I'm not an expert at LS-Dyna, but I can share some information about Ansys Static Structrual that is relevant.  I'm responding here rather than your new post since this post has the images.

In a nonlinear (large deflection ON) static structural analysis, you can turn on the Inverse Option. That takes an input mesh shape that assumes a load (gravity load in your case) is present at Time=0 and solves for the shape of the mesh that would exist when the gravity load is removed at Time=1. To perform inverse solving, simply set the Inverse Option property to Yes in the Advanced category of the Analysis Settings.

Another way to have the shape of the soil surface remain unchanged when a gravity load is applied is to use the INISTATE command. This command applies a stress to every element in the model at Time=0. If you apply the correct values of stress and a gravity load, then the mesh will be in equilibrium and the mesh won't move at the end of the first step at Time=1.