Polygonal Research Articles

A genetic algorithm's novel rainfall distribution method for optimized hydrological modeling at basin scales

ABSTRACT Rainfall has a dominant role in rainfall-runoff models, with the rendering of these models depending both on the accuracy of the data and on the way that rainfall is spatially allocated. The research proposes a methodological framework where a genetic algorithm (GA)-based method responsible for the spatial distribution of gauge observations at the basin scale is coupled with the HEC-HMS hydrological model to produce simulated discharges of high accuracy. The custom-developed GA is used to divide a 2D space, adhering to specific criteria, into polygonal geometries that represent gauge zones of influences, similar to the Thiessen polygon method concept. Consequently, a collection of vectorial polygonal areas, equivalent in number to the employed monitoring stations, is produced with the areal weights to be used for distributing the rainfall across the case study basin and subsequently to force the hydrological simulations. The generated gauge weights are validated for a different temporal precipitation event. The final outputs expressed through a series of statistical measures, clearly demonstrate the effectiveness of the specific methodology (e.g. R2 and Nash–Sutcliffe are larger than 0.83 and 0.73). The methodology can foster accurate hydrological simulations, especially in cases where there is a limited number of rainfall stations and corresponding observations.

Dynamics of 137Cs content in higher aquatic plants of Kyiv and Kaniv Reservoirs

The study aimed to determine the temporal parameters of 137Cs specific activity in higher aquatic plants of Kyiv and Kaniv reservoirs. Parameters characterizing the dynamics of 137Cs content for hydrophytes (Potamogeton perfoliatus and Ceratophyllum demersum) and helophytes (Phragmites australis and Typha angustifolia) were determined based on the results of 137Cs specific activity determination in the aboveground plant organs. Plant samples were collected in 1989–2021 from a polygonal area in the upper part of Kyiv reservoir (shallow waters near Strakholyssia village) and in 2000–2021 from a polygonal area in the middle part of the Kaniv reservoir (shallow waters near Rzhyschiv and the Borispol islands). A model of specific activity of aquatic organisms was substantiated (or developed) in case of exponential decrease in radionuclide levels in the environment. The main parameter of the model is the period of resulting halving of organism specific activity due to the decrease in radionuclide concentration in water, its irreversible fixation in bottom sediments, and radioactive decay. During 1989–2021, the half-life period of 137Cs specific activity in Potamogeton perfoliatus, Ceratophyllum demersum, and Phragmites australis of Kyiv reservoir was 5.4 years. Two time intervals were identified, characterized by different intensities of radionuclide specific activity decrease. During the periods of 1989–1996 (1998), the specific activity of 137Cs in the studied plant species decreased by half in approximately 2 years. For the time interval of 2011(2012)–2021, the half-life period of 137Cs content in Potamogeton perfoliatus and Ceratophyllum demersum increased on average to 12 years. A tendency towards a slowdown in the rate of specific activity decrease was observed for Phragmites australis. During 1987–1996, the half-life period of maximum 137Cs specific activity in higher aquatic plants of Kaniv reservoir was 2.1 years. Thus, during the first decade after the accident, the rates of 137Cs content decrease in plants of Kyiv and Kaniv reservoirs did not differ. In 2000–2021, the rate of 137Cs specific activity decrease in Potamogeton perfoliatus and Ceratophyllum demersum of Kaniv reservoir corresponded on average to a half-life period of about 20 years. For Phragmites australis and Typha angustifolia, only a tendency to decrease in 137Cs activity was noted, corresponding to a half-life period of about 40 years. Over time, the half-life period of 137Cs specific activity in higher aquatic plants of Kyiv and Kaniv reservoirs increases, which is explained by the slowdown in the rate of decrease in volumetric activity of 137Cs in the waters of the Dnipro and Pripyat rivers. A model of 137Cs specific activity in higher aquatic plants of reservoirs for periods exceeding 20 years after accidental radionuclide entry into ecosystems of large plain reservoirs has been proposed. The parameters of the model describing the dynamics of 137Cs specific activity can be used for predictive assessments of 137Cs content in higher aquatic plants in the event of accidental radionuclide entry into ecosystems of large plain reservoirs.

