Subdivision Scheme Research Articles

A numerical comparative analysis of methods for solving fractional differential equations

In this paper, we present two collocation methods utilizing subdivision schemes and Bernstein polynomials as their basis functions to solve Caputo-type fractional differential equations. These methods transform fractional differential equations into systems of linear equations, which can be solved using various suitable techniques. Notably, these methods are versatile, making them applicable to both boundary value problems and initial value problems. To demonstrate their effectiveness, we applied these methods to three test problems of fractional differential equations, including the Bagley-Torvik equation and fractional oscillation equation. Our results reveal that both methods provide a more accurate approximation to the exact solution when compared to various algorithms found in the existing literature, showcasing their efficiency, accuracy, and consistency.

Exploring the advection-diffusion equation through the subdivision collocation method: a numerical study

The current research presents a novel technique for numerically solving the one-dimensional advection-diffusion equation. This approach utilizes subdivision scheme based collocation method to interpolate the space dimension along with the finite difference method for the time derivative. The proposed technique is examined on a variety of problems and the obtained results are presented both quantitatively in tables and visually in figures. Additionally, a comparative analysis is conducted between the numerical outcomes of the proposed technique with previously published methods to validate the correctness and accuracy of the current approach. The primary objective of this research is to investigate the application of subdivision schemes in the fields of physical sciences and engineering. Our approach involves transforming the problem into a set of algebraic equations.

Multivariate compactly supported C∞ functions by subdivision

This paper discusses the generation of multivariate C∞ functions with compact small supports by subdivision schemes. Following the construction of such a univariate function, called Up-function, by a non-stationary scheme based on masks of spline subdivision schemes of growing degrees, we term the multivariate functions we generate Up-like functions. We generate them by non-stationary schemes based on masks of three-directional box-splines of growing supports. To analyze the convergence and smoothness of these non-stationary schemes, we develop new tools which apply to a wider class of schemes than the class we study. With our method for achieving small compact supports, we obtain in the univariate case, Up-like functions with supports [0,1+ϵ] in comparison to the support [0,2] of the Up-function. Examples of univariate and bivariate Up-like functions are given. As in the univariate case, the construction of Up-like functions can motivate the generation of C∞ compactly supported wavelets of small support in any dimension.

HSS-progressive interpolation for Loop and Catmull–Clark Subdivision Surfaces

Loop and Catmull–Clark subdivision schemes are the most popular and commonly employed approximation subdivision schemes. However, some features of the given mesh are lost in their refinement process. Thus, the subdivision surfaces of Loop and Catmull–Clark schemes does not interpolate the vertices of the given mesh and shrink in some cases. This paper proposed new progressive iterative approximation (PIA) formats based on Hermitian and skew-Hermitian splitting iteration technique (HSS). The proposed method named H-PIA and its weighted version called WH-PIA force the limit surface of Loop and Catmull–Clark subdivision schemes to interpolate the vertices of the given mesh. The approximate optimal weight of WH-PIA is given, and the convergence of H-PIA and WH-PIA are proved. Various test examples are provided to illustrate the efficiency and effectiveness of the proposed H-PIA and WH-PIA. Experimental results demonstrate that the rate of convergence of H-PIA and WH-PIA are faster than that of the PIA and weighted PIA (W-PIA), respectively.

A regularization–correction approach for adapting subdivision schemes to the presence of discontinuities

Linear approximation methods suffer from Gibbs oscillations when approximating functions with jumps. Essentially non oscillatory subcell-resolution (ENO-SR) is a local technique avoiding oscillations and with a full order of accuracy, but a loss of regularity of the approximant appears. The goal of this paper is to introduce a new approach having both properties of full accuracy and regularity. In order to obtain it, we propose a three-stage algorithm: first, the data is smoothed by subtracting an appropriate non-smooth data sequence; then a chosen high order linear approximation operator is applied to the smoothed data and finally, an approximation with the proper jump or corner (jump in the first order derivative) discontinuity structure is reinstated by correcting the smooth approximation with the non-smooth element used in the first stage. This new procedure can be applied as subdivision scheme to design curves and surfaces both in point-value and in cell-average contexts. Using the proposed algorithm, we are able to construct approximations with high precision, with high piecewise regularity, and without smearing nor oscillations in the presence of discontinuities. These are desired properties in real applications as computer aided design or car design, among others.

A 3-point point quinary approximating subdivision schemes and its application in geometric modeling and computer graphics

This article discusses the significance of a subdivision scheme with shape parameters in geometric modeling and computational geometry. A new recursive method for generating the mask of 3-point quinary approximating subdivision schemes (ASSs) is presented. The proposed subdivision scheme exhibits all the geometric properties and offers better shape control than non-parametric subdivision schemes. The article includes several numerical examples to demonstrate the high practical value of the proposed schemes in geometric modeling and computer graphics.

