The Force Method Version of Goodman's Joint Element with Convergence Proof

Abstract

Interfaces widely present in nature and civil engineering are the most fundamental elements causing discontinuous deformation and failure. The Goodman joint element is incorporated into the vast majority of commercial and proprietary finite element programs due to its simplicity. However, the numerical properties of the Goodman element are not ideal. A lot of effort has gone into enhancing its numerical properties, only to make some empirical suggestions for adjusting mesh and computation parameters, yet there are still many examples that fail to reach convergence or incur drastic numerical oscillations of contact stress. Considering that the Goodman element is implemented within the framework of the displacement method, and poorlyposed problems in the displacement method are usually well-posed problems in the force method, this study proposes a force method version of the Goodman element, abbreviated as FMVGE. The primal unknowns in FMVGE are the contact stresses on the interface rather than nodal deformations. The core problem in FMVGE represents the contact conditions in the form of a quasi-variational inequality (QVI). By employing process iteration rather than state iteration used in the displacement method, at the same time, FMVGE precisely satisfies the contact conditions, with convergence being theoretically guaranteed. Analysis of classic examples and engineering cases indicates that the numerical properties of FMVGE are significantly superior to the widely adopted master–slave block method. Appendices A and B provide the proof of the convergence of the FMVGE solution and the core Matlab code for solving the QVI, respectively.