Path-following methods

Abstract

Until now, the determination of equilibrium paths of complex structures with strongly nonlinear behavior is a challenge for path-following schemes. Structural problems with close path sections or structures exhibiting brittle softening behavior may lead to experiment failure due to control type properties. From the early seventies, a range of methods have been developed, that are able to trace a path with snap-backs and snap-throughs in load and displacement. These arc-length methods have been modified and enhanced since then. Typical problems arising from this kind of constraint equations are loss of convergence, bisectioning while reducing step size to computer precision and unloading until reaching the undeformed state of the structure – either purely elastic or with secant stiffness due to softening behavior. In the past, arc-length methods have been applied to trace softening problems. However, they are not reliable in the post-critical range due to potential artificial unloading. For a consistently linearized cylindrical arc-length method, artificial unloading could be seen for softening behavior in several examples. Some applications of finite element programs require user intervention after computation has been started. For example, when static analysis does not converge, it has to be manually switched to a dynamic analysis. Particularly for softening behavior, a variety of methods have been developed that directly influence the softening process. Often, these methods turn out to be computationally intensive. The aim of this work is to develop a self-adapting, robust and efficient path-following technique for elastic and in particular for softening structural behavior, based on the finite element method. These control methods are derived from properties of the equilibrium path. Therefore, the main focus of this work is spilt into two parts: first, characteristic properties of equilibrium paths of elastic and softening behavior are identified. Second, control methods arising from these properties are developed by choosing appropriate control parameters. User intervention is not required. For elastic structural behavior, adaptivity criteria are presented choosing and controlling relevant degrees of freedom of the cylindrical arc-length equation. To avoid artificial unloading of softening problems, the phenomenon has first to be described: if both, elastic and dissipative energy decrease, artificial unloading can be detected. As a strategy to prevent artificial unloading, an active process zone is selected as the control region. Control parameters are evaluated in this control region. The identification of the control region is based on material parameters. A locally formulated, strain-based continuum damage model is used here for discussion. In order to be able to describe path-following methods in detail, it is necessary to distinguish between load case and control parameter. The load case is connected with the modelled structure. The control parameter is chosen independently and should be able to assess the full path. Comparing numerical and real experiments, it becomes obvious, that failure of the contol method for both types arises from the same reasons: either the control quantity decreases or the combination of control directive and equilibrium state is not unique. The form of collapse of a control method does not rely on the fact of whether the experiment is realized in reality or numerically. Decisive is the possibility of developing dynamic effects. In case of a static numerical analysis, oscillations cannot develop and the computation will not converge. Undesirable phenomena arising from failure of the control method are grouped into three kinds of mechanisms: First, dynamical snap-through or loss of convergence respectively. Second, elastic unloading on the elastic loadig path and third, unloading with secant stiffness, if softening has started. Another similarity of numerical and real experiments is the distinction between load case and control quantity. A prerequisit for this is a controlled testing machine. Continuously growing control parameters with physical meaning can trace full path until a desired deformation or damage state is reached. For complex structures with elastic or even softening behavior it is often not possible to detect suitable control parameters in advance. In adaptive schemes, constantly changing increments are therefore controlled instead of monotonically growing quantities. In this work, three adaptively modified cylindrical arc-length equations are presented. The adaptivity criterion Maximum has turned out to be the most efficient of those three. The maximum displacement increment of the previous load step enters the constraint equation quadratically. For elastic structural behavior, this method has proven to be efficient, but has shown artificial unloading in the context of softening. Alternatively, equilibrium points can be computed with an optimization algorithm or with a dynamic analysis with loading and unloading branches. The area between these branches can be understood as the energy absorbed by damping. In order to be able to reliably control the damage process of a structure, it is necessary to avoid artificial unloading. If the damage process progesses, energy plots show increasing elastic or dissipative energy. If both energies decrease, artificial unloading is detected. To avoid unloading, the control quantities are evaluated in the control region. The control region is identified in the process zone by monitoring equivalent strains of all gauss points. The Gauss point fulfilling all requirements identifying the process zone, defines the control region. Starting from this idea, three control methods for softening behavior are proposed: The first implementation restricts all possible solution points of a cylindrical arc-length method by multiplying the increment of equivalent strains of the gauss point defining the process zone. As this method is very step size sensitive and computationally intensive it will not be found in all numerical examples. The second method is based on this constrained arc-length method, just eliminating the arc-length part of the constraint equation. Only the increment of equivalent strains is prescribed in the equation. This method requires only a small number of steps compared with the methods proposed in other publications. An advantage of this adaptive strain control is the ability to trace dissipative as well as elastic parts of the equilibrium path without necessitating a switch in control types. The last proposal separates the control quantity and the parameter identifying the control region. When the control region is determined, the maximum displacement increment of the related element is chosen as the control parameter. This increment is measured as the change of a displacement in the direction of the same degree of freedom in the last converged step. Within this adaptive displacement control in process zone, the constraint equation and its linearizations have become decoupled from the choice of the equivalent strain measure. Finally, the performance of methods from other publications and the methods proposed here are predicted in relation to specific structural problems. In order to do this, structural problem types are grouped into compounds with similar properites regarding their demands on path-following schemes. For each of these categories a recommendation is given, regarding which control methods are expected to be suitable. In conclusion, the adaptive strain control as well as the adaptive displacement control in process zone have proven to be effective and robust path-following techniques in the context of softening.