Natural Fractures Help Unlock Tight Carbonates in Oman

_ This article, written by JPT Technology Editor Chris Carpenter, contains highlights of paper SPE 211336, “Natural Fractures Help Unlock Tight Carbonates: Redeveloping the Natih B Reservoir in a Giant Field in the Sultanate of Oman,” by Loic Bazalgette, Nawal Aamri, SPE, and Chris Field, Petroleum Development Oman, et al. The paper has not been peer reviewed. _ Within the stratigraphy of one of the largest fields in north Oman is the Natih B reservoir. The development of the reservoir has undergone several challenges over the past decades because of the complexity of the reservoir setting and difficulties in locating the natural fractures. Success in developing such tight carbonates depends on the integration of high-quality seismic, borehole imagery, petrophysics, and dynamic data. The objective of the complete paper is to develop an integrated technical approach that can be used to unlock one of the largest undeveloped resources in the operator’s current portfolio. Introduction With approximately 1 billion bbl of oil originally in place in the development area of interest, Natih B is a significant undeveloped resource in the operator’s portfolio. Attempts were made during the 1980s and 1990s to produce from Natih B, but production was uneconomical because of low productivity or early gas breakthrough. The low matrix permeability of Natih B organic-rich carbonates negatively affected the flow of the early Natih B producers, but the presence of large-scale, vertically persistent fracture corridors and fault-damage zones allowed gas from the Natih A reservoir above to shortcut into these wells. The opportunity to redevelop Natih B was initiated in late 2018. The initiative involved a tight collaboration between the operator’s fractured-reservoir specialist and subject-matter experts of the structural geology of fractured reservoirs and of carbonate reservoir geology. Overview of Natih B Properties The Natih carbonate reservoir is composed of multiple units named Natih A, B, C, D, E, F, and G from top to bottom. In the field of interest, these are structured as a fault-bounded, northeast-dipping monocline (Fig. 1). In this structure, the reservoir is shallow. The initial reservoir pressure is 61 bar at 400 m true vertical depth subsea (TVDSS). All reservoir units are in pressure communication through cross-cutting fracture corridors and have a common free water level at approximately 550 m TVDSS. Current production is from all reservoir units with the exception of Natih B. In the outcrops of the Salakh Arch area (north Oman), the Natih B unit is divided into five subunits. Under Natih B reservoir conditions, this subdivision scheme can be simplified to four well-correlatable subunits with relatively small thickness variations (Natih B1 to B4). In the area of interest, the subunit thickness is approximately 15 m for B1, 30 m for B2, 10–15 m for B3, and 15–20 m for B4. Facies show a cyclicity of clean limestone and organic-rich layers, which is more pronounced in the B1 and B2 subunits. Natih B4 isolates the Natih B reservoir unit from the underlying Natih C reservoir. Above the free water level, the porosity of Natih B subunits B1, B2, and B3 ranges generally between 0.15 and 0.30. The permeability of Natih B is low and ranges from 0.2 to 5 md, with an average of approximately 1 md.

Advancing Curve and Surface Images Modeling with Two-Parameters Polynomial-Based Quaternary Subdivision Schemes

In the context of this paper, we introduce a novel polynomial function that relies on two parameters. This polynomial enables the creation of a family of quaternary subdivision schemes for curve and surface modeling. One of these parameters is responsible for determining the specific member of the family while the other parameter provides the means to finely control the shape of the resulting curve or the regular surface images. This two-parameter approach adds significant versatility to the subdivision schemes to meet specific requirements and preferences. The exploration of various family members within this class of quaternary schemes is a focal point of our research. By adjusting the parameters, we investigate and delineate the distinctive characteristics of specific family members. This provides valuable insights into how these schemes can be harnessed to achieve various modeling goals. This insight empowers users to select the most suitable family members in accordance with their specific needs and design objectives.

Subdivision collocation method for numerical treatment of regularized long wave (RLW) equation

This research project introduces a novel computational approach for solving the regularized long wave equation. The proposed method utilizes a subdivision scheme with appropriate basis functions to transform the equation into a system of linear algebraic equations. A suitable numerical technique is employed to compute the solution of the transformed equations. Theoretical analysis of stability and error for the proposed method is also conducted. Furthermore, the invariants of three physical properties, waves, mass (M), momentum (P), and energy (ɛ), are calculated. Additionally, numerical evidence is presented to demonstrate the effectiveness and accuracy of the method. The results of the numerical experiments confirm the efficiency and high accuracy of the proposed method. Moreover, the numerical results of the invariants validate the conservation laws and align with the theoretical results.

