wiki:sns:intactgh:resolution_and_convergence []

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====== Resolution and Convergence ======

===== Quick “Rule of Thumb” Tips =====
    * For quick **design-stage results**, low resolutions  (<100,000) with linear elements will generally solve problems in less than 1-2 minutes on most computers. For increased accuracy and validation, increase this resolution gradually up to 1,000,000+ cells until you have consistent results. 
    * Cell size is a **more advanced setting**. One common way to use this setting is making sure you are capturing your model’s small features. To do this with linear elements, set the cell size to 1/3 the thickness of the smallest desired feature. Note that **setting too small of a cell size can quickly result in long solve times or failures.** 
    * Quadratic elements will need **far lower resolution** than with linear elements. A good starting resolution is around 1,000 cells increasing in increments up to 50,000+ cells, again until consistent.  

===== Resolution =====

Resolution is the maximum number of finite element “cells” used to compute an approximate solution of the given analysis problem. These cells create a “solution grid” where only the cells that intersect the solid are included in the actual resolution count. Each finite element cell is a cube with side length equal to the “cell size”. Due to the cubic scaling of resolution w.r.t cell size it is easy to select a cell size that yields an unexpectedly large resolution. 

{{ :wiki:sns:intactgh:cellsizevsresolution.png?500 }}

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 [[wiki:sns:intactgh:select_resolution|document]].**

{{ :wiki:sns:intactgh:resolutiongridgraphic.png?500 }}

===== Convergence =====

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.

{{ :wiki:sns:intactgh:displacement_convergence.png?500 }}

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 Intact.Simulation for Grasshopper) 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 Intact.Simulation. This is particularly common for models with small features and large size to scale ratio. See section on [[wiki:sns:intact.gh:linear_static_analysis#known limitations & common problems|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 Intact.Simulation for Grasshopper 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.