Monoreactive multimass oscillator of non-fixed frequency

Introduction. The work relates to the field of mechanical engineering, namely to oscillating mechanical systems. The relevance of the study is determined by the fact that vibrations of inertial masses occur everywhere.Target. Development of a mathematical model of a monoreactive multimass oscillator of non-fixed frequency.Research methods. It is proved that the points that are the coordinates of the end of an arbitrary vector in the coordinate system are the vertices of a regular polygon. The shape and dimensions of the polygon are not related to the coordinates of the vector, i.e. unchangeable. The center of a regular polygon in all cases coincides with the middle of the vector. In the considered (idealized) case, the polygon, at the vertices of which oscillating loads of masses m are located, lies in the Z plane. In technical applications, the loads should not interfere with each others movements, therefore, each load should have its own plane, and all planes should be parallel (like multi-piston mechanism).Results. The condition for the occurrence of free harmonic oscillations is the equality to zero of the total energy of the system, which in the case under consideration is exclusively kinetic, which determines the monoreactive nature of the oscillator. In the considered multidimensional flat monoreactive oscillator, free harmonic linear oscillations of weights can occur.Conclusion. Only kinetic energy is involved in energy exchange. There is no need for elastic elements. The oscillator does not have a fixed natural frequency of oscillation. The frequency depends on the initial speeds and positions of the weights. A regular polygon undergoes a double rotation - around point 0 and around point r. At the same time, the loads carry out linear harmonic oscillations with amplitude . The use of a crank-slider or crank-rod mechanism will allow for parallel movement of loads.

Folding Auxetic Polygonal Kirigami Tubes

Abstract Tubular auxetic structures have wide-ranging applications including medical stents, collapsible energy absorbers, and novel fasteners. To accelerate the development in these areas, and open up new application directions, an expanded range of design and construction methods for auxetic tubes is required. In this study, we propose a new method to construct polygonal cross-sectional auxetic tubes using the principles of origami and kirigami. These tubes exhibit useful global auxetic behavior under axial extension, despite the individual polygon faces not being auxetic themselves. In general, a flat kirigami sheet cannot be simply folded into a polygonal tube since this creates kinematic incompatibilities along the polygon edges. We resolve this issue by replacing the edge folds with an origami mechanism consisting of a pair of triangular facets. This approach eliminates the incompatibilities at the edges while maintaining a connection between faces. The proposed edge connection also introduces additional control parameters for the tube kinematics: for example, introducing a kinematic limit on tube extension and enabling non-uniform behavior along the length of the tube. The rich kinematic behavior possible with polygonal cross-sectional kirigami tubes has potential applications ranging from soft robotics to energy-dissipating devices.

A recursive trapezoid-based algorithm designed to compute the strength- and stiffness-related geometric properties of beams with polygonal cross-sections

A new recursive trapezoid-based algorithm is proposed for calculating the strength- and stiffness-related geometric characteristics of polygonal beam sections. By taking as input the coordinate vertices of the polygon, the approach may effectively be applied to determine the cross-section area and the position of its centroid along with the axial area moments of inertia and product of inertia relative to the cross-sections’ centroidal axes. As the method combines the theory of planar moments with the well-known trapezoid numerical method, it provides a straightforward tabular procedure that can be easily implemented using any spreadsheet software. In addition, aiming to determine the principal area moments of inertia, a computation matrix formalism based on the theory of orthogonal diagonalization of symmetric matrices is put forward as an alternative to the classical method of vanishing the first derivative and solving for the principal directions. Some benchmark examples are worked out to prove the relevance and accuracy of the proposed approach for several beam sections ranging from the most common triangular and quadrilateral ones to very complex open and closed polygonal profiles. The solution presented here appears to be virtually foolproof compared to the conventional one, as it does not necessitate the decomposition of complex polygonal cross-section area into elementary surfaces of different shapes, then use the parallel axes theorem, and so forth; rather, it requires only insertion of the coordinates of the vertices of the polygon, turn by turn in the clockwise order, relative to an arbitrary cartesian coordinate system, and no further steps.

Dwell time for optical fabrication using the modified discrete convolution matrix method

More accurate dwell time calculation methods are necessary to achieve superior error convergence in producing optically critical components. Although the discrete convolution matrix method finds widespread application, it still has approximate errors in the non-uniform discrete form of tool paths. To address this issue, this paper introduced a modified matrix elements method and presented the general Voronoi polygon area weight calculation forms under different tool path discretization forms. The mechanism is explained through analysis and the validity is verified by numerical simulation. The modified method significantly improved uniformity distribution and accuracy in computation of surface residuals. This improvement holds promise as a guiding principle for the fabrication of ultra-precision optical components.

