Development and Application of an ALE Large Deformation Formulation

Document Type : Research Article



This paper presents a complete derivation and implementation of the Arbitrary Lagrangian Eulerian (ALE) formulation for the simulation of nonlinear static and dynamic problems in solid mechanics. While most of the previous work done on ALE for dynamic applications was mainly based on operator split and explicit calculations, this work derives the quasi-static and dynamic ALE equations in its simple and correct form, using a fully coupled implicit approach. Full expression for the ALE virtual work equations is given. Time integration relations for the dynamic equations are also derived. Examples of quasi-static and dynamic large deformation applications are presented.


[1]     M.S. Gadala, "Recent trends in ALE formulation and its applications in solid mechanics," Appl. Mech. Engrg. vol. 193, pp. 4247-4275, 2004. 
[2]     Benson DJ., "An efficient, accurate, simple ALE method for nonlinear finite element programs," Comput. Meth. Appl.Mech. Eng. Vol. 72, pp. 305–350, 1989.
[3]     Haber RB., "A mixed Eulerian-Lagrangian displacement model for large-deformation analysis in solid mechanics," Comput. Meth. Appl. Mech. Eng. Vol. 43, pp. 277–292, 1984.
[4]     Hue´tink J, Vreede PT, van der Lugt J, "Progress in mixed Eulerian-Lagrangian finite element simulation of forming processes", Int. J. Numer. Meth. Eng. Vol. 30, pp. 1441–1457, 1990.
[5]     Kennedy JM, Belytschko TB, "Theory and application of a finite element method for arbitrary Lagrangian-Eulerian fluids and structures, "  Nuc. Eng. Des. Vol. 68, pp. 129–146, 1981.
[6]     Schreurs PJG, Veldpaus FE, Brekelmans WAM, "Simulation of forming processes using the arbitrary Eulerian- Lagrangian formulation," Comput. Meth. Appl. Mech. Eng. Vol. 58, pp. 19–36, 1986.
[7]     T.J.R. Hughes, W.K. Liu, T.K. Zimmermann, "Lagrangian-Eulerian finite element formulation for incompressible viscous flows," Comput. Methods Appl. Mech. Engrg. Vol. 29, pp. 329-349, 1981.
[8]     J. Wang, M.S. Gadala, "formulation and survey of ALE method in nonlinear solid mechanics," Finite Element  Anal. Des. Vol. 24, pp. 253-269 , 1997.
[9]     Gadala, M.S. and Wang, J., "Apractical procedure for mesh motion in ALE method," Eng. With Computers, Vol. 14, pp. 223-234 , 1998.
[10]  M.S. Gadala, M.R. Movahhedy, J. Wang, "On the mesh motion for ALE modeling of metal forming processes," Finite Elements in Analysis and Design, Vol. 38, pp. 435-459, 2002.
[11]  R. Haber, M.S. Shepard, J.F. Abel, R.H. Gallagher, D.P. Greenberg, "A general two-dimentional, graphical finite element preprocessor utilizing discrete transfinite mapping," Int. J. Numer. Methods Engrg. Vol. 17, pp. 1015-1044, 1981.
[12]  A. Huerta, F. Casadei, "New ALE applications in non-linear fast-transient solid dynamics, "  Engineering Computations, Vol. 11, pp. 317-345, 1994.
[13]  H.N. Bayoumi, M.S. Gadala, "A complete finite element treatment for the fully coupled implicit ALE formulation, " Computational Mechanics, Vol. 33, pp. 435-452, 2004..