Highly accurate symplectic element based on two variational principles

For the stability requirement of numerical resultants, the mathematical theory of classical mixed methods are relatively complex. However, generalized mixed methods are automatically stable, and their building process is simple and straightforward. In this paper, based on the seminal idea of the generalized mixed methods, a simple, stable, and highly accurate 8-node noncompatible symplectic element(NCSE8) was developed by the combination of the modified Hellinger-Reissner mixed variational principle and the minimum energy principle. To ensure the accuracy of in-plane stress results, a simultaneous equation approach was also suggested. Numerical experimentation shows that the accuracy of stress results of NCSE8 are nearly the same as that of displacement methods, and they are in good agreement with the exact solutions when the mesh is relatively fine. NCSE8 has advantages of the clearing concept, easy calculation by a finite element computer program, higher accuracy and wide applicability for various linear elasticity compressible and nearly incompressible material problems. It is possible that NCSE8 becomes even more advantageous for the fracture problems due to its better accuracy of stresses.

Numerical Studies of Vanka-Type Smoothers in Computational Solid Mechanics

In this paper multigrid smoothers of Vanka-type are studied in the context of Computational Solid Mechanics(CSM).These smoothers were originally developed to solve saddle-point systems arising in the field of Computational Fluid Dynamics(CFD),particularly for incompressible flow problems.When treating(nearly)incompressible solids,similar equation systems arise so that it is reasonable to adopt the‘Vanka idea’for CSM.While there exist numerous studies about Vanka smoothers in the CFD literature,only few publications describe applications to solid mechanical problems.With this paper we want to contribute to close this gap.We depict and compare four different Vanka-like smoothers,two of them are oriented towards the stabilised equal-order Q_(1)/Q_(1)finite element pair.By means of different test configurations we assess how far the smoothers are able to handle the numerical difficulties that arise for nearly incompressible material and anisotropic meshes.On the one hand,we show that the efficiency of all Vanka-smoothers heavily depends on the proper parameter choice.On the other hand,we demonstrate that only some of them are able to robustly deal with more critical situations.Furthermore,we illustrate how the enclosure of the multigrid scheme by an outer Krylov space method influences the overall solver performance,and we extend all our examinations to the nonlinear finite deformation case.