  Chameleon gravity is an example of a model that gives rise to interesting
phenomenology on cosmological scales while simultaneously possessing a
screening mechanism, allowing it to avoid solar system constraints. Such models
result in non-linear field equations, which can be solved analytically only in
simple highly symmetric systems. In this work we study the equation of motion
of a scalar-tensor theory with chameleon screening using the finite element
method. More specifically, we solve the field equation for spherical and
triaxial NFW cluster-sized halos. This allows a detailed investigation of the
relationship between the NFW concentration and the virial mass parameters and
the magnitude of the chameleon acceleration, as measured at the virial radius.
In addition, we investigate the effects on the chameleon acceleration due to
halo triaxiality. We focus on the parameter space regions that are still
allowed by the observational constraints. We find that given our dataset, the
largest allowed value for the chameleon-to-NFW acceleration ratio at the virial
radius is $\sim 10^{-7}$. This result strongly indicates that the chameleon
models that are still allowed by the observational constraints would not lead
to any measurable effects on galaxy cluster scales. Nonetheless, we also find
that there is a direct relationship between the NFW potential and the
chameleon-to-NFW acceleration ratio at the virial radius. Similarly, there is a
direct (yet a much more complicated) relationship between the NFW
concentration, the virial mass and the acceleration ratios at the virial
radius. Finally, we find that triaxiality introduces extra directional effects
on the acceleration measurements. These effects in combination could
potentially be used in future observational searches for fifth forces.
