Approximal Research Articles

Generation of an optimal triangulated irregular network for topographic surface via optimal transport theory

ABSTRACT A digital elevation model (DEM) is widely recognized as the most effective digital representation of the Earth’s surface and serves as the fundamental platform for simulating various Earth systems. Extensive efforts have been devoted to exploring methods for generating high-fidelity DEM datasets that are computationally efficient for diverse applications. However, the existing methods do not guarantee the optimal digital representation of the Earth’s surface. This study proposed a novel curvature-based geodesic centroidal Voronoi tessellation method for generating a topographic triangulated irregular network (TIN) DEM based on optimal transport theory. This study is the first to present a globally optimized digital representation of the Earth’s surface with a predetermined number of vertices, which is crucial for computational feasibility. This study achieves the optimal TIN by measuring mean curvature and introducing geodesic distances on the topographic surface. Representative vertices that best adapt to the topography are identified through an optimal surface approximation process. Experimental results confirm that the proposed method effectively generates the optimal digital representation of the topographic surface with the lowest elevation errors and minimal deviations from the original topographic features. By generating optimal TIN DEM with any desired number of vertices, the proposed method not only balances high-precision representation and computational efficiency but also offers a novel approach to deepening the understanding of topographic structures. Furthermore, it provides an effective solution for compressing extensive topographic data and facilitating multiscale representation of the Earth’s surface.

Comparison of Methods for Calculating Volumes of Mineral Raw Materials During Surface Exploitation

The aim of this paper is to compare two geodetic methods - GNSS and aerial photogrammetry in the context of calculating the volumes of mineral resources in open-pit mines. The advantages and weaknesses of both methods are analyzed, as well as their impact on accuracy, efficiency, and costs. The results show that the GNSS method provides high accuracy on simpler and flatter terrains but demonstrates weaknesses on more complex terrains due to the approximation of the actual surface using breakpoints obtained by the GNSS method. The aerial photogrammetry method enables fast and efficient data collection and provides a detailed 3D model, making it particularly useful in hard-to-access or hazardous areas. Additionally, it significantly saves time during data collection and reduces the demand for geodetic professionals. The paper also compares volume calculation methods, emphasizing how technological advancements allow many software solutions to treat the 3D model of the subject area as a whole rather than as segments such as profiles, prisms, etc.

SP-TSA Spherical Projections and Tubular Surface Approximation for UAV Object Trajectory Estimation

SP-TSA Spherical Projections and Tubular Surface Approximation for UAV Object Trajectory Estimation

Neutron stars as dense liquid drop at equilibrium within the effective surface approximation

Neutron stars as dense liquid drop at equilibrium within the effective surface approximation

Evaluation of static bending caused damage of glass-fiber composite structure using terahertz inspection

Abstract Composite materials find increasing applications in modern industry and transport. However, they may lose desirable mechanical properties due to external mechanical excitations, ultraviolet radiation, moisture penetration, or other factors. Therefore, an effective way to assess the condition of materials is necessary. In this article, a nondestructive evaluation of glass fiber-reinforced composite subjected to five-stage static bending is presented. For this reason, pulsed excitation terahertz imaging was utilized, and a data processing/exploration scheme was proposed. The proposed, novel approach consists of an efficient data registration algorithm based on surface approximation (for surface roughness and unevenness elimination) and a parametrization scheme applied for the signals gained from the time response of the glass fiber-reinforced polymer layer. Obtained parameters enable the global description of the evaluated material state and prediction of failure effectively, even in the early stages of destruction.

AI-Powered Approaches for Hypersurface Reconstruction in Multidimensional Spaces

The present article explores the possibilities of using artificial neural networks to solve problems related to reconstructing complex geometric surfaces in Euclidean and pseudo-Euclidean spaces, examining various approaches and techniques for training the networks. The main focus is on the possibility of training a set of neural networks with information about the available surface points, which can then be used to predict and complete missing parts. A method is proposed for using separate neural networks that reconstruct surfaces in different spatial directions, employing various types of architectures, such as multilayer perceptrons, recursive networks, and feedforward networks. Experimental results show that artificial neural networks can successfully approximate both smooth surfaces and those containing singular points. The article presents the results with the smallest error, showcasing networks of different types, along with a technique for reconstructing geographic relief. A comparison is made between the results achieved by neural networks and those obtained using traditional surface approximation methods such as Bézier curves, k-nearest neighbors, principal component analysis, Markov random fields, conditional random fields, and convolutional neural networks.

