Implicit surface reconstruction methods have been shown to address these issues well, including hole-filling, reconstructing surfaces from noisy samples, reconstructing sharp corners and edges, and reconstructing surfaces without normal vectors in the point cloud. Basis functions are commonly used to define the space of implicit functions for implicit surface reconstruction. Basis functions are constructed from a discrete set of points scattered throughout the domain, whose distribution and locations play an important role to defining the implicit function. Examples of these points include control points for B-splines, centers for radial basis functions, and shifts for wavelets. Implicit surface reconstruction methods distribute these points in various ways. Heuristics include point density, error-controlled, and curvature-based subdivisions. Octrees are notable because the error of the surface reconstruction decays with the sampling width between control points, which decreases exponentially with respect to the octree depth. Additionally, the neighborhoods of control points from octrees can be solved for and evaluated in parallel using graphics processing units , which allows for on-demand surface reconstruction as demonstrated in [43]. Another approach for distributing the points that control the implicit function is to locate them directly on the points in the point cloud. In the formulation by Carr et al., a chosen subset of points in the point cloud and points projected in the direction of the normal vectors are used to place the radial basis function centers, rolling grow tables resulting in fewer centers than octrees that are still distributed near the surface. The explicit formulation by Hicken and Kaur uses all points in the point cloud to define the implicit function and shows favorable decay in surface reconstruction error as the number of points in the point cloud NΓ increases.

This structure has been used in combination with RBFs for hole-filling in [37] and anisotropic basis functions for representing sharp corners in [40]. Another approach is to construct a uniform grid of points to control the implicit function. Unlike the aforementioned approaches, the distribution of points is decoupled from the resolution of the point cloud. As a result, deformations to the geometric shape can be represented without loss in accuracy near the surface as shown by Zhao et al.. This makes it a popular structure in partial differential equation based reconstruction methods that evolve the surface during reconstruction, such as in [47, 48]. In general, more points representing the implicit function are required to achieve the same level of accuracy to other approaches. As a result, implicit functions defined by a uniform grid are more computationally expensive to solve for in both time and memory usage than the aforementioned approaches, as experienced by Sibley and Taubin, but can be reduced by a GPU-based multigrid approach as implemented by Jakobsen et al..The signed distance function presents an ideal candidate for implicit surface reconstruction and geometric non-interference constraints. It is known that the zero level set of the SDF is a smooth representation of the points in a point cloud, and its gradient field is a smooth representation of the normal vector field from the normal vectors in a point cloud. As a result, many formulations to approximate the SDF have been researched for implicit surface reconstruction. We note that there exists other methodologies, such as wavelet and a Fast Fourier Transform based method, that fit a smooth indicator function instead, but are less applicable for non-interfernce constraints where a measurement of distance is desired. We identify four categories that approximate the SDF in some way: explicit formulations, interpolation formulations with RBFs, PDE-based formulations, and energy minimization formulations.Explicit formulations use the data defined in the point cloud to define linear approximations to the SDF.

These formulations then apply smoothing to these linear approximations in order to define the level set function. Risco et al. present the simplest approach which uses the nearest edge and normal vector to define the function explicitly. The resultant constraint function is piecewise continuous but non-differentiable at points where the nearest edge switches. Belyaev et al. derive a special smoothing method for defining signed Lp-distance functions, which is a continuous and smooth transition between piecewise functions. Hicken and Kaur use modified constraint aggregation methods to define the function in a smooth and differentiable way. Upon the investigation of Hicken and Kaur, the signed Lp-distance functions give poor approximations of the surface. Additionally, Hicken and Kaur’s formulation is shown to increase in accuracy as the data in the point cloud, number of points NΓ, increases. We identify Hicken and Kaur’s explicit formulation as a good candidate for enforcing non-interference constraints,as it is continuous and differentiable with good accuracy.Another method to construct the level set function is to solve an interpolation problem given an oriented point cloud P. Because the data points of P always lie on the zero contour, nonzero interpolation points for the implicit function can be defined on the interior and exterior, as originally done by Turk and O’Brien. Radial basis functions are then formulated to interpolate the data. To avoid overfitting, thinplate splines can be used to formulate the smoothest interpolator for the data, as noted in [37, 45]. Solving for the weights of a RBF involves solving a linear system, which is often dense and very computationally expensive due to their global support. Turk and O’Brien solve up to 3,000 RBF centers, and improvements by Carr et al. allow up to 594,000 RBF centers to be constructed in reasonable time . The vector field is then integrated and fit, usually by a least squares fitting, to make the zero level set fit the point cloud. We classify the methods that solve for the vector field as a solution to a partial differential equations as PDE-based methods. Poisson’s method uses variational techniques to Poisson’s equation to construct a vector field. Improvements to this method add penalization weights to better fit the zero contour to the point cloud in [54].

