  We study the asymptotic behaviour, as a small parameter $\varepsilon$ tends
to zero, of minimisers of a Ginzburg-Landau type energy with a nonlinear
penalisation potential vanishing on a compact submanifold $\mathcal{N}$ and
with a given $\mathcal{N}$-valued Dirichlet boundary data. We show that
minimisers converge up to a subsequence to a singular $\mathcal{N}$-valued
harmonic map, which is smooth outside a finite number of points around which
the energy concentrates and whose singularities' location minimises a
renormalised energy, generalising known results by Bethuel, Brezis and H\'elein
for the circle $\mathbb{S}^1$. We also obtain $\Gamma$-convergence results and
uniform Marcinkiewicz weak $L^2$ or Lorentz $L^2$ estimates on the derivatives.
We prove that solutions to the corresponding Euler-Lagrange equation converge
uniformly to the constraint and converge to harmonic maps away from
singularities.
