Solution of non-linear problems in structural mechanics by optimization methods

Abstract

Finite element simulation of failure is a useful tool for the structural engineer for two main reasons. First of all, the ultimate loads are readily available from the results. The second, and maybe more important reason, is that the engineer can actually see how the structure breaks down, allowing for a more rational design of the structure. Simulation of failure involves a situation of unstable equilibrium that may pose convergence problems to numerical calculations. The focus of this thesis is hence to investigate methods by which convergence might be improved. Two main ideas have been investigated: 1) Include dynamics since failure is in reality a dynamic process, and 2) Use methods from the related field of optimization. For this purpose, the mechanical problem is posed as an optimization problem and restricted step (also called trust region) methods are used, for which strong convergence proofs hold. We have also developed and implemented a material model suitable for failure simulation. The model is a 3D damage model, and detailed derivation and explicit formulas are given. Simulations with the material model have proved to give realistic results. However, in the case of non-convergence, implementation of the Levenberg-Marquardt method (a particular trust region method) together with the energy functionals, led only to minor improvements in the behavior of the computations. Some reasons for this are given in the conclusions. Complete understanding of failure of structures is a challenge for the future. Whether or not optimization techniques will be used for this purpose, we still do not know, the window is open.