A Hamiltonian Reduction Method for Nonlinear Dynamics

Abstract

Nonlinear analysis of dynamic problems has become important for modern industrial design applications. The increasing pressure on airlines to decrease fuel costs demands the design of more efficient aircraft. This requires aircraft manufacturing companies to design lighter structural components. The result is the need for more realistic and accurate modelling of critical structural components. Over the years, more powerful finite element discretization methods and improved numerical methods and programming techniques for dynamic analyses of structures have been introduced. Despite these advances and the increase in available computer power, the analysis of nonlinear dynamic problems is yet a computationally demanding task, implying it is very expensive. To reduce the computational time of nonlinear finite element analyses, reduction methods have been developed. These methods have as aim to reduce the number of degrees of freedom, while retaining sufficient accuracy of the solution.

Recently, a new reduction method, applicable to nonlinear static stability problems, has been developed at Delft University of Technology. The aim of this thesis is to extend the reduction method for statics to nonlinear dynamics. This is achieved by using the Hamiltonian formulation to describe the motion of a system. A reduced order model (ROM) is constructed for free vibrations, forced vibrations and damped vibrations, using Hamilton’s equations of motion. These are integrated to obtain the response of the ROM, in terms of displacements and momenta. The displacements of the full finite element model are computed by back-substituting the reduced response into the displacement expansion. The ROM is implemented in a finite element framework.

The ROM is applied to beams, to plates as well as to shells. Overall, good agreement is found between the ROM and Abaqus. The big advantage of the ROM is found when the computational times for beams and plates are compared to that of Abaqus. A drastic reduction in time is observed for the ROM, while still maintaining accurate results. The ROM thus saves valuable computational time.