Heat Source Calculation
Consider a ray j transferring an amount of power Qj through a domain. During a small time interval Δt, the ray propagates a short distance Δs from initial position qj to position qj+Δqj and its power decreases by an amount ΔQ. Then conservation of energy dictates that the deposited ray power in the domain should increase by ΔQ:
where δ is the Dirac delta function. In the limit as this time derivative becomes arbitrarily small, this expression can be rewritten as
In this expression, every ray is treated as an infinitesimally small point source. This can lead to numerical issues because the heat source becomes infinitely large at the location of each point source. In the following section an alternative expression for the domain heat source is given.
Creating a Bounded Heat Source Term
The Ray Heat Source node defines a variable for the contribution to the heat source by rays in each mesh element. This variable uses constant shape functions, so it may be discontinuous across boundaries between elements. For a mesh element j with volume Vj the average heat source Qsrc,j changes according to the expression
The integral on the right-hand side is a volume integral over element j. The resulting time derivative of the heat source is the average value over the mesh element, which may be written more concisely as
where the sum is taken over all rays that are within mesh element j.