Hybrid elements, which are based on a two-field variational formulation
with the displacements and stresses interpolated separately, are known to
deliver very high accuracy, and to alleviate to a large extent problems of
locking that plague standard displacement-based formulations. The choice of
the stress interpolation functions is of course critical in ensuring the high
accuracy and robustness of the method. Generally, an attempt is made to
keep the stress interpolation to the minimum number of terms that will
ensure that the stiffness matrix has no spurious zero-energy modes, since
it is known that the stiffness increases with the increase in the number of
terms. Although using such a strategy of keeping the number of interpolation
terms to a minimum works very well in static problems, it results either in
instabilities or fails to converge in transient problems. This is because choosing
the stress interpolation functions
merely on the basis of removing spurious
energy modes can violate some basic principles that interpolation functions
should obey. In this work, we address the issue of choosing the interpolation
functions based on such basic principles of interpolation theory and mechanics.
Although this procedure results in the use of more number of terms than the
minimum (and hence in slightly increased stiffness) in many elements, we show
that the performance continues to be far superior to displacement-based
formulations, and, more importantly, that it also results in considerably increased
robustness.
