Avoiding negative elastic moduli when using Lagrange interpolation for material grading in finite element analysis

Polynomial interpolations, one of the most common interpolations used in finite element methods (FEMs), are a workhorse of many FEM codes. These interpolations are readily available for all kinds of elements, and using them for modeling the variation of elastic moduli in graded elements is thus both convenient and natural. Yet, like all polynomial interpolations, they can be prone to oscillations that can result in regions of negative elastic modulus in the element, even with only positive nodal values of elastic moduli. The result of these negative modulus regions, even if the region is small, can be unexpected singularities in the solution. This defeats the purpose of using polynomial interpolations for capturing material grading in the element. We demonstrate the issue using three-node quadratic Lagrange interpolations of material grading in otherwise isoparametric p-type elements and show how to avoid this problem.