Optimal Surrogate Boundary Selection and Scalability Studies for the Shifted Boundary Method on Octree Meshes

TL;DR

This paper improves the Shifted Boundary Method (SBM) for PDE simulations on complex geometries by optimizing surrogate boundary selection, proving convergence, and demonstrating scalability on octree meshes with diverse applications.

Contribution

It introduces an optimal surrogate boundary selection strategy for SBM, proves its convergence, and extends the method's scalability on octree meshes for complex geometries.

Findings

Optimal surrogate boundary significantly reduces numerical error.

Mathematical proof of SBM convergence with the optimal boundary.

Successful application to complex shapes and topologies in large-scale simulations.

Abstract

The accurate and efficient simulation of Partial Differential Equations (PDEs) in and around arbitrarily defined geometries is critical for many application domains. Immersed boundary methods (IBMs) alleviate the usually laborious and time-consuming process of creating body-fitted meshes around complex geometry models (described by CAD or other representations, e.g., STL, point clouds), especially when high levels of mesh adaptivity are required. In this work, we advance the field of IBM in the context of the recently developed Shifted Boundary Method (SBM). In the SBM, the location where boundary conditions are enforced is shifted from the actual boundary of the immersed object to a nearby surrogate boundary, and boundary conditions are corrected utilizing Taylor expansions. This approach allows choosing surrogate boundaries that conform to a Cartesian mesh without losing accuracy or stability. Our contributions in this work are as follows: (a) we show that the SBM numerical error can be greatly reduced by an optimal choice of the surrogate boundary, (b) we mathematically prove the optimal convergence of the SBM for this optimal choice of the surrogate boundary, (c) we deploy the SBM on massively parallel octree meshes, including algorithmic advances to handle incomplete octrees, and (d) we showcase the applicability of these approaches with a wide variety of simulations involving complex shapes, sharp corners, and different topologies. Specific emphasis is given to Poisson's equation and the linear elasticity equations.