Hourglass control in linear and nonlinear problems

Mesh stabilization techniques for controlling the hourglass modes in under-integrated hexahedral and quadrilateral elements are described. It is shown that the orthogonal hourglass techniques previously developed can be obtained from simple requirements that insure the consistency of the finite element equations in the sense that the gradients of linear fields are evaluated correctly. It is also shown that this leads to an hourglass control that satisfies the patch test. The nature of the parameters which relate the generalized stresses and strains for controlling hourglass modes is examined by means of a mixed variational principle and some guidelines for their selection are discussed. Finally, effective means of implementing these hourglass procedure in computer codes are described. Applications to both the Laplace equation and the equations of solid mechanics in 2 and 3 dimensions are considered.