I asked this on mathoverflow as well and apologies for cross-posting. I am trying to compute this so-called bending energy matrix. The bending energy of a thin plate in 3D is given by:
$$ BE = \int_0^{X}\int_0^{Y}\int_0^{Z} \sum_{d=1}^{3}\left\{\left(\frac{\partial^2u_d}{\partial x^2}\right)^2+\left(\frac{\partial^2u_d}{\partial y^2}\right)^2+\left(\frac{\partial^2u_d}{\partial z^2}\right)^2 + 2\left[\left(\frac{\partial^2u_d}{\partial x \partial y}\right)^2 + \left(\frac{\partial^2u_d}{\partial x \partial dz}\right)^2 + \left(\frac{\partial^2u_d}{\partial y \partial z}\right)^2\right]\right\}dx dy dz $$
Now, I am trying to embed this so that the above equation would be equal to
$$ BE = u'\Sigma u $$
where u would be a vector that defines the components of the field $u_d$ in the bending energy equation. Now, I need to find this matrix $\Sigma$ so that the above equality can be specified. This should be a large and sparse matrix. However, I have been struggling all day to formulate the form of this matrix and what it's entries should be. I was hoping someone here might give me some pointers on how to construct this matrix.
I would really appreciate any help you can give me.