Построение параметрического семейства вейвлетов и использование его в обработке изображений

<p>This article is devoted to the construction of a parametric family of biorthogonal wavelets according to the lifting scheme and subdivision schemes, and the use of such a family in the problem of image rendering when part of the pixel data in the image is missing or overwritten in some way. The parametric family of wavelets provides a parametric family of filters for restoring damaged images. With such a recovery, the desired wavelet is selected not from any general considerations, but from a parametric family in the process of solving an optimization problem.</p>

MTH subdivision scheme with sharp and semi-sharp features

In this paper, a new non-stationary bivariate subdivision scheme is proposed to obtain a more flexible limit surface. By rewriting the univariate subdivision scheme (Fakhar et al., 2022) under two different methods, the tensor product and the subdivision rules for extraordinary points/faces are constructed. The new scheme produces continuous limit surfaces of higher order, except at extraordinary points/faces where the continuity is G1. Algorithms for generating sharp and semi-sharp features on the hybrid subdivision surface are presented. Numerical examples are also given to show the performance of these new schemes.

A Simple Method for Constructing Symmetric Subdivision Schemes with High-Degree Polynomial Reproduction

In this paper, we present an efficient method for constructing symmetric subdivision schemes reproducing high-degree polynomials, without solving a system of linear equations. Original symmetric subdivision and its deduced subdivisions have similar increasing characteristics to the family of pseudo-splines from the polynomial reproduction point of view. Several examples are given to illustrate the efficiency of the method.

An Unified λ -subdivision Scheme for Quadrilateral Meshes with Optimal Curvature Performance in Extraordinary Regions

We propose an unified λ -subdivision scheme with a continuous family of tuned subdivisions for quadrilateral meshes. Main subdivision stencil parameters of the unified scheme are represented as spline functions of the subdominant eigenvalue λ of respective subdivision matrices and the λ value can be selected within a wide range to produce desired properties of refined meshes and limit surfaces with optimal curvature performance in extraordinary regions. Spline representations of stencil parameters are constructed based on discrete optimized stencil coefficients obtained by a general tuning framework that optimizes eigenvectors of subdivision matrices towards curvature continuity conditions. To further improve the quality of limit surfaces, a weighting function is devised to penalize sign changes of Gauss curvatures on respective second order characteristic maps. By selecting an appropriate λ , the resulting unified subdivision scheme produces anticipated properties towards different target applications, including nice properties of several other existing tuned subdivision schemes. Comparison results also validate the advantage of the proposed scheme with higher quality surfaces for subdivision at lower λ values, a challenging task for other related tuned subdivision schemes.

Acoustic simulation using singular boundary method based on loop subdivision surfaces: A seamless integration of CAD and CAE

This paper firstly combines the Burton-Miller-type singular boundary method (BMSBM) with the Loop subdivision surfaces (LSS) for the acoustic simulation of 3D complicated structures. As a modern modeling technique, the LSS provides a flexible and simple means to represent model surfaces by controlling meshes and subdivision rules. Thus, it can achieve arbitrary topology and multi-resolution structures. As a semi-analytical and boundary-type meshless technique, the BMSBM using the fundamental solutions is simple, accurate, and straightforward for solving acoustic radiation and scattering problems. The new approach constructs a geometry model using a subdivision scheme, and therefore can create a multi-resolution hierarchical model meeting various accuracy requirements without a repetitive meshing procedure. The proposed method effectively breaks through the technical bottleneck that the BMSBM makes it difficult to collate boundary nodes of complex structures and achieves a seamless integration between CAD and CAE. The numerical results of three examples demonstrated the effectiveness and accuracy of the proposed method for acoustic simulation. The present study provides an innovative idea for the subsequent optimization of materials and shapes of acoustic structures, which has the potential to enhance precision and efficiency significantly.

Rational reparameterization of unstructured quadrilateral meshes for isogeometric analysis with optimal convergence

Unstructured quadrilateral meshes are widely used in representing surfaces of arbitrary topology. However, in connection with isogeometric analysis, unstructured meshes suffer from sub-optimal convergence rates in extraordinary regions, especially for solving higher-order PDEs (partial differential equations). In this work, a rational reparameterization method is proposed to achieve optimal convergence rates for isogeometric analysis using Catmull-Clark surfaces. The scheme lifts the convergence of analysis solutions by tuning the weights of control vertices of the input geometry in extraordinary regions, which are often neglected in constructing analysis-suitable parameterizations. Preferred tuning parameters with optimal convergence rates in L2-norm are determined statistically based on a test suite composed of typical testing solutions with a range of parameters. Numerical experiments confirm the validity of the proposed method for PDEs up to the fourth-order. Comparisons with the original Catmull-Clark scheme and a state-of-the-art tuned subdivision scheme also show the superiority of the proposed method in terms of both optimal convergence and reduced absolute errors of resulting analysis solutions.

