Obtain element boundary forces in Finite Element Method Suppose that we have a rectangular domain discretized using 16 voxels (4x4 voxels) and I apply a Finite Element Analysis with arbitrary boundary conditions in the domain. As a result, I obtain the nodal displacements of every node in the domain; with those displacements it is possible to calculate the nodal forces in every node in the domain.
My question is: Is it possible to calculate, with those same nodal displacements (or forces) the forces that are applied on the boundary of the $i$th-voxel (element)? 
 A: The basic problem is that there are patterns of distributed loading around the circumference of the element which are self-equilibrating, in the sense that they produce nodal forces of zero.
For a particular element type (especially if it has simple geometry like a rectangle) you might be able to define some general form of load distribution around the element which avoids self-equilibrating load patterns, and then match it to the nodal forces.
Another approach is to use the element stresses, but since the stress field has a higher order of accuracy at some "special" points inside the element (depending on the element formulation) just finding the strains from the element shape functions at a point on the boundary, and then finding the stresses, may not give good results. In particular, the stresses along the boundaries of two adjacent elements will NOT be equal if you calculate them in that way.
Commercial FE post processors usually do some sort of ad hoc interpolation between the stresses at the "high accuracy" internal points in the elements to produce nice looking plots. If you use that idea and then interrogate the plotted data to get what you want, bear in mind that the results will probably not be consistent with the underlying FE formulation for any well-defined meaning of "consistent" - though of course they are usually good enough to use to make engineering decisions.
Yet another idea would be to use a so-called "stress based" FE formulation where the fundamental variables in the model are stresses, not displacements. There are also hybrid formulations where the variables are a mix of stresses and displacements. Those formulatons may give you consistent stresses across element boundaries, and you could then calculate the edge loads directly from the stress field. 
However that path may lead you into doing research into the FE method itself, rather than using it as a tool to do something else! Mainstream FE continuum mechanics is almost entirely done using displacement formulations.
