Adaptive path following methods

Static equilibrium paths of structures beyond critical points help understanding and evaluating experimental results and related phenomena. A variety of control types can be applied for tracing static nonlinear equilibrium paths. The choice of an appropriate constraint equation for the incremental iterative scheme is crucial and affects the convergence properties decisively.

Control limit points and sharp snap-backs, artificial unloading against the background of material path dependency or inefficiency due to small time steps are problems evolving from highly nonlinear computations. The aim of this research is the development adaptive schemes that increase numerical robustness, minimize user interference and circumvent artificial unloading.

Each incremental step allows choosing a new constraint equation. Thus, for m steps, m constraint equations are used during computation. The constraint or control equation filters equilibrium points out of the infinite number of equilibrium points of an equilibrium path. The points displayed satisfy prescribed selection criteria. Displacement control, for instance, selects points in a fixed distance and direction from the current equilibrium point. It is also very common to increment the load factor λ. A popular control function for geometrically nonlinear computations is the arc length control, f = s-ŝ. The arc length s is not a physical but a purely geometric quantity. As it is not linked to a certain direction, each predictor step plays a fundamental role concerning the convergence properties of the problem.

Our current research contains the development of methods selecting the controlling parameters based on criteria, that have been evaluated by monitoring fundamental phenomena in the structural response. Control region and control parameter(s) are separated. The control region is where a particular quantity is controlled and control parameter(s) form the constraint equation. The control region is chosen to be the process zone and can be identified by some simple requirements. These specifications refer to damage related parameters identifying the process of damage at a material point. This permits setting up requirements to locate the crack tip and to follow this spot adaptively. The chosen control parameters may or may not be the parameters monitored.Any other quantity being dedicated to the control region and to the damage process can be controlled just as well.

Project titel:
Adaptive continuation methods for highly nonlinear equilibrium paths including softening
Researcher:
Tanja Pohl