Mesh Simplification for Centralized Algorithms - IEEE Xplore

IEEE TRANSACTIONS ON INDUSTRIAL INFORMATICS, VOL. 9, NO. 3, AUGUST 2013

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Wireless Sensor Network Configuration—Part I: Mesh Simplification for Centralized Algorithms Kurt Derr, Member, IEEE, and Milos Manic, Senior Member, IEEE

Abstract—This is the first of a two-part investigation of centralized and decentralized approaches for determining the optimal configuration of a sensor network. In this first part, we present a centralized approach for the generation of mesh (wireless sensor) network configurations that provide complete sensing coverage and communication connectivity of a domain. A challenging problem in deploying wireless sensor networks is maximizing coverage in irregular shaped polygonal areas while maintaining a high degree of node connectivity. The novelties presented in this paper are: 1) a centralized mesh simplification technique, the Iterative Node Removal with Constrained Delaunay Triangulation and Smoothing (INRCDTS) algorithm, and 2) a centralized mesh generation approach with INRCDTS that may be used for any nonintersecting closed polygonal area. Additionally, we provide a comparison of two centralized mesh generation techniques. The INRCDTS was built and tested as an enhancement of two traditional mesh generation techniques: advancing front technique and MATLAB partial differential equation toolbox. The INCRCDTS introduces the ability to tune the generated mesh configuration to the number of nodes and nodal spacing. The INRCDTS enhancement has proven to increase the uniformity of the mesh in an irregular shaped polygonal area relative to advancing front and MATLAB partial differential equation algorithms by 23% and 41%, respectively. Index Terms—Delaunay triangulation, mesh generation, mesh simplification, sensor node, wireless sensor network (WSN).

I. INTRODUCTION

A

WIRELESS sensor network (WSN) senses the environment as well as communicates collected data to one or more base stations (BSs) outside of the domain being monitored. The sensor network deployment is a critical issue that affects both cost and detection capability, requiring consideration of both coverage and connectivity. A sensor node (SN) may perform the dual function of sensing the environment and acting as a relay node (RN), communicating sensed data to a BS for monitoring and analysis. An alternative approach is to deploy relay nodes for communications along with sensor nodes that only perform sensing. In this approach, the SNs provide sensed data to the RNs, which perform multihop wireless communication through other RNs to export the data to one or more BSs. Manuscript received May 08, 2012; revised November 19, 2012; accepted January 24, 2013. Date of publication February 08, 2013; date of current version August 16, 2013. Paper no. TII-12-0349. K. Derr is with the Idaho National Laboratory, Idaho Falls, ID 83415 USA (e-mail: [email protected]; [email protected]). M. Manic is with the University of Idaho, Idaho Falls, ID 83402 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TII.2013.2245906

In either case, the deployment algorithm must ensure sufficient sensing coverage to avoid sensing voids, as well as sufficient connectivity to ensure that all sensor data may be transmitted to the BSs through a backbone network. In this paper, we assume that both the SN and RN are integrated into one platform. Sensor coverage is related to the traditional art gallery (AG) problem in computational geometry, which looks to determine the minimum number of cameras that need to be placed in an art gallery to ensure that all points within the gallery area are covered [1]. Traditional algorithms for solving the AG problem, where a good prior model of the environment exists, are to triangulate the polygonal area and place a camera in each triangulation. This approach is not viable for a sensor network deployment requiring multiple communications pathways between SNs to ensure robust communications in case of sensor failure. Communications between guards is not addressed in the AG problem. Also, sensors have limited range while a guard can watch a point in the AG as long as line-of-sight vision or camera view exists. Other properties of SNs, such as a limited energy supply and ad hoc topology, negate the direct application of AG techniques to SN coverage [2]. A SN deployed to provide monitoring of the environment forms a mesh network. The terms mesh network or mesh generation will be used throughout this paper to describe a WSN configuration or the generation of a WSN configuration, respectively. The mesh network typically has a partial topology, where each node is connected to a few other nodes, providing multiple paths for propagating data and enhancing the reliability of the network. The mesh topology may be created using centralized algorithms that have a priori information about the environment or decentralized algorithms with no a priori information but capable of automatically deploying to any environment using only local information. This paper presents a novel mesh simplification algorithm, Iterative Node Removal with Constrained Delaunay Triangulation and Smoothing (INRCDTS), used to enhance the configuration of the mesh sensor network generated by two software tools: 1) NETGEN developed by Schöberl and 2) MATLAB partial differential equation (PDE) toolbox algorithms. The NETGEN tool is used for the advancing front mesh generation algorithm and the MATLAB PDE toolbox mesh generation algorithm is for Delaunay triangulation. The major contributions of this paper are: 1) novel mesh simplification algorithm, INRCDTS, for improving the quality of meshes generated by a centralized algorithm; 2) a centralized mesh generation approach with INRCDTS that may be used for any nonintersecting closed polygonal area; and 3) a comparison of two centralized mesh generation techniques for configuration