SORM‐Enhanced Inverse Reliability Analysis for Geotechnical Multiobjective Reliability‐Based Design Optimization

ABSTRACTThe first‐order reliability method (FORM) is mostly employed in the existing geotechnical reliability‐based design (RBD) methods due to its computational simplicity and efficiency. However, the first‐order Taylor approximation of the limit state surface (LSS) may result in significant errors, especially in cases of highly nonlinear LSS characterized by substantial curvatures. Therefore, FORM‐based RBD methods require a modification of the curvatures to enhance the accuracy of the probabilistic constraints, specifically by converting the target reliability index into a more precise target failure probability. Correspondingly, reliability index‐based design is converted into failure probability‐based design. In this study, the parabolic second‐order reliability method (SORM), which avoids the Hessian calculations, is adopted to improve the accuracy of probabilistic constraints beyond what is achievable with FORM. The proposed SORM‐enhanced RBD method accounts for the curvature information of the nonlinear LSS, modifying the target reliability index to align with the exact target failure probability through the application of SORM. Moreover, by incorporating an implicit coupling function, multiobjective RBD can be effectively implemented without any additional surrogate model. Furthermore, the proposed RBD method is readily extended to reliability‐based design optimization (RBDO) through integration with an optimization strategy. The proposed RBDO method demonstrates a more precise convergence of the probabilistic constraints, surpassing the accuracy of FORM‐based RBDO methods. Notably, the proposed SORM‐enhanced RBDO method not only significantly improves accuracy but also bypasses the necessity for Hessian computation, which remains both the second‐order accuracy and first‐order efficiency. The feasibility of the proposed method is demonstrated through a mathematical example and three practical geotechnical design examples.

Kernel multigrid on manifolds

Kernel methods for solving partial differential equations work coordinate-free on the surface and yield high approximation rates for smooth solutions. Localized Lagrange bases have proven to alleviate the computational complexity of usual kernel methods for data fitting problems, but the efficient numerical solution of the ill-conditioned linear systems of equations arising from kernel-based Galerkin solutions to PDEs is a challenging problem which has not been addressed in the literature so far. In this article we apply the framework of the geometric multigrid method with a τ≥2-cycle to scattered, quasi-uniform point clouds on the surface. We show that the resulting solver can be accelerated by using the Lagrange function decay and obtain satisfying convergence rates by a rigorous analysis. In particular, we show that the computational cost of the linear solver scales log-linear in the degrees of freedom.

Optimum geometry of the Hydraulic Machinery impeller based on the response surface approximation surrogate model

Optimum geometry of the Hydraulic Machinery impeller based on the response surface approximation surrogate model

Design and optimization of the exhaust system for an aviation piston engine

Abstract To further enhance the performance of an aviation piston engine, an engine performance simulation analysis model was constructed by using GT-Power software to design improvements to its exhaust system. The impacts of different exhaust system schemes on the exhaust process, the in-cylinder combustion, and the heat release process at a cruising speed of 5500 RPM were analyzed. Based on the results, under cruising speed conditions, a new exhaust system was designed for the engine by referring to the structural form of a resonant exhaust system. The D-optimal Latin hypercube sampling method was used to obtain sample data for the experimental design of the structural parameters of the designed exhaust system. The response surface approximation models characterizing power and fuel consumption rate were established by using the least squares method. The significance of the impact of each structural parameter on engine performance was analyzed through a multifactor analysis of variance. The key structural parameters of the designed exhaust system were then subjected to multi-objective optimization. The results indicate that at 5500 RPM, the designed exhaust system can increase engine power by 2.88%; after optimization, engine power can be further increased by 3.15%, and the fuel consumption rate is reduced by 4.07%.

Biquartic Hesitant Fuzzy Bézier Surface Approximation Model with Its Visualization

Geometric modeling has evolved significantly since its inception, with pioneers like Pierre Bézier laying the foundation for curve modeling. Concurrently, fuzzy set theory, introduced by Lotfi A. Zadeh in 1965, addressed uncertainty in decision-making processes. Integrating fuzzy set theory with geometric modeling has led to advancements in handling imprecise data and uncertainty. This paper proposes a novel approach, the hesitant fuzzy Bézier surface (HFBS) approximation model, which combines geometric modeling with hesitant fuzzy sets to address uncertainty in surface approximation. The model utilizes hesitant fuzzy control net relations to construct HFBSs, enabling visualization of surfaces under varying degrees of uncertainty. A biquartic HFBS example is presented, demonstrating the model’s ability to handle hesitancy among experts’ opinions. The paper discusses the properties of HFBS and suggests its extension to interpolation models for broader applicability. Ultimately, the HFBS model offers an approach to geometric modeling in situation where uncertainty is inherent, which aim to handle complex data in real-world applications.

