Introduction

An essential idea in finite element modelling (FEM) is convergence, the idea that as you make the simulation more and more lifelike, its results will converge to the real-life stress state. In most simulations, which mainly use single-point or rigid-body constraints, the main variable that must be converged is element size: as you make the elements smaller and smaller, the simulation will become truer and truer to reality. In this case, the procedure to obtain convergence is fairly simple:

• run successive simulations
• reducing the element size each time — for example, by halving it
• until the results — for example, the maximum stress
• converge within some amount — for example ±5%.

Recently I was trying to do contact simulations, which requires converging three main variables: mesh size, and also contact stiffness and clearance adjustment. Contact stiffness describes the pressure that pushes apart two intersecting bodies in the simulation; in real life, contact pressure is infinite, and things can’t intersect at all. Clearance adjustment describes how much the FEM is able to slightly warp the bodies’ geometry to make the mesh solve; in real life clearance adjustment is zero.

This blog post presents some studies I did to determine how the three variables affect each other in a simple contact simulation. The simulation consisted of a spherical indenter, 5 mm in diameter, pressed with a force of 100 N onto a flat plate, both made of 6061-T6 aluminium (E = 68.90 GPa, ν = 0.33). In this situation, Hertzian contact theory predicts a maximum shear stress of 505 MPa. In the simulation, the mesh was refined in the small circular region surrounding the contact areas of both indenter and plate. Simulations were done in the FreeCAD FEM workbench.

Results

From this, we can see that clearance adjustment is the single most important variable influencing convergence of the whole model, and must be the variable that is converged first: at overly-high clearance adjustments, the stresses for small elements and large clearance adjustments go through the roof. As clearance adjustment goes lower and lower, the stress eventually converges with zero error. In fact, this converged stress is equal to the stress with a clearance adjustment of zero.

If we examine the stress distributions of simulations with high and low (i.e. converged; equivalent to zero) clearance adjustments, we see that at high clearance adjustment, the part gets warped to produce high stresses at the edges of the contact area. This is completely different from the real-life stress distribution, which is closely approximated by the simulation with converged clearance adjustment.

There’s a relationship between element size and clearance adjustment: if element size is too large and clearance adjustment is too small, then the simulation will not solve. If element size is sufficiently small, then the simulation will solve with zero clearance adjustment. This critical element size is positively related to the degree of deformation sustained by the parts; as force decreases, the bodies will deform less, and a smaller element size is needed to solve the model at zero clearance adjustment.

In some cases it might be necessary to decrease element size beyond the critical size in order to achieve stress convergence. However, in this study, the element size was already converged at the critical size.

Finally, we can see that contact stiffness doesn’t influence the simulation’s ability to solve. Instead, a high contact stiffness increases stress for all combinations of element size and clearance adjustment. However, the effect of contact stiffness on maximum stress decreases as clearance adjustment decreases; this means that contact stiffness should be converged last, so that an overly-high stiffness is not selected based on an unconverged clearance adjustment value.