Highly efficient solid and solid-shell finite elements with mixed strain–displacement assumptions specifically set up for explicit dynamic simulations using symbolic programming

• Incompatible mode (IM) elements based on corresponding enhanced strain (EAS) elements in explicit time integration.
• Efficient IM mode elements and combined IM/EAS elements for problems in structural dynamics.
• Highly efficient program achieved using symbolic programming with AceGen.

The method of incompatible modes is applied to volumetric elements and so-called solid-shell elements with linear interpolation as an alternative to the well known method of enhanced assumed strains in order to reduce artificial stiffness effects. The incompatible modes are directly adjoint to known EAS elements. In the context of explicit time integration, the incompatible degrees of freedom are treated as regular unknowns with associated modal masses and incorporated into the global equations instead of being condensed-out locally decreasing the computational effort. The issue of artificial kinetic energy is briefly discussed. Further the implementation concept applying symbolic programming is described.