A mixed method for 3D nonlinear elasticity using finite element exterior calculus

• Published in: International journal for numerical methods in engineering 2022-12, Vol.123 (23), p.5801-5825
• Main Authors: Dhas, Bensingh, Nagaraja, Jamun Kumar, Roy, Debasish, Reddy, Junuthula N.
• Format: Article
• Language: English
• Subjects: Approximation; Compatibility; Configurations; Constitutive relationships; Critical point; differential forms; Elasticity; finite element exterior calculus; Finite element method; Hu–Washizu principle; Mathematical analysis; Mixed methods research; Newton methods; nonlinear elasticity; Principles; Three dimensional models
• ISSN: 0029-5981; 1097-0207
• DOI: 10.1002/nme.7089

In this study, a mixed finite element model for 3D nonlinear elasticity using a Hu–Washizu (HW) type variational principle is presented. This mixed variational principle takes the deformed configuration and sections from its cotangent bundle as the input arguments. The critical points of the proposed HW functional enforce compatibility of these sections with the configuration, in addition to mechanical equilibrium and constitutive relations. Using this variational principle we construct a mixed FE approximation that distinguishes a vector from a 1‐form, a feature not commonly found in FE approximations for nonlinear elasticity. This distinction plays a pivotal role in identifying suitable FE spaces for approximating the 1‐forms appearing in the variational principle. These discrete approximations are constructed using ideas borrowed from finite element exterior calculus, which are in turn used to construct a discrete approximation to our HW functional. The discrete equations describing mechanical equilibrium, compatibility, and constitutive rule, are obtained by seeking extremum of the discrete functional with respect to the respective degrees of freedom. The discrete extremum problem is then solved numerically; we use Newton's method for this purpose. This mixed FE technique is then applied to a few benchmark problems wherein conventional displacement based approximations encounter locking and checker boarding. These studies help establish that our mixed FE approximation, which requires no artificial stabilizing terms, is free of these numerical bottlenecks.