Polyhedral approximation and uniformization for non-length surfaces

We prove that any metric surface (that is, metric space homeomorphic to a 2 -manifold with boundary) with locally finite Hausdorff 2 -measure is the Gromov–Hausdorff limit of polyhedral surfaces with controlled geometry. We use this result, together with the classical uniformization theorem, to prove that any metric surface homeomorphic to the 2 -sphere with finite Hausdorff 2 -measure admits a weakly quasiconformal parametrization by the Riemann sphere, answering a question of Rajala–Wenger. These results have been previously established by the authors under the assumption that the metric surface carries a length metric. As a corollary, we obtain new proofs of the uniformization theorems of Bonk–Kleiner for quasispheres and of Rajala for reciprocal surfaces. Another corollary is a simplification of the definition of a reciprocal surface, which answers a question of Rajala concerning minimal hypotheses under which a metric surface is quasiconformally equivalent to a Euclidean domain.

SuperUDF: Self-Supervised UDF Estimation for Surface Reconstruction.

Learning-based surface reconstruction based on unsigned distance functions (UDF) has many advantages such as handling open surfaces. We propose SuperUDF, a self-supervised UDF learning which exploits a learned geometry prior for efficient training and a novel regularization for robustness to sparse sampling. The core idea of SuperUDF draws inspiration from the classical surface approximation operator of locally optimal projection (LOP). The key insight is that if the UDF is estimated correctly, the 3D points should be locally projected onto the underlying surface following the gradient of the UDF. Based on that, a number of inductive biases on UDF geometry and a pre-learned geometry prior are devised to learn UDF estimation efficiently. A novel regularization loss is proposed to make SuperUDF robust to sparse sampling. Furthermore, we also contribute a learning-based mesh extraction from the estimated UDFs. Extensive evaluations demonstrate that SuperUDF outperforms the state of the arts on several public datasets in terms of both quality and efficiency. Code will be released after accteptance.

Local spline refinement driven by fault jump estimates for scattered data approximation

We present new fault jump estimates to guide local refinement in surface approximation schemes with adaptive spline constructions. The proposed approach is based on the idea that, since discontinuities in the data should naturally correspond to sharp variations in the reconstructed surface, the location and size of jumps detected in the input point cloud should drive the mesh refinement algorithm. To exploit the possibility of inserting local meshlines in one or the other coordinate direction, as suggested by the jump estimates, we propose a quasi-interpolation (QI) scheme based on locally refined B-splines (LR B-splines). Particular attention is devoted to the construction of the local operator of the LR B-spline QI scheme, which properly adapts the spline approximation according to the nature and density of the scattered data configuration. A selection of numerical examples outlines the performance of the method on synthetic and real datasets characterized by different geographical features.

Contact angle hysteresis on nonwetting microstructured surfaces: Effect of randomly distributed pillars or holes.

We present a numerical study of the advancing and receding apparent contact angles for a liquid meniscus in contact with an ultrahydrophobic surface with randomly distributed microsized pillars or holes in the Cassie's wetting regime. We study the Wilhelmy plate system in the framework of the full capillary model to obtain these angles using the heterogeneous surface approximation model for a broad interval of values of pillar or hole concentration and for both square and circular shapes of the pillars or holes cross-section. Three types of random placing of defects on the plate are investigated, i.e., two with restrictions: (1) with maximum and (2) with minimum distance between the defects (in these cases the defects are isolated), and (3) without restrictions (the defects can overlap). The results show that the type of defect distribution and also the type of the defects shape (circular or square) does not affect the magnitude of the two angles. The results of the numerical simulations showed that the retention force for a plate with randomly located defects is not greater, and for larger concentrations of pillars or holes, it is smaller than that for periodically spaced ones. Comparisons with experimental results for the receding contact angle on surfaces with pillars and with the advancing contact angle on surfaces with periodically arranged holes is carried out.

