Prediction — How much does it cost to divide h by two?
If we refined a line, we would double its number of elements. But our beam occupies three
dimensions. By dividing the size by two we double simultaneously
div_x, div_y and div_z:
2 × 2 × 2 = 8
The calculation time does not have to grow exactly with that proportion: it also intervenes the degrees of freedom, connectivity, memory and the solver. But the geometric growth explains why “refining just in case” is an expensive strategy.
Mental model — Three different questions
| Concept | Question | Evidence |
|---|---|---|
| Refinement | Have we increased the resolution? | Divisions, nodes and elements. |
| Quality | Are the elements geometrically acceptable? | SHPP,SUMMARY and CHECK. |
| Convergence | Is a relevant physical quantity stabilized? | Variation of displacement between meshes. |
None of the three tests replace the others
A million distorted elements do not make a good mesh. A perfect mesh but too thick can give a stiff response. Two equal results do not prove either nothing if the length or load were changed simultaneously.
Anatomy of a comparable family
A convergence study changes only one decision: discretization. In all three cases we will keep exactly:
- geometry
1.0 × 0.10 × 0.05 m; SOLID185with the same formulation;- steel with
EX=210E9 PaandPRXY=0.30; - fixed support, the total force of
-1000 Nand its direction; - the coordinate of the node used to measure the displacement.
We will also preserve the element family and meshing strategy. Compare tetrahedra thick linear lines with thin hexahedrons would mix two changes and make it difficult to interpret the cause.