Four-Field Mixed Finite Element Methods for Incompressible Nonlinear Elasticity
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We introduce conformal mixed finite element methods for 2D and 3D incompressible nonlinear elasticity in terms of displacement, displacement gradient, the first Piola-Kirchhoff stress tensor, and pressure, where finite elements for the curl and the div operators are used to discretize strain and stress, respectively. These choices of elements follow from the strain compatibility and the momentum balance law. Some inf-sup conditions are derived to study the stability of methods. By considering 96 choices of simplicial finite elements of degree less than or equal to 2 in 2D and 3D, we conclude that 28 choices in 2D and 6 choices in 3D satisfy these inf-sup conditions. The performance of stable finite element choices are numerically studied. Although the proposed methods are computationally more expensive than the standard two-field methods for incompressible elasticity, they are potentially useful for accurate approximations of strain and stress as they are independently computed in the solution process.
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