Elastic potential energy of polyhedra

Hi Vasileios,

good point. The equations [2,3,4] are wrong. In general, elastic energy is
int F(u) du
in case F=k*u, int F du = int k*u du = 0.5*k*u^2 = 0.5*F^2/k
But it is not the case of polyhedrons interaction, where the function F(u) is actually unknown.
But probably some estimation could be done, e.g. F=k*V and assume V being proportional to u^3.
Or the approaches mentioned by you (today is too late for me to investigate them :-)

> Btw, looking at the calculation of the equivalentPenetrationDepth [5,6], is the exact "area" calculated or is an estimation made?

estimation [7]

cheers
Jan

[7] https://gitlab.com/yade-dev/trunk/blob/master/pkg/dem/Polyhedra_Ig2.cpp#L116

Hi Jan,

Thanks for taking a look at this.
I think assuming that V is proportional to u^3 using a fixed estimation can be risky, since it depends highly on the geometry of the overlap region. :p

I like the idea of an equivalentPenetrationdepth, since it simplifies our problem in that the elastic potential work is F*equivalentPenetrationDepth (the integral of the force along the distance it acts on).

Currently, this distance is estimated by dividing the penetrationVolume by the area perpendicularly to the contact normal direction.
I think it would be more straightforward if we calculated two opposite points of the overlap volume, along the normal direction, passing from the contact point, and calculate the equivalentPenetrationDepth as the distance of these points.

Calculating the equivalentPenetrationDepth deterministically can give us an exact calculation for the "area" as well (=penetrationVolume/equivalentPenetrationDepth), in case you have any other uses for this parameter.

All the best,
Vasileios

> I think assuming that V is proportional to u^3 using a fixed estimation can be risky, since it depends highly on the geometry of the overlap region. :p
> the function F(u) is actually unknown.

this is what I meant :-)

> two opposite points of the overlap volume, along the normal direction, passing from the contact point, and calculate the equivalentPenetrationDepth as the distance of these points.

then it would be needed to distinguish equivalentPenetrationDepth and this "estimatedPEnetrationDepth"

cheers
Jan

Good point, thanks Jan!

Regards,
Vasileios

Hi Jan,

I am coming back to this after giving it some thought.

In retrospect, I think assuming an equivalent penetrationDepth or an equivalent contact area is a matter of perspective (i.e. subjective to interpretation) for polyhedra.

What if instead, we calculated u_n incrementally by integrating the normal relative velocity? I am just not sure how this will be affected by the change of contact normals during a contact.

Hope you are keeping well,
Vasileios

The incremental approach sounds reasonable.

Another idea: energy should be conserved, so for a contact, the potential energy is equal to the kinetic energy if the two interacting bodies would repulse. Don't know if it helps much for the value determination, but could be used for verification.

cheers
Jan

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