Convexity and Monotonicity Preservation of Ternary 4-Point Approximating Subdivision Scheme

This work discusses a ternary 4-point approximation subdivision technique with two properties, namely, convexity and monotonicity preservation. The fundamental contribution of this research article is to extract the conditions that assure the suggested subdivision scheme’s convexity and monotonicity. The methodology for extracting these conditions is explained in two theorems. These theorems prove that if the initial data is strictly convex and monotone and the derived conditions are satisfied, then the limiting curve generated by the proposed subdivision scheme will also be convex and monotone. To show the graphical simulations of results, 2D graphs are plotted. Curvature plots are also drawn to fully comprehend the derived conditions. The entire discourse is backed up by convincing examples.

A Surface Subdivision Scheme Based on Four‐Directional S13 Non‐Box Splines

AbstractIn this paper, we propose a novel surface subdivision scheme called non‐box subdivision, which is generalized from four‐directional S13 non‐box splines. The resulting subdivision surfaces achieve C1 continuity with the convex hull property. This scheme can be regarded as either a four‐directional subdivision or a special quadrilateral subdivision. When used as a quadrilateral subdivision, the proposed scheme can control the shape of the limit surface more flexibly than traditional schemes due to the natural introduction of auxiliary face control vertices.

A New Definition of the Dual Interpolation Curve for CAD Modeling and Geometry Defeaturing

The present paper provides a new definition of the dual interpolation curve in a geometric-intuitive way based on adaptive curve refinement techniques. The dual interpolation curve is an implementation of the interpolatory subdivision scheme for curve modeling, which comprises polynomial segments of different degrees. Dual interpolation curves maintain various desirable properties of conventional curve modeling methods, such as local adaptive subdivision, high interpolation accuracy and convergence, and continuous and discontinuous boundary representation. In addition, the dual interpolation curve is mainly applied to solve the difficult geometry defeaturing problems for curve modeling in existing computer-aided technology. By adding fictitious and intrinsic nodes inside or at the vertices of interpolation elements, the dual interpolation curve is flexible and convenient for characterizing a set of ordered points or discrete segments. Combined with the Lagrange interpolation polynomial and meshless method, the proposed approach is capable of characterizing the non-smooth boundary for geometry defeaturing. Experimental results are given to verify the validity, robustness, and accuracy of the proposed method.

Vector subdivision schemes and their convergence for arbitrary matrix masks

Employing a matrix mask, a vector subdivision scheme is a fast iterative averaging algorithm to compute refinable vector functions for wavelet methods in numerical PDEs and to produce smooth curves in CAGD. In sharp contrast to the well-studied scalar subdivision schemes, vector subdivision schemes are much less well understood, e.g., Lagrange and (generalized) Hermite subdivision schemes are the only studied vector subdivision schemes in the literature. Because many wavelets used in numerical PDEs are derived from refinable vector functions whose matrix masks are not from Hermite subdivision schemes, it is necessary to introduce and study vector subdivision schemes for any general matrix masks in order to compute wavelets and refinable vector functions efficiently. For a general matrix mask, we show that there is only one meaningful way of defining a vector subdivision scheme. Motivated by vector cascade algorithms and recent study on Hermite subdivision schemes, we shall define a vector subdivision scheme for any arbitrary matrix mask and then we prove that the convergence of the newly defined vector subdivision scheme is equivalent to the convergence of its associated vector cascade algorithm. We also study convergence rates of vector subdivision schemes. The results of this paper not only establish intrinsic links between vector subdivision schemes and vector cascade algorithms but also improve known results on (generalized) Hermite subdivision schemes and lead to new special types of vector subdivision schemes. Several examples are provided to illustrate the results in this paper on various types of vector subdivision schemes with convergence rates.

A neural builder for spatial subdivision hierarchies

Spatial data structures, such as k-d trees and bounding volume hierarchies, are extensively used in computer graphics for the acceleration of spatial queries in ray tracing, nearest neighbour searches and other tasks. Typically, the splitting strategy employed during the construction of such structures is based on the greedy evaluation of a predefined objective function, resulting in a less than optimal subdivision scheme. In this work, for the first time, we propose the use of unsupervised deep learning to infer the structure of a fixed-depth k-d tree from a constant, subsampled set of the input primitives, based on the recursive evaluation of the cost function at hand. This results in high-quality upper spatial hierarchy, inferred in constant time and without paying the intractable price of a fully recursive tree optimisation. The resulting fixed-depth tree can then be further expanded, in parallel, into either a full k-d tree or transformed into a bounding volume hierarchy, with any known conventional tree builder. The approach is generic enough to accommodate different cost functions, such as the popular surface area and volume heuristics. We experimentally validate that the resulting hierarchies have competitive traversal performance with respect to established tree builders, while maintaining minimal overhead in construction times.