Core Losses

There are three main losses in an un-gapped cores:

Copper loss. In low frequency, pretty much follows DC resistance, at higher frequency skin effect and proximity effect complicate resistance calculations.

Hysteresis, with goes up with gauss.

A bigger core or more turns reduce gauss, thus hysteresis loss. But, those increase copper loss. You can also use better core material, that has less loss at the gauss you are operating at.

If the waveform is not balanced, you will end up with a DC bias that will lower the gauss in one half of the cycle, and increase it in the other, net result being higher gauss. In that case a gapped core will help.

Eddy current loss depends on the lamination thickness (metal), or grain size (powdered iron or ferrite). Reducing eddy current losses almost always increases the core size due to the increased number of gaps. For example, thinner lamination will have the same insulation thickness on the sides, the same stack will have less iron, higher gauss.

A 4th loss occurs in gapped cores, I refer to that as flux cutting. The winding next to the gap will see a lot of flux. If the flux on one side of a wire is different than the other side, you will have circulating current in that wire. Sheet copper is worse, Litz wire is better, keeping copper away from this flux is best. One way is to leave that part of the winding empty, another is to break up one gap into many gaps will reduce the distance dangerous levels of flux travel, or switching from a large gap to a powdered iron core (it has a distributed gap).

If your losses are large, the heat increases the losses in copper and core (except for some power ferrites, that have a dip in losses above room temp.) In extreme cases this results in thermal runaway, the heating increases losses, that increase heating, etc. until the insulation fails.

Flyback topology is used because it has a low parts count and inexpensive, but also only used at lower power. But that is also because the fly back requires more filtering, especially on the output.

The flyback has higher loss because it is the inductor (gapped) as well as the transformer. In discontinuous operation the peak current (I^2 R losses) can be twice what are in a push pull.

In my experience, for a successful design one needs dedicated optimization tools that have access to databases of cores, materials, and wires (filtered by manufacturability and commonality rules). I have never been able to detect any useful general rules from the results. Essentially, it comes down to a giant brute-force search of all possible combinations. (However, not having any knowledge of magnetic components design will produce dreadful results, as one must know how to set up a proper "cost" function).

A (weak) common factor of all the commercial designs I am familiar with is that they tend to have the highest possible temperature the materials allow (100 - 120 deg C). This seems to ensure that they have the smallest size and use the least amount of material (bringing down cost). As soft-ferrites are optimized to have their minimum loss above 85 degrees, high temperature does not mean that efficiency is necessarily low. This rule will become invalid when reliability is added to the cost function.