# Unclear definition of objective function for MPM

While reading this paper on MPM (page 48), the definition of the objective function $E(\boldsymbol v_{\boldsymbol i})$ didn't make sense to me, although I completely understand the definition of (what I suppose to be) it's derivative $\boldsymbol h(\boldsymbol v^{n+1})$.

Excerpt from the paper:

We will show that solving

$$\boldsymbol h(\boldsymbol v^{n+1}) = \boldsymbol M \boldsymbol v^{n+1} -\Delta t \boldsymbol f(\boldsymbol x^n + \Delta\boldsymbol v^{n+1}) - \boldsymbol M \boldsymbol v^n = \boldsymbol 0$$

is equivalent to minimizing the following objective function:

$$E(\boldsymbol v_{\boldsymbol i}) = \sum_{\boldsymbol i} \frac{1}{2} m_{\boldsymbol i}\lVert \boldsymbol v_{\boldsymbol i} - \boldsymbol v_{\boldsymbol i}^n \rVert^2 + e(\boldsymbol x_{\boldsymbol i}^n + \Delta t \boldsymbol v_{\boldsymbol i}).$$

I believe $\boldsymbol v^{n+1}$ and $\boldsymbol v^{n}$ are velocities of all grid nodes of cartesian grid, whereas $\boldsymbol v^{n+1}_{\boldsymbol i}$ and $\boldsymbol v^{n}_{\boldsymbol i}$ are the velocities of one particular grid node.

## The questions:

1. Why is $E(\boldsymbol v_{\boldsymbol i})$ taking only single velocity vector $\boldsymbol v_{\boldsymbol i}$ as its parameter ($\boldsymbol v_{\boldsymbol i}$ = the velocity of the grid node at position $\boldsymbol i$) while it's summing over all grid nodes $\boldsymbol i$ (the sum $\sum_{\boldsymbol i}$) – isn't that nonsense? Shouldn't it take all grid nodes' velocities $\boldsymbol v$ instead?
2. Isn't it nonsense to sum $e(\boldsymbol{\hat x})$ $i$-times, given that:
• $e(\boldsymbol{\hat x}) = \sum_p V^0_p \Psi(\boldsymbol{\hat F}_p(\boldsymbol{\hat x}) )$ - i.e. sum over all particles $p$ (page 45/equation 191)
• supposing $E(\boldsymbol v_{\boldsymbol i}) = \sum_{\boldsymbol i} \left(\frac{1}{2} m_{\boldsymbol i}\lVert \boldsymbol v_{\boldsymbol i} - \boldsymbol v_{\boldsymbol i}^n \rVert^2 + e(\boldsymbol x_{\boldsymbol i}^n + \Delta t \boldsymbol v_{\boldsymbol i})\right)$ (note the parentheses)
3. I'd suggest to define $E$ instead as $$E(\boldsymbol v) = e(\boldsymbol x^n + \Delta t \boldsymbol v) + \sum_{\boldsymbol i} \frac{1}{2} m_{\boldsymbol i}\lVert \boldsymbol v_{\boldsymbol i} - \boldsymbol v_{\boldsymbol i}^n \rVert^2$$ Is this redefinition of $E$ correct? If not, why is the original formulation correct?
• This will take some time to look through. I'm not very familiar with the visual effects community's approach to MPM. But, superficially, you appear to be correct. Feb 15 '17 at 20:01
• Thank you, @BiswajitBanerjee. I really appreciate your comment – great to know my approach doesn't look completely wrong. Feb 15 '17 at 20:25