Wind Farm Layout Optimization Problem (2D) with boundary constraints
by
Risco et al. {cite:p}Risco_et_al
Robotic Design Optimization (3D) with anatomical constraints
by
Lin et al. {cite:p}Lin_et_al
and
Bergeles et al. {cite:p}Bergeles_et_al
Aerodynamic Shape Optimization (3D) with spatial integration constraints
by
Brelje et al. {cite:p}Brelje_et_al
Our method is based on an implicit surface reconstruction method, Smooth Signed Distance (SSD) by Calakli and Taubin {cite:p}Calakli_and_Taubin
We compare ourselves to an explicit formulation for the signed distance function by Hicken and Kaur {cite:p}Hicken_and_Kaur
.
Their method is continuous and differentiable but much like many other methods, scales in computational complexity with the number of points in the input point cloud.
The function interpolates data points via piecewise linear signed distance functions defined by local hyperplanes.
The piecewise functions are then smoothly combined with KS-aggregation.
$$
\phi_{H}(\mathbf{x}) = \frac{\sum^{N_\Gamma}{i=1} d_i(\mathbf{x})e^{-\rho (\Delta_i(\mathbf{x}) - \Delta{\text{min}})}}
{\sum^{N_\Gamma}{j=1} e^{-\rho (\Delta_j(\mathbf{x}) - \Delta{\text{min}})}}
$$
where
Three main formulations were previously used in gradient-based optimization.
Risco et al. {cite:p}Risco_et_al
presents a generic 2D formulation that is continuous and non-differentiable, because it uses the nearest neighbor to calculate the distance.
While it has worked in practice, it is not considered a sufficient solution because it is non-differentiable, not generic to 3D shapes, and scales with Brelje_et_al
presents a generic 3D constraint function that is continuous and differentiable, but scales with Lin_et_al
presents a generic 3D constraint function that is continuous and differentiable.
Their formulation prioritizes smoothness over accuracy in the function, making ti a poor representation of the signed distance.
Additionally, the constraint function scales with