Bicubic Spline Research Articles

Automated high-precision alignment of geologic maps with well orientation

The 3D reflection seismic interpretation entails the creation of structure maps of subsurface horizons across extensive areas, often spanning tens of thousands of square kilometers within sedimentary basins. To be correlated with well measurements, maps made from 3D reflection seismic are routinely assigned the correct depth at well control points. However, integrating orientation information measured from wells remains a challenge due to the absence of upscaling guidelines. We address the scale-dependent nature of geologic structure and well orientation information in subsurface mapping, particularly focusing on integrating kilometer-scale seismic interpretation with centimeter-scale well-measured dip and azimuth data. Our pioneering methodology streamlines this integration process through three essential steps: first, transforming well dip and azimuth into a gradient representation; second, smoothly extending gradient differences between the map and well within a user-defined radius of influence; and third, constructing an updated horizon map by bicubic spline interpolation to ensure automated and high-precision alignment. In addition, a structure-oriented interpolation technique is introduced to preserve faults and local structures during orientation correction. Application of the methodology to a 3D reflection seismic horizon demonstrates its effectiveness in automating the complex task of tying subsurface maps to well orientation information. This not only reduces the need for manual interventions but also introduces a new perspective on subsurface mapping. It provides a robust framework that enhances the accuracy and reliability of geologic interpretation, offering significant advancements beyond previous efforts in the literature.

Construction of compositional geological models using the variational grid method of geomapping and the object-hierarchical approach

The variational grid method of geomapping is based on approximation by bicubic splines, the permissible grid size of which is limited by the amount of computer RAM. The paper describes an approach to solving problems of constructing maps of geological parameters, using this method, when computer resources are insufficient. The approach is based on the sequential mapping for a set of fragments of the entire mapping area. The implementation of the approach is considered for two options. The first, if the data density is sufficiently high, consists of sequential calculations over partially overlapping “bands”. If the data distribution is significantly uneven, the second option is implemented, which consists in calculating a detailed grid for local areas and smoothly pasting it into the general map. Smoothness at the boundaries of bands and insets is ensured through the condition of equality of spline coefficients for coinciding nodes, which is set by adjusting the system of linear algebraic equations when solving the mapping problem. Based on these approaches, a multiscale structural-geological model of the sedimentary cover over the territory of Western Siberia was implemented. The formation of a compositional model is carried out on the basis of an object-hierarchical approach, which ensures consistency in the calculations of all its elements, as well as automation of constructions.

Sensitivity Analysis of the Inverse Distance Weighting and Bicubic Spline Smoothing Models for MERRA-2 Reanalysis PM2.5 Series in the Persian Gulf Region

Various studies have proved that PM2.5 pollution significantly impacts people’s health and the environment. Reliable models on pollutant levels and trends are essential for policy-makers to decide on pollution reduction. Therefore, this research presents the sensitivity analysis of the Bicubic Spline Smoothing (BSS) and Inverse Distance Weighting (IDW) models built for the PM2.5 monthly series from MERRA-2 Reanalysis collected during January 2010–April 2017 in the region of the Persian Gulf, in the neighborhood of the United Arab Emirates Coast. The models’ performances are assessed using the Mean Absolute Error (MAE), Root Mean Square Error (RMSE), and Mean Absolute Percentage Error (MAPE). RMSE, Mean Bias Error (MBE), and Nash–Sutcliff Efficiency (NSE) were utilized to assess the models’ sensitivity to various parameters. For the IDW, the Mean RMSE decreases as the power parameter increases from 1 to approximately 4 (the optimal beta value) and then stabilizes with a further increase. NSE values close to 1 indicate that the model’s predictions are very efficient in capturing the variance of the observed data. NSE is almost constant as a function of the number of neighbors and the parameter when β > 4. In BSS, the RMSE and NBE plots suggest that incorporating more points into the mean calculation for buffer points leads to a general decrease in model accuracy. Moreover, the MBE plot shows that the mean bias error initially increases with the number of points but then starts to plateau. The increasing trend suggests that the model tends to systematically overestimate the PM2.5 values as more points are included. The leveling-off of the curve indicates that beyond a certain number of points, the bias introduced by including additional points does not significantly increase, suggesting a threshold beyond which further inclusion of points does not markedly change the mean bias. It was also proved that the methods’ generalizability may depend on the dataset’s specific spatial characteristics.

