A variationally consistent nodal integration for cubic serendipity finite elements with optimal convergence in explicit transient heat conduction analysis

• The convergence deterioration in the explicit transient heat conduction analysis is observed for the 13-node quadrilateral and 39-node hexahedral cubic serendipity elements with lumped heat capacity matrices. • The underlying cause for this convergence decay is found to be the violation of variational integration consistency. • A variationally consistent nodal integration (VCNI) is developed for the cubic serendipity elements in the context of the boundary enhanced Galerkin weak form. • VCNI identically fulfills the variational integration consistency and thus enables an optimal convergence for the explicit transient heat conduction analysis. • The optimal convergence and accuracy superiority of VCNI are consistently confirmed by numerical results. The 13-node quadrilateral and 39-node hexahedral cubic serendipity elements produce nodally integrated positive-definite lumped heat capacity matrices in higher-order finite element analysis. However, these elements display severe convergence deterioration in explicit transient heat conduction analysis with lumped heat capacity matrices. This convergence decay is due to the violation of variational integration consistency by the standard Galerkin formulation with lumped heat capacity matrices. This issue is resolved by introducing the boundary-enhanced Galerkin weak form that incorporates the elemental boundary contribution in the discrete finite element formulation. Subsequently, it is theoretically proven that a direct nodal integration identically fulfills the variational integration consistency in the context of the boundary-enhanced Galerkin weak form. The proposed variationally consistent nodal integration therefore enables optimal convergence for explicit transient heat conduction analysis with lumped heat capacity matrices. The efficacy of the proposed variationally consistent nodal integration formulation for the 13-node quadrilateral and 39-node hexahedral cubic elements is thoroughly demonstrated via numerical examples.