  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.
