A Sequential Sampling Generation Method for Multi-fidelity Model Based on Voronoi Region and Sample Density

Multi-fidelity surrogate model-based engineering optimization has received much attention because it alleviates the computational burdens of expensive simulations or experiments. However, due to the nonlinearity of practical engineering problems, the initial sample set selected to produce the first set of data will almost inevitably miss certain features of the landscape, and thus the construction of a useful surrogate often requires further, judicious infilling of some new samples. Sequential sampling strategies used to select new infilling sample during each iteration can gradually extend the dataset and improve the accuracy of the initial model with an acceptable cost. In this paper, a sequential sampling generation method based on the Voronoi region and the sample density, terms as SSGM-VRDS, is proposed. First, with a Monte Carlo-based approximation of a Voronoi tessellation for region division, Pearson correlation coefficients and cross validation (CV) are employed to determine the candidate Voronoi region for infilling a new sample. Then, a relative sample density is defined to identify the position of the new infilling point at which the sample are the sparsest within the selected Voronoi region. A correction of this density is carried out concurrently through an expansion coefficient. The proposed method is applied to three numerical numerical functions and a lightweight design problem via finite element analysis (FEA). Results suggest that the SSGM-VRDS strategy has outstanding effectiveness and efficiency in selecting a new sample for improving the accuracy of a surrogate model, as well as practicality for solving practical optimization problems.

Approximation of Gaussian Curvature by the Angular Defect: An Error Analysis

It is common practice in science and engineering to approximate smooth surfaces and their geometric properties by using triangle meshes with vertices on the surface. Here, we study the approximation of the Gaussian curvature through the Gauss–Bonnet scheme. In this scheme, the Gaussian curvature at a vertex on the surface is approximated by the quotient of the angular defect and the area of the Voronoi region. The Voronoi region is the subset of the mesh that contains all points that are closer to the vertex than to any other vertex. Numerical error analyses suggest that the Gauss–Bonnet scheme always converges with quadratic convergence speed. However, the general validity of this conclusion remains uncertain. We perform an analytical error analysis on the Gauss–Bonnet scheme. Under certain conditions on the mesh, we derive the convergence speed of the Gauss–Bonnet scheme as a function of the maximal distance between the vertices. We show that the conditions are sufficient and necessary for a linear convergence speed. For the special case of locally spherical surfaces, we find a better convergence speed under weaker conditions. Furthermore, our analysis shows that the Gauss–Bonnet scheme, while generally efficient and effective, can give erroneous results in some specific cases.

Identification, Computational Examination, Critical Assessment and Future Considerations of Spatial Tactical Variables to Assess the Use of Space in Team Sports by Positional Data: A Systematic Review

The aim of the review was to identify the spatial tactical variables used to assess the use of space in team sports using positional data. In addition, we examined computational methods, performed a critical assessment and suggested future considerations. We considered four electronic databases. A total of 3973 documents were initially retrieved and only 15 articles suggested original spatial variables or different computation methods. Spatial team sport tactical variables can be classified into 3 principal types: occupied space, total field coverage by several players; exploration space, the mean location (±standard deviations in X- and Y-directions) of the player/team during the entire game; and dominant/influence space, the region the players can reach before any other players. Most of the studies, i.e., 55%, did not include goalkeepers (GKs) and total playing space to assess occupied space, however, several proposed new variables that considered that all playing space could be “played” (i.e. effective free-space, normalized surface area). Only a collective exploration space variable has been suggested: the major range of the geometrical centre (GC). This suggestion could be applied to assess collective exploration space variables at a sub-system level. The measurement of the dominant/influence space has been based on the Voronoi region (i.e. distance d criteria), but several studies also based their computation on the time (t). In addition, several weighted dominant areas have been suggested. In conclusion, the use of spatial collective tactical variables considering the principal structural traits of each team sport (e.g. players of both teams, the location of the space with respect to the goal, and the total playing space) is recommended.

