Precomputed Radiative Heat Transport for Efficient Thermal Simulation

Abstract Architectural design and urban planning are complex design tasks. Predicting the thermal impact of design choices at interactive rates enhances the ability of designers to improve energy efficiency and avoid problematic heat islands while maintaining design quality. We show how to use and adapt methods from computer graphics to efficiently simulate heat transfer via thermal radiation, thereby improving user guidance in the early design phase of large‐scale construction projects and helping to increase energy efficiency and outdoor comfort. Our method combines a hardware‐accelerated photon tracing approach with a carefully selected finite element discretization, inspired by precomputed radiance transfer. This combination allows us to precompute a radiative transport operator, which we then use to rapidly solve either steady‐state or transient heat transport throughout the entire scene. Our formulation integrates time‐dependent solar irradiation data without requiring changes in the transport operator, allowing us to quickly analyze many different scenarios such as common weather patterns, monthly or yearly averages, or transient simulations spanning multiple days or weeks. We show how our approach can be used for interactive design workflows such as city planning via fast feedback in the early design phase.


S.1. FEM discretization
Here, we provide a short summary on the derivation of Eq. ( 10) following the finite element approach.Starting from Eq. ( 8), we first multiply by a test function (for which we choose any interpolation basis function I j according to Galerkin's method): d T dt I j (x) = ε(x) cp(x)ρ(x)h(x) E(x) − σT (x) 4 I j (x).
Next, we integrate over the entire domain, and substitute the piecewise-constant approximation for the temperature field.We also assume that the material properties are similarly given as piecewise-constant data, consequently: On the left-hand side, clearly only the temperature coefficients T i are time-dependent, while only the indicator functions vary spatially.We can therefore re-arrange terms and exchange the order of integration, summation, and differentiation.Noting that the product I i I j is 1 when i = j and 0 otherwise, only one term of the sum remains and the left-hand side simplifies as follows: Similarly, the fourth-order term on the right-hand side simplifies because at any point x only one term of the sum is non-zero (also note I 4 i = I i as the indicator is either 0 or 1): Note that, choosing piecewise-constant interpolation and test functions here conveniently simplifies the fourth power of the sum.Had we used more common piecewise-linear functions instead, the sum would consist of three terms per triangle, and the fourth-power would generate a total of 15 terms (including 12 additional mixed terms, where temperature variables at the corners of the triangle interact in its interior due to linear interpolation).Furthermore, with linear functions I i I j would also be non-zero for any pair of adjacent nodes i and j.
Combining the aforementioned simplifications, index i no longer occurs.For consistency of notation, we replace j with i in the following, so we now have We then split the integral on the right-hand side and rearrange constants to obtain where we again use A i = Ω I i (x) d x on the right-most term.Finally, we divide by A i and also note that the remaining integral results in the incident flux on the support region of I i , which we denote as Φi = Ω E(x)I i (x) d x, to arrive at Eq. ( 10):

S.2. Boundary conditions
Here, we briefly outline how to incorporate the Dirichlet boundary conditions, describing external (solar) irradiation, into the temperature simulation.As discussed in the main paper, Eq. ( 10) (restated above) can be concisely expressed in matrix-vector notation as Eq. ( 14), i.e., d T/ dt = T T •4 .Recall that T is a vector of all (pervertex) temperature variables, and superscript ª•4º refers to taking the component-wise fourth power of this vector.We now outline the derivation of Eq. ( 18) from this starting point.Our boundary data specifies effective temperature values at the vertices of the emitter geometry.Therefore, some entries in the temperature vector T become known due to boundary conditions (T D ), while most entries remain unknown variables (T U ). Theoretically, we can now sort T such that all unknown components are listed before known data, i.e., we assume without loss of generality T = [T T U T T D ] T .Applying the same sorting and partitioning to the transport operator T expands the temperature equation to d dt As the given boundary data, T D , is known throughout the simulation, we can discard the second row of this block-vector equation.
Retaining only the first row reads Denoting the last term as b = T UD T •4 D results in Eq. ( 18).

S.3. Material Parameters
Table S1 summarizes material parameters for our various numerical experiments.

Table S1 :
Material properties and simulation parameters used for the simulations shown in the various Figures.Emissivity ε, diffuse reflectivity r d , specular reflectivity rs, mass density ρ [kg / m 3 ], specific heat capacity cp [J / (kg K)], shell thickness h [m], and fixed boundary temperature T [K] for emitting objects.