A boundary element analysis of quasi-potential inviscid incompressible flow in multiply connected airfoil wing

AbstractThe quasi-potential flow in a doubly linked airfoil wing with a sharp trailing edge can be solved using a high-order boundary-element method (BEM). A bicubic Hermite spline interpolation is used to express the unknown, with node derivatives given by finite difference method. The revolutionary traits apply to bleeding-edge settings; specifically, challenges relating to the three-dimensional problem of multiply connected domains are investigated. Numerical validation examples were utilized to assess the implemented BEM approach.

Numerical subgrid Bi-cubic methods of partial differential equations in image segmentation

Image segmentation is a core research in the image processing and computer vision. In this paper, we suggest a Bi-cubic spline phase transition potential and elaborate a Bi-Cubic spline phase transition potential development. In the image segmentation, we develop the new approach to apply the novel computational fluid dynamics in the boundary with subgrid. The numerical subgrid Bi-cubic method with Bi-Cubic spline for minimizing the piecewise constant energy functional is very efficient, robust and fast in the image segmentation with a multispecies multiphase segmentation models. The subgrid Bi-cubic spline is applied on the boundary with subgrid and the regular grid is applied on the non-boundary. The model generates a multispecies multiphase distribution with Bi-Cubic spline and we can extract the image segments with multispecies multiphase. Finally, we analyze the models and show the numerical results. Numerical results are presented with OCR (Optical Character Recognition) and the medical image.

Downscaling MERRA-2 Reanalysis PM2.5 Series over the Arabian Gulf by Inverse Distance Weighting, Bicubic Spline Smoothing, and Spatio-Temporal Kriging.

This study offers a detailed analysis of the fine particulate matter (PM2.5) series in the Arabian Gulf zone, employing three interpolation models, Inverse Distance Weighting (IDW), Bicubic Spline Smoothing (BSS) and Spatio-Temporal Kriging (STK). Unique advancements include the use of complete temporal records in IDW, the management of edge effects in S with synthetic buffer points, and the application of STK to detrended data residuals. The results indicated that the BBS, particularly adept at handling boundary conditions, significantly outperformed the other methods. Compared to IDW, the Mean Absolute Error (MAE), Root Mean Square Error (RMSE), and Mean Absolute Percentage Error (MAPE) decreased by 21%, 15%, and 21%, respectively, in BSS. Compared to STK, MAE, RMSE, and MAPE were lower with around 60%, 61%, and 58%, respectively in BSS. These findings underscore the efficacy of the BSS method in spatial interpolation for environmental monitoring, contributing to enhanced PM2.5 analysis and public health management in the region.

Physical and mechanical modeling of shaping pneumatic formwork by bicubic splines

The article deals with the hierarchical design structure, consisting of four stages of the stress state of pneumatic shells. The main tasks of designing and constructing pneumatic formworks during the construction of thin-walled architectural shells are considered. The introduction of pneumatic formwork into architectural and construction practice has a significant impact on the ability of the architect, builder and designer in the search for new and more advanced forms, as well as the builder in translating the creative ideas of the architect into steel and modern concrete. Thanks to the flexibility and elasticity of the material, it allows the construction of curvilinear spatial shells of a new elastic quality level. The proposed geometric model makes it possible to obtain a variety of software as well as the formation of a discrete-point frame of a finite shape by bicubic splines, which provides a predetermined accuracy for solving subsequent software design and construction problems.

Polyhedral control-net splines for analysis

Generalizing tensor-product splines to smooth functions whose control nets can outline more general topological polyhedra, bi-cubic polyhedral-net splines form a first-order differentiable piecewise polynomial space. Each polyhedral control net node is associated with one bi-cubic function. A polyhedral control net admits grid-, star-, n-gon-, polar- and three types of T-junction configurations. Analogous to tensor-product splines, polyhedral-net splines can both model curved geometry and represent higher-order functions on the geometry. This paper explores the use of polyhedral-net splines for solving elliptic partial differential equations on curved smooth free-form surfaces without additional meshing.