Investigation on the correlation between grain boundary character distribution and precipitate behavior in T91 ferritic/martensitic steel

In the present study, the relationship between microstructure, grain boundary character distribution (GBCD) and precipitate behavior in T91 steel was investigated via a series of thermomechanical treatments (TMTs). The experimental results revealed that subjecting the steel to cold-rolling (CR) with a deformation of 15–30% followed by tempering at 750 °C for 1 h caused the microstructure of T91 steel to transition from the typical lath martensite structure to a uniform polygonal ferrite, resulting in a change in the GBCD. Specifically, the proportion of random grain boundaries for the ferritic structure was significantly higher than that for the martensitic structure. The M23C6 carbide appeared as irregular polygons and precipitated within the ferrite grains, while it exhibited a rod-like shape and precipitated at the martensitic boundaries. Additionally, the orientation relationship (OR) between M23C6 carbide and the adjoining matrix transformed from the double Kurdjumov-Sachs (DKS) relation to (112) M23C6 // (110) ferrite and 201¯M23C6 // [012] ferrite. The change of OR resulted in a higher coarsening rate of M23C6 carbides precipitated in ferrite compared to those precipitated at martensitic boundaries.

Reusable energy-absorbers design harnessing snapping-through buckling of tailored multistable architected materials

Abstract Multistable metamaterials are artificially engineered materials that possess microarchitectures capable of maintaining multiple stable configurations. However, the realization of mechanical metamaterials with numerous programmable stable configurations using Double-Curved Beam (DCB) elements remains an ongoing challenge. In this study, we exploit the snapping-through buckling phenomenon exhibited by architected DCB structures to devise a mechanical metamaterial with a unique deformation mode, encompassing Multi-stability, Multi-path, Multi-platform, and Multi-step characteristics, hence referred to as a 4M mechanical metamaterial. By employing DCB as fundamental building block elements, architected metamaterials with two-dimensional (2D) series or parallel lattices are successfully constructed, as well as three-dimensional (3D) tubular geometries, denoted as DCB-n-m-C and DCB-n-m-M metamaterials, respectively. These metamaterials exhibit reversible energy absorption characteristics and the stiffness can be transformed from positive to negative under both small and large elastic deformations. Functional gradient design and tailored deformation capability are given by adjusting the wall thickness of each layer of double-curved beams, thereby demonstrating the multi-path deformation features inherent in 4M metamaterials. DCB-n-m-M metamaterials has multiple energy platforms in the process of snapping-through, which reflects the multi-platform characteristics of 4M metamaterial. Consequently, novel properties such as multistability, programmability, and reusable energy absorption characteristics are achieved. To comprehensively understand the mechanical response of the metamaterials, a thorough investigation into the influence of geometric parameters is conducted, including the number of polygonal edges, the number of the layers, and the aspect ratio Q. This investigation involves a combination of theoretical analyses, numerical simulations, and experimental verifications. The introduced design strategy paves a way for the innovative design of multistable, multi-step, tailored, and reversible deformation metamaterials.

Security Improvement of UAV Group IoT in Polygonal Area Based on Similarity Clustering Algorithm

Abstract Currently, unmanned aerial vehicle (UAV) group Internet of Things (IoT) has poor stability, poor endurance, and low security. In this regard, we report a polygon area UAV group IoT clustering algorithm for large-scale, high-speed mobile IoT environment. First, the UAV group IoT was divided into a polygon area and a few cluster heads were used to cover the polygon area uniformly. Then, the clustering index in the maximum speed similarity clustering algorithm was induced into the weighted clustering algorithm. Simultaneously, the link retention rate, relative node degree, and node residual energy ratio were improved to select the IoT node with the maximum weight as the cluster head. Finally, the IoT security was improved by reputation calculation. The results show that the clustering algorithm in this paper can reduce the number of clusters and lower the handover rate between clusters, improve the stability of clustering, prolong the survival time of the minimum node, improve the overall endurance of the IoT, and have high security.

Dispersion problem on a convex polygon

Given a set P={p1,p2,…,pn} of n points in R2 and a positive integer k(≤n), we wish to find a subset S of P of size k such that the cost of a subset S, cost(S)=min⁡{d(p,q)|p,q∈S}, is maximized, where d(p,q) is the Euclidean distance between two points p and q. The problem is called the max-min k-dispersion problem. In this article, we consider the max-min k-dispersion problem, where a given set P of n points are vertices of a convex polygon. We refer to this variant of the problem as the convex k-dispersion problem.We propose an 1.733-factor approximation algorithm for the convex k-dispersion problem. In addition, we study the convex k-dispersion problem for k=4. We propose an iterative algorithm that returns an optimal solution of size 4 in O(n3) time.

