# 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. – Biswajit Banerjee Feb 15 '17 at 20:01
• Thank you, @BiswajitBanerjee. I really appreciate your comment – great to know my approach doesn't look completely wrong. – sarasvati Feb 15 '17 at 20:25