Uugh... I don't envy this task.
Many companies use FEA in two ways.
1. Design engineers create simple models to optimize the parts they design in conjunction with hand calculations. FEA is a tool that allows design engineers to produce more robust components. Design engineers also work with full time FEA analysts to plan more complex models.
2. FEA analysts (also engineers) work full-time creating complex models of systems of parts and models requiring advanced analysis techniques. FEA analysts usually belong to a separate "analysis group" who specialize in performing complex analyses. Often analysts specialize in things like thermal analysis, structural analysis, etc, especially in larger companies.
The process generally works something like this: The systems engineering group employs the project management group to coordinate the design of a module of a device (for example a suspension system in a car). The project management group goes to the design engineering group and requests a number of engineers to create a design. The first phase of the design (concepts, spreadsheet analysis, basic FEA, etc.) is carried out by the design engineers.
Once a preliminary design is established, the project management group, the analysis group, and the design group work together to determine areas requiring advanced analysis. The project management group requests analysis support for the required analyses. The analysis group validates the preliminary design produced by the design group. If issues arise, design iteration is required to correct the problems.
While the analysis group is working, design engineers begin coordinating drafting efforts, prioritizing parts by risk of modification (similitude to previous successful designs, etc). At this point design engineers spend a lot of time focusing on stackup analysis and the like, but must also address any potential problems identified by the analysis group (high stresses, thermal issues, etc). Particularly challenging design issues may require the analysis group to evaluate the resolution.
Once the design is deemed acceptable, analysis is no longer necessary. Although, thermal analysis is often calibrated to TC measurements on actual hardware in critical applications (aircraft turbine engines, etc). Sometimes failures are also evaluated with FEA to help determine root cause.
The whole design process involves a series of technical reviews (conceptual design review, preliminary design review, etc) which vary from place to place. These reviews involve presenting designs and design validations - which often involve presenting FEA.
I don't know if all that is helpful or confuses things... but maybe a few block diagrams later it will all make sense on a your flow chart.
