Vtyjkyuk

Suppose I have an undirected graph $G = (V,E)$, suppose $|V| = m$ and for each such vertex we associate a point mass $Q_1,ldots Q_m$. Now we connect $Q_i$ and $Q_j$ with a spring if and only if $(Q_i,Q_j) in E$.

Now imagine you have such system (call it a mesh) in space, no movement at first. So basically the kinetic energy of the system is given by

$$
T = sum_Q_i frac12lVert Q_i' rVert^2
$$

Assume we are in a vector field defined as

$$
U(Q_i) = k_r sqrtx_i^2 + y_i^2 + k_g z_i
$$

the total amount of potential energy is given by

$$
V = sum_i U(Q_i) + k_e sum_i sum_j in mathcalN(i) left(lVert Q_i - Q_j rVert - l_ij right)^2
$$

The motion equation can be derived by setting
$$
0 = fracpartial Vpartial Q_i + fracddtfracpartial Tpartial Q_i'
$$

And such equation leads me to

$$
0 = k_g hate_3 + k_r fractextProj_span(e_1,e_2) Q_ilVert textProj_span(e_1,e_2) Q_i rVert + k_e sum_j in mathcalNi left(lVert Q_i - Q_j rVert - l_ij right) fracQ_i - Q_jlVert Q_i - Q_j rVert + Q_i''
$$

And this lead me to the motion equation

$$
Q_i'' = -k_g hate_3 - k_r fractextProj_span(e_1,e_2) Q_ilVert textProj_span(e_1,e_2) Q_i rVert - k_e sum_j in mathcalNi left(lVert Q_i - Q_j rVert - l_ij right) fracQ_i - Q_jlVert Q_i - Q_j rVert
$$

Is the equation correct?
(I'm currently simulating it and I'm getting weird results, like instabilities).

What I'm trying to simulate is essentially like stretching some elastic material, assuming it is modelled with a triangular mesh where each triangle's edge has a spring connecting the two adjacent vertices.

The assumptions are uniform mass, I think even considering it it would cancel out.

Yes, a few things. First off, you used the potential energy of a deflected linear spring to be $k Delta x^2$, but it's actually $frac12 k Delta x^2$. Secondly, I think deriving the equation of motion for an individual vertex using $F=ma$ is much easier here. You can then just write the equation of motion for an individual particle as:

$$-nabla U(Q_i) + sum_j in mathcalN(i) k_ij (||Q_i-Q_j||-l_ij) fracQ_j-Q_i = m_i Q''_i
$$

Then, if you're encountering stability issues, I would recommend checking that you're using a stable integrator. For equations like this, the Leapfrog or Verlet integrator is common. Leapfrog is easy to implement (explicit), accurate, and stable.