1551-3203 © 2013 IEEE

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of SN deployment in an irregular shaped polygonal area. INRCDTS was shown to significantly improve the uniformity of these techniques while introducing the ability for the user to constrain the number of nodes and node spacing in the generated meshes. A number of simulation test cases are subsequently described in this paper. All simulations are run on a Dell computer, Intel Core 2 Quad CPU 2.33 GHz. MATLAB student version 7.8 was used to build all simulation models. The remainder of this paper is organized as follows. Section II discusses related work. Section III provides a detailed formulation of the problem being solved and preliminaries. Section IV presents the centralized mesh generation algorithms (MATLAB PDE toolbox and NETGEN). Section V discusses the INRCDTS algorithm for mesh simplification. Section VI discusses coverage and connectivity concepts for a wireless sensor network. Section VII discusses mesh generation metrics. Section VIII discusses the simulation test case results, and Section IX presents our conclusions. II. RELATED WORK Coverage and connectivity are key issues in a wireless sensor network. Algorithms for wireless sensor networks that address coverage and connectivity issues may be categorized as centralized versus distributed, deterministic versus nondeterministic or single tier versus two tier networks. Wang et al. [3] present a centralized approach to the deployment of SNs by partitioning the domain into polygonal areas and deploying sensors in each polygon. The polygons are triangulated, and sensors are placed at the intersection of the perpendicular bisectors of the triangle edges (the triangle in the center) to satisfy both coverage and connectivity. Wang et al. [4] presentss the Configuration Change Protocol (CCP) integrated with the Span coordination algorithm for topology maintenance to provide integrated coverage and connectivity configuration for a WSN. Span [5] is a distributed energy-efficient algorithm where nodes make decisions on when to sleep or join the network. CCP configures a network to provide different degrees of coverage. Wang et al. [6] use Voronoi diagrams to find coverage holes after an initial deployment. The range of a sensor must cover the Voronoi region where a sensor is located, otherwise a coverage hole exists. The locations to place sensors to fill the coverage holes are determined by one of three algorithms: a vectorbased algorithm that pushes sensors away from one another, the voronoi-based algorithm that pulls sensors into the coverage hole, or the Minimax algorithm which moves nodes closer to the farthest Voronoi vertex to fill the coverage holes. Ke et al. [7] presents an evaluation of SN deployments using a grid-based configuration. A 50 50 grid area is used to evaluate the minimum number of required active SNs in a WSN. Optimality conditions are derived for providing complete coverage with a subset of working SNs. Other SN algorithms include using techniques such as providing coverage around specific points of interest [8]; genetic algorithms with multi-objective fitness functions [9] to address connectivity, energy, and coverage design parameters, constructing a WSN on grid points [10], [11], hybrid genetic programming and genetic algorithms to optimize total coverage

Fig. 1. Simple irregular polygon domain.

