Mixed Variational Approaches in Nonlinear Continuum Mechanics and Discretization by Finite Element Exterior Calculus

Abstract

Numerical solution techniques for nonlinear elasticity have been an active area of research for a while. Right from the early days, it was recognized that constructing well-performing FE approximations for nonlinear elasticity was a challenging task. Earlier attempts to construct purely displacement-based formulations often resulted in poor quality of approximation. Stabilization methods based on assumed strain and enhanced strain were introduced to circumvent this problem. These stabilization terms had little to do with the underlying physics and were introduced purely from a numerical standpoint.

In the present thesis, a new geometric perspective for nonlinear analysis is adopted where the quantities defined on the co-tangent bundles of reference and deformed configurations are treated as primary unknowns alongside deformation. Such a treatment invites compatibility of the fields (defined on co-tangent bundles) with the base-spaces (configurations of the solid body) so that the configurations can be appropriately realized, often as a subset of the Euclidean space (as in the present work). Using the moving frame, the metric and connection are expressed in terms of differential forms. The geometric understanding of stress as a co-vector valued 2-form fits squarely within our overall program. Based on a Hu-Washizu (HW) type variational principle, a mixed FE approximation for nonlinear elasticity is proposed and applied to 2D hyperelasticity, 3D hyperelasticity, beams and shells. This version of HW variational principle uses kinetics and kinematics written in terms of vector fields and differential forms. Discrete approximations for the differential forms appearing in the HW functional are constructed with ideas borrowed from finite element exterior calculus (FEEC). These approximations are in turn used to construct a discrete approximation to the HW functional. The discrete equations describing mechanical equilibrium, compatibility and constitutive relations, are obtained by seeking an extremum of the discrete functional with respect to appropriate degrees of freedom. The discrete extremum problem is then solved numerically using Newton's method. As evidenced through a swathe of numerical tests, the proposed mixed FE technique is locking free and offers high accuracy without the need for re-meshing.