With the use of a k-d tree structure, the computational complexity of the k-nearest neighbor search scales better than linearly, O) on average, but the structure is not suitable for gradient-based optimization because the derivatives are discontinuous when the set of k-nearest neighbors switches. Outside the domain of non-interference constraint formulations currently employed in optimization, we discovered a significant body of research conducted on a remarkably similar problem by the computer graphics community. Surface reconstruction in the field of computer graphics is the process of converting a set of points into a surface for graphical representation. Implicit surface reconstruction methods such as Poisson , Multi-level Partition of Unity, and Smooth Signed Distance, to name a few, construct an implicit function from a point cloud to represent a surface. We observed that some of these distance-based formulations can be applied to overcome prior limitations in enforcing geometric non-interference constraints in gradient-based optimization. The first objective of this thesis is to devise a general methodology based on an appropriate surface reconstruction method to generate a smooth and fast-to-evaluate geometric non-interference constraint function from an oriented point cloud. It is desired that the function locally approximates the signed distance to a geometric shape and that its evaluation time scales independently of the number of points sampled over the shape NΓ.
The function must also be an accurate implicit representation of the surface implied by the given point cloud. The contribution of this paper is a new formulation for representing geometric non-interference constraints in gradient-based optimization. We investigate various properties of the proposed formulation, vertical drying racks commercial its efficiency compared to existing noninterference constraint formulations, and its accuracy compared to state-of-the-art surface reconstruction methods. Additionally, we demonstrate the computational speedup of our formulation in an experiment with a path planning optimization and shape optimization problem. This section, in full, is currently being prepared for submission for publication of the material. Anugrah J. 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 energy is a sustainable method for electric power generation that mitigates greenhouse gas emissions from other power generation resources, such as with fossil fuels. Predictions show that the climate change mitigation from wind energy development ranges from 0.3◦C to 0.8◦C by 2100. Off-shore wind farms can also mitigate the impacts of hurricanes for coastal communities. As such an impactful energy resource, the field of wind farm optimization has gained recent attention to maximize the energy production and economic feasibility of developing wind farms. The increased adoption of multidisciplinary design optimization techniques by the wind energy community has produced many recent works including the optimization of wind turbine designs, wind farm layouts, and active wind farm control. In general, turbine design, wind turbine layout, and active turbine control strategies are the three main methods to increase wind farm efficiency by reducing the wake interaction between turbines .
Although these methods individually may increase the net efficiency, it has been shown that considering multiple or all three methods can further the a more optimal model. Recent simultaneous optimization studies include control and layout optimization and turbine design and layout optimization. Numerical optimization, as an important design tool to solving these problems, has been widely used for wind farm optimization. Gradient-based and gradient-free algorithms are the two main algorithms to perform optimization. Historically, gradient-free algorithms have been used for wind farm optimization problems due to the high multi-modality in the design space of these problems. Gradient-free optimizers are robust to local minima, while gradient-based optimizers often converge to a local optima. However,as these problems increase in scale and the number of disciplines, the dimensionality of the design space may become impractical for gradient-free optimization. Gradient-free optimizers scale poorly in the number of function evaluations as the number of design variables increase in these complex wind farm problems. Gradient-based optimization, especially with analytic gradients, scales better in the number of function evaluations over gradient-free optimizers in these cases. In addition, recent developments have added methods for gradient-based optimizers to navigate the multi-modal design space of these problems. As a result, gradient-based optimization continues to play a key role in optimizing wind farms. When modeling wind farms for gradient-based optimization, it is important to consider the computational speed and differentiability of the models. High fidelity models are often very computational expensive to evaluate, and these models must be evaluated up to hundreds of times during optimization. Therefore, lower fidelity models that are less computationally expensive are often considered for use in gradient-based optimization.
