  Many solid mechanics problems on complex geometries are conventionally solved
using discrete boundary methods. However, such an approach can be cumbersome
for problems involving evolving domain boundaries due to the need to track
boundaries and constant remeshing. In this work, we employ a robust smooth
boundary method (SBM) that represents complex geometry implicitly, in a larger
and simpler computational domain, as the support of a smooth indicator
function. We present the resulting equations for mechanical equilibrium, in
which inhomogeneous boundary conditions are replaced by source terms. The
resulting mechanical equilibrium problem is semidefinite, making it difficult
to solve. In this work, we present a computational strategy for efficiently
solving near-singular SBM elasticity problems. We use the block-structured
adaptive mesh refinement (BSAMR) method for resolving evolving boundaries
appropriately, coupled with a geometric multigrid solver for an efficient
solution of mechanical equilibrium. We discuss some of the practical numerical
strategies for implementing this method, notably including the importance of
grid versus node-centered fields. We demonstrate the solver's accuracy and
performance for three representative examples: a) plastic strain evolution
around a void, b) crack nucleation and propagation in brittle materials, and c)
structural topology optimization. In each case, we show that very good
convergence of the solver is achieved, even with large near-singular areas, and
that any convergence issues arise from other complexities, such as stress
concentrations. We present this framework as a versatile tool for studying a
wide variety of solid mechanics problems involving variable geometry.
