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# How Can Explicit Solvers Help With Stubborn Nonlinear Statics Models?

August 1, 2014 By: Steven Hale

Picture this: you've been working hard on your analysis for an FEA consulting project. You have a stubborn nonlinear statics model that won’t converge despite your best efforts.  What do you do?  You’ve tried adding substeps, modifying contact settings, and even changing solvers with no luck.  An explicit solver might be the answer.

As I discussed in an earlier post, explicit solvers are the best choice for simulating high-energy, short-duration dynamic events such as impact, drop testing, and blast analysis.  However, under certain circumstances they can also be useful for static analyses.  Explicit solvers rely on the assumption that model properties are linear within each time step and matrices are updated at the end of each step.  This assumption is considered to be accurate because only very small, conditionally stable time steps are used.  The assumption is significant because it eliminates the need for convergence iterations which can often prevent highly nonlinear implicit analyses from solving.  This means that explicit solvers can be used to handle highly nonlinear statics problems that either will not solve with an implicit solver due to convergence problems or solve very slowly because many iterations are required.

The typical types of highly-nonlinear analyses that explicit dynamics solvers are particularly well-suited to solving include complex contact interfaces with significant sliding, and models with nonlinear material properties such as plasticity, viscoplasticity, or hyperelasticity.  Metal forming, buckling, and parts with soft materials such as foams and elastomers are common examples.  Commercial explicit dynamics codes such as LS-Dyna and ANSYS-Explicit include robust contact algorithms and material models that can handle such problems without numerical difficulties.  In addition, large element distortions are well-tolerated because default single-point integration elements are typically used.

As an example, a two-dimensional static analysis of a staple compressed by a punch into a die was solved using an explicit solver (LS-Dyna).  This geometry is shown in Figure 1.

While seemingly simple, the FEA model contains frictional contact, significant sliding, and considerable material plasticity.  This type of problem can be solved implicitly but may run into convergence difficulties, requiring adjustments to convergence and solver settings, multiple trials, numerous equilibrium iterations, and long run-times.  The explicit solver, on the other hand, is able to march through the problem quickly without any modifications to the solver settings.  The final solution for the explicit solver is shown in Figure 2.  Comparing this result to an implicit solution solved in ANSYS shows good correlation in both deformed shape and plastic strain distribution.

While a solution to a complex nonlinear statics analysis can be obtained with an explicit solver, it’s not without its challenges.  As with all finite element analyses, you need the model to produce an accurate solution in a reasonable amount of time.  Because the explicit time steps are so small, it’s only practical for short transients on the order of less than 0.1 seconds for medium to large-size meshes.  For the staple example, the transient was set to 0.001 seconds.  Transients on this order can cause dynamic effects to become significant which reduce the accuracy of a static analysis.  To remove dynamic effects, system damping can be added.  An initial run without any damping can be used to get the dominant frequency response.  Critical damping values can be calculated from this response and applied to the model to remove kinetic energy.  Ultimately, the kinetic energy should be a small percentage of the total energy.  In the staple example, an alpha damping value of 300 s-1 was used to remove kinetic energy.

## staple_compress.gif

To conclude, explicit dynamics solvers provide an alternate approach to solving highly nonlinear static analyses that will not converge or converge very slowly with an implicit solver.  Explicit solvers are very robust at handling models that include complex contact with significant sliding and models with highly-nonlinear material behavior and element distortion.  While implicit solvers are still the first choice for statics problems, explicit solvers with appropriate transient time and damping settings can be used to provide robust solutions to particularly stubborn statics problems.

Head to the comments below if you have seen similar benefits using explicit dynamics tools to solve challenging static problems.  I look forward to hearing about your experiences!