Filtering a design response for a shape optimization

You can apply a filter to a design response that will smooth out local peaks for a shape optimization. The filter is defined as

Φ_{j} = \frac{\overset{N}{\sum_{i - 1}} Φ_{i} B_{i j}}{\overset{N}{\sum_{i - 1}} B_{i j}},

B_{i j} = {(r_{j} - d_{j i})}^{p},

r_{j} = r_{max} e^{- 0.5 (k_{max} \int σ)},

k_{max} = max (| n_{j} \times n_{k} |),

where

Φ_{j}

is the filter function for node j. The main filter function,

B_{i j}

, decreases with the distance, d, between nodes i and j. The maximum radius of influence,

r_{m a x}

, defines the maximum distance from node j for the nodes i to influence the filter value. The local curvature,

κ_{m a x}

, is approximated by the vector product of the node normal,

n_{j}

, and the neighboring nodes,

n_{k}

. The curvature dependency,

r_{j}

, defines a weight function that reduces the radius at higher local curvature.