Inverse finite element analysis using a simple reduced integration hexahedral solid-shell element

This paper introduces the inverse finite element method using simple brick elements that can be used for shell analysis. The proposed element is the inverse counterpart of an existing Lagrangean-based direct trilinear hexahedral finite element that uses the approaches of reduced integration, assumed natural strains and enhanced assumed strain to prevent locking defects in shell modeling. Like the standard trilinear hexahedral element, this locking-free element has eight vertex nodes and three displacement degrees-of-freedom per node. It also has one scalar enhanced-strain degree-of-freedom, which is eliminated at the element level. Both inverse and direct finite element formulations are identical up to the definition of the Lagrangean-based equilibrium equations. For the inverse approach, these equations have as unknowns the positions of the nodes in the undeformed configuration. The current approach is particularly well suited for a category of inverse problems where a given shape must be attained after large elastic deformations. This is the case in the design of turbine blades, to be developed here.