  Strong Disorder Renormalization is an energy-based renormalization that leads
to a complicated renormalized topology for the surviving clusters as soon as
$d>1$. In this paper, we propose to include Strong Disorder Renormalization
ideas within the more traditional fixed cell-size real space RG framework. We
first consider the one-dimensional chain as a test for this fixed cell-size
procedure: we find that all exactly known critical exponents are reproduced
correctly, except for the magnetic exponent $\beta$ (because it is related to
more subtle persistence properties of the full RG flow). We then apply
numerically this fixed cell-size procedure to two types of renormalizable
fractal lattices (i) the Sierpinski gasket of fractal dimension $D=\ln 3/\ln
2$, where there is no underlying classical ferromagnetic transition, so that
the RG flow in the ordered phase is similar to what happens in $d=1$ (ii) a
hierarchical diamond lattice of fractal dimension $D=4/3$, where there is an
underlying classical ferromagnetic transition, so that the RG flow in the
ordered phase is similar to what happens on hypercubic lattices of dimension
$d>1$. In both cases, we find that the transition is governed by an Infinite
Disorder Fixed Point : besides the measure of the activated exponent $\psi$, we
analyze the RG flow of various observables in the disordered and ordered
phases, in order to extract the 'typical' correlation length exponents of these
two phases which are different from the finite-size correlation length
exponent.
