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Arbitrary Lagrangian-Eulerian formulation for quasi-static and dynamic metal forming simulation Bayoumi, Hassan N.


Many engineering problems involve large material deformation, large boundary motion and continuous changes in boundary conditions. The Arbitrary Lagrangian Eulerian (ALE) formulation has emerged in recent years as a technique that can alleviate many of the shortcomings of the traditional Lagrangian and Eulerian formulations in handling these types of problems. Using the ALE formulation the computational grid need not adhere to the material (Lagrangian) nor be fixed in space (Eulerian) but can be moved arbitrarily. Two distinct techniques are being used to implement the ALE formulation, namely the operator split approach and the fully coupled approach. A survey of the ALE literature shows that the majority of ALE implementations for quasi-static and dynamic analyses are based on the computationally convenient operator split technique. In addition, all previous dynamic ALE formulations are based on explicit time integration where no linearization is needed. This thesis presents a fully coupled implicit ALE formulation for the simulation of quasi-static and dynamic large deformation and metal forming problems. ALE virtual work equations are derived from the basic principles of continuum mechanics. A new method for the treatment of convective terms that sidesteps the computation of the spatial gradients of stresses is used in the derivation. The ALE virtual work equations are discretized using isoparametric finite elements. Full expression for the resulting ALE finite element matrices and vectors are given. A new relation that relates grid displacements with material displacements is introduced. The ALE finite element equations are implemented into a 2-D computer code for plane stress, plane strain and axisymmetric problems. The transfinite mapping method is used as the mesh motion scheme for internal nodes. A new treatment for mesh motion on material boundaries is introduced and implemented. Implicit, explicit and mixed implicitexplicit time integration schemes are implemented in the code. A line search technique is employed to accelerate the convergence of implicit calculations. Several quasi-static and dynamic large deformation applications are solved using the developed code. Experimental analysis of a simple V-bending process is conducted for comparison. Comparison of ALE predictions for deformed shapes, equivalent stress and plastic strain distributions and loading curves with analytical, numerical and experimental results are presented. ALE results are in good agreement with other methods of analysis. ALE is shown to prevent mesh distortion and eliminate the need for special contact treatments for problems with corner contact.

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