Issues in convergence improvement for non-linear finite element programs

Systems of non-linear equations as they arise when analysing various physical phenomena and technological processes by the implicit Finite Element Method (FEM) are commonly solved by the Newton-Raphson method. The modelling of sheet metal forming processes is one example of highly non-linear problems where the iterative solution procedure can become very slow or diverge. This paper focuses on techniques to overcome these numerical difficulties. Several methods to generate initial guesses within the radius of convergence are proposed. Appropriate stopping criteria for the iterative procedure are discussed. A combination of various line search methods with the continuation method is proposed. The efficiency and robustness of these numerical procedures are compared based on a set of test examples. A particular form of line search was identified which allows the stable and efficient solution of highly non-linear sheet metal forming problems. Even though the present investigations were motivated by the application of the implicit FEM to the simulation of sheet metal forming processes, the findings are general enough to be applicable to a wide spectrum of non-linear FEM applications.