  For a compact $ d $-dimensional rectifiable subset of $ \mathbb{R}^{p} $ we
study asymptotic properties as $ N\to\infty $ of $N$-point configurations
minimizing the energy arising from a Riesz $ s $-potential $ 1/r^s $ and an
external field in the hypersingular case $ s\geq d$. Formulas for the weak$ ^*
$ limit of normalized counting measures of such optimal point sets and the
first-order asymptotic values of minimal energy are obtained. As an
application, we derive a method for generating configurations whose normalized
counting measures converge to a given absolutely continuous measure supported
on a rectifiable subset of $ \mathbb{R}^{p} $. Results on separation and
covering properties of discrete minimizers are given. Our theorems are
illustrated with several numerical examples.
