I have just the opposite worry about my new (inherited) induction range: I am worried that it will destroy my pans and pots. ... destroyed pans and the part that was directly above the heating element became convex.
Does anybody know what is the danger of this happening on induction range?
Yes, this can happen. I have first hand experience with warping a matfer bourgeat 12 in skillet and a 7 ply demeyere 12.6 in skillet. I'm not 100% it was the burner with the demeyere, it might have been de-glazing. Here is how to warp a pan:
1) Heat up a cold pan on max
2) add cold or even room temperature ingredients. Use a liquid (water) if you want to maximize the chance of warping.
The warping I encounter is that the pan becomes convex and do not sit flat on the stove top. This is particularly annoying on induction because heating here is highly dependent on distance from the source of magnetic field (1/r^3 if its a coil vs 1/r for conduction)
While it is probably true that a thicker pan is more resistant to deformation than a thinner pan, even a demeyere 7 ply and 3mm of carbon steel can easily be deformed.
I've never warped a cast iron pan. I bet it is because I don't cook on them as much compared to carbon steel or stainless clad rather than some inherent superiority of cast iron.
I have a few tips to prevent warping:
1) Preheat on medium (5 on my stove that goes up to 10). If you need high heat (stir fry), increase after the pan is already fairly hot.
2) Deglaze with a small (1/4 cup) at a time until the liquid stops evaporating immediately. Don't deglaze with liquid straight from the fridge.
3) DO NOT clean the pan immediately after you finish cooking. Wait for it to cool completely. I've never had a huge problem with cast iron or carbon steel when cleaning and bar keeper's friend will strip even the worst crud off stainless pans. I used to soak the pans in vinegar because I assumed the mild acid won't attack stainless much. This is true, except that a salty acid solution may cause pitting and we often cook with salt. So, I don't recommend that. If you really want to keep the finish on your stainless, maybe pre-wash to get rid of salt and then soak in vinegar.