Arbitrary Lagrangian-Eulerian method and its application in solid mechanics
Creators: name: Wang, Jin
Publisher: name: University of British Columbia
Year: 2009

description: The applicability and accuracy of existing finite element formulation methods for finite strain deformation and metal forming problems are investigated. It is shown that the existing formulation methods, both Lagrangian and Eulerian type, are not suitable nor efficient for large deformation problems especially when boundary conditions change during deformation, as is the case in most metal forming problems. This creates a need for a more general and efficient type of formulation. An Arbitrary Lagrangian-Eulerian (ALE) method is presented for the general application in solid mechanics and large deformation problems. A consistent ALE formulation is developed from the virtual work equation transformed to an arbitrary computational reference configuration. The formulation presents a general approach to ALE method in solid mechanics applications. It includes load correction terms and it is suitable for both rate-dependent as well as rate-independent material constitutive laws. The proposed formulation reduces to both updated Lagrangian and Eulerian formulations as special cases. The formulation is presented in a form that makes the programming an extension to existing Lagrangian and Eulerian type programs. An efficient mesh motion scheme for ALE formulation is developed with a procedure for handling boundary motion within the scheme, which can ensure homogeneous mesh results. A practical and more efficient numerical method is presented to handle supplementary constraint equations on element level rather than on the global level. Different numerical algorithms for the integration of the rate type constitutive equation are investigated and coupled with the return mapping algorithm to provide plastic incremental consistency. A numerical procedure for stress integration is developed based on the physical meaning of stress. Jaumann and Truesdell rates are taken as the objective stress rates in the constitutive equation. An alternative numerical treatment for rate of deformation tensor [sup t]D[sub ij] is presented to maintain incremental objectivity of the tensor. It is shown by numerical examples that the use of Truesdell stress rate with a developed numerical integration procedure gives consistently more accurate results than other procedures presented. An algorithm for updating material associated properties is presented and applied in simulation of various metal forming problems. A 2D finite element program, ALEFE, based on the presented formulation is developed and tested. The program may reduce to an updated Lagrangian or Eulerian methods as special cases. The mesh motion for the whole domain is controlled by the motion of the boundary nodes. The program can handle unsymmetric stiffness matrices and coupled displacements/velocity boundary conditions. The input data is designed to be similar to available commercial finite element codes, so that the data generation phase may be directly imported from these programs. The output data format is designed to be compatible with general graphic simulation and data processing commercial softwares, so that contour, x-y and deformed mesh plots may be easily created from the output data of ALEFE. Various benchmark and practical problems are simulated by the developed program. Practical simulation cases include flat punch forging process, sheet metal extrusion process, necking bifurcation of a bar in tension, a steady strip rolling and compression between wedge-shaped dies. Numerical results are compared with analytical solutions or experimental results available in literature.