Subdivision Algorithm Research Articles

Research on Technology of Building Three-Dimensional Geological Simulation Model by Octree Algorithm Method

Accurate terrain data should be obtained in order to research geological structure modeling problem, and it is difficult to truly reflect the form of three-dimensional geological body by traditional three-dimensional spatial data model method. By taking geological structure modeling problem as research object, this paper analyzes geological modeling principles, builds the volume element model of octree subdivision algorithm, proposes to build a three-dimensional geological body by octree subdivision algorithm based on multi-scale, meets the needs of multi-scale organization of data as well as multi-scale division of geological bodies and attributes based on the geological object expression of octree subdivision algorithm, establishes the links between various geological objects through spatial volume element, lays a good foundation for spatial analysis and provides data support for the three-dimensional visualization of geology. The results of simulation experiment show that the proposed geological modeling method through octree subdivision algorithm is accurate and efficient, and it can adapt to the construction of three-dimensional complex geological model under common complex geological conditions and reflect the true form of three-dimensional geological body.

CIRCLE APPROXIMATION BY QUARTIC G2SPLINE USING ALTERNATION OF ERROR FUNCTION

In this paper we present a method of circular arc approximation by quartic Bezier curve. Our quartic approximation method has a smaller error than previous quartic approximation methods due to the alternation of the error function of our quartic approximation. Our method yields a closed form of error so that subdivision algorithm is available, and curvaturecontinuous quartic spline under the subdivision of circular arc with equal-length until error is less than tolerance. We illustrate our method by some numerical examples.

Numerical Solutions to the Bellman Equation of Optimal Control

In this paper, we present a numerical algorithm to compute high-order approximate solutions to Bellman’s dynamic programming equation that arises in the optimal stabilization of discrete-time nonlinear control systems. The method uses a patchy technique to build local Taylor polynomial approximations defined on small domains, which are then patched together to create a piecewise smooth approximation. The numerical domain is dynamically computed as the level sets of the value function are propagated in reverse time under the closed-loop dynamics. The patch domains are constructed such that their radial boundaries are contained in the level sets of the value function and their lateral boundaries are constructed as invariant sets of the closed-loop dynamics. To minimize the computational effort, an adaptive subdivision algorithm is used to determine the number of patches on each level set depending on the relative error in the dynamic programming equation. Numerical tests in 2D and 3D are given to illustrate the accuracy of the method.

A unified method for hybrid subdivision surface design using geometric partial differential equations

Surface subdivision gains popularity in surface design owing to its flexible applications for geometry with complicated topology and simple computational scheme, and the geometric partial differential equation (GPDE) method is an advanced technology for constructing high-quality smooth surfaces. In this paper, we composite these two ingredients to form a unified method for freeform surface design. We choose the mean curvature flow and Willmore flow as our driven GPDEs, and the finite element method coupled with a hybrid Loop and Catmull–Clark subdivision algorithm as the numerical simulation method. This research presents a novel technique to evaluate the finite element basis functions and the first attempt for constructing the GPDE subdivision surface with hybrid control meshes consisting of triangles and quadrilaterals. Numerical experiments show that the construction method is efficient and robust, yielding high-quality hybrid subdivision surfaces.

An Efficient Data Structure for Representing Trilateral/Quadrilateral Subdivision Surfaces

Abstract With the increase of subdivision depths, some problems of common data structures for representing the subdivision surfaces appear, such as excessive computer memory consumption and low efficiency of the data query which restrict the popularization and application of subdivision surfaces in more fields. By utilizing the topological characteristics of the subdivision surface, a two-layer data structure named CELL is presented in order to better realize the piecewise representation of trilateral/quadrilateral subdivision surfaces. The inner structure of CELL represents the subdivision surface patch by using arrays, and the outer structure of CELL represents the topological relations between the subdivision surface patches. Based on Catmull-Clark subdivision scheme, the structural compositions of CELL and the realization mechanism of the subdivision algorithm are proposed. Additionally, sharp and semi-sharp features are constructed, and a primary study on amalgamation of the image/Z-map model and subdivision surface is presented. The results of the experimental and theoretical analysis show the superior performance of CELL with relation to computer memory consumption, data query, subdivision surface computation and algorithm development.

Study on Algorithm for Grid Subdivision and Encryption Based on Technology of Three Dimensional Geological Modeling

To achieve 3D grid models which have a non-uniform size and varying properties, we proposed the algorithm of grid subdivision and encryption by human-computer interaction. This algorithm was the technology based on 3D geological modeling, and achieving process has following three steps. Firstly, we converted many 2D cross sections to 3D space, and reconstructed 3D vector models using the algorithm of optimal path suture, and set the property of abnormal body and surrounding rock. Then, achieving 3D grids subdivided according to the relationship between the center of 3D grid and 3D vector models, the properties of 3D grids were determined. Finally, we encrypted grids in the survey area and expansion area, and modified the properties. The results show that the algorithm can realize the conversion from 3D vector models to 3D grid models, and this process is reliable and efficient.

