Resolution and Convergence

Resolution is the maximum number of finite elements used to compute an approximate solution of the stress analysis problem. Only those elements that intersect the solid are included in the count. Generally speaking, higher resolution increases accuracy of the solution but also requires more memory and longer computation times. In the context of stress analysis, increased resolution should make the system more flexible, generally increasing the computed displacements.

So why not just always use the maximum resolution? There are at least two important reasons:

• You may run out of memory and/or will have to wait for a long time to get your solution, which will still only be a numerical approximation of an idealized model of physical reality.
• To see if the solution is converging, you need to compare the solutions at several different resolutions.

To learn more on how to pick the proper resolution for analysis, read this document.

Convergence is a key concept in understanding and interpreting the solutions computed by any numerical approximation software. Numerical simulation approximates an idealized theoretical model by breaking it, or the space around it, into small pieces called finite elements. In principle, as elements get smaller and smaller (increasing their number and resolution), the numerical simulation should get closer and closer to the theoretically exact answer. At some point, the simulation gets so close to the exact answer that increasing resolution does not visibly improve the results. In technical jargon, we say that the numerical solution has “converged”. In this sense, there are no “correct” solutions, but only converged solutions.

To establish that the solution converged, solve the same problem a number of times, gradually increasing the resolution, until displacement values stay approximately in the same range. If displacement does not converge, there is no guarantee that the numerical solution is accurate.

If computed displacement values did converge, one can also study convergence of stresses. But it is important to remember that the linear theory of elasticity (used by every structural analysis software, including the present version of Scan&Solve™) predicts infinite stresses near “wedges,” re-entrant corners, interfaces between different materials, and other singularities. In physical reality, this cannot happen, because the material simply deforms more “plastically” (as opposed to “elastically”). This means that at some points in a model, stresses may never converge – they will just get bigger and bigger as you increase the resolution. The more complex your model is, the more likely you will have some singularities like that. It does make sense to study convergence of stress values at particular locations in the model that are away from singularities.

The solution of the problem may not always converge, even for displacement. There are several reasons for this.

• There may not be enough resolution, even at the maximum resolution allowed by Scan&Solve™. This is particularly common for models with small features and large size to scale ratio. See section on limitations for more details.
• The model is physically unstable, and small changes in the geometry, material, or boundary conditions lead to large changes in displacements and/or stresses. Numerical approximation may simulate such small changes during the solution process.
• Are converged solutions always correct? No. Every numerical procedure has its limitations, and Scan&Solve™ is no exception. The correctness of computed results depends on specific elements used in the solution procedure and individual steps in the procedure process, including function and solid sampling, surface and volume integration, and the solution of linear system of equations. To validate the solution procedure, it is highly advised to test it against a variety of similar and different problems, as well as on problems with known solutions.

The convergence checking process can be automated in SnSScript. Read this document to learn how to easily create a convergence plot with SnSScript.