  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 sufficient 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.