About polygon area uncertainty in GIS and its implications on agro-forestry estimates

Error affecting calculation of a polygon area from a digital map is an issue that is commonly neglected by remote sensing and Geographic information system (GIS) users. In this work, a method is presented aimed at estimating the uncertainty related to area calculation of polygons in a vector map. Additionally, area error relationship with polygon geometric features was analyzed, as well. A multivariate regression-based approach was applied for this task. After presenting the method, to demonstrate its operational capabilities, it was applied to 2 case studies corresponding (i) to a forest map and (ii) to the map from the Geo Spatial Aid Application used for in the framework of the EU Common Agriculture Policy. Estimated uncertainty (percentage) median values were found to be 0.01% and 0.02% for (i) and (ii), respectively. It was also demonstrated that the same polygon shape built using less vertices (longer vectors) generates area estimates that are less accurate than the ones from polygons built using a higher number of vertices (shorter vectors).

Synthesis of 1-DOF mechanisms for exact regular polygonal path generation based on non-circular gear transmissions

Design of one-degree-of-freedom (1-DOF) mechanisms is of paramount interest. This work deals with the generation of exact regular polygonal paths by using two particular 1-DOF mechanisms. The design of two-bar and three-bar mechanisms including non-circular gears is presented and evaluated. Differences in the kinematics of both mechanisms are discussed. The three-bar mechanism allows derivable velocity and acceleration curves during the whole trajectory, including the polygon vertices, a feature that cannot be achieved with the simpler two-bar mechanism. Hence, the three-bar mechanism makes it possible to generate perfect vertices using non-circular gears. To the best of the authors’ knowledge, this mechanism is the first 1-DOF mechanism with articulated links and a non-circular gear transmission that can generate exact regular polygonal paths. The proposed mechanisms can also be applied to generate two straight lines with a given angle, which is another novel contribution of this work. A prototype of the three-bar mechanism has been developed for experimental validation.

Multi-granularity learning of explicit geometric constraint and contrast for label-efficient medical image segmentation and differentiable clinical function assessment

Automated segmentation is a challenging task in medical image analysis that usually requires a large amount of manually labeled data. However, most current supervised learning based algorithms suffer from insufficient manual annotations, posing a significant difficulty for accurate and robust segmentation. In addition, most current semi-supervised methods lack explicit representations of geometric structure and semantic information, restricting segmentation accuracy. In this work, we propose a hybrid framework to learn polygon vertices, region masks, and their boundaries in a weakly/semi-supervised manner that significantly advances geometric and semantic representations. Firstly, we propose multi-granularity learning of explicit geometric structure constraints via polygon vertices (PolyV) and pixel-wise region (PixelR) segmentation masks in a semi-supervised manner. Secondly, we propose eliminating boundary ambiguity by using an explicit contrastive objective to learn a discriminative feature space of boundary contours at the pixel level with limited annotations. Thirdly, we exploit the task-specific clinical domain knowledge to differentiate the clinical function assessment end-to-end. The ground truth of clinical function assessment, on the other hand, can serve as auxiliary weak supervision for PolyV and PixelR learning. We evaluate the proposed framework on two tasks, including optic disc (OD) and cup (OC) segmentation along with vertical cup-to-disc ratio (vCDR) estimation in fundus images; left ventricle (LV) segmentation at end-diastolic and end-systolic frames along with ejection fraction (LVEF) estimation in two-dimensional echocardiography images. Experiments on nine large-scale datasets of the two tasks under different label settings demonstrate our model’s superior performance on segmentation and clinical function assessment.

Maximizing the Area of Polygons via Quasicyclic Polygons

Based on Peter’s work from 2003, quadrilaterals can be characterized in the following way: “among all quadrilaterals with given side lengths 𝑎, 𝑏, 𝑐 and 𝑑, those of the largest possible area are exactly the cyclic ones”. In this paper, we will give the corresponding characterization for every polygon, by means of quasicyclic polygons properties.

Herbo-Mineral Medicine, Lithom Exhibits Anti-Nephrolithiasis Activity in Rat Model of Hyperoxaluria by Attenuating Calcium Oxalate Crystal Formation and Oxidative Stress.

