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](https://is.muni.cz/th/183259/prif_d/diss.pdf).



Construction of combinatorial model categories with all objects fibrant
There are a ton of papers about what you are asking. Another is the thesis of Richard Williamson (arXiv:1304.0867v1). Also, Valery Isaev has a paper that produces a model structure with all objects fibrant, given some cylinder or path object information (<https://arxiv.org/pdf/1312.4327.pdf>). The thesis of Remy Tuyeres produces a model structure given some even more general category theoretic information. Maybe check the references of those three sources; I'll bet there are many others.


