The tractions that cells exert on a gel substrate from the observed displacements is
an increasingly attractive and valuable information in biomedical experiments. The computation
of these tractions requires in general the solution of an inverse problem. Here, we resort to the
discretisation with finite elements of the associated direct variational formulation, and solve the
inverse analysis using a least square approach. This strategy requires the minimisation of an
error functional, which is usually regularised in order to obtain a stable system of equations with
a unique solution. In this paper we show that for many common three-dimensional geometries,
meshes and loading conditions, this regularisation is unnecessary. In these cases, the computational
cost of the inverse problem becomes equivalent to a direct finite element problem. For the
non-regularised functional, we deduce the necessary and suffcient conditions that the dimensions
of the interpolated displacement and traction fields must preserve in order to exactly satisfy or
yield a unique solution of the discrete equilibrium equations. We apply the theoretical results to
some illustrative examples and to real experimental data. Due to the relevance of the results for
biologists and modellers, the article concludes with some practical rules that the finite element
discretisation must satisfy.