Let $\Omega \subset R^3$ be a bounded open region. It is well known that there exists a smooth surface $\Gamma$ with minimum area and constant mean curvature which is orthogonal to $\partial \Omega$ and divides $\Omega$ into two regions of equal volume. I wonder if there is a numerical algorithm for constructing such $\Gamma$. 



Convergent algorithm for dividing a body into two regions of equal volume
There is an algorithm, based on a $\Gamma$-convergence result, which can find the minimal surface which divides a shape into two regions of equal volume. More details can be found [in this link](http://www-ljk.imag.fr/membres/Edouard.Oudet/research/Kelvin/optimal_cutting_n.php) and [this other link](http://www-ljk.imag.fr/membres/Edouard.Oudet/research/Kelvin/kelvin_n.php). 


The idea is that instead of looking for a minimal surface, we minimize the sum of the perimeters of the two regions determined by $\Gamma$. Thus the problem is transformed into an optimal partitioning problem. In the link, the examples in $3D$ are only for a division of a cube, but they can be generalized to a general shape, like the ones presented in 2D for the circle and the equilateral triangle. The case needed here is relatively simple, since we search an optimal partition containing only two parts.


Alternatively, you could use the software [Evolver](http://www.susqu.edu/brakke/evolver/evolver.html), which can treat perimeter optimization problems.


