# A Common Case of Divergence of Nonlinear Studies

Divergence of Nonlinear Studies can occur due to excessive loads or other nonlinearities. If loads are increasingly incremented, the solution is expected to diverge at some point. This posting explains a common case where the user applies loads that exceed the buckling loads.

First, let us provide a quick review of the solution control methods that are available to solve nonlinear static studies in SolidWorks Simulation:

# 1. Force Control Method

This is the default and most commonly used method. Loads are incremented by auto-stepping, or as otherwise specified by the pseudo time curve, to seek equilibrium and evaluate the response at each solution step. This method is used by default because it is the most efficient method in most practical cases. However, this method fails when the slope of the Force-Displacement curve becomes zero (flat), which happens at buckling as shown below:

This method, therefore, can be used to predict buckling but cannot be used to evaluate the post-buckling response as the resistance (stiffness) becomes singular, leading to predicting infinite displacements by the Force Control Method . The solver stops with a message indicating the possibility of the occurrence of buckling.

# 2. Displacement Control Method

The Displacement Control Method is based on knowing the Force-Displacement curve (represented by the displacement versus time curve) for a certain degree of freedom. The specified displacement is incremented and a loading parameter is evaluated for each step. The solution singularity faced by the Force Control Method is resolved by knowing the prescribed displacement of the controlling DOF, enabling the solver to evaluate equilibrium and calculate the response for the whole model. However, the method fails due to singularity when the slope of the Force-Displacement curve becomes vertical (infinite) as shown below:

# 3. Arc Length Control Method

The Arc Length Control Method seeks equilibrium by sensing the slope of the Force-equilibrium without prescribing displacement or force at a solution step. It is the most powerful but it is slower than the two other methods. The method can be used to study post-buckling and snap back buckling as shown by the Force-Displacement method shown below:

## An Illustrative Example

Consider a column made of Alloy Steel with dimensions 1”X1”X100”. The column is fixed at one side and a compressive force of 1000 Lbs is applied to the other end as shown.

A buckling study calculates the buckling load factor to be: 0.62655 or a buckling load of 0.62655x1000=626.55 Lbs (closely agreeing with the theoretical Euler solution).

A Nonlinear study with same loads and fixtures and default settings (force control and default linear time curve) stops at the same buckling load:

Giving the message:

# Recommendation

Before running nonlinear studies for problems with slender geometries and compressive loads, it is recommended to run a buckling study to make sure that the applied loads are smaller than the minimum buckling loads. If the applied loads exceed buckling, the Force Control Method will fail. In most cases, it is not practical to allow buckling under the applied loads and therefore geometry or material changes are needed to make sure that the applied loads are smaller than buckling.

However, buckling is sometimes a desired feature of the design and studying the post-buckling response of the structure is needed. In such cases, the Arc Length Control Method can be used. If displacement history can be prescribed for a DOF, the Displacement Control Method can be used instead.

For a practical example of the use of the Arc Length Control Method, refer to the **Snap_Through/Snap-Back of a cylindrical Sheet** in the SolidWorks Simulation Nonlinear Tutorials.

**To access the Arc Length Tutorial:**

- Click
**Help, SolidWorks Simulation, Tutorials.** - From the list of Tutorials, click (note that Simulation Premium license is required to create/run Nonlinear studies).
- Click the
- Follow the step-by-step instructions.

The information above is included in the attached Word document for convenience.

This could spark an intriguing conversation!