FEM Thermal

FEM Thermal#

The following tutorial assumes that a sufficient-quality volumetric mesh has been generated with segmented regions to apply boundary conditions.

Formulation#

Thermal Diffusion is currently fit for isotropic materials. A Continuous Galerkin approach constructs a piecewise solution using shapefunctions, the linear system solved with a hardware-accelerated conjugate gradient method. Such heat transfer problems are posited in weak form using Temperature as the single degree-of-freedom (DOF).

Materials#

Thermal FEM requires the following material properties:

  • Mass|Density

  • SpecificHeat

  • ThermalConductivity

Varying material properties can be applied to different volumes. If materials are not defined for a volume region, properties default to the physical constants for that system.

Initial Conditions#

For transient problems, an initial temperature state must be defined.

Boundary Conditions#

For Thermal FEM, boundary conditions are not required on all surfaces. Free boundaries experience no gradients normal to the surface (and thus also act as a symmetry).

Dirichlet (value)#

Temperature asserts a scalar value onto a surface in units of Kelvin.

Neumann (gradient)#

Heat Flux asserts a scalar heat flow per unit area (W/m2) normal to the surface. Positive is heat gain into the domain.

Thermal Radiation (beta) asserts Stefen-Boltzmann emission normal to the surface based on local temperature and emissivity. Due to additional temporal stability requirements, this boundary condition requires a time-varying analysis and Linear Newmark integrator, which is a work-in-progress.

Robin (composite)#

Thermal Convection asserts both a scalar reference temperature (K) and scalar heat transfer coefficient (W/m2-K) heat flow normal to the surface.

Solvers#

Once the physics and conditions are defined, an adjacency matrix A is assembled to represented the behavior of the system. The residual vector b is defined, and in preparation for solution v, linear operations are performed to manipulate the matrix and residual vector into a solvable system-of equations of form A*v = b.

The Conjugate Gradient method solves such 1-DOF formulations on the available GPU within a tolerance defined by Convergence in compute.cfg maximum double-precision tolerance on order of 1e-16.

Steady-State#

Linear Steady State converges upon the solution for t->infinity.

Transient#

Linear Euler converges upon the solution iteratively in time with timestep defined in the solver settings.