Ill-posed boundary conditions encountered in 3D and plate finite element simulations

The setting of 3D variational elasticity requires a regularity of the displacement field which is not sufficient to allow the definition of the displacement value along a line and a fortriori at a point. Therefore, prescribing displacements along lines and applying lineic forces result in ill-posed problems. In the same way, pointwise loads are ill-posed boundary conditions in the Reissner-Mindlin bending plate theory, but not in the Love-Kirchhoff one. The consequence on finite element simulations is immediate: the strain energy does not converge with mesh refinement for practical mesh sizes.