Additionally, the differentiability of the models is a requirement in order to perform gradient-based optimization. The ability to calculate derivatives within the model has not always been readily available. Oftentimes, significant effort must be made to hand derive the derivatives, or in the worst case, using the finite difference method for derivatives, which is on the same order of function evaluations as gradient-free optimization. Current state-of-the-art gradient-based optimizations are performed using automatic differentiation, however it still requires a level of effort to implement into new models, especially when local smoothing techniques are required. A notable research problem in wind farm layout optimization is the representation of wind farm boundary constraints. Boundary constraints in wind farm layout optimization prevent the placement of a wind turbine on regions outside of the permitted zone. Examples of exclusion zones for off-shore wind farms include unsuitable seabed gradients, shipwrecks, and shipping lanes. These zones are often disjoint, non-convex, and highly irregular shapes represented in 2D. There exists a lack of a generic method to represent these boundaries in the wind farm optimization community. Additionally, the state-of-the art methods suffer from the same problems noted in Section 1.1, where the computational complexity scales with the number of points representing the polygonal wind farm boundary. Conveniently, the first contribution of this thesis addresses this issue. The new geometric non-interference constraint formulation provides a smooth, differentiable, and fast-to-evaluate constraint function that represents the wind farm boundary suitable for gradient-based optimization. Another tool that may show to benefit gradient-based wind farm optimization is a new modeling code language called the computational system design language. CSDL is an algebraic modeling language for defining numerical models that fully automates adjoint-based sensitivity analysis. Additionally, CSDL contains a three-stage compiler system that constructs an optimized computational graph representation of the models. As a new design language, it shows potential to improving the convenience and speed of developing the models to perform gradient-based wind farm optimization. The second objective of this thesis is to implement the two aforementioned tools–the geometric non-interference constraint formulation and the computational system design language –and perform optimization studies on multiple wind farm optimization problems. We conduct optimization studies on turbine hub heights, turbine yaw misalignment, and wind farm layout, and investigate their properties as it pertains to gradient-based optimization. These three problems demonstrate the potential of gradient-based optimization in turbine design, wind farm control, and wind farm layout optimization problems. Using well know analytical models, we conduct multiple optimization studies using CSDL as a modeling paradigm and verify its accuracy with other industry-leading optimization frameworks. Additionally, vertical grow racks we perform a wind farm layout optimization with a real-world wind farm, highlighting the accuracy and efficiency of the geometric non-interference constraint formulation. This section, in full, is currently being prepared for submission for publication of the material. Anugrah J. Joshy and John T. Hwang. The thesis author was a contributor to this material. We identify two preexisting methods for enforcing geometric non-interference constraints in gradient-based optimization that are both continuous and differentiable. Previous constraint formulations that utilize the nearest neighbor distance, e.g., Risco et al. and Bergeles et al. , have been used in optimization, but we note the that they are non-differentiable and may incur numerical difficulties in gradient-based optimization. Brelje et al. implement a general mesh-based constraint formulation for noninterference constraints between two triangulations of objects. Two nonlinear constraints define their formulation. The first constraint is that the minimum distance of the design shape to the geometric shape is greater than zero, and the second constraint is that the intersection length between the two bodies is zero, i.e., there is no intersection.
A binary check, e.g., ray tracing, must be used to reject optimization iterations where the design shape is entirely in the infeasible region, where the previous two constraints are satisfied. As noted by Brelje et al., this formulation may be susceptible to representing very thin objects, where the intersection length is very sensitive to the step size of the optimizer. Additionally, the constraint function has a computational complexity of O, which may be addressed by the use of graphics processing units . Lin et al. implement a modified signed distance function, making it differentiable throughout. Using an oriented set of points to represent the bounds of the feasible region, the constraint function is a distance-based weighted sum of signed distances between the points and a set of points on the design shape. This representation is inexact and is found to compromise accuracy for a smoothness in the constraint representation in practice. Additionally, their formulation has a computational complexity of O.Our first objective—to derive a smooth level set function from a set of oriented points—closely aligns with the problem of surface reconstruction in computer graphics. Surface reconstruction is done in many ways, and we refer the reader to for a full survey on surface reconstruction methods from point clouds. We, in particular, focus on surface reconstruction with implicit function representations from point clouds. Implicit surface reconstruction is done by constructing an indicator function between the interior and exterior of a surface, whose isocontour represents a smooth surface implied by the point cloud. The methodologies for surface reconstruction use implicit functions as a means to an end; however, the focus of our investigation is on the implicit function itself for enforcing non-interference constraints. We identify that the direct connection between non-interference constraints and implicit functions in surface reconstruction is that the reconstructed surface represents the boundary between the feasible and infeasible region in a continuous and differentiable way. The surface reconstruction problem begins with a representation of a geometric shape. Geometric non-interference constraints may be represented by geometric shapes using scanned samples of the surface of an anatomy, outer mold line meshes, user defined polygons, and a sampled set of points of seabed depths. Many geometric shape representations, including those mentioned, can be sampled and readily converted into an oriented point cloud and posed as a surface reconstruction problem. The construction of any point cloud comes with additional complexities. For example, machine tolerance of scanners introduce error into scans, and meshing algorithms produce different point cloud representations for the same geometric shape. As a result, implicit surface reconstruction methods often take into consideration nonuniform sampling, noise, outliers, misalignment between scans, and missing data in point clouds. 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. One approach is to adaptively subdivide the implicit function’s domain using an octree structure. Octrees, as used by, recursively subdivide the domain into octants using various heuristics in order to form neighborhoods of control points near the surface. 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, resulting in fewer centers than octrees that are still distributed near the surface.