Calcium oxalate monohydrate (COM) forms the most common type of kidney stones observed in clinics, elevated levels of urinary oxalate being the principal risk factor for such an etiology. The objective of the present study was to evaluate the anti-nephrolithiatic effect of herbo-mineral formulation, Lithom. The in vitro biochemical synthesis of COM crystals in the presence of Lithom was performed and observations were made by microscopy and Scanning Electron Microscope (SEM) based analysis for the detection of crystal size and morphology. The phytochemical composition of Lithom was evaluated by Ultra-High-Performance Liquid Chromatography (UHPLC). The in vivo model of Ethylene glycol-induced hyperoxaluria in Sprague-Dawley rats was used for the evaluation of Lithom. The animals were randomly allocated to 5 different groups namely Normal control, Disease control (ethylene glycol (EG), 0.75%, 28 days), Allopurinol (50 mg/kg, q.d.), Lithom (43 mg/kg, b.i.d.), and Lithom (129 mg/kg, b.i.d.). Analysis of crystalluria, oxalate, and citrate levels, oxidative stress parameters (malondialdehyde (MDA), catalase, myeloperoxidase (MPO)), and histopathology by hematoxylin and eosin (H&E) and Von Kossa staining was performed for evaluation of Lithom. The presence of Lithom during COM crystals synthesis significantly reduced the average crystal area, feret's diameter, and area-perimeter ratio, in a dose-dependent manner. SEM analysis revealed that COM crystals synthesized in the presence of 100 and 300 μg/mL of Lithom exhibited a veritable morphological transition from irregular polygons with sharp edges to smoothened smaller cuboid polygons. UHPLC analysis of Lithom revealed the presence of Trigonelline, Bergenin, Xanthosine, Adenosine, Bohoervinone B, Vanillic acid, and Ellagic acid as key phytoconstituents. In EG-induced SD rats, the Lithom-treated group showed a decrease in elevated urinary oxalate levels, oxidative stress, and renal inflammation. Von Kossa staining of kidney tissue also exhibited a marked reduction in crystal depositions in Lithom-treated groups. Taken together, Lithom could be a potential clinical-therapeutic alternative for management of nephrolithiasis.

A compact and efficiently designed two-port MIMO antenna for N78/48 5G applications

A tightly packed irregular polygon with a distinctive protruding strip, specifically tailored for the N78/48 application, enhances a highly compact and effectively designed two-port MIMO (multiple-input, multiple-output) antenna. Its dimensions measure 20x31.5 × 1.6 mm3, with εr being 4.4 of FR-4 substrate, and it impressively delivers an extensive impedance bandwidth (Sxx < −10 dB) spanning the 3.25–3.85 GHz range. The design incorporates a MIMO antenna with closely spaced elements, merely 1.5 mm (0.012 λ0) apart. The microstrip inserts a feeding line with a partially truncated ground, a grounded stub, and a side stubs embedded in the ground plane, which improve isolation. Positioning T shaped decoupling elements between radiators helps the antenna's bandwidth enhance and improves isolation (S21) across the band. Extensive validation of the antenna's performance has been carried out through comprehensive s-parameter analysis, closely mirroring the results obtained through measurements. Despite its compact form, this antenna efficiently minimizes coupling, achieving S21 levels exceeding −19.25 dB throughout the band. Notably, the antenna attains an impressive peak gain of 3.3 dBi and exhibits a radiation efficiency of 92%. A total affective reflection coefficient (TARC) that starts at-10 dB, a MEG of 0.481 dB, and a channel capacity loss (CCL) of 0.3016 bits/s/Hz are some of the things that make up its MIMO diversity performance. The envelope correlation coefficient (ECC) is 0.0089. It's particularly suitable for 5G-NR sub-6GHz requirements, offering an efficient and compact solution.

A Convex Polygon of Twist Influenced by Nandakumar and Ramana Rao

In this study, the researchers address the conundrum: To divide a 2-dimensional convex polygonal region Q into a specified natural number n of convex pieces, all of which have equal areas although perimeters could be different, such that the total length of cuts used is maximized. With the aid of AutoCAD software, the results show that a convex polygon Q divided into n=2 equal area partitions must have its cut line λ¬ or its area bisector originate in some vertex "v" _"α" of Q with the least measure. We claim that this cut line λ is the maximum possible cut length that divides polygon Q into n=2 equal area partitions. The resulting proof of this conjecture can be used recursively for a convex polygon Q divided into n=2^k equal area partitions, where k∈N

MRI Extracellular Volume Fraction in Liver Fibrosis-A Comparison of Different Time Points and Blood Pool Measurements.

