Convergence Considerations for Small Displacement Nonlinear Analysis

However, convergence is not guaranteed under all circumstances.

Contact problems, especially those with friction, often cause convergence difficulties. In order to improve the chances of a successfully converged solution, methods have been built in to help problems converge that would otherwise oscillate back-and-forth and never converge. One method involves a "sticky gap", wherein a residual stickiness is introduced to prevent the "undecided" nodes from bouncing in and out of contact. Another method is gap/contact status freezing where, after a number of oscillating iterations, gap/contact elements are not allowed to change their open/closed status.

Note: These methods are activated only for near-converging yet stagnated problems, and do not interfere with converging (or radically diverging) cases.

Problem Setup

Make sure that the nonlinear problem represents a realistic physical situation for which a feasible solution exists. In particular, special care needs to be taken in selecting the proper orientation of gap elements. This is especially important when using a specified gap coordinate system. See the description of the CGAP and CGAPG elements for more details.

Sufficient Support

Since gap/contact elements only provide one-way support, it is possible to formulate the problem in such a way that the individual components will have rigid body freedom under certain loading conditions. This will manifest as zero pivot in the solution process. To avoid such situations, it is advisable to provide sufficient support to all components so that, even without gap/contact elements, there are no rigid body modes. If "solid" supports are not feasible for all parts (the part needs to move), a very weak set of springs can be used to prevent the part from "flying away" when gap/contact elements are not engaged. The stiffness of such auxiliary springs can be selected so as to allow for large motion of the part, compatible with the overall size of the model. If the gap elements and contact interfaces are properly set up, such weak springs will exert virtually no effect when the solution has converged.

Reasonable Gap Stiffness

The gap stiffness values

K A

and

K T

essentially represent penalty springs that are hard enough to prevent perceptible penetration of contacting nodes. While, theoretically, higher stiffness values enforce the contact conditions more precisely, excessively high values may cause difficulties in convergence or poor conditioning of the stiffness matrix (this is especially true for

K T

). If any such symptoms are observed, it may be beneficial to reduce the value of gap stiffness. As a baseline recommendation, a reasonable range of gap stiffness is of the order of:(1)

(10^{3} t o 10^{6}) \cdot E \cdot h

Where, $E$ is the typical value of elastic modulus and $h$ is the typical element size in the area surrounding the gap elements. Such range will generally keep the gap penetration below one thousandth/one millionth of the element size, respectively. A good value for $K T$ is of the order of $0.1 \cdot K A$ .

To facilitate reasonable values of

K A

and

K T

, OptiStruct supports the automatic calculation of these parameters, specifically:

Option $K A$ =AUTO determines the value of $K A$ for each gap element using the stiffness of surrounding elements. Additional options SOFT and HARD create respectively softer or harder penalties. SOFT can be used in cases of convergence difficulties and HARD can be used if undesirable penetration is detected in the solution.
Option $K T$ =AUTO automatically calculates the value of $K T$ . If MU1>0, the result here is the same as with blank $K T$ -- its value is calculated as $M U 1 \cdot K A$ . However, if MU1=0 or blank, $K T$ =AUTO produces a non-zero value of $K T$ , calculated as $K T$ = $0.1 \cdot K A$ . Therefore, $K T$ =AUTO can be used to specify enforced stick conditions.

Friction

The presence of friction, due to its strong nonlinear, non-conservative nature, may cause difficulties in nonlinear convergence, especially when sliding is present. Therefore, solving the problem without friction can often provide convergence in otherwise failing problems. Or, in cases when presence of frictional resistance is necessary and minimal sliding is expected, enforcing a stick condition may be a viable solution, and will often lead to a better convergence than Coulomb friction (refer to the PGAP and PCONT Bulk Data Entries for details). In cases of larger sliding motions, the stick condition may lead to divergence through a "tumbling" mode.

Gap Offset

In order to provide theoretical correctness, friction produces bending moments in gap/contact elements of non-zero length (this results from the transfer of frictional force from the contact surface to the end nodes). This offset operation can, however, cause convergence problems and counter-intuitive results. In problems with friction, it may be advisable to turn off the offset operation via a parameter:

GAPPRM,GAPOFFS,NO

This will produce more intuitive results in the presence of friction. However, it may violate the rigid body balance of the body, and should therefore be used with caution, especially for problems without full SPC support. Refer to the PGAP and PCONT Bulk Data Entries for details.

Incremental Loading

If the nonlinear procedure diverges, in spite of taking the measures described above, the incremental loading procedure (applying the total load in a number of increments) can be used to achieve convergence. Refer to the NLPARM Bulk Data Entry for details. Note, however, that if the problem is incorrectly formulated (the solution exhibits excessive deformations, free rigid body motions, an ill-conditioned stiffness matrix, extremely high nonlinear error, etc.), incremental loading cannot be counted on to provide a converged solution.

Nonlinear Expert System

In some difficult to converge cases, an expert system can be used to achieve convergence:

PARAM,EXPERTNL,YES

The expert system will try to adjust the load increment and other nonlinear parameters to achieve convergence. However, if the problem is incorrectly formulated (the solution exhibits excessive deformations, free rigid body motions, an ill-conditioned stiffness matrix, extremely high nonlinear error, etc.), expert system cannot be counted on to provide a converged solution.

Moreover, in some cases it can lead to long computational times without success. This may be due to using very small load increments or re-running the solution with modified nonlinear parameters.

Arc-Length Method

The arc-length method is available for post-buckling problems in nonlinear static analysis. This can be activated via the NLPCI Bulk Data Entry.