On referential and spatial formulations of inverse elastostatic analysis

• Proved that the spatial and referential formulations of inverse elastostatic analysis are equivalent.
• Showed that the spatial and referential constitutive equations of hyperelastic materials can be made form-invariant by invoking a covariant construction. Consequently, the stress functions of anisotropic materials can always be represented in spatial form.
• Discussed the update of material symmetry representations in the inverse solution process.
• Delineated conditions under which material routines can be preserved in the inverse analysis.

There are two families of element formulation for inverse elastostatic analysis in the literature. They employ different computation procedures at the element and material levels. It has been suggested that one of them can preserve material library while the other requires redeveloping the material functions. In this article, we show that these two formulations are completely equivalent. Under certain conditions both can preserve the material library while under other conditions neither can. We show that the modification of material functions is caused by the need of updating material symmetry in the inverse solution process. Several theoretical results were obtained while establishing the equivalence, including an identity relating the inverse and forward tangent tensors for isotropic materials. A comprehensive documentation of these two formulations is provided.