Generalized mixed variational principles and solutions of ill-conditioned problems in computational mechanics: Part I. Volumetric locking Original Research Article

Although the finite element method (FEM) has been extensively applied to various areas of engineering, the ill-conditioned problems occurring in many situations are still thorny to deal with. This study attempts to provide a high-performing and simple approach to the solutions of ill-conditioned problems. The theoretical foundation of it is the parametrized variational principles, called the generalized mixed variational principles (GMVPs) initiated by Rong in 1981. GMVPs can solve many kinds of ill-conditioned problems in computational mechanics. Among them, four cases are investigated in detail: the volumetric locking, the shear locking, the inhomogeneousness and the membrane locking problems, composing four parts of the study, Part I–IV, respectively. This paper is Part I, wherein a GMVP specially suited to the nearly incompressible and incompressible materials is constructed. A derived condition to overcome the rank deficiency of the global matrix of FEM for the perfect incompressibility is put forward. Based on these, a new approach, named the order-one shooting method, is proposed to avoid the spurious checkerboard pressure modes produced by certain FEM formulations.