Purely in itself, moving from DX9 to DX11 will have basically little effect on the look of the game, or its performance. Each version of DX is a superset of the previous one - it just adds new options and capabilities. In fact DX11 will still run on DX9 GPUs, as long as you don't use any of the new features. The API changed significantly from DX9 to 10 (and a little from 10 to 11), so DX9 titles need major rewrites of their graphics code to move to DX11, which takes time. At the end you'll have a DX11 title that takes no advantage of the new features, but after that you can start to make improvements.
For a flavour of what could be improved, some of the headline features of DX11 over DX9 are instancing, tessellation and compute shaders (plus a bunch of other smaller things I've probably forgotten).
'Instancing' means drawing multiple copies of objects in one go. In DX9 if you want to draw e.g. ten tanks, you set up the position/rotation of a tank and draw it, and repeat this ten times. In DX11 you can set up an array of transforms for each tank and draw ten at once, potentially saving a load of CPU time (each draw call is expensive). This may well be of use here.
'Tessellation' can be used to dynamically generating new vertices depending on where the camera is. It's useful for height-mapped ground terrains to generate the optimum number of vertices for the given camera angle, and avoid copying so much data to the GPU (again slow). I don't know if this is applicable at the moment, but it's pretty much a requirement for doing good looking dynamic water.
'Compute shaders' are for doing general purpose (not necessarily graphics) parallelisable work on the GPU. It's still mainly used to do graphics effects though and is great for doing super-quick particle systems, hundreds of dynamic lights, post-processing effects, stuff like that. Again I don't know how useful it'll be for this engine.
So, DX12. The main advantage of DX12 is that it can be much more efficient on the CPU, by stripping away most of the driver and giving direct access to the hardware. It's Windows 10 only though, so most (or all?) existing titles add DX12 in addition to DX11 support, as otherwise you're cutting your customer base. That doubles up your work though each time you add a new graphics feature, so personally I think DX11-only is the way to go for now.
At the end of the day though I doubt graphics rendering is actually the bottleneck, so there will be limitations on what a change of DX version can achieve for performance. I suspect that most of the CPU is used for AI, so even if you removed all CPU usage from the graphics, I doubt it would have that great an effect. The other coming improvements around AI efficiency will likely do more .
DirectX 12 now lets developers get closer to the “metal” of their users’ graphics hardware, providing access to features previously unavailable for developers. Exposing lower level hardware means that skilled developers can write more efficient code that executes faster. Specifically, developers can improve optimization with faster multi-threaded performance and more efficient CPU utilization. This could potentially allow for low-level optimizations similar to those we see in console gaming where games are tuned for extremely specific mid-range hardware, eeking out performance that would be impossible without such hyper-specific optimizations.
Previously, DirectX 11 only offered a fairly high level of hardware abstraction, which vastly simplifies the process of coding. In fact, that’s the whole idea behind what makes hardware APIs so powerful. But it also meant that opportunities for fine-tuning code against specific hardware features were less frequent. Microsoft’s Developer Blog provides some great insights into how, exactly, DirectX 12 accomplishes this.
https://blogs.msdn.microsoft.com/directx/2014/03/20/directx-12/#where-doesWelcome to the real world!
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