Keywords: solid-shell elements, explicit time integration, efficient implementation, symbolic programming.

For the implementation of element routines designed for explicit time integration algorithms efficient programming regarding the number of operations performed on the element level is of particular importance. The limitation of the time step size and the absence of global equation solving compared to implicit algorithms leads to a domination of element operations for example in a central difference scheme.

The paper presents an implementation concept for element routines based on the application of the symbolic programming tool AceGen [1], a plug-in for the computer algebra software Mathematica. The operations necessary for the computation of the internal nodal force vector, which is the most time-consuming part of an explicit analysis, are implemented symbolically. This means that vector and matrix operations and differentiations do not have to be computed in advance in order to realize a conversion into a programming language. Consequently, programming errors can be avoided almost completely and less time is required for the implementation. Program code in Fortran is generated and simultaneously optimized automatically, which leads to very efficient routines compared to a manually implemented code.

The implementation concept is presented for volumetric shell elements with only displacement degrees of freedom, the so-called solid-shell elements [2,3], using linear/quadratic interpolation of geometry and displacements in the in-plane direction together with a linear interpolation in thickness direction. Besides the pure displacement formulation with standard (full) numerical integration, the implementation of different approaches in order to reduce the so-called locking phenomena are discussed: the method of assumed natural strains (ANS) [4], and the enhanced assumed strain (EAS) method [5]. Both ANS and EAS, as well as a 'Mortar'-type contact formulation with analytical description of the contact surface are implemented into the in-house finite element code FEAP-MeKa [6] using AceGen. The specifics of the implementation concept are discussed and the efficiency and functionality of the element formulations are presented on numerical examples.

1

J. Korelc, http://www.fgg.uni-lj.si/Symech/, 2010.

2

H. Parisch, "A continuum-based shell theory for non-linear applications", Int. J. Num. Meth. Eng., 38, 1855-1883, 1995. doi:10.1002/nme.1620381105

3

R. Hauptmann, K. Schweizerhof, "A systematic development of 'solid-shell' element formulations for linear and non-linear analyses employing only displacement degrees of freedom", Int. J. Num. Meth. Eng., 42(1), 49-69, 1998. doi:10.1002/(SICI)1097-0207(19980515)42:1<49::AID-NME349>3.3.CO;2-U

4

K.J. Bathe, E. Dvorkin, "A formulation of general shell elements - the use of mixed interpolation of tensorial components", Int. J. Num. Meth. Eng., 22, 697-722, 1986. doi:10.1002/nme.1620220312

5

J. Simo, M. Rifai, "A class of mixed assumed strain methods and the method of incompatible modes", Int. J. Num. Meth. Eng., 29, 1595-1638, 1990. doi:10.1002/nme.1620290802

6

K. Schweizerhof, co-workers, "FEAP-MeKa, Finite Element Analysis Program", Karlsruher Institut für Technologie, based on Version 1994 of R. Taylor, "FEAP - A Finite Element Analysis Program", University of California, Berkeley.

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