Explicit finite element and finite difference methods are used to solve a wide variety of transient problems in industry and academia. Unfortunately, explicit methods are rarely discussed in detail in finite element text books. This paper reviews the basic explicit finite element and finite difference methods that are currently used to solve transient, large deformation problems in solid mechanics. A special emphasis has been placed on documenting methods that have not been previously published in journals.

Computational methods in Lagrangian and Eulerian hydrocodes

/pdf/feap-a-finite-element-analysis-program-qvz36simr5.pdf

FEAP - - A Finite Element Analysis Program

In this paper, a formulation is presented for the finite element treatment of multibody, large deformation frictional contact problems. The term multibody is used to mean that when two bodies mechanically contact, both may be deformable. A novel aspect of the approach advocated is that the equations governing contact are developed in the continuum setting first, before deriving the corresponding finite element equations This feature distinguishes the current work from many earlier treatments of contact problems and renders it considerably more general. In particular, the approach yields a characterization of the frictional constraint (assuming a Coulomb law) suitable for arbitrary discretizations in either two or three dimensions. A geometric framework is constructed within which both frictionless and frictional response are naturally described, making subsequent finite element discretization a straightforward substitution of finite-dimensional solution spaces for their continuum counterparts. To our knowledge, this general formulation and implementation of the frictional contact problem in a finite element setting has not been reported previously in the literature.

The development includes exact linearization of the statement of virtual work, which enables optimal convergence properties for Newton-Raphson solution strategies, and which appears to be highly desirable (if not essential) for the general robustness of implicit finite element techniques. Since the theory and subsequent linearization require no limitations on the amount of deformation or relative sliding that can occur, the resulting treatment of frictional contact is suitable for a wide range of examples displaying significant non-linear behaviour. This assertion is substantiated through presentation of a variety of examples in both two and three dimensions.

A continuum-based finite element formulation for the implicit solution of multibody, large deformation-frictional contact problems

Formulation and analysis of conserving algorithms for frictionless dynamic contact/impact problems

This paper proposes a formulation of dynamic contact problems which enables exact algorithmic conservation of linear momentum, angular momentum, and energy in finite element simulations. It is seen that a Lagrange multiplier enforcement of an appropriate contact rate constraint produces these conservation properties. A related method is presented in which a penalty regularization of the aforementioned rate constraint is utilized. This penalty method sacrifices the energy conservation property, but is dissipative under all conditions of changing contact so that the global algorithm remains stable. Notably, it is also shown that augmented Lagrangian iteration utilizing this penalty kernel reproduces the energy conserving (i.e. Lagrange multiplier) solution to any desired degree of accuracy. The result is a robust, stable method even in the context of large deformations, as is shown by some representative numerical examples. In particular, the ability of the formulation to produce accurate results where more traditional integration schemes fail is emphasized by the numerical simulations. © 1997 by John Wiley & Sons, Ltd.

/pdf/design-of-energy-conserving-algorithms-for-frictionless-3t4doeg7wd.pdf

Design of energy conserving algorithms for frictionless dynamic contact problems

A new approach to enforce surface contact conditions in transient non-linear finite element problems is developed in this paper. The method is based on the Lagrange multiplier concept and is compatible with explicit time integration operators. Compatibility with explicit operators is established by referencing Lagrange multipliers one time increment ahead of associated surface contact displacement constraints. However, the method is not purely explicit because a coupled system of equations must be solved to obtain the Lagrange multipliers. An important development herein is the formulation of a highly efficient method to solve the Lagrange multiplier equations. The equation solving strategy is a modified Gauss-Seidel method in which non-linear surface contact force conditions are enforced during iteration. The new surface contact method presented has two significant advantages over the widely accepted penalty function method: surface contact conditions are satisfied more precisely, and the method does not adversely affect the numerical stability of explicit integration. Transient finite element analysis results are presented for problems involving impact and sliding with friction. A brief review of the classical Lagrange multiplier method with implicit integration is also included.

/pdf/lagrange-constraints-for-transient-finite-element-surface-1p9u5brai3.pdf

Lagrange constraints for transient finite element surface contact

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Lagrange constraints for transient finite element surface contact