JuliaFEM.jl/docs/src/thermo_elastic_walkthrough.md

[Thermo-elasticity: multi-field walkthrough](@id thermo_elastic_walkthrough)

This page is a narrative companion to test/domains/thermo_elastic/test_thermo_elastic_kernel.jl, which is the executable contract for the coupled u + T path through the DOF-based assembler.

What you assemble

ThermoElasticKernel couples vertex displacement u (3 DOFs per node) with vertex temperature T (1 DOF per node). The global stiffness has blocks K_uu, K_TT, and (when the coupling coefficient β ≠ 0) off-diagonal K_uT / K_Tu. With β = 0 the kernel is block-diagonal and matches independent elasticity and heat solves on the same mesh.

Minimal code (block-diagonal smoke)

The following uses the same material stack as the regression test, a small Hex8 box, and DOFBasedCOOCache (the multi-field kernel is wired through this cache type rather than the narrow create_cache overload used for ContinuumKernel alone).

using JuliaFEM
using SparseArrays

mesh = create_structured_box_mesh(Hex8;
    xmin = 0.0, xmax = 1.0, nx = 2,
    ymin = 0.0, ymax = 1.0, ny = 2,
    zmin = 0.0, zmax = 1.0, nz = 2,
)

mech  = LinearElastic(E = 210e9, ν = 0.3)
therm = HeatConductivity(k = 50.2)
β     = 0.0
kernel = ThermoElasticKernel(ContinuumFormulation{FullThreeD}(), mech, therm, β)

S = @DOFSet{u::DOF{Displacement{3}, Vertex},
            T::DOF{Temperature, Vertex}}
elements, handler = create_elements!(mesh, Element{Hex8, Lagrange{1}, S})

asm   = DOFBasedCOOAssembler()
cache = DOFBasedCOOCache(elements, handler, mesh, kernel)
assemble!(cache, asm, kernel, mesh)
K, f = extract_system(cache)

@assert size(K, 1) == size(K, 2) == 108
@assert nnz(K) == 5488

Where to read next

• Kernel source and coupling equations: src/domains/thermo_elastic/kernel.jl.
• Compile-time DOF layout: local_dof_layout(Element{Hex8, Lagrange{1}, S}).
• Matrix-free parity with K * x: same test file as above.