedTuple())
+end
+
+@inline function update_qpoint_buffer!(
+    buffer::AbstractVector{SymmetricTensor{4,3,Float64,36}},
+    workspace::AssemblyMaterialWorkspace{FieldType, StateType},
+    ::HellingerReissnerKernel,
+) where {FieldType, StateType}
+    fields = getfield(workspace, 1)
+    extract_tangent!(buffer, fields, FieldType)
+    return nothing
+end
+
+@inline function _hr_B_symmetric(∇N::Vec{3,Float64}, comp_u::Int)
+    eα = basevec(Vec{3,Float64}, comp_u)
+    h = 0.5
+    B = h * (∇N ⊗ eα + eα ⊗ ∇N)
+    return symmetric(B)
+end
+
+"""
+    evaluate_entry(kernel::HellingerReissnerKernel, geometry_cache, 𝔻_vec, layout_i, layout_j, elem_id::Int)
+
+Volume kernel; `elem_id` is unused.
+
+| (field_i, field_j) | block | contribution |
+| ------------------ | ----- | ------------ |
+| (1, 1)             | `K_uu` | `0` |
+| (1, 2)             | `K_uσ` | `+ Σ_q E_j : ε(N_i, α_i) detJ_w` |
+| (2, 1)             | `K_σu` | `+ Σ_q E_i : ε(N_j, α_j) detJ_w` |
+| (2, 2)             | `K_σσ` | `− (Σ_q detJ_w) · (G M⁻¹ G)_{i,j}` |
+
+`E_k` is the unit Voigt basis stress for component `k`; `ε(N,α)` the
+symmetric gradient of shape function `N` for displacement component `α`.
+"""
+@inline function evaluate_entry(
+    kernel::HellingerReissnerKernel,
+    geometry_cache,
+    𝔻_vec::AbstractVector{<:SymmetricTensor{4,3}},
+    layout_i::DOFLayoutEntry,
+    layout_j::DOFLayoutEntry,
+    ::Int,
+)
+    fi = field_idx(layout_i)
+    fj = field_idx(layout_j)
+    node_i = entity_local(layout_i)
+    node_j = entity_local(layout_j)
+    comp_i = component(layout_i)
+    comp_j = component(layout_j)
+
+    n_ips = length(geometry_cache.detJ_w)
+    K_ij = 0.0
+
+    if fi == 1 && fj == 1
+        return 0.0
+
+    elseif fi == 1 && fj == 2
+        Eσ = _hr_unit_stress(comp_j)
+        @inbounds for q in 1:n_ips
+            ∇N_i = geometry_cache.∇N_data[q, node_i]
+            detJw = geometry_cache.detJ_w[q]
+            B = _hr_B_symmetric(∇N_i, comp_i)
+            K_ij += dcontract(Eσ, B) * detJw
+        end
+
+    elseif fi == 2 && fj == 1
+        Eσ = _hr_unit_stress(comp_i)
+        @inbounds for q in 1:n_ips
+            ∇N_j = geometry_cache.∇N_data[q, node_j]
+            detJw = geometry_cache.detJ_w[q]
+            B = _hr_B_symmetric(∇N_j, comp_j)
+            K_ij += dcontract(Eσ, B) * detJw
+        end
+
+    else  # fi == 2 && fj == 2
+        vol = 0.0
+        @inbounds for q in 1:n_ips
+            vol += geometry_cache.detJ_w[q]
+        end
+        K_ij = -vol * kernel.σσ[comp_i, comp_j]
+    end
+
+    return K_ij
+end
+
+@inline evaluate_mass_entry(
+    ::HellingerReissnerKernel,
+    geometry_cache,
+    qp_buffer,
+    layout_i::DOFLayoutEntry,
+    layout_j::DOFLayoutEntry,
+) = 0.0