Multi-objective optimization design of NPR protection shell for hydrogen storage tank

In order to improve the safety of on-board hydrogen storage tanks during collision, a protective shell based on a negative Poisson’s ratio (NPR) core is designed in this paper. After analyzing and comparing the crashworthiness of three typical honeycomb structures, the concave hexagonal negative Poisson’s ratio honeycomb is selected as the energy-absorbing inner core of the hydrogen storage tank protection structure. The sensitivity analysis of the structural parameters of the NPR shell is conducted through orthogonal tests to identify parameters with a significant impact on crash performance. These parameters are then used as experimental variables for subsequent optimization design. Subsequently, a response surface approximation model between the optimization objective and the structural parameters is established based on the response surface method. Finally, the adaptive simulated annealing algorithm (ASA), neighborhood cultivation genetic algorithm (NCGA), and non-dominated sorting genetic algorithm-II (NSGA-II) are used to optimize the structural parameters, and the optimization results are verified by the whole-vehicle crash simulation. The simulation results demonstrate that all three algorithms achieve better optimization results, among which NCGA is more advantageous in improving the overall performance of the protective shell. After NCGA optimization, the specific energy absorption of the protective shell increases by 19.75%, the maximum collision force of the rigid wall decreases by 23.63%, and the maximum stress of the hydrogen storage tank body decreases by 22.77%. These findings indicate that the designed protection shell effectively improves the crashworthiness of the hydrogen storage tank during collisions.

The technique for approximation of three-dimensional surfaces in order to synthesize algorithms for preventing aircraft collisions with obstacles

The work marks the beginning of the practical application of algorithms for making the aircraft recovery maneuver from three-dimensional constraint surfaces, which are a combination of terrain and artificial obstacles. An analysis of events leading to aviation accidents was carried out, and a comparison of on-board systems for preliminary notification of aircraft crews about collisions with natural or artificial obstacles was made. It is shown that such systems are insufficient due to their passive-recommendatory nature of issuing warnings. A question has been raised about the need to implement an active automatic collision avoidance system with spatial obstacles. In order to apply existing algorithms for the aircraft recovery maneuver from spatial constraint surfaces, a technique has been developed for approximating three-dimensional surfaces (obstacles) specified on a digital terrain map in the form of discrete height readings with a certain step on a coordinate grid. A paraboloid of revolution was chosen as a continuous 2nd order surface approximating the obstacle, and its characteristic parameters were determined. To determine the characteristic parameters of the paraboloid, an algorithm for determining the intersection of a three-dimensional surface and a plane, based on the principle of determining the intersection of triangles in space, as well as a method for selecting the inflection point of the terrain, based on determining the value of the terrain height gradient, are proposed for use. The construction of an approximating paraboloid using the example of a natural obstacle in the form of a mountain range is given. When synthesizing algorithms for preventing collisions of aircraft with obstacles, the need to take into account not only the parameters of the constraint surfaces and dynamic characteristics of aircraft, but also the accuracy characteristics of data sources about their position is noted. Promising application areas of the developed methodology are shown.

An improved approximate integral method for nonlinear reliability analysis

In order to evaluate the failure probability corresponding to the Limit State Function (LSF) in structural reliability, the First Order Reliability Method (FORM) linearizes the LSF and directly calculates the failure probability based on the Most Probable Point (MPP). But this method is unable to effectively handle nonlinear problems. The Second Order Reliability Method (SORM), on the other hand, utilizes the Hessian matrix to obtain curvature information at the MPP and computes the failure probability with Breitung's second-order approximation formula. However, SORM is unable to maintain satisfactory computational accuracy in highly nonlinear problems due to its lack of detailed analysis on the shape of the limit state surface. To address this issue, this paper proposes an Improved Approximation Integration Method (IAIM). The initial approximation of the limit state surface is based on the intersection region between the hypersphere and the n-dimensional paraboloid. Subsequently, a statistical analysis of the deviation between the aforementioned region and its projected area is conducted to construct a corresponding fitting function, which is then utilized for dimensionality reduction in the multidimensional integration of the approximation area. Finally, several examples are presented to demonstrate the accuracy and feasibility of the proposed IAIM.

Approximation of Freeform Surfaces with Polyhedra Composed of Congruent Triangles

Approximation of Freeform Surfaces with Polyhedra Composed of Congruent Triangles

B‐splines image approximation using resampled chordal parameterization

AbstractImage processing often requires filtering, which can be effectively performed by using B‐spline surface approximation. This article presents a fast method of approximating data in rectangular (image) form using such surfaces. The method considers dynamic changes of data by representing rows and columns in chordal parameterized form. To improve performance, after parametrization, lines are uniformly resampled using linear interpolation. It allows using the same basis functions with uniform parameterization for approximation of all processed lines. After approximation, reconstruction of original parameterization is required, but its computational complexity is also linear.