Quasi-quadratic elements for nonlinear compressible and incompressible elasticity

This work deals with novel triangular and tetrahedral elements for nonlinear elasticity. While it is well-known that linear and quadratic elements perform, respectively, poorly and accurately in this context, their cost is very different. We construct an approximation that falls in-between these two cases, which we refer to as quasi-quadratic. We seek to satisfy the following: (1) absence of locking and pressure oscillations in the incompressible limit, (2) an exact equivalence to quadratic elements on linear problems, and (3) a computational cost comparable to linear elements on nonlinear problems. Our construction is formally based on the Hellinger-Reissner principle, where strains and displacement are interpolated linearly on nested meshes, but it can be recast in a pure displacement form via static condensation. We show that (1) and (2) are fulfilled via numerical studies on a series of benchmarks and analyze the cost of quadrature in order to show (3).

[1]  Arzhang Angoshtari,et al.  A geometric structure-preserving discretization scheme for incompressible linearized elasticity , 2013 .

[2]  Ying Dai,et al.  The Brezzi-Pitkäranta stabilization scheme for the elasticity problem , 2015, J. Comput. Appl. Math..

[3]  Zhimin Zhang,et al.  A New Finite Element Gradient Recovery Method: Superconvergence Property , 2005, SIAM J. Sci. Comput..

[4]  Ted Belytschko,et al.  Assumed strain stabilization of the 4-node quadrilateral with 1-point quadrature for nonlinear problems , 1991 .

[5]  J. Ball.  Convexity conditions and existence theorems in nonlinear elasticity , 1976 .

[6]  Peter Wriggers,et al.  A stabilization technique to avoid hourglassing in finite elasticity , 2000 .

[7]  Ferdinando Auricchio,et al.  Mixed Finite Element Methods , 2004 .

[8]  M. Suri.  Stable hp mixed finite elements based on the Hellinger-Reissner principle , 2005 .

[9]  G. A. Frazier,et al.  Treatment of hourglass patterns in low order finite element codes , 1978 .

[10]  Alexander Linke,et al.  On the role of the Helmholtz decomposition in mixed methods for incompressible flows and a new variational crime , 2014 .

[11]  A. Ern,et al.  A Hybrid High-Order method for the incompressible Navier-Stokes equations based on Temam's device , 2018, J. Comput. Phys..

[12]  Charles E. Augarde,et al.  The use of Timoshenko's exact solution for a cantilever beam in adaptive analysis , 2008 .

[13]  P. Tallec.  Numerical methods for nonlinear three-dimensional elasticity , 1994 .

[14]  O. C. Zienkiewicz,et al.  Incompressibility without tears—HOW to avoid restrictions of mixed formulation , 1991 .

[15]  C. Felippa.  A study of optimal membrane triangles with drilling freedoms , 2003 .

[16]  Michael A. Puso,et al.  A highly efficient enhanced assumed strain physically stabilized hexahedral element , 2000 .

[17]  Alessio Quaglino,et al.  A framework for creating low‐order shell elements free of membrane locking , 2016 .

[18]  L. P. Bindeman,et al.  Assumed strain stabilization of the eight node hexahedral element , 1993 .

[19]  S. Timoshenko,et al.  Theory of Elasticity (3rd ed.) , 1970 .

[20]  Frédéric Hecht,et al.  New development in freefem++ , 2012, J. Num. Math..

[21]  C. Cyron,et al.  Volume-averaged nodal projection method for nearly-incompressible elasticity using meshfree and bubble basis functions , 2015 .

[22]  Stefanie Reese,et al.  A new stabilization technique for finite elements in non-linear elasticity , 1999 .

[23]  Gabriel Wittum,et al.  Mass Transport in Porous Media with Variable Mass , 2012 .

[24]  F. Brezzi,et al.  On the Stabilization of Finite Element Approximations of the Stokes Equations , 1984 .

[25]  Dominique Chapelle,et al.  Detailed reliability assessment of triangular MITC elements for thin shells , 2008 .

[26]  M. Fortin,et al.  Mixed Finite Element Methods and Applications , 2013 .

[27]  Max Gunzburger,et al.  Finite Element Methods for Viscous Incompressible Flows: A Guide to Theory, Practice, and Algorithms , 1989 .

[28]  Thomas Fritz,et al.  Verification of cardiac mechanics software: benchmark problems and solutions for testing active and passive material behaviour , 2015, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[29]  Michel Fortin,et al.  Mixed and Hybrid Finite Element Methods , 2011, Springer Series in Computational Mathematics.

[30]  S. Timoshenko,et al.  Theory of elasticity , 1975 .

[31]  D. Malkus,et al.  Mixed finite element methods—reduced and selective integration techniques: a unification of concepts , 1990 .

[32]  K. Schweizerhof,et al.  On a systematic development of trilinear three-dimensional solid elements based on Simo's enhanced strain formulation , 1996 .