  We have performed numerical simulation of a 3-dimensional elastic medium,
with scalar displacements, subject to quenched disorder. We applied an
efficient combinatorial optimization algorithm to generate exact ground states
for an interface representation. Our results indicate that this Bragg glass is
characterized by power law divergences in the structure factor $S(k)\sim A
k^{-3}$. We have found numerically consistent values of the coefficient $A$ for
two lattice discretizations of the medium, supporting universality for $A$ in
the isotropic systems considered here. We also examine the response of the
ground state to the change in boundary conditions that corresponds to
introducing a single dislocation loop encircling the system. Our results
indicate that the domain walls formed by this change are highly convoluted,
with a fractal dimension $d_f=2.60(5)$. We also discuss the implications of the
domain wall energetics for the stability of the Bragg glass phase. As in other
disordered systems, perturbations of relative strength $\delta$ introduce a new
length scale $L^* \sim \delta^{-1/\zeta}$ beyond which the perturbed ground
state becomes uncorrelated with the reference (unperturbed) ground state. We
have performed scaling analysis of the response of the ground state to the
perturbations and obtain $\zeta = 0.385(40)$. This value is consistent with the
scaling relation $\zeta=d_f/2- \theta$, where $\theta$ characterizes the
scaling of the energy fluctuations of low energy excitations.
