My goal is to: 

 - Flatten out a cube 

 - Keep the cube's base on the same position of z-axis (i.e. flat on the ground plane)

 - Ideally, the volume of the cube should remain constant

Think of it like a block of clay slowly flowing out to the sides:  

(High quality 2d sketch:)

![enter image description here][3]


So here is my scene: I have a *"cube"* (black object) and a lattice (orange object) that encapsulates the cube:

<img src="https://i.sstatic.net/b9Hjq.png" width="400" />

Then I: 

 1. Selected the lattice with <kbd>B</kbd>

 2. <kbd>Ctrl</kbd><kbd>P</kbd> -> Lattice Deform

 3. Added Shape Keys under menu *Data: Object Data* 

  ![enter image description here][1]

 4. Then I tried to transform the cube with the lattice. Here is the result:

 ![enter image description here][2]

The way the cube is deformed is perfect, **EXCEPT the base of the cube does NOT stay on the ground plane** -- it gets pushed down below it.

So, my question is: how can I deform the cube like this, yet keep the base on the ground plane?


  [1]: https://i.sstatic.net/Gpvbt.png
  [2]: https://i.sstatic.net/NzeMQ.png
  [3]: https://i.sstatic.net/E7sDb.png