A quasi-static problem is usually formulated by the equations of static equilibrium and a load parameter which shows the level of loading. Path-following methods are widely used to analyze these kinds of problems. These methods add a constraint function to the equilibrium equations in order to determine the loading evolution. There is a parameter in the constraint function, the step-length, which should be positive in each analysis step and which is determined by a step-length adaptation law. Different adaptation laws control the step-length growth differently, and thus, they influence the performance of the solution. We propose two novel types of adaptation laws based on (a) a local degree of smoothness and (b) global performance measures of the solution. The former uses the angle between the linearized solution path and the tangent to the analytical solution curve while the latter employs simple prediction models for the future evolution of two performance measures. Moreover, appropriate constraint functions for the latter are suggested. Example problems of structural damage are solved by path-following methods utilizing the proposed adaptation laws as well as a conventional one. Results show that the new laws raise distinct possibilities to have solutions with an improved performance.@en

Using a path-following algorithm to analyze a quasi-static nonlinear structural problem involves selecting an appropriate constraint function. This function should improve the desired performance targets of the path-following algorithm such as robustness, speed, accuracy, and smoothness. In order to be able to draw a fair objective selection of a constraint function, it is necessary to collect adequate constraint equations as well as to define the performance of nonlinear methods. In this paper, three new path-following constraints applicable for damage analysis of quasi-brittle materials are proposed. Additionally, performance criteria and their numerical measures for a posteriori assessment of robustness, smoothness, accuracy, and speed of solving nonlinear problems by a path-following method are proposed. Based on the proposed criteria, the performance of the three new constraints and two existing ones is compared for two example problems. As a result, the performance measures are shown to possess an ability to clearly explore the strengths of each constraint. They establish a firm basis for the assessment of not only path-following methods but also other methods for solving nonlinear structural problems.@en