Contour detection with bicubic spline surfaces

ABSTRACT Contour detection has a rich history in multiple fields such as geography, engineering, and earth science. The predominant approach is based on piecewise planar tessellation and now being challenged concerning the extraction of contour objects for non-linear elevation functions, particularly with respect to bicubic spline functions. A storage-efficient method was developed in previous research, but the detection of the complete set of contour objects is yet to be realized. Although intractable, theoretical underpinnings pertinent to curvature resulted in an approach to realize the complete detection of objects. Given a digital elevation model dataset, in this study, a bicubic spline surface function was first determined. Thereafter, candidate initial points on the edges across the region of interest were identified, and the recursive disaggregation of rectangles was repeated if the non-existence of a solution could not be assured. A developed tracking method was then applied. During advancement, other initial points on the same contour curve were identified and eliminated to circumvent duplicate detection. The completeness of the outlets provides analytical tools for elevation and other geographical assessments. Demonstrative experiments included the development of a three-dimensional contour-based network and slope assessments. The latter application transforms the slope analysis type from raster-based to vector-based. Highlights Detection of a complete set of contour objects amenable to bicubic spline surfaces. Small closure inside a single patch is detectable if size exceeds the standard. Curvature & tolerances central to step length adjustment and tangent angle determination. Redundant initial points are identified and eliminated during the tracking process. Various potential applications in addition to geographical elevations.

Algorithm 1032: Bi-cubic Splines for Polyhedral Control Nets

For control nets outlining a large class of topological polyhedra, not just tensor-product grids, bi-cubic polyhedral splines form a piecewise polynomial, first-order differentiable space that associates one function with each vertex. Akin to tensor-product splines, the resulting smooth surface approximates the polyhedron. Admissible polyhedral control nets consist of quadrilateral faces in a grid-like layout, star-configuration where n ≠ 4 quadrilateral faces join around an interior vertex, n -gon configurations, where 2n quadrilaterals surround an n -gon, polar configurations where a cone of n triangles meeting at a vertex is surrounded by a ribbon of n quadrilaterals, and three types of T-junctions where two quad-strips merge into one. The bi-cubic pieces of a polyhedral spline have matching derivatives along their break lines, possibly after a known change of variables. The pieces are represented in Bernstein-Bézier form with coefficients depending linearly on the polyhedral control net, so that evaluation, differentiation, integration, moments, and so on, are no more costly than for standard tensor-product splines. Bi-cubic polyhedral splines can be used both to model geometry and for computing functions on the geometry. Although polyhedral splines do not offer nested refinement by refinement of the control net, polyhedral splines support engineering analysis of curved smooth objects. Coarse nets typically suffice since the splines efficiently model curved features. Algorithm 1032 is a C++ library with input-output example pairs and an IGES output choice.

Study on Dynamic Prediction and Visualization Simulation of Fire Temperature Field Based on UAVs

In the process of petrochemical fire rescue, temperature is an important reference index, which can be used to analyze the possibility of domino accident and tank collapse. A method for predicting temperature field by bicubic spline interpolation is proposed in this paper. It collects data from unmanned aerial vehicles (UAVs), and the accuracy of prediction is influenced by the sampling strategy. It's not that the more UAV routes are, the higher the overall prediction accuracy of temperature field will be. And the smaller the data interval, the more conservative the prediction results in the high- temperature zones.

Interpolation of gas-liquid multicomponent fluid properties for piping fluid flow analysis

The article proposes the use of bicubic spline interpolation to optimize hydraulic and thermal calculations while processing thermodynamic library data. Research was carried out both in single-phase gas and liquid regions, as well as in the two-phase gas-liquid region. The accuracy of bicubic spline interpolation was tested for various thermodynamic libraries on multi-component mixtures. Various coordinate systems (P, x), (P, T), (P, H), (P, S) were tested, optimal coordinates and interpolation grids were selected based on the nature of changes in various thermophysical properties. Experiments were conducted in the range of up to 80% of critical pressure, since in the near-critical region the properties behavior changes sharply. Plots of properties and interpolation absolute and relative errors were also produced for a more detailed study and determination of problem areas where property functions undergo sharp changes that make interpolation difficult. The results obtained demonstrate an acceptable level of accuracy for engineering calculations in the approximation of data from thermodynamic libraries.

Reconstruction of local volatility surface from American options

Abstract In this paper, we discuss the reconstruction of a local volatility surface from American option prices. First of all, the American option prices are calculated by an accurate and fast finite difference scheme. Then the local volatility is obtained by minimizing the distance between theoretical prices and market option prices, which yields an optimization problem. The Bicubic spline regularization technique is used to overcome the ill-posedness of the reconstruction problem. We solve the nonlinear optimization problem by using a gradient-based optimization algorithm. Finally, we test our model with numerical examples and real market American put option data. The results show the good performance of our method.

