The program uses the Newton-Raphson approach to solve nonlinear problems. The load is subdivided into a series of load increments which can be applied over several load steps. The following figure illustrates the use of Newton-Raphson equilibrium iterations in a single-degree-of-freedom nonlinear analysis.
Before each solution, the Newton-Raphson method evaluates the out-of-balance load vector, which is the difference between the restoring forces (the loads corresponding to the element stresses) and the applied loads. The program then performs a linear solution, using the out-of-balance loads, and checks for convergence. If convergence criteria are not satisfied, the out-of-balance load vector is reevaluated, the stiffness matrix is updated, and a new solution is obtained. This iterative procedure continues until the problem converges.
A number of convergence-enhancement and recovery features, such as line search, automatic load stepping, and bisection, can be activated to help the problem to converge. If convergence cannot be achieved, then the program attempts to solve with a smaller load increment.
In some nonlinear static analyses, if you use the Newton-Raphson method alone, the tangent stiffness matrix may become singular (or non-unique), causing severe convergence difficulties. Such occurrences include nonlinear buckling analyses in which the structure either collapses completely or "snaps through" to another stable configuration. For such situations, you can activate an alternative iteration scheme, the arc-length method, to help avoid bifurcation points and track unloading.
The arc-length method causes the Newton-Raphson equilibrium iterations to converge along an arc, thereby often preventing divergence, even when the slope of the load vs. deflection curve becomes zero or negative. This iteration method is represented schematically in Figure 8.4: Traditional Newton-Raphson Method vs. Arc-Length Method.
To summarize, a nonlinear analysis is organized into three levels of operation:
The "top" level consists of the load steps that you define explicitly over a "time" span (see the discussion of "time" in Loading in the Basic Analysis Guide). Loads are assumed to vary linearly within load steps (for static analyses).
Within each load step, you can direct the program to perform several solutions (substeps or time steps) to apply the load gradually.
At each substep, the program performs a number of equilibrium iterations to obtain a converged solution.
Figure 8.5: Load Steps, Substeps, and Time illustrates a typical load history for a nonlinear analysis. Also see the discussion of load steps, substeps, and equilibrium iterations in Loading in the Basic Analysis Guide.
The program gives you a number of choices when you designate convergence criteria: you can base convergence checking on forces, moments, displacements, or rotations, or on any combination of these items. Additionally, each item can have a different convergence tolerance value. For multiple-degree-of-freedom problems, you also have a choice of convergence norms.
You should almost always employ a force-based (and, when applicable, moment-based) convergence tolerance. Displacement-based (and, when applicable, rotation-based) convergence checking can be added, if desired, but should usually not be used alone.
If all energy put into a system by external loads is recovered when the loads are removed, the system is said to be conservative. If energy is dissipated by the system (such as by plastic deformation or sliding friction), the system is said to be nonconservative. An example of a nonconservative system is shown in Figure 8.6: Nonconservative (Path-Dependent) Behavior.
An analysis of a conservative system is path independent: loads can usually be applied in any order and in any number of increments without affecting the end results. Conversely, an analysis of a nonconservative system is path dependent: the actual load-response history of the system must be followed closely to obtain accurate results. An analysis can also be path dependent if more than one solution could be valid for a given load level (as in a snap-through analysis). Path dependent problems usually require that loads be applied slowly (that is, using many substeps) to the final load value.
When using multiple substeps, you need to achieve a balance between accuracy and economy: more substeps (that is, small time-step sizes) usually result in better accuracy, but at a cost of increased run times. The program provides automatic time-stepping designed for this purpose.
Automatic time-stepping adjusts the time-step size as needed, gaining a better balance between accuracy and economy. Automatic time-stepping activates the bisection feature.
Bisection provides a means of automatically recovering from a convergence failure. This feature cuts a time-step size in half whenever equilibrium iterations fail to converge and automatically restart from the last converged substep. If the halved time step again fails to converge, bisection again cuts the time-step size and restart, continuing the process until convergence is achieved or until the minimum time-step size (specified by you) is reached.
Consider what happens to loads when your structure experiences large deflections. In many instances, the loads applied to your system maintain constant direction no matter how the structure deflects. In other cases, forces change direction, "following" the elements as they undergo large rotations.
The program can model both situations, depending on the type of load applied. Accelerations and concentrated forces maintain their original orientation, regardless of the element orientation. Pressure loads always act normal to the deflected element surface, and can be used to model "following" forces. Figure 8.7: Load Directions Before and After Deflection illustrates constant-direction and following forces.
Nodal coordinate system orientations are not updated in a large deflection analysis. Calculated displacements are therefore output in the original directions.
Applying an imposed rotation (D command) in a load step is done in a "rotation vector" form, where the magnitude and rotation direction are given by the values of the ROTX, ROTY, and ROTZ components on the D command(s). For step loading (KBC,1), the imposed rotation values are applied at the beginning of the load step. For ramped loading (KBC,0), the values are linearly ramped over the load step.
For compound rotations imposed over multiple load steps, each set of rotations is applied sequentially to the previous deformed configuration. For example, rotating a body about a nodal x-axis first then rotating it about a nodal y-axis is done simply as:
| D,node,ROTX,value |
| D,node,ROTY,0.0 |
| D,node,ROTZ,0.0 |
| SOLVE |
| D,node,ROTY,value |
| D,node,ROTX,value |
| D,node,ROTZ,0.0 |
| SOLVE |
This simplifies the specification of compound motion such as a robotic arm. See examples presented in Appendix A: Example: Analyses with Multiple Imposed Rotations to find how different ways of specifying imposed rotations affect the final configuration.
The sequence in which the rotations are applied determine how the constrained node rotates from its initial configuration to its final configuration. The examples presented in Appendix A: Example: Analyses with Multiple Imposed Rotations demonstrate the effect of the order of imposed rotations on the final configuration.
The rotation components of the output displacements (ROTX, ROTY, and ROTZ) are the sum of all incremental rotations. They generally cannot be interpreted as a pseudovector or rotation vector.
The procedure for analyzing nonlinear transient behavior is similar to that used for nonlinear static behavior: you apply the load in incremental steps, and the program performs equilibrium iterations at each step. The main difference between the static and transient procedures is that time-integration effects can be activated in the transient analysis. Thus, "time" always represents actual chronology in a transient analysis. The automatic time-stepping and bisection feature is also applicable for transient analyses.