Computational strategies for the solution of large nonlinear problems via quasi-newton methods

The usefulness of quasi-Newton methods for the solution of nonlinear systems of equations is demonstrated. After a review of the Newton iterative method, several quasi-Newton updates are presented and tested. Special attention is devoted to the solution of large sparse systems of equations such as those issued from spatial discretization of continua by finite elements. The numerical examples presented comprise static and dynamic analyses of geometrical, material and combined nonlinear structural problems and a model fluid flow problem with different levels of nonlinearity. All the results are assorted with a complete discussion of the different methods used, of the convergence rates and of the associated computer costs. From the present studies, it can be concluded that computational costs for the solution of large nonlinear systems of equations can be reduced drastically by using convenient quasi-Newton updates or by adequate combined Newton/quasi-Newton strategies.