  The main purpose of the present paper is to establish a link between
quadrature surfaces (potential theoretic concept) and sandpile dynamics
(Laplacian growth models). For this aim, we introduce a new model of Laplacian
growth on the lattice $\mathbb{Z}^d$ $(d\geq 2)$ which continuously deforms
occupied regions of the \emph{divisible sandpile} model of Levine and Peres, by
redistributing the total mass of the system onto $\frac 1m$-sub-level sets of
the odometer which is a function counting total emissions of mass from lattice
vertices. In free boundary terminology this goes in parallel with singular
perturbation, which is known to converge to a Bernoulli type free boundary.
  We prove that models, generated from a single source, have a scaling limit,
if the threshold $m$ is fixed. Moreover, this limit is a ball, and the entire
mass of the system is being redistributed onto an annular ring of thickness
$\frac 1m$. By compactness argument we show that, when $m$ tends to infinity
sufficiently slowly with respect to the scale of the model, then in this case
also there is scaling limit which is a ball, with the mass of the system being
uniformly distributed onto the boundary of that ball, and hence we recover a
quadrature surface in this case.
  Depending on the speed of decay of $m$, the visited set of the sandpile
interpolates between spherical and polygonal shapes. Finding a precise
characterisation of this shape-transition phenomenon seems to be a considerable
challenge, which we cannot address at this moment.
