Scalable adaptive PDE solvers in arbitrary domains

TL;DR

This paper introduces a scalable octree-based adaptive PDE solver capable of handling complex geometries with high adaptivity, validated through fluid flow simulations and a COVID-19 transmission risk assessment.

Contribution

The work presents a novel octree-based adaptive discretization method for PDEs that efficiently manages incomplete octrees in complex geometries, enabling accurate simulations.

Findings

Validated convergence analysis of the framework.

Successfully computed drag coefficients across a wide Reynolds number range.

Applied the method to real-world COVID-19 transmission risk assessment.

Abstract

Efficiently and accurately simulating partial differential equations (PDEs) in and around arbitrarily defined geometries, especially with high levels of adaptivity, has significant implications for different application domains. A key bottleneck in the above process is the fast construction of a `good' adaptively-refined mesh. In this work, we present an efficient novel octree-based adaptive discretization approach capable of carving out arbitrarily shaped void regions from the parent domain: an essential requirement for fluid simulations around complex objects. Carving out objects produces an $\textit{incomplete}$ octree. We develop efficient top-down and bottom-up traversal methods to perform finite element computations on $\textit{incomplete}$ octrees. We validate the framework by (a) showing appropriate convergence analysis and (b) computing the drag coefficient for flow past a sphere for a wide range of Reynolds numbers ($\mathcal{O}(1-10^6)$) encompassing the drag crisis regime. Finally, we deploy the framework on a realistic geometry on a current project to evaluate COVID-19 transmission risk in classrooms.