Path integrals formulations leading to propagator evaluation for coupled linear physics in large geometric models

Abstract

Reformulating linear physics using second kind Fredholm equations is very standard practice. One of the straightforward consequences is that the resulting integrals can be expanded (when the Neumann expansion converges) and probabilized, leading to path statistics and Monte Carlo estimations. An essential feature of these algorithms is that they also allow to estimate propagators for all types of sources, including initial conditions. The resulting practice is a single Monte Carlo run, for one given set of sources, producing propagators that can later be used with any other set of sources for fast simulations, typically as parts of optimization, inversion, sensitivity analysis and command control algorithms. The present paper illustrates how this practice can be extended to problems involving several interacting physics, provided that their coupling is only at the boundary of the system or at interfaces between sub-parts, and may itself be given the form of a second kind Fredholm equation. A full practical implementation is described as part of the Stardis code, with the example of transferring heat via the coupling of radiation, reaction-diffusion and convection as typically expected in the multidisciplinary context of urban climate modeling. Besides, we show how recent advances in computer graphics indicate that these algorithms can be made numerically extremely efficient when facing large CAD geometries: computing the propagator becomes strictly independent of the geometry refinement, i.e. is identical whatever the number of triangles and tetraedra used to numerize the surface and volume descriptions. To the best of our knowledge this is the first report of propagator computations that remains practical for coupled physics in large CAD geometries. Program summaryProgram Title: Stardis 0.7.2 (built on stardis-solver 0.12.3)CPC Library link to program files:https://doi.org/10.17632/k76zrx4n6b.1Developer's repository link:https://www.meso-star.com/projects/stardis/stardis.htmlLicensing provisions: GPLv3Programming language: ANSI C and PythonNature of problem: Estimating temperatures in coupled heat transfer systems involving large CAD and/or large numbers of spatially distributed sources.Solution method: Stardis uses the Monte Carlo Method. Each temperature estimate constructs a propagator of each of the energy sources within the system: initial temperature, temperature boundary conditions, volume powers and surface fluxes. The propagator can be stored for further use outside the Monte Carlo code.Additional comments including restrictions and unusual features: The Stardis estimates of propagators are only reliable when radiative transfer can be linearized around a reference temperature. Stardis can deal with the nonlinearity of radiation when computing temperatures, but then nothing can be interpreted as a meaningfull propagator for any external usage.Python scripts were used to execute the different configurations, post-process the results and generate the different graphs.