FEM Thermal

FEM Thermal#

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

Formulation#

Thermal FEM is a specialization of the 1-DOF diffusion model, whereby spatial integration is performed using a Continuous Galerkin technique (constructing the solution piecewise using shapefunctions in a transformed basis). Such heat transfer problems are solved using Temperature as the single degree-of-freedom (DOF).

Materials#

Thermal FEM requires the following material properties:

  • Mass|Density

  • SpecificHeat

  • ThermalConductivity

If materials are not defined for a volume region, properties default to the physical constants.

Initial Conditions#

For transient problems, an initial state must be defined.

Boundary Conditions#

For Thermal FEM, it is not required to set boundary conditions on all surfaces. Free boundaries will experience no gradients normal to the surface.

Dirichlet (state)#

Temperature asserts values onto a surface in units of Kelvin.

Neumann (gradient)#

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

Thermal Radiation asserts Stefen-Boltzmann emission normal to the surface based on local temperature and emissivity. It is only compatitble with transient solvers (such as Linear Euler) and radiation requires a sufficiently-small timestep to prevent numerical overshoot.

Robin (composite)#

Thermal Convection asserts both a reference temperature (K) and heat transfer coefficient (W/m2-K) to model surface convection (as driven by fluid flow, etc).

Solvers#

Once the physics and conditions are defined, an adjacency matrix A is assembled to represented the behavior of the system. A residual vector b is also defined. 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.

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.