Efficient and robust path-following solution methods and a framework for performance assessment of numerical solutions

Abstract

The most effective and wide-spread methods of solving a quasi-static structural problem are continuation or path-following methods. They add equations, called constraint functions, to those of a quasi-static problem. A constraint function introduces a parameter, called a step-length, which monotonically increases during the solution procedure. In this thesis, we improve the efficiency of path-following methods in two directions: a) by proposing new formulations for the constraint function and b) by proposing new laws for the evolution of the step-length. Furthermore, as a second contribution line of the thesis, we propose a framework for the performance assessment of numerical methods in order to be able to compare the methods of solving a problem and to select the best one. In a structural analysis, it is desired to understand and predict the behavior of structures in order to design them according to engineering needs. Quasi-static analysis is one of the main types of structural analysis which deals with dynamic phenomena occurring during static loading. This type of analysis neither can be solved in one step like a static analysis nor requires to be formulated as a dynamic problem. Thus, it should be solved by a step-bystep evolution of the loads in an artificial/pseudo time. This evolution is usually formulated by the parametrization of the static equation of the problem under analysis and thus, the evolution of the parameters in the artificial time is traced. Path-following methods are usually utilized to find this evolution by adding constraint functions to the equations of the problem. The greatest advantage of utilizing a path-following method is that it is not necessary to know the evolution of the parameters of the problem; it is only needed to increase the step-length. A first issue of a path-following method is to design the mathematical formulation of the constraint function. Experts of path-following analysis of structural quasi-static problems have suggested many constraint functions for solving different problems. The focus have been on enhancing the robustness of the solution method. However, they are not explicitly tailored to improve other performance criteria such as speed and smoothness. We propose four new constraint functions applicable for solving a quasi-static structural damage problem which improve the performance of path-following methods. A first constraint function exploits the elastic unloading angles in the space of load and displacements. This function is easily understandable from a graphical point of view and simply calculated for only the degrees of freedom with external loads. A second constraint function is designed based on the second law of thermodynamics. For a structural problem with an isotropic damage model for its materials behavior, this constraint function is composed of the history parameter of the damage model. It benefits from the non-decreasing nature of the history parameter. A third and a fourth constraint function are proposed based on the same idea of the second one but with a different distribution of the history parameter and the rate of change of history parameter over damaged material points in the body of the structure under analysis,vi respectively. All of the four constraint functions automatically follow a dissipative path and prevent global artificial unloading, which strengthens the robustness of path-following methods. In addition, we discuss necessary conditions to construct constraint functions suitable for damage models with a single history parameter. According to the conditions, we proposed a simple generic constraint function with an adjustable function. Another issue of a path-following method is how to increase the step-length in each analysis step. A literature review of researches on this subject reveals the fact that most of the works recommend using a measure of local nonlinearity to construct a step-length adaptation law. In this thesis, we propose a new local step-length adaptation law by multiplying a modification factor to a conventional law. The modification factor is a function of the angle between the tangent to the analytical curve and the linearized solution curve. This factor is designed to locally keep the smoothness of the linearized solution curve around a desired value. This adaptation law as well as the adaptation laws in the literature look at the solution locally and do not consider the global solution (i.e. the entire solution up to a desired stage). Thus, we propose a new global step-length adaptation law so as to, again, improve the performance of path-following methods from a global point of view. Since this law is based on the overall performance measures of the complete solution found by path-following methods, and every adaptation law is used to adapt the step-length in each increment of an analysis, approximation or prediction of some of responses is required. We also propose two models to predict two performance measures of the method and to utilize them in the global step-length adaptation law. New constraint functions and step-length adaptation laws, in the thesis, are designed to enhance the performance of path-following methods. For a proper understanding of what performance is and refers to, we define clear and distinct performance criteria which express the concept of quality of a broad class of numerical methods including path-following methods. These criteria are the robustness, accuracy, speed, and smoothness a method provides by solving each problem. To quantitatively assess these criteria, we also defined measures to give the degree of ‘goodness’ for each of criteria. We also propose the conditions of constructing appropriate objective combinations of these measures in order to obtain overall performance of numerical methods. The objective performance measures are employed to select the best method of solving a problem based on the desired importance of each criterion. We specifically design a visualization space to clearly show the regions of dominance of each method of solving a problem. Together with each performance measure, we also propose a similarity measure to calculate the ‘similarity’ between two methods. The performance criteria and their measures, the objective performance measures, and the similarity measures compose a framework of performance assessment of numerical methods. The performance of new constraint functions and step-length adaptation laws are assessed and compared to conventional ones in some numerical examples. The results show a) the improvements they provide in comparison to the conventional ones and b) the reliability of the performance assessment by the new framework.