Impact localization with a weighted spectral cross correlation method

The impact localization is a typical inverse problem to identify the damage source of structures. To overcome the positioning problems of the low resolution caused by sparse reference points, a weighted spectral cross-correlation (WSCC) method is proposed. First, a mathematical model with a series of Fiber Bragg Grating (FBG) sensors is built to generate a database of transient frequency responses of each reference point when random shock happens. Second, to identify the spectrum difference between the impact point and the reference point, a cross correlation model is built by mean and normalization operations on the frequency domain. Third, to figure out the impact point among the adjacent reference points, bicubic spline interpolation is realized by subdividing the grid of the WSCC. To validate the accuracy of our suggested method, an impact localization system is designed with a multi FBG sensor network and a composite laminate. The experimental result shows that the error of our suggested method is almost ten times lower than that of other traditional methods. The weighted spectral cross-correlation has advantages in the field of structural health monitoring.

Содержание курса «Геометрическое моделирование» для направления подготовки «Математика и компьютерные науки»

In this paper has been considered the main content and distinctive features of the “Geometric Modeling” training course for the “Mathematics and Computer Science” training program 02.03.01 (“Mathematical and Computer Modeling” specialization).
 The goal of the “Geometric Modeling” course study is the assimilation of mathematical methods for construction of geometric objects with complex curved shapes, and techniques for their computer visualization by using polygons of curves and surfaces. Methods for construction of structures’ curved shapes using spline representations, as well as techniques for construction of surfaces and volumetric geometries using motion operations and basic logical operations on geometric objects are considered. The spline representations include linear and bilinear splines, Hermite cubic splines and Hermite surfaces, natural cubic and bicubic interpolation splines, Bezier curves and surfaces, rational Bezier splines, B-splines and B-spline surfaces, NURBS-curves and NURBS-surfaces, transfinite interpolation methods, and splines of surfaces with triangular form. Logical operations for intersection of two spline curves, and intersection of two parametric surfaces are considered. The principles of scientific visualization and computer animation are considered in this course as well.
 Some examples for visualization of initial data and results of curves and surfaces construction in two- and three-dimensional spaces through the software shell developed by authors and used by students while doing tests have been demonstrated. The software shell has a web interface with the WebGL library graphic support. Tasks for four practical studies in a computer classroom, as well as several variations of homework are represented.
 The problems occurring in preparation materials for some course sections are discussed, as well as the practical importance of acquired knowledge for the further progress of students.
 The paper may be interesting for teachers of “Geometric Modeling” and “Computer Graphics” courses aimed to students with a specialization in mathematics and information, as well as to those who independently develop software interfaces for algorithms of geometric modeling.

Идентификация квадратичного ядра уравнения Вольтерра для моделирования нелинейных динамических систем

In the last two decades, integral models have been used to describe the dynamics of stationary nonlinear systems in which the terms of the Volterra series are the kernels. The most commonly used are the linear term (the impulse transition function depends on one variable) and the quadratic term (depending on two variables). An active experiment in which a special combination of rectangular pulses is fed to the input of the system is carried out to select two of its components in the output signal of the identified system - the output of the linear "subsystem" and the output of the "quadratic" subsystem. After isolating the output of the "quadratic" subsystem, the identification of the quadratic term of the Volterra series is reduced to solving a two-dimensional integral equation of the first kind. In the literature, inversion formulas are given in which the quadratic kernel function is obtained as a result of arithmetic operations with second-order derivatives of the output signal. Differentiation of functions is an incorrectly posed problem, when small errors in the assignment of a function (measurement noise) cause large errors in derivatives (especially in second-order derivatives). The paper proposes the use of smoothing cubic splines for stable calculation of derivatives. To calculate the mixed second-order derivative, a spline with two variables is built - a smoothing bicubic spline. The main problem that arises in practice when processing the data of a real experiment is the selection of a smoothing parameter, on the value of which a smoothing error of noisy data depends. As a rule, the value of the variance of the measurement noise is unknown in the experiment. Therefore, in this work, it is proposed to use an algorithm based on the L-curve method to select the smoothing parameter in the constructed splines (especially in the bicubic one), which does not require setting the variance of the measurement noise. The proposed identification algorithm has a high computational efficiency. The performed computational experiment showed a small methodical error (about 1%) and good resistance to noise in measurements of the output signals of the identified system. To reduce the random component of the identification error, it is proposed to use post-processing with a local-spatial compound filter.