Extracellular volume (ECV) correlates with the degree of liver fibrosis. To analyze the performance of liver MRI-based ECV evaluations with different blood pool measurements at different time points. Prospective. 73 consecutive patients (n = 31 females, mean age 56 years) with histopathology-proven liver fibrosis. 3T acquisition within 90 days of biopsy, including shortened modified look-locker inversion recovery T1 mapping. Polygonal regions of interest were manually drawn in the liver, aorta, vena cava, and in the main, left and right portal vein on four slices before and after Gd-DOTA administration at 5/10/15 minutes. ECV was calculated 1) on one single slice on portal bifurcation level, and 2) averaged over all four slices. Parameters were compared between patients with fibrosis grades F0-2 and F3-F4 with the Mann-Whitney U and fishers exact test. ROC analysis was used to assess the performance of the parameters to predict F3-4 fibrosis. A P-value <0.05 was considered statistically significant. ECV was significantly higher in F3-4 fibrosis (35.4% [33.1%-37.6%], 36.1% [34.2%-37.5%], and 37.0% [34.8%-39.2%] at 5/10/15 minutes) than in patients with F0-2 fibrosis (33.3% [30.8%-34.8%], 33.7% [31.6%-34.7%] and 34.9% [32.2%-36.0%]; AUC = 0.72-0.75). Blood pool T1 relaxation times in the aorta and vena cava were longer on the upper vs. lower slices at 5 minutes, but not at 10/15 minutes. AUC values were similar when measured on a single slice (AUC = 0.69-0.72) or based on blood pool measurements in the cava or portal vein (AUC = 0.63-0.67 and AUC = 0.65-0.70). Liver ECV is significantly higher in F3-4 fibrosis compared to F0-2 fibrosis with blood pool measurements performed in the aorta, inferior vena cava, and portal vein at 5, 10, and 15 minutes. However, a smaller variability was observed for blood pool measurements between slices at 15 minutes. 1 TECHNICAL EFFICACY: Stage 3.

Gradient-based wind farm layout optimization with inclusion and exclusion zones

Abstract. Wind farm layout optimization is usually subjected to boundary constraints of irregular shapes. The analytical expressions of these shapes are rarely available, and, consequently, it can be challenging to include them in the mathematical formulation of the problem. This paper presents a new methodology to integrate multiple disconnected and irregular domain boundaries in wind farm layout optimization problems. The method relies on the analytical gradients of the distances between wind turbine locations and boundaries, which are represented by polygons. This parameterized representation of boundary locations allows for a continuous optimization formulation. A limitation of the method, if combined with gradient-based solvers, is that wind turbines are placed within the nearest polygons when the optimization is started in order to satisfy the boundary constraints; thus the allocation of wind turbines per polygon is highly dependent on the initial guess. To overcome this and improve the quality of the solutions, two independent strategies are proposed. A case study is presented to demonstrate the applicability of the method and the proposed strategies. In this study, a wind farm layout is optimized in order to maximize the annual energy production (AEP) in a non-uniform wind resource site. The problem is constrained by the minimum distance between wind turbines and five irregular polygon boundaries, defined as inclusion zones. Initial guesses are used to instantiate the optimization problem, which is solved following three independent approaches: (1) a baseline approach that uses a gradient-based solver; (2) approach 1 combined with the relaxation of the boundaries, which allows for a better design space exploration; and (3) the application of a heuristic algorithm, “smart-start”, prior to the gradient-based optimization, improving the allocation of wind turbines within the inclusion polygons based on the potential wind resource and the available area. The results show that the relaxation of boundaries combined with a gradient-based solver achieves on average +10.2 % of AEP over the baseline, whilst the smart-start algorithm, combined with a gradient-based solver, finds on average +20.5 % of AEP with respect to the baseline and +9.4 % of AEP with respect to the relaxation strategy.

Harmonic-measure distribution functions of simply connected and doubly connected polygonal domains

The h-function encodes aspects of the behaviour of Brownian particles released from a specified fixed point inside the region. In turn, this behaviour is influenced by the shape of the region's boundary and the location of the fixed point. We compute the h-functions of numerous simply connected and doubly connected polygonal regions. The Schottky–Klein prime function plays a key role in computing h-functions of these regions. Also, in the computation, we use the Schwarz–Christoffel mappings, the Cayley-type mappings, and their generalisations, which are expressed in terms of the prime function. We also cross-check the validity of all the h-functions we find by rederiving them via numerical simulation of Brownian motion.