area, number of nodes deployed, and energy utilization [12], and two-tiered constrained relay node placement where the SN consists of SNs forwarding data to RNs which transport the data to BSs [13]–[15]. These related techniques all provide a fixed degree of coverage with no adaptive reconfiguration, with the exception of the CCP, to meet the requirements of the domain. CCP requires nodes to broadcast HELLO messages which include the locations of all known multihop neighbors. The newly developed mesh simplification algorithm, INRCDTS, presented in this paper differs from the above approaches in that INRCDTS allows for adjustment of the mesh by user-specified parameters (number of nodes and desired distance between nodes). The INRCDTS algorithm takes the mesh networks generated by the advancing front and MATLAB PDE toolbox algorithms and adjusts those mesh networks based on these user-specified parameters. III. PROBLEM FORMULATION AND PRELIMINARIES Consider an example where a WSN is used to monitor and measure pollutants in a lake. We are given a set of sensors initially placed in close proximity of one another within the boundary of the lake. Every sensor in has a sensing range and a communications range . Each sensor is initially placed at a known coordinate. The initial deployment of all sensors does not provide full sensing coverage of the lake. The problem is to determine the location of SNs that will provide complete area sensing coverage and proper formation (equilateral triangular mesh with specific internodal spacing) with multiple communications paths between neighboring sensors. The SN will forward acquired data to one or more sinks, or BSs, outside the periphery of the lake. The area of interest or domain representing the lake is an arbitrary irregular shaped polygon shown in Fig. 1. This domain is comprised of line segments at arbitrary orientations. The coverage requirement does not change after the network has been deployed due to environmental or other conditions.

DERR AND MANIC: WIRELESS SENSOR NETWORK CONFIGURATION—PART I: MESH SIMPLIFICATION FOR CENTRALIZED ALGORITHMS

TABLE I SN ALGORITHM TERMS AND NOTATION

We will use the term nodes to refer to sensor nodes (SNs) throughout this paper. The problem of optimal placement of nodes in an SN necessary to completely cover the entire monitoring area was proven to be NP-complete [16]–[18] for most formulations of WSN deployments. Additionally, given possible locations for nodes and nodes, there are possible mesh networks. The goal of the algorithmic approach in this paper is to determine an optimal solution that satisfies two requirements with a fixed number of nodes and specific internodal spacing: 1) complete coverage and 2) maximum formation. The terms and notation used in this paper are defined in Table I. Parameter default values are noted in parenthesis. Section IV presents two centralized approaches for mesh generation. The boundary of the domain and maximum edge length are user-specified configuration parameters for these algorithms. IV. CENTRALIZED MESH GENERATION ALGORITHMS The problem of finding the discretization of a domain is known as mesh generation. A triangulation of the closed bounded domain is referred to as a mesh of . The connectivity of the mesh is the connection between the vertices as

(1) where represents a set of nodes and the set of edges or connections between those nodes. The geometric quality of the mesh, the shape of the domain, and the size of the elements determine the computational complexity and discretization errors (difference between desired and actual shapes within the mesh) [19]. A number of classical approaches to centralized mesh generation can be applied to the problem of finding the optimal location of nodes in the domain. This section presents two centralized approaches for mesh generation: 1) the MATLAB PDE toolbox mesh generator and 2) the NETGEN open source automatic mesh generator.

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The mesh generation algorithms covering the domain , represented by the closed polygon in Fig. 1, have the following requirements for generating triangles [20]. C1: , , where is the domain, is a single triangle in a triangulation spanned by three vertices , , and , and is the set of triplets representing all triangles in the triangulation that entirely cover . C2: The interior of every element , , in is nonempty. C3: The intersection of the interior of two elements is , i.e., C4: The intersection of two elements is either , a vertex/node, or an edge connecting two vertices. C1 implies that the entire domain is covered by triangles. The triangles may not be degenerate (C2), meaning that the points in a triangle cannot be collinear. The interiors of two triangles cannot intersect (C3). Two triangles may only intersect at a vertex or an edge (C4). A mesh generating algorithm generally works as follows. Step 1) First generate node locations within the geometry; Step 2) Mesh the boundary of the geometry to provide an initial set of nodes; Step 3) Triangulate existing nodes, and Step 4) Insert nodes incrementally into the mesh, refining existing nodes as new nodes are inserted to meet the mesh quality metrics criterion. The results of mesh generation using the MATLAB PDE toolbox, Section IV-A, and NETGEN, Section IV-B, are presented next. Centralized mesh generation tools create as many nodes as necessary for discretization of the domain. A post-processing optimization method is then used to optimize the mesh. This is the aim of the novel mesh simplification algorithm, INRCDTS, discussed in Section V. The MATLAB PDE or NETGEN algorithms used in combination with the INRCDTS algorithm provide a centralized approach to mesh generation that can be tailored to both edge length and the number of nodes.