GPU-based collision analysis between a multi-body system and numerous particles

The use of a graphics processing unit (GPU) is an ideal solution for problems on data-parallel computations. The serial CPU-based program for collision analysis between a multi-body system and numerous particles is rebuilt as a parallel program that uses the advantages of a GPU. In this study, a GPU is used to effectively perform multi-body dynamic simulation with particle dynamics. The multibody system has 20 circular objects, 19 spring-damper force elements, and 2 revolute joints. The motion equations are formulated using the Cartesian coordinate system, and the implicit Hilber-Hughes-Taylor integration algorithm is used for the integral equation. To detect collisions between a multi-body system and particles or between particles, a spatial subdivision algorithm and a discrete element modeling are used. The developed program is verified by comparing the results with ADAMS. The numerical efficiencies of the serial program using CPU and the parallel program using GPU are compared according to the number of particles. The results show that the greater the number of particles, the more computing time can be saved. For example, when the number of particles is 900, the computing speed of the parallel analysis program is about five times faster than that of the serial analysis program.

Generalized Lane–Riesenfeld algorithms

The Lane–Riesenfeld algorithm for generating uniform B-splines provides a prototype for subdivision algorithms that use a refine and smooth factorization to gain arbitrarily high smoothness through efficient local rules. In this paper we generalize this algorithm by maintaining the key property that the same operator is used to define the refine and each smoothing stage. For the Lane–Riesenfeld algorithm this operator samples a linear polynomial, and therefore the algorithm preserves only linear polynomials in the functional setting, and straight lines in the geometric setting. We present two new families of schemes that extend this set of invariants: one which preserves cubic polynomials, and another which preserves circles. For both generalizations, as for the Lane–Riesenfeld algorithm, a greater number of smoothing stages gives smoother curves, and only local rules are required for an implementation.

Non-uniform non-tensor product local interpolatory subdivision surfaces

In this paper we exploit a class of univariate, C1 interpolating four-point subdivision schemes featured by a piecewise uniform parameterization, to define non-tensor product subdivision schemes interpolating regular grids of control points and generating C1 limit surfaces with a better behavior than the well-established tensor product subdivision and spline surfaces. As a result, it is emphasized that subdivision methods can be more effective than splines, not only, as widely acknowledged, for the representation of surfaces of arbitrary topology, but also for the generation of smooth interpolants of regular grids of points.To our aim, the piecewise uniform parameterization of the univariate case is generalized to an augmented parameterization, where the knot intervals of the kth level grid of points are computed from the initial ones by an updating relation that keeps the subdivision algorithm linear. The particular parameters configuration, together with the structure of the subdivision rules, turn out to be crucial to prove the continuity and smoothness of the limit surface.

Algrithm for detecting homoclinic orbits of time-continuous dynamical system and its application

Detecting homoclinic orbits is a key problem in nonlinear dynamical systems, especially in the study of bifurcation and chaos. In this paper, we propose a new method to solve the problem with trajectory optimization. By defining a distance between a saddle point and its near trajectories, the problem becomes a common problem in unconstrained nonlinear optimization to minimize the distance. A subdivision algorithm is also proposed in this paper to improve the integrity of results. By applying the algorithm to the Lorenz system, the Shimizu-Morioka system and the hyperchaotic Lorenz system, we successfully find many homoclinic orbits with the corresponding parameters, which suggests that the method is effective.

A Randomized Subdivision Algorithm for Determining the Topology of Nodal Sets

Topology is a natural mathematical tool for quantifying complex structures. In many applications, such as, for example, in the context of phase-field models in materials science, the structures of interest arise as sub- or superlevel sets of continuous functions, i.e., as nodal domains. From a computational point of view, any attempt at constructing a truthful representation of the topology of nodal domains has to involve a discretization step, and it is natural to wonder whether this step introduces topological artifacts. In this paper, we present a randomized subdivision algorithm which, given a smooth function, constructs an adaptive rectangular grid containing the essential information necessary for approximating nodal domains. Furthermore, under mild regularity assumptions the algorithm will also provide a computer-assisted proof for the correctness of the approximation by showing that the rectangular grid can be used to construct rectangular complexes which are homotopy equivalent to the nodal domai...

Discrete Gradient Flows on surfaces using subdivision

AbstractGradient flows on surfaces play a role in a wide range of applications, for example in biological modelling, computer graphics and shape optimization. We present an approach to compute a discrete gradient flow by using a newly developed C2‐smooth subdivision algorithm. We will briefly present the subdivision algorithm and how we can use it to compute the mean curvature flow. This method can be used similarly for many other gradient flows. (© 2012 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Interproximate curve subdivision

This paper presents a new curve subdivision algorithm called interproximate subdivision for generating curves that interpolate some given vertices and approximate the other vertices. By the interproximate subdivision, only the vertices specified to be interpolated are fixed and the other vertices are updated at each refinement step. The refinement rules are derived to ensure that the eigenvalues of the refinement matrix satisfy the necessary condition of C2 continuity. The interproximate subdivision also contains tension parameters assigned to vertices or edges for shape adjustment. Compared to the 4-point interpolatory subdivision scheme, the interproximate subdivision does not force the new inserted vertices to be interpolated and is thus expected to have improved behavior; and compared to the cubic B-spline refinement scheme, the interproximate subdivision is able to generate curves interpolating user-specified vertices. In addition, the paper also presents two extensions of the interproximate subdivision: one automatically adapts the tension parameters locally according to the geometry of the control polygon during the refinement to achieve convexity preservation and the other automatically relaxes the interpolating property of some vertices to achieve better shape behavior.

