# The derivation of thin plate spline interpolation energy function？ [closed]

I am trying to derive the "thin plate energy functional". Given a thin plate $$z = z(x,y)$$, how does one derive easily the energy functional

$$\iint_{\mathbb{R}^2} \,\left[\left(\frac{\partial ^2z}{\partial x^2}\right)^2+2\left(\frac{\partial^2 z}{\partial x\partial y}\right)^2+\left(\frac{\partial^2 z}{\partial y^2}\right)^2\right]\mathrm{d}x\,\mathrm{d}y\quad ?$$

• crossposted at physics.stackexchange.com/q/594389/38462 --- that site seems to be the appropriate site, rather than here; in any case, please don't cross-post without disclosing this, to avoid duplication of efforts. Nov 18 '20 at 9:23
• you can find a derivation here: homepages.engineering.auckland.ac.nz/~pkel015/… --- the equation you wrote down is for young modulus $\nu=0$. Nov 18 '20 at 10:43

It very much depends on what you consider a "derivation" and what you consider an "easy" one. You can start assuming that the energy density depends on $$(\kappa_1^2+\kappa_2^2)g^{\frac 12}$$, where $$\kappa_i$$ are the principal curvatures, which can be considered quite natural from a physical point of view (in one dimension this corresponds to assuming that the energy density of a 1D elastic continuum depends quadratically on the curvature, and it leads to the non-linear theory of the Euler Elastica). Then you can deduce your integral functional as a linearized form of the problem, as is done in most textbooks (and also sketched in the wiki page you linked).
Btw: I would not say "Given a thin plate $$z=z(x,y)$$", as $$z$$ is in fact the placement function.