Tasdizen et al. prioritize minimal curvature and minimal error in the vector field by solving a set of coupled second order PDEs to derive their level set function. Zhao et al. use the level set method, originally introduced by Osher and Sethian, for surface reconstruction, with the advantage of modeling deformable shapes. In the aformentionedPDE-based methods, the setup for the implicit function reduces to solving a PDE by time-stepping or a sparse linear system in the case of Poisson’s equation. In the analysis done by Calakli and Taubin, they found that Poisson’s method often over-smooths some surfaces. We also note that solutions to PDEs are more difficult to implement than other methods in practice.Another methodology is to solve an optimization problem that minimizes some energy function with respect to the values of the basis function directly. The smooth signed distance surface reconstruction method minimizes an energy function with three terms. Minimizing these three terms maximizes smoothness and minimizes the approximation error of the zero level set and the gradient field to the data in P, all in a least squares sense. Alternative forms, such as in [18, 44], propose a different energy term to this formulation, which does a direct least squares fit to the approximate signed distance function. We perform a more thorough discussion of the four energy terms in Chapter 3, as our method also poses an energy minimization problem. The energy minimization problem posed by these papers is a well-posed unconstrained quadratic programming problem. The solution to these unconstrained QP problems reduces to the solution of a linear system. Making use of hierarchical structures, such as octrees, and compactly supported basis functions, the linear system is sparse and recursively solved at increasing depths of the structure. These advantages allow for fast solutions on the order of minutes as reported by [19, 44]. It should be noted that the time and space consumed by hierarchical approaches grows exponentially with the depth of the octree, so many implementations limit the depth up to 11. The resultant number of control points in Tang and Feng are on the order of 106 .We note that interpolation formulations with RBFs, growing rack PDE-based formulations, and energy minimization formulations are different approaches to the same problem of approximating the SDF. The primary differences lie within the derivation and implementation of such methods. The energy minimization formulation by Calakli and Taubin performs a least squares fit to the data in the point cloud. Thin-plate spline RBFs are an exact solution to the same energy minimization formulation to interpolate the data and maximize smoothness, as derived by [56]. The two-step energy minimization formulation by Sibley and Taubin follows the same approach as PDE-based methods, where a vector field is solved for and then a least squares fit is done to fit the surface. We recommend the interested reader to Calakli and Taubin who discuss the similarities and differences between SSD and Poisson surface reconstruction methods.We summarize the context for all the methods in Table 2.1, highlighting the main differences in their formulation, basis function representation, and distribution of points controlling the function. We note our method is an energy minimization formulation, which uses the same energy terms as Calakli and Taubin, but with a different basis function and different distribution of control points. This section, in part, is currently being prepared for submission for publication of the material. The authors of this work are Ryan C. Dunn, Anugrah Jo Joshy, Jui-Te Lin, C´edric Girerd, Tania K. Morimoto, and John T. Hwang. The thesis author was the primary investigator and author of this material.Wind farm optimization problems contain an objective to be minimized with respect to design variables.

The objective function is important to the optimization problem because its minimum defines the optimal design of the wind farm. When an objective function is narrow in scope, the optimal result often compromises other aspects of the design. We tabulate the different objective functions in Table 2.2. Historically, many wind farm optimizations have focused on the annual energy production of a wind farm, which is considered to be too narrow of a scope. When optimizing AEP, solutions may be difficult to manufacture, economically impractical, and over-reliant on the model’s assumptions making it impractical in the real-world. The levelized cost of energy objective function is larger in scope and emphasizes efficient and economically feasible solutions, and is often considered a better objective than AEP. Recent work has converted an existing hub height optimization using AEP to improve the economic viability and competitiveness using LCoE as the objective function. Other optimizations such as cable length, noise, and mass are considered more narrow in scope. In some cases, multiple objective functions are considered in an optimization. For further conversation on the selection of an appropriate objective function, we direct the reader to [57].The design variables of the wind farm optimization are also important to defining the problem. In terms of decreasing the wake interactions of wind turbines, three main methods emerge that define the design variables of each problem. The first method uses the turbine positions as design variables and is called the wind farm layout optimization problem . The WFLOP has a highly multi-modal design space that is especially challenging for gradient-based optimization. The second method uses the turbine’s tower, blade, and rotor-nacelle assembly as design variables and is the turbine design optimization problem. The third method uses the turbine’s control to affect its wakes and is the turbine control optimization problem. We refer the reader to two reviews on wind turbine controls strategies. A summary of the design variables considered in previous gradient-based optimization studies is in Table 2.3. Note that some recent studies consider two optimization problems at once: simultaneous layout and control optimization and simultaneous layout and turbine design optimization. In these studies, the optimal results were improved compare to solving the optimization problems individually.An important problem in the wind farm layout optimization problem is the representation of the wind farm boundary. The wind farm boundary defines the feasible regions of a wind farm, and can be limited due to the geographic features of the land or sea. In optimization, these zones must be enforced as constraints to ensure turbines are placed in a feasible zone. For gradient-free optimization, the representation of these boundaries are simple. Wind farm boundaries may be enforced by a binary function or by discretizing the domain and excluding the points outside of the feasible space. However, a gradient-based boundary constraint function must be continuous and differentiable.