Finite Element Technology for Tetrahedral Elements

• Development of improved tetrahedral elements for structurally optimized, threedimensional components
• Improving efficiency and predictability

Tetrahedral finite elements are extensively used in finite element analyses owing to their geometric versatility in discretizing bodies with complex geometries. Nevertheless, these elements are prone to volumetric locking, an artificially increased element stiffness that becomes particularly pronounced in the nearly incompressible material regime. This deficiency causes degraded convergence rates, and inaccurate stress field predictions. From a theoretical standpoint, volumetric locking can be attributed to an over-constraining of the kinematic field within the element domain.

Numerous methods have been proposed in the literature to remove volumetric locking in tetrahedral elements, such as mixed displacement-pressure formulations, Cosserat point-based methodologies, and enhanced assumed strain approaches. Although these techniques significantly remove the volumetric locking, they are generally accompanied by inherent drawbacks, such as the introduction of additional field unknowns, elevated computational cost, or considerable reformulation of the classical displacement-based finite element framework. Within the context of large-scale explicit dynamic simulations, where computational efficiency and numerical robustness are important, the adoption of such methods may be impractical.

The research objective is the development of novel element formulations with a special modification of the material law for certain parts of element stiffness matrix, with which the approximation quality of the element can be improved. This approach has been proven to be effective for the isotropic linear elastic material law near incompressible limit.

The present work addresses the extension of the aforementioned approach to an elastoplastic material model, with particular focus on the widely used von Mises plasticity model. A fundamental characteristic of this model is the isochoric nature of plastic flow, which implicitly enforces volumetric incompressibility in the plastic regime, thereby reintroducing the volumetric locking. The extension of the proposed method to this context, however, is not straightforward. This complexity arises from the fact that the return mapping direction in stress space is governed by the deviatoric stress tensor, which is inherently dependent on the current stress state itself. This return mapping direction is consistent with the isochoric plastic flow assumption, and consequently, material incompressibility in the plastic regime is not imposed through a fixed material parameter, as is the case in the elastic formulation, but rather emerges implicitly from the stress state evolution.