A revisit to smoothness preserving fractal perturbation of a bivariate function: Self-Referential counterpart to bicubic splines

Construction of fractal interpolation surfaces has recently been considered in the standpoint of a parameterized class of fractal (self-referential) functions corresponding to a given bivariate continuous function. In this paper, we describe a procedure so that the elements in this parameterized class preserve smoothness (C(2,2)-regularity) of the original bivariate function defined on a rectangle. As a consequence, we generalize the bicubic spline by means of a two-parameter family of fractal functions, which we call bicubic fractal splines. Under certain hypotheses, upper bounds for the interpolation error for the bicubic fractal spline and its derivatives are obtained. A detailed exposition of C(2,2)-regular self-referential functions is provided not only as a prelude to the bicubic fractal splines, but also to elucidate the study of smoothness preserving bivariate self-referential functions appeared recently in the fractal literature.

Comparison of negative blended lenticular lens design methods for high myopic spectacles

The power distribution, high-order polynomial, bicubic spline interpolation, and geometric construction methods were used to reduce the edge thickness and weight of negative blended lenticular lenses. Four lenses with the same optical parameters were designed, simulated, machined, and evaluated using the four methods. Spherical and cylindrical power maps were generated and compared. The methods reduced the edge thickness by 78.73% for power distribution method, 77.40% for high-order polynomial method, 78.12% for bicubic spline interpolation method and 79.86% for geometric construction method for a 35 mm radius compared to that with the spherical surface. Geometric construction, power distribution, high-order polynomial, and bicubic spline interpolation reduced the lens weight by 43.86%, 35.98%, 33.06%, and 29.76%, respectively. The center optical area for the power distribution and geometric construction methods were larger than those with the other methods.

Synthesis of a smoothing bicubic spline for differentiating experimental data of nonparametric identification

Over the past two decades, so-called Voltaire series have been used to describe the dynamics of nonlinear systems in terms of input-output. Nonparametric identification of models using Voltaire series consists in constructing estimates for impulse transition functions (IPFs) depending on two or more arguments, which naturally makes identification algorithms much more complicated than in the one-dimensional case. So, in order to identify the two-dimensional IPF (corresponding to the quadratic term of the Voltaire series), it is necessary to calculate the second-order mixed derivatives of the output two-dimensional signal of the system, when a series of rectangular pulses of different amplitudes at different times are fed to its input. Everyone knows, the problem of differentiation is an ill-posed problem and one of the manifestations of incorrectness is poor resistance to errors in the initial data. It is proposed to use two-dimensional smoothing cubic (bicubic) spline (abbreviated SBS) to overcome this problem. The two tasks that constitute SBS synthesis: assignment and implementation of different types of boundary conditions at the border of the rectangular region where SBS is determined; optimal values estimation of two smoothing parameters due to the different “smoothness” of IPF for different two arguments. An acceptable solution to this synthesis problem is proposed in the paper. Our performed computational experiment showed the efficiency of the proposed algorithm for calculating second-order mixed derivative from noisy initial data.

A novel approach to reconstruct residual stress fields induced by surface treatments in arbitrary 3D geometries using the eigenstrain method

A novel approach has been proposed to reconstruct residual stress fields after surface treatments in arbitrary 3D geometries using the eigenstrain method based on bicubic spline interpolation surface and finite element (FE) method. Firstly, the complex curved surface is represented by piecewise interpolation of the smooth surface using explicit bicubic functions, then the computation of the distance and corresponding orientation are calculated by solving a constrained nonlinear multivariable problem using MATLAB, which is insensitive to the complexity of the curved treated surface and reduces the errors induced by differentiation operations when using Wall distance module in COMSOL in previous works. Three instances of 3D geometries after surface treatment studied previously in literature and finally an instance of a real turbine blade after LSP treatment conducted by ourselves are analyzed, which validate the proposed approach and illustrate its effectiveness in the reconstruction of residual stress fields for complex 3D geometries in various cases. The proposed approach provides a general and flexible way to the implementation of eigenstrain method in arbitrary 3D geometries using data interchange between the mathematical calculation software and FE analysis software for the design in industrial applications.