Inferences of the lunar interior from a probabilistic gravity inversion approach

<p>The shape and location of density anomalies inside the Moon provide insights into processes that produced them and their subsequent evolution. Gravity measurements provide the most complete data set to infer these anomalies on the Moon [1]. However, gravity inversions suffer from inherent non-uniqueness. To circumvent this issue, it is often assumed that the Bouguer gravity anomalies are produced by the relief of the crust-mantle or other internal interface [2]. This approach limits the recovery of 3D density anomalies or any anomaly at different depths. In this work, we develop an algorithm that provides a set of likely three-dimensional models consistent with the observed gravity data with no need to constrain the depth of anomalies a priori.</p><p>The volume of a sphere is divided in 6480 tesseroids and n Voronoi regions. The algorithm first assigns a density value to each Voronoi region, which can encompass one or more tesseroids. At each iteration, it can add or delete a region, or change its location [2, 3]. The optimal density of each region is then obtained by linear inversion of the gravity field and the likelihood of the solution is calculated using Bayes’ theorem. After convergence, the algorithm then outputs an ensemble of models with good fit to the observed data and high posterior probability. The ensemble might contain essentially similar interior density distribution models or many different ones, providing a view of the non-uniqueness of the inversion results.</p><p>We use the lunar radial gravity acceleration obtained by the GRAIL mission [4] up to spherical harmonic degree 400 as input data in the algorithm. The gravity acceleration data of the resulting models match the input gravity very well, only missing the gravity signature of smaller craters. A group of models show a deep positive density anomaly in the general area of the Clavius basin. The anomaly is centered at approximately 50°S and 10°E, at about 800 km depth. Density anomalies in this group of models remain relatively small and could be explained by mineralogical differences in the mantle. Major variations in crustal structure, such as the near side / far side dichotomy and the South Pole Aitken basin are also apparent, giving geological credence to these models. A different group of models points towards two high density regions with a much higher mass than the one described by the other group of models. It may be regarded as an unrealistic model. Our method embraces the non-uniqueness of gravity inversions and does not impose a single view of the interior although geological knowledge and geodynamic analyses are of course important to evaluate the realism of each solution.</p><p>References: [1] Wieczorek, M. A. (2006), Treatise on Geophysics 153-193. doi: 10.1016/B978-0-444-53802-4.00169-X. [2] Izquierdo, K et al. (2019) Geophys. J. Int. 220, 1687-1699, doi: 10.1093/gji/ggz544, [3]  Izquierdo, K. et al., (2019) LPSC 50, abstr. 2157. [4] Lemoine, F. G., et al. ( 2013), J. Geophys. Res. 118, 1676–1698 doi: 10.1002/jgre.20118.</p><p> </p>

Abstract Voronoi Diagrams from Closed Bisecting Curves

We present the first algorithm for constructing abstract Voronoi diagrams from bisectors that are unbounded or closed Jordan curves. It runs in expected [Formula: see text] many steps and [Formula: see text] space, where [Formula: see text] is the number of sites, [Formula: see text] denotes the average number of faces (connected components) per Voronoi region in any diagram of a subset of [Formula: see text] sites, and [Formula: see text] is the maximum number of intersection points between any two related bisectors.

High Speed Measuring of a Grinding Tool Surface Topography by a Voronoi Diagram

It is very important that a grinding tool surface topography is measured and analyzed. In this paper, a line scan camera is installed in our three dimensional measurement system by shortening a measurement time. In this experiment, a diamond grinding wheel SD140Q100M is employed. As a result, the time can be reduced to about 24[s] from 1380[s]. On the other hand, a voronoi diagram is introduced as an evaluation method of a distribution of cutting edges on the grinding tool. According this method, the dispersion can be visualized. Furthermore, it is found that the voronoi region has a relation to a protrusion height of the cutting edge.

ABSTRACT VORONOI DIAGRAMS WITH DISCONNECTED REGIONS

Voronoi diagrams, AVDs for short, are based on bisecting curves enjoying simple combinatorial properties, rather than on the geometric notions of sites and distance. They serve as a unifying concept. Once the bisector system of any concrete type of Voronoi diagram is shown to fulfill the AVD axioms, structural results and efficient algorithms become available without further effort; for example, the first optimal algorithms for constructing nearest Voronoi diagrams of disjoint convex objects, or of line segments under the Hausdorff metric, have been obtained this way. One of these axioms stated that all Voronoi regions must be pathwise connected, a property quite useful in divide&conquer and randomized incremental construction algorithms. Yet, there are concrete Voronoi diagrams where this axiom fails to hold. In this paper we consider, for the first time, abstract Voronoi diagrams with disconnected regions. By combining a randomized incremental construction technique with trapezoidal decomposition we obtain an algorithm that runs in expected time [Formula: see text], where s is the maximum number of faces a Voronoi region in a subdiagram of three sites can have, and where mj denotes the average number of faces per region in any subdiagram of j sites. In the connected case, where s = 1 = mj , this results in the known optimal bound [Formula: see text].