Convexity preserving interpolatory subdivision with conic precision

The paper is concerned with the problem of shape preserving interpolatory subdivision. For arbitrarily spaced, planar input data an efficient non-linear subdivision algorithm reproducing conic sections and respecting the convexity properties of the initial data, is here presented. Significant numerical examples are included to illustrate the effectiveness of the proposed method and the smoothness of the limit curves.

Non-uniform interpolatory subdivision via splines

We present a framework for deriving non-uniform interpolatory subdivision algorithms closely related to non-uniform spline interpolants. Families of symmetric non-uniform interpolatory 2n-point schemes of smoothness Cn−1 are presented for n=2,3,4 and even higher order, as well as a variety of non-uniform 6-point schemes with C3 continuity.

Adaptive mesh subdivision for efficient light baking

Light baking has long been a popular technique for real-time rendering. It usually precomputes and bakes the global lighting effects as vertex attributes or textures. Vertex baking requires less memory but can cause artifacts for large triangles. Texture baking can avoid this and generate a high-quality visual effect in real-time rendering. However, it requires significant memory consumption, which may limit the real-time performance and usage. To address this problem, we propose an adaptive mesh subdivision algorithm for memory-efficient light baking, including a fast triangle subdivision level determination method and an optimized solution to calculate vertex colors. Only the subdivided mesh is required during the real-time rendering. Therefore, memory requirements can be significantly reduced while keeping the visual effect. Besides, the subdivision level is allowed to be intuitively controlled by users with a specified parameter. Our algorithm can be easily implemented on commodity graphics hardware and integrated in existing real-time applications such as online preview systems.

An alternative method for constructing interpolatory subdivision from approximating subdivision

This paper presents a new perspective for constructing interpolatory subdivision from primal approximating subdivision. The basic idea is constructing the subdivision rule for new inserted vertices of a new interpolatory subdivision scheme based on an approximating subdivision algorithm applied to a local configuration of the mesh with one vertex updated for interpolation of the vertex. This idea is demonstrated by presenting two new interpolatory subdivision schemes based on Catmull–Clark subdivision for an arbitrary polygonal mesh and Loop subdivision for a triangular mesh, respectively. These algorithms are simple and have a small stencil for computing new points. The new perspective also shows a link between those classic approximating and interpolatory subdivision algorithms such as cubic B-spline curve subdivision and the four-point interpolatory subdivision, Catmull–Clark subdivision and Kobbeltʼs interpolatory scheme, and Loop subdivision and the butterfly algorithm.

Subdivision algorithm for optimal control

SUMMARYWe present a novel algorithm for optimal control of nonlinear systems based on a subdivision algorithm. The algorithm presented in this paper is an alternative to a set‐oriented approach for optimal feedback stabilization. We compare the proposed algorithm to the set‐oriented approach, contrast these two approaches, and use examples to show that the new algorithm produces comparable results. Also, we demonstrate by example that we receive a precomputed optimal solution. The main contribution of the paper is understanding how cost function improves with further subdivision of state space and smaller memory footprint of the final solution in comparison with set‐oriented approach. Copyright © 2012 John Wiley & Sons, Ltd.

The analysis of subdivision measurement based on sinusoidal magnetic field model

A new space subdivision algorithm for the application object like linear motor whose magnetic flux density of magnetic field is sinusoidal was proposed. Space sinusoidal and cosine signals can be obtained by two sensors whose distance equals a quarter of magnetic polar pitch. By continuously doubling the measured signals’ frequency by sinusoidal and cosine operation, a set of orthogonal signals were produced whose equivalent pole pitch equals half of previous signals’. Using an electronic comparator to detect zero crossings, quadrature binary pulses may in turn be obtained from the frequency doubled signals and high-speed, high-precision displacement measurement can be achieved. By analyzing the signal’s signal to noise ratio, subdivision multiples’ constraints when zero crossings detection is available can be obtained. Experimental results showed the effectiveness of this subdivision method.

Long-time behavior of the proper orthogonal decomposition method

We present explicit error bounds concerning the behavior of the proper orthogonal decomposition (POD) method when the data are drawn from long trajectories. We express the error of the POD method in terms of the canonical angle for systems with exponentially decaying behavior. We test our theoretical bounds numerically using a linear parabolic equation. The considerations are motivated by a subdivision algorithm for the computation of invariant measures in discrete dynamical systems using the POD method as a model reduction tool. Copyright © 2011 John Wiley & Sons, Ltd.