  Motivated by Ball and Majumdar's modification of Landau-de Gennes model for
nematic liquid crystals, we study energy-minimizer $Q$ of a tensor-valued
variational obstacle problem in a bounded 3-D domain with prescribed boundary
data. The energy functional is designed to blow up as $Q$ approaches the
obstacle. Under certain assumptions, especially on blow-up profile of the
singular bulk potential, we prove higher interior regularity of $Q$, and show
that the contact set of $Q$ is either empty, or small with characterization of
its Hausdorff dimension. We also prove boundary partial regularity of the
energy-minimizer.
