J. Jirousek, VARIATIONAL FORMULATION OF 2 COMPLEMENTARY HYBRID-TREFFTZ FE MODELS, Communications in numerical methods in engineering, 9(10), 1993, pp. 837-845
The paper presents two complementary variational formulations of Treff
tz-type elements wherein the piece-wise assumed displacement approxima
tions a priori fulfil the governing differential equations over the el
ement subdomain. The two formulations are such that either the displac
ement conformity or the reciprocity of the conjugate tractions is enfo
rced at the element interfaces, one more strongly than the other. This
is achieved through linking the piece-wise approximations either thro
ugh an auxiliary displacement or a boundary traction frame. The two fo
rmulations concern solid mechanics and structures. Although for hybrid
elements monotonic convergence in energy cannot be warranted, in prac
tice the two presented formulations yield opposite bounds on the energ
y provided that the number of internal Trefftz functions is sufficient
ly large.