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

Dhas, B and Nagaraja, JK and Roy, D and Reddy, JN (2022) A mixed method for 3D nonlinear elasticity using finite element exterior calculus. In: International Journal for Numerical Methods in Engineering .

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Official URL: https://doi.org/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.

Item Type: Journal Article
Publication: International Journal for Numerical Methods in Engineering
Publisher: John Wiley and Sons Ltd
Additional Information: The copyright for this article belongs to the Authors.
Keywords: Calculations; Degrees of freedom (mechanics); Differentiation (calculus); Elasticity; Finite element method; Variational techniques, 1-forms; Differential forms; Discrete approximation; Finite element exterior calculus; Hu�washizu principle; Mechanical equilibrium; Mixed finite elements; Mixed method; Nonlinear elasticity; Variational principles, Newton-Raphson method
Department/Centre: Division of Chemical Sciences > Materials Research Centre
Date Deposited: 14 Sep 2022 10:24
Last Modified: 14 Sep 2022 10:24
URI: https://eprints.iisc.ac.in/id/eprint/76458

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