What is a good way to mount / secure a regular backpack, while commuting?
Option 1

Your bike has a parcel rack on the rear - I'd suggest exploring modifications to that rack to hold your existing backpack more securely.

Bungee cords are good because they have tension, holding the hooks on.  And the length will somewhat adjust for bigger or smaller loads.

Downside - they do slip and move under vibration.

Answer - either make or buy a mesh of bungees to hold your existing bag on top of your rack.  Something like this:



I'd secure the rear edge and one side to your rack and organise the hooks so they secure to your seatpost, and to the other side of your rack, forming a nice mesh.  Each hole should be too small for your bag to slip through even at full stretch.

The shoulder straps on your bag may hang off the side.  Just position it so they don't.  If the straps hang through the structure of your rack then line the underside of the rack with some firm PVC scrap plastic (also, that's a decent mudguard too.  Or you might want to drape the shoulder straps over your seatpost before securing the mesh down.

Perhaps use carabiners instead of bent wire hooks, for added security.



Option 2

You have a rear rack, but its not good enough.  Use the rack to carry a cage like this:



Picture shows two, but you could easily just ahve one rack on whatever side you prefer.  The pictured bags aren't required either - they're more for grocery shopping.

Advantage is that it takes a heck of a bump to throw something up and out of the cage.   They will also "concertina" when empty and take up much less space.  Also good for keeping long skirts out of the rear wheel!

Downsides are added weight and they're not rated for heavy loads.



Option 3

You have a rear rack, why not add a matching front rack too ?



This shows a front basket on a rigid MTB.  These work well and can carry a lot more weight because of the bottom struts.  A bike with front suspension can't carry as much cos the struts would change length.

Advantage, your stuff is right there in sight at all times, its not going to jump out of the basket, and also you can access it easier. 

Downside, the handling of the bike might change - it depends how close you can get it to your head tube and how long your stem is.  Plus the brake cables can sometimes foul the basket.