Compatible-strain mixed finite element methods forincompressible nonlinear elasticity

JOURNAL OF COMPUTATIONAL PHYSICS

卷:361

Article

Shojaei, Mostafa Faghih¹  Yavari, Arash^1,2 

[1] Georgia Inst Technol, Sch Civil & Environm Engn, Atlanta, GA 30332 USA

[2] Georgia Inst Technol, George W Woodruff Sch Mech Engn, Atlanta, GA 30332 USA

关键词: Mixed finite element methods;    Finite element exterior calculus;    Nonlinear elasticity;    Incompressible elasticity;    Hilbert complex;   

DOI  :  10.1016/j.jcp.2018.01.053

来源: Elsevier

【 摘 要 】

We introduce a new family of mixed finite elements for incompressible nonlinear elasticity - compatible-strain mixed finite element methods(CSFEMs). Based on a Hu-Washizu-type functional, we write a four-field mixed formulation with the displacement, the displacement gradient, the first Piola-Kirchhoff stress, and a pressure-like field as the four independent unknowns. Using the Hilbert complexes of nonlinear elasticity, which describe the kinematics and the kinetics of motion, we identify the solution spaces of the independent unknown fields. In particular, we define the displacement in H-1, the displacement gradient in H(curl), the stress in H(div), and the pressure field in L-2. The test spaces of the mixed formulations are chosen to be the same as the corresponding solution spaces. Next, in a conforming setting, we approximate the solution and the test spaces with some piecewise polynomial subspaces of them. Among these approximation spaces are the tensorial analogues of the Nedelec and Raviart-Thomas finite element spaces of vector fields. This approach results in compatible-strainmixed finite element methods that satisfy both the Hadamard compatibility condition and the continuity of traction at the discrete level independently of the refinement level of the mesh. By considering several numerical examples, we demonstrate that CSFEMs have a good performance for bending problems and for bodies with complex geometries. CSFEMs are capable of capturing very large strains and accurately approximating stress and pressure fields. Using CSFEMs, we do not observe any numerical artifacts, e.g., checkerboarding of pressure, hourglass instability, or locking in our numerical examples. Moreover, CSFEMs provide an efficient framework for modeling heterogeneous solids. (C) 2018 Elsevier Inc. All rights reserved.

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