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Stress Analysis Convergence Tips for "Dummies"

September 19, 2014 By: Peter Barrett

How many times have you picked up one of those yellow “for Dummies" books and gotten some key tips? There is even a "Fantasy Football for Dummies" book that for \$17, may pay dividends if your star quarterback doesn't get hurt. The goal of this blog is to provide some guidance for understanding and more efficiently achieving convergence to make your nonlinear stress analyses converge - maybe even on the first try!

First a few definitions:

Convergence Criterion: With a nonlinear analysis there will always be inherent error in your model.  The convergence criterion is the allowance of error.  This is often defined as a percent of the applied load where "load"  is all the external forces applied to the model, which can also be reaction forces where displacement loading is used (Newton-Raphson restoring forces).  Typical values for force convergence criterion range from .1 to .5% of the applied load.  The user can change this value, but typically the default should be used since non-convergence is more likely caused by the model setup.

Force Convergence Value (FCV) or Residual: This is the unbalanced "error" force that is a result of the changing stiffness of your model caused by either geometric, material or contact nonlinearities. The residual is the difference between the applied load and the summation of internal forces of the equilibrium iteration.  The program calculates this automatically with no user input required. Tracking rate of change in residual vs. changes in solution settings is an excellent way of optimizing solution efficiency.

Convergence: Convergence occurs when the FCV or Residual is less than the Convergence Criterion.  It is usually followed by a fist pump or loud “YES” yelled from the bowels of the dark corner cubicle.  Ninety percent of the time, force convergence is the only criteria needed for stress analyses. The other 10 percent might require additional displacement convergence (one use is concrete cracking), volumetric compatibility of u-P elements (used with rubber/seal materials), max plastic strain increment, or creep strain rate limits.

Rigid Body Motion (RBM):  For static analyses of assemblies, non-convergence is often a result of one or more bodies "flying" off into outer space. Very large or near infinite displacement is caused by RBM. If the only "support" preventing unlimited motion is another part of the assembly it is often possible that the contact support does not engage and thus part(s) of the model  "fly" through the other bodies.

Now for the fun part! Everyone that has run more than one nonlinear analysis has experienced the frustration of non-convergence. Three tips for efficiently overcoming these problems are:

1. Find the source!

a. Non-convergence is caused by either RBM or too large FCV.  One needs to diagnose the cause in order to find the cure!

b. Determine the cause by reviewing the output and/or plotting the unconverged displacements.  Large displacements or displacement increments indicate RBM.  If displacements are small, the problem is unconverged residuals.

c. Sometimes isolating a sub-region of the model and/or adding temporary displacement boundary conditions can help isolate the problem.

d. Try changing all contact surfaces to bonded and then convert one-by-one back to standard contact to uncover the rigid body motion/convergence issue.

e. Plotting force residuals can sometimes be used to find the non-convergence contact pair(s) in large assemblies.  Be careful however, since these plots are only useful when the force convergence value is similar in magnitude to the criterion.  A force residual plot with values of 10e+30 compared to a criterion of 500 can often point to a non-problem area.

2. Fix the Rigid Body Motion

a. Adding boundary constraints, weak springs, or converting the loading to displacements rather than forces, are techniques commonly used to eliminate RBM.  Apply displacement loads via a rigid "block".  While not always required, if you can analyze all the parts of your model independently of each other, you will never have rigid body motion problems.

b. Start with all parts in the assembly touching. This can be achieved by moving bodies, adding contact offsets, or adding stabilization damping.

c. Add friction to the contact surfaces.

d. Run the analysis dynamically.

3. Overcome Non-Convergence

a. Ramp on the load slowly.

b. Reduce the stiffness of the contact elements. (My experience has shown that ramping loads slowly  and/or lowering contact stiffness will solve 90% of convergence issues)

c. Specify a reasonable minimum criterion when the unbalance forces are small.  If the loading is zero and the criterion is a percent of the load, the model will never converge without a minimum criterion.

d. Refine the mesh in the contact region to reduce the percentage of elements flipping in and out of contact.

e. Modify the mesh to reduce element distortion.  Mesh such that the deformed elements become more uniform in shape.

f. Adjust the material curves to extend beyond the anticipated strain levels and/or increase the tangent stiffness slope.

g. Change the material law for Hyperelastic models. Test with 1 element problems (see my blog - The Value of a Single Element Model for Complex FEA)

Help is available in most software manuals and tutorials, or from FEA consultantsTraining classes can also be a great avenue to enhance your analysis skills. Now, please root for Aaron Rodgers (Cal Berkeley Alumni) to dominate my Fantasy Football league this year!