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I've just finished modeling some object in Fusion 360, exported it as an .STL file and opened it up in Blender in order to make it smoother. So I applied Shade Smooth with Auto smooth enabled on 45 degrees. Surfaces got smooth enough as I need them to be.

After that I tried exporting my model again as an .STL, but after exporting, it didn't export smoothness I applied. I tried exporting many times, also with .OBJ and .STL and nothing. I've searched similar problem on many other forums and communities but nothing helpful was found. I've tried Meshmixer but it distorts my surfaces instead of smoothing them out.

I got to the point where I'm not able to perform even that simple step in Blender. There should be a simple answer on what I'm doing wrong or why Blender won't do what I want.

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Shade smooth is only shading done in software and doesn't change the geometry of your model. If you want it smoother you need to add a subdivision surface modifier which will add the "smoothness" to your geometry.

As you have imported this as an STL, the topology will be pretty bad and the subdivision surface modifier may not work well (which may be why meshmixer is giving you trouble too). Ideally, you want to remesh it first. Decimate might work depending on the model and then add subdivision surface modifier afterwards to smooth it.

Remember when modelling for 3d printing, model with shade flat, not shade smooth.

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