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

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