FEM Thermal

FEM Thermal#

This tutorial assumes a sufficient-quality volumetric mesh has been generated with segmented regions ready for boundary conditions.

Formulation#

Thermal diffusion is modeled for isotropic materials using a continuous Galerkin approach. The linear system is solved via a hardware-accelerated conjugate gradient method. Heat transfer is posited in weak form using Temperature as the single degree-of-freedom (DOF).

Material Properties#

  • Mass|Density

  • SpecificHeat

  • ThermalConductivity

Properties can be assigned to specific volumes. Unassigned volumes default to system physical constants.

Initial Conditions#

Transient problems require an initial temperature state definition.

Boundary Conditions#

Not all surfaces require boundary conditions. Free boundaries experience zero normal gradient (acting as symmetry planes).

Dirichlet (Value)#

  • Temperature: Scalar value in Kelvin applied to a surface.

Neumann (Gradient)#

  • Heat Flux: Scalar heat flow per unit area (W/m²) normal to the surface. Positive indicates heat gain.

  • Thermal Radiation: Stefan-Boltzmann emission based on local temperature and emissivity. Requires time-varying analysis and Linear Newmark integration (work-in-progress).

Robin (Composite)#

  • Thermal Convection: Combines reference temperature (K) and heat transfer coefficient (W/m²·K) for surface heat exchange.

Solvers#

An adjacency matrix A and residual vector b are assembled and transformed into A*v = b. The Conjugate Gradient solver resolves 1-DOF formulations on the GPU to ~1e-16 tolerance.

Steady-State#

Linear Steady State converges to the solution as t → ∞.

Transient#

Linear Euler provides time-iterative solutions with user-defined timesteps.