Complete transient heat workflow

examples/transient_heat_decay.py is a runnable 2D/3D example using the public Study, model, field, constraint, step and result APIs. It includes initialization, time advancement, center-temperature history, field output and an analytical comparison. The application does not manually copy time-step arrays or assemble a new matrix in a Python loop; the selected platform procedure owns that work.

python examples/transient_heat_decay.py --dimension 2 --cells 12 --steps 72 --output heat_2d.xdmf
python examples/transient_heat_decay.py --dimension 3 --cells 12 --steps 72 --output heat_3d.xdmf

Use an installed development environment, or run from the source parent as described in the repository workflow. The script does not modify Python paths.

The domain is the unit square/cube. Conductivity, density and heat capacity are one. All faces are fixed at 300 K; the initial excess temperature is the product of sin(pi*x_i). The exact temperature is 300 + exp(-dimension*pi**2*t) * product(sin(pi*x_i)). The reported relative L2 error is normalized by the excess temperature, not by the 300 K background, which would conceal error in the decaying signal.

The example uses implicit Euler and first-order tensor-product elements. Refine both mesh and time step to check combined convergence; this alone does not measure the spatial or temporal convergence order separately.

expressions.interpolate initializes the field using a coordinate expression; pi is built in, and must not be passed as a user parameter. The material's reference temperature is absolute kelvin. In this thermal-only Study the mechanical coefficients of the shared thermoelastic material are not used.

The XDMF/HDF5 pair contains the computed fields. The neighboring result JSON contains the center probe history and the reported relative error. Failed or non-finite runs exit unsuccessfully. A finite result is not by itself a claim that a chosen grid meets an engineering tolerance: inspect the reported error.