Finite calculus formulation for incompressible solids using linear triangles and tetrahedra

Published in ''Int. Journal for Numerical Methods in Engineering'' Vol. 59 (11), pp. 1473-1500, 2004<br />
doi: 10.1002/nme.922
== Abstract ==

Many finite elements exhibit the so-called ‘volumetric locking’ in the analysis of incompressible orquasi-incompressible problems.  In this paper, a new approach is taken to overcome this undesirableeffect. The starting point is a new setting of the governing differential equations using a finite calculus(FIC) formulation. The basis of the FIC method is the satisfaction of the standard equations for balanceof momentum (equilibrium of forces) and mass conservation in a domain of finite size and retaininghigher order terms in the Taylor expansions used to express the different terms of the differentialequations over the balance domain. The modified differential equations contain additional terms whichintroduce the necessary stability in the equations to overcome the volumetric locking problem. The FICapproach has been successfully used for deriving stabilized finite element and meshless methods for awide range of advective–diffusive and fluid flow problems. The same ideas are applied in this paperto derive a stabilized formulation for static and dynamic finite element analysis of incompressiblesolids using linear triangles and tetrahedra. Examples of application of the new stabilized formulationto linear static problems as well as to the semi-implicit and explicit 2D and 3D non-linear transientdynamic analysis of an impact problem and a bulk forming process are presented. 

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