Abstract and Applied Analysis

Volume 2015 (2015), Article ID 720249, 7 pages

http://dx.doi.org/10.1155/2015/720249

## A Novel Model of Conforming Delaunay Triangulation for Sensor Network Configuration

College of Automation, Harbin Engineering University, Harbin 150001, China

Received 19 August 2014; Accepted 10 September 2014

Academic Editor: Zheng-Guang Wu

Copyright © 2015 Yan Ma et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Delaunay refinement is a technique for generating unstructured meshes of triangles for sensor network configuration engineering practice. A new method for solving Delaunay triangulation problem is proposed in this paper, which is called endpoint triangle’s circumcircle model (ETCM). As compared with the original fractional node refinement algorithms, the proposed algorithm can get well refinement stability with least time cost. Simulations are performed under five aspects including refinement stability, the number of additional nodes, time cost, mesh quality after intruding additional nodes, and the aspect ratio improved by single additional node. All experimental results show the advantages of the proposed algorithm as compared with the existing algorithms and confirm the algorithm analysis sufficiently.

#### 1. Introduction

Recently, the concept of intelligent network system is very popular in the world. Actually, how can we deploy and optimize the sensor? It is still a difficult issue to scientists that affects both cost and detection capability, which are required considerations of both coverage and connectivity. A sensor node may perform the dual function of sensing the environment and acting as a relay node. In a real sensor network system, all sensor nodes distribute as a discrete data set, which will form a mesh network to provide monitoring of the environment. The terms mesh network will be used throughout this paper to describe a sensor network configuration [1].

Delaunay triangulation (DT) is an effective method to carve up a discrete data region, which is especially widely used in sensor network configuration engineering field [2–5]. In most cases, there is a constricting relationship among the discrete data. The discrete data may comprise some vector lines and close polygons, which must be included in the result of partition. In general, the Delaunay triangulation will not contain all edges of the graph. So far there are two types of DT algorithm: constrained Delaunay triangulation [6, 7] and conforming Delaunay triangulation [8, 9]. The former method is a best approximation of the Delaunay triangulation, given that it must contain all features in the graph. Generally, the DT property cannot be preserved and the quality of the mesh declines in constrained Delaunay triangulation, which will influence the stability and convergence of finite element numerical calculation. In the meanwhile, the conforming DT method can be considered as a degenerate Delaunay triangulation, whose relationships to the constrained graph are that each vertex of the graph is a vertex of the triangulation and each edge of the graph is a union of edges of the triangulation and also satisfies the DT property. But any introduced new node will lead the original graph to change. Therefore, the method should be used restrainedly depending on the actual situation.

Usually, constructing conforming DT is more difficult than constructing constrained DT, as it requires a number of points to achieve conformity. The core technique of the conforming DT is to subdivide the constraints. This paper presents a novel node refinement algorithm, which has better triangulation quality and fewer additional nodes than other algorithms.

The rest of this paper is organized as follows. In Section 2, the basic Delaunay triangulation problem is briefly introduced and the main idea of endpoint triangle’s circumcircle model (ECTM) is described in detail in Section 3. Then the convergence and complexity of ECTM are analyzed in Sections 4 and 5. Section 6 gives the simulation experiments’ results of the new ECTM with other methods. Finally, a conclusion is drawn in Section 7.

#### 2. Problem Description

Suppose is a planer straight line graph, where , is exterior feature constraint, are interior feature constraints, and is the collection of all discrete points and endpoints of feature lines.

If is a close constraint, describes the single connected domain of ; should satisfy the following conditions.

All defined regions by interior constraints are in the defined region by exterior constraints; namely, .

The mutual parts of feature constraints are the finite points in collection ; namely, , where .

For any with the above conditions, inserting some additional points on the features, how can we get a geometrical equivalent DT graph by using the whole discrete data and additional points corresponding to former graph? Two types of refinement algorithms for conforming DT are considered for solving the problem. One is refining feature lines at first and then executing DT, such as in the literature of [10–12]. Others have the different ideas exactly. Firstly they execute DT and then refine the feature lines. References [10, 13–17] are the representative algorithms.

Inspired by the algorithms of [14, 16, 17] an improved feature refinement algorithm named endpoint triangle’s circumcircle model (ETCM) is proposed in this paper.

#### 3. Endpoint Triangle’s Circumcircle Model

##### 3.1. Basic Idea of ETCM Definition

Endpoint’s triangle containing a feature is such a triangle which uses one of the endpoints as a vertex and intersects one edge of the triangle with the feature simultaneously.

Suppose is a feature that is not contained in DT meshes, is the endpoint’s triangle of , and is the endpoint’s triangle of ; and are the circumcircles of and , respectively. The basic idea of ETCM is as follows.

Let , . If , choose the longer one between and as the line to be inserted; the corresponding point of intersection is as additional point. Then let the shorter one be the remainder feature line; execute the approach to the shorter one as mentioned above. It will stop execution until and take the midpoint of as additional point at that time. In particular if the feature line being treated influences the feature line which has been inserted, the influenced feature line segment should be transacted again.

##### 3.2. Description of ETCM

For designing the ECTM algorithm, we must confirm a data structure at first. There are mainly four structures being considered as shown in Algorithm 1.