By abstract construction of a combinatorial model category, I mean starting from a locally presentable category satisfying some assumptions, e.g. equipped with a cylinder or a cocylinder satisfying some special hypothesis, and from these data build a model category structure. The question now is:
What are the known abstract constructions of a combinatorial model category with all objects fibrant and such that not all maps are fibrations (to rule out the case of the discrete model structure) ?
The only example I am aware of is the third section of Marc Olschok's PhD, Model structures from balls.