Numerical solutions of nonlinear problems of continua—I: Survey of formulation methods and solution techniques

Abstract

A comprehensive survey is provided for the formulation and solution techniques of finite element applications in nonlinear continuum mechanics problems. The survey discusses the Lagrangian, the updated Lagrangian, and the Eulerian formulations. It is shown that many analysts describe relative or updated Lagrangian formulation under the name of Eulerian formulation. Hence, little effort has been devoted to the development of a consistent Eulerian formulation. The applications, limitations and suitability of each formulation to both material and geometrical nonlinear problems are discussed. In the solution methods, exact equilibrium, approximate equilibrium and self-correcting techniques are discussed. An emphasis is given to the applicability of these methods to particular nonlinear problems and to recent developments and modifications of each method to suit a particular nonlinear field.