# Effects of volume fraction and filter radius

This blog post explains the significance of the volume fraction and filter radius in the Topology optimization (TO).

### Volume fraction (volfrac)

Volume fraction is the fraction of the original volume that the geometry of the optimized structure will have. Since, weight reduction is the major goal and density of the material remains constant therefore volume become the deciding criteria in our analysis. Even though lowest possible volume that can perform the intended work is preferred, but reduction of too much volume in the design criteria can affect the feasibility of the structure as well.
Here, in this case two scenarios are taken both having the same mechanical loading but with different volume fractions 0.4 and 0.6. It is quite visible that the volume fraction = 0.4 i.e. top one is not feasible at all after density filtration. The effects of volume fraction becomes more significant when structure is subjected to multiple types of loading like thermal and mechanical as the conductance of material gets affected by the amount and distribution of material within the structure.

### Filter radius ($r_{min}$)

$r_{min}$ i.e. filtering radius is used to avoid the check-boarding pattern in TO.  Check-boarding pattern basically means alternative void and solid elements in the geometry as shown in figure. With the help of filtering radius i.e. $r_{min}$ we are taking the weighted affects of the surrounding elements in the optimal value of each elements thus generating less discretized structure and producing a more feasible structure.

Here, $ele_a$ is the element whose optimal function value will be modified and $ele_b$ are the surrounding atoms and $W_b$ is the weighted factor that is decaying linearly with distance.

Check-boarding pattern can also be avoided by using higher order elements or finer mesh, but these two can significantly increase the computational cost.