A. MATLAB PDE Toolbox Mesh Generation The PDE toolbox in MATLAB (Release 2009a, PDE Toolbox 1.0.16) may be used to generate mesh structures. The PDE toolbox uses a Delaunay triangulation algorithm to create a triangular mesh using a specified geometry. The Delaunay triangulation connects lines to neighboring vertices and forms triangles. The shape of the geometry and maximum edge length, or NSL, determines the size of the mesh as well as the number of triangles and nodes. The intersection of the lines in the mesh diagram represents the vertices or SNs. A Delaunay triangle by definition must adhere to the circle criterion: of nodes from the set in (1) of of the circumcircle (2)

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Fig. 3. AFMG example.

B. NETGEN Advancing Front Mesh Generator

Fig. 2. Mesh generation with PDE maximum edge length of 40; 204 nodes.

A Delaunay triangulation of a set of node locations is a triangulation where none of the node locations are contained within the interior of the circumcircle of any triangle . Although the Delaunay triangulation technique provides a method for connecting nodes or points, the method does not provide a mechanism to generate points. Procedures for point insertion are typically based on either the Bowyer-Watson [21], or Green-Sibson algorithm [22]. The steps used by the PDE toolbox to generate a mesh are as follows. Step 1) The PDE toolbox creates the first set of nodes at the intersection of line segments on the boundary of the domain. Step 2) The interior of the closed polygon is then subdivided into triangles by elimination of protruding points. A point is a protruding point if: 1) the angle formed by and adjacent nodes and is less than 180 degrees and 2) the triangle formed by these points contain no other points from the polygon [34]. Step 3) Create additional nodes on the edges to adhere to the maximum edge length constraint provided to the algorithm. Step 4) Recursively refine the largest triangles inside the geometry by choosing the center of each triangle’s circumscribed circle, adding points, removing large edges within the largest triangles, and connecting the new points to nearby points on the periphery of the large triangle. This step is repeated recursively until all of the triangles meet the specified size constraint. Fig. 2 depicts the mesh generated by the PDE toolbox with a maximum edge length or NSL of 40 m. An NSL of 40 m, which is an achievable outdoor wireless communications range, will be used throughout this paper unless otherwise noted. There are 204 nodes in the generated mesh. The PDE algorithm produces a nonuniform mesh using an NSL value of 40. A smaller edge length would produce a more uniform mesh but would require many more SNs, i.e., smaller triangles provide a more optimal fit to the lake geometry.

NETGEN is an open-source automatic 2-D/3-D mesh generation tool distributed under the conditions of the LGPL (GNU Lesser General Public License). Triangles are meshed by an advancing front surface mesh generator. The Advancing Front Mesh Generation (AFMG) algorithm starts from an initial front, the boundary of the domain, and constructs the mesh element by element sweeping a front across the domain. The steps for creating a mesh by the classic Advancing Front Mesh Generation (AFMG) algorithm are given here. Step 1) Start at the boundary of the domain and create an initial front of nodes that meet some spacing criteria based on edge length. Connect the boundary nodes. Step 2) Create new points (nodes) based on optimal spacing criteria and connect those nodes to nodes of the current front. The newly added nodes become the current front. Step 3) Repeat step 2) until the front is empty and an entire mesh has been formed. If the desired mesh distribution is not achieved, such as a uniform mesh distribution, then a post processing mesh optimization may be performed. Some classical techniques for mesh optimization are node relocation, edge collapsing, and edge swapping [23]. Fig. 3 shows three diagrams as an example of mesh generation using the advancing front method. Diagram 1 shows the initial boundary discretization for the advancing front method. The boundary is a rectangular area with nodes fairly equally spaced based at the desired edge length size. The front is represented in diagram 2 as dotted line segments. The solid lines represent the edges of the new triangles that have been formed in the mesh. In diagram 3 several of the edges are no longer part of the front and have formed part of a new triangle. The front may be thought of as a stack of edges, where edges are pushed and popped. The domain is completely covered when the stack is empty and the fronts have merged together. Fig. 4 shows the results of a mesh generated by NETGEN for our domain. The mesh generation results in a nonuniform mesh with 126 nodes. However, note that NETGEN produces a more uniform mesh than the PDE toolbox with 35% less nodes. AFMG techniques are known to produce higher quality node distributions than techniques based on Delaunay triangulation due to the optimum positioning of new nodes [24]. Section V discusses the novel mesh simplification algorithm. This algorithm optimizes a mesh within a nonintersecting closed


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coverage overlap. is some fraction of the desired spacing between nodes: in

if

remove

(3)

Step 2) Iterative removal of internal nodes with CDT. The next step after removal of boundary nodes is the removal of internal nodes. Edge collapsing or vertex removal of an edge AB replaces this edge by only one point, leading to positioning A on B, or B on A: find

Fig. 4. Mesh generation with NETGEN, maximum edge length of 40; 126 nodes.

polygon to conform to the configuration parameters of number of nodes, distance between nodes, and geometry of the domain. V. INRCDTS ALGORITHM FOR MESH SIMPLIFICATION Our newly developed mesh simplification algorithm INRCDTS improves the quality of an existing mesh within a polygonal area. This section discusses the INRCDTS algorithm. The MATLAB PDE and NETGEN algorithms create triangles of varying sizes to accommodate the boundary geometry of the domain. These techniques may create more nodes than are desired for a wirelsss sensor network. The INRCDTS algorithm simplifies the mesh to conform to the number of nodes desired by the user and improves the quality of the mesh. This reduction of nodes in the mesh is known as mesh simplification. Our INRCDTS algorithm for mesh simplification involves both cleanup, Laplacian smoothing, and constrained Delaunay triangulation. Cleanup removes both boundary and internal nodes when the edge length between nodes is less than a desired threshold, effectively changing the node connectivity and reducing the number of nodes. Laplacian smoothing relocates an internal node to the average location of all nodes adjacent to the internal node [25]. A Delaunay triangulation algorithm that takes into account constrained edges, such as roads, bridges, and lake boundaries, is known as a constrained Delaunay triangulation (CDT) algorithm. The following notation is used to describe the mesh simplification algorithm more precisely. • . • of . • and from (1) represent the sets of nodes and edges in the mesh, respectively, where the total number of nodes is composed of the boundary nodes and the internal nodes . The node set is ordered sequentially in a counter clockwise direction for . The three steps of the mesh simplification algorithm previously discussed are given here. Step 1) Removal of unnecessary boundary nodes. The boundary nodes in the mesh that are separated by a distance of less than are removed to minimize

is

(4)

If , remove , else if , remove Internal nodes are removed by iterating though all of the edges of the mesh till the smallest length edge is found. One of the vertices or of this edge is removed as long as the vertex is not on a boundary. If both vertices of are on the boundary, then the next smallest edge in the mesh is processed in the same manner. After an internal node is removed, the entire mesh is retriangulated using the CDT algorithm. The internal node removal and retriangulation process continues until the desired number of nodes in the mesh is reached. Step 3) Smoothing. Smoothing is an iterative process for repositioning internal vertices to improve the quality of the mesh. Global or local techniques can be used to address this optimization problem. Global techniques simultaneously find the optimal location for each internal node in the mesh. Local optimization techniques iterate through all internal nodes, calculating the optimal position of one node while keeping all others fixed. A local optimization technique for the smoothing problem is used in this paper. The area of a nonself-intersecting closed polygon defined by vertices with coordinates , is (5) The , coordinates

,

of the centroid of this polygon are (6) (7)

The smoothing algorithm, listed in Table II, iterates through all of the internal vertices of the mesh calculating the polygonal area of the vertices adjacent to , the centroid point of the polygon, and the distance from the node to the centroid , . If is greater than , node is moved to , . The smoothing algorithm terminates when the average internal node movement is less than or equal to , which is the movement threshold. Section VI discusses coverage and connectivity for a WSN.

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TABLE II SMOOTHING ALGORITHM

VI. COVERAGE AND CONNECTIVITY CONCEPTS The nodes in a WSN must maintain sufficient sensing coverage and network connectivity for quality of service. Here, we explore the relationships between coverage and connectivity demonstrating that coverage implies connectivity.

Fig. 5. Optimal sensor layout with sensor spacing of

.

A. Sensing Coverage and Network Connectivity Concepts Communications range of the sensor affects the connectivity of the mesh network. Sensing range of the sensor affects the sensing coverage of the mesh network. Effective sensor deployment algorithms must consider both coverage and network connectivity [26]. We use isotropic sensing and communications models, as well as a 2-D equilateral triangular mesh deployment that provides maximum coverage with the smallest overlap and gap in coverage [2], [27], [28]. Each sensor in our mesh network has the same communications and sensing ranges. Two sensors and are connected if within and across the boundary of the lake, i.e., the boundary represents the shore line and not an impenetrable object. Fig. 5 shows the maximum coverage and no gap layout for nodes with . Note that this is our optimal sensor layout, resulting in full coverage with no sensing gaps in the sensor field and with minimal overlap in coverage. The area of coverage overlap between two sensors is the area of the intersection of two circles of radius . Fig. 6 shows the effect of , i.e., sensor . This sensor spacing also results in no gap coverage. However, an SN deployment with these configuration parameters will require a larger number of sensors to cover the same area as in Fig. 5 because of greater coverage overlap and closer sensor spacing. With , communications connectivity of the network is automatically guaranteed. If , additional sensing nodes will be required to ensure that the mesh network is completely. Therefore, the desired relationship between and is (8) Based on the relationships depicted in Figs. 5 and 6 and previous research [2]–[4], [27], the following are minimized in a domain when (8) holds: • sensing gaps; • overlap in coverage;

Fig. 6. Nonoptimal sensor layout with sensor

.

• number of nodes. Therefore, our simulations choose to follow relationship (8). B. K-Coverage and K-Connectivity Degree of coverage and degree of connectivity are important concepts in the design of wireless sensor network configurations. K-coverage, or K-covered, is the degree of coverage. K-coverage refers to the number of nodes , that provide sensing coverage for every location of the domain. 1-coverage implies the locations within the domain are only monitored by one sensor. The concept of K-connectivity, or K-connected, applies to the communications paths between nodes. If a sensor is 2-connected, the sensor has communications paths to two other nodes. A goal of the sensor deployment is to tailor the number of nodes that are K-connected to the application. Some applications will require to maintain a highly reliable and robust sensor network adaptable to node failures. The relationship between coverage and connectivity is that each node must be able to find at least one route to a BS, otherwise known as network connectivity. Full area coverage requires that every location in the domain be within range of a node. Therefore, connectivity does not imply coverage. Two


nodes can be connected through multihop transmissions with data relayed by intermediate nodes. C. Geometric Analysis of Coverage and Connectivity Although connectivity does not imply coverage, 1-coverage does imply that the communications graph is connected. The following description and theorem demonstrates how coverage implies connectivity. The communications graph of a set of nodes from (1) is defined as follows: (9) For any nodes and in , there exists an edge in if and only if the distance between nodes and is less than or equal to the communications range . We also define a path from the vertex in to a vertex in as an alternating sequence of vertices and edges:

(10) (11) In (11), vertex is reachable (shown with the symbol ) from vertex if and only if there exists a path from to . The following theorem presents the conditions for which a covered network guarantees communications connectivity. Theorem 1: If each location in a domain is uniquely covered by one sensor, then the communications graph is connected if [4]. Proof: Assume that a node has a sensing circle of radius with any point less than covered by the node. Let a line segment join two nodes contained within some convex polygonal area with each point on being covered by at least one sensor. Sensor coverage overlaps so that each point on is less than from a sensor. Multiple contiguous line segments may be connected together where each point on the entire contiguous line will be covered by one or more sensors. A point on the line segment must be less than from a sensor to be covered, by definition. Let the distance between two sensors at points and in two different contiguous line segments and , . The point at the intersection of these two sensing circles would be at a distance from both points and and hence not covered by either sensing circle. Therefore, the required distance between points and in contiguous line segments and must be less than . Since is less than ,a communication edge must exist between the sensors at points and in different line segments. Therefore, a communications path exists across multiple line segments between multiple sensors, and, as a result, the SN coverage also provides communications connectivity. As long as the topology of the SN ensures full coverage, communications connectivity is also guaranteed. Section VII addresses the topic of mesh generation metrics. Multiple metrics are defined to evaluate and compare the performance of mesh generation algorithms.

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VII. MESH GENERATION METRICS Several metrics used to evaluate the performance of mesh generation algorithms are discussed here. Average Equilateral Volume Skewness. The average equilateral volume skewness [29] of the mesh is

where (12) is the total volume skewness, is the number of triand angle units in the mesh, is the optimal triangle size, and is the actual triangle size. Average Equilateral Angle Skewness. The average equilateral angle skewness [29] of the mesh is

(13) where is the largest angle in triangle, is the smallest angle in triangle, and is the equilateral angle of 60 . Values from 0 to 0.25 are considered excellent for . The optimal value of equilateral angle skew is 0, meaning that the triangles are as well formed as possible. Average Aspect Ratio. The average aspect ratio [20], , of the mesh is (14) is the length of the longest side of a triangle and where is the length of the shortest side of the triangle. The ideal aspect ratio for an equilateral triangle is 1. Triangle Quality. The triangle quality metric [30], [32] is (15) where is the area of the triangle, and , , and are the lengths of triangle sides. A triangle is considered good if , fair if , and poor if .A triangle is equilateral, which is the ideal case, when equals 1. Coverage Effectiveness. Coverage effectiveness [23] provides a measure of how much of a given area will be covered by the mesh network

(16) is the where is the total number of node triangular areas, triangle size in the mesh, and is the size, in square meters, of the total domain minus fixed obstacle size within the area minus area within of walls and obstacles.

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Formation Effectiveness. Formation effectiveness [23] is a measure of the ideal spacing arrangement between nodes (17) where is the triangle size in mesh, is the ideal triangle area (all sides of ), and is the number of triangles in mesh. Average Degree of the Network. The degree of a node is the number of edges incident with . is the degree of all nodes averaged over the collection of nodes . Sensor Coverage Overlap (SCO). The greater the degree of K-Coverage, the greater the area of sensor coverage overlap. The requirements of a node deployment may dictate a high SCO value for reliability and survivability purposes or a low SCO value to minimize cost while still maintaining complete coverage. Section VIII of the paper discusses and contrasts the simulation performance of the novel mesh simplification technique and the mesh generation algorithms.

Fig. 7. PDE mesh simplification with INRCDTS algorithm.

VIII. SIMULATION TEST CASES RESULTS Here, we provide a comparative analysis between existing approaches and a newly developed algorithm presented in this paper (INRCDTS). First, two known and widely used types of available mesh generation algorithms were selected to create an initial mesh. These two known algorithms, Advancing Front and Delaunay Triangulation, were executed by the following tools: NETGEN (AFMG) and MATLAB PDE (Delaunay triangulation). Neither one is meeting the requirements of the SN deployment for sensory coverage of a specific area (such as the one in Fig. 1). These requirements are to minimize the number of nodes and to provide spacing that complies with sensing ranges of commercially available sensors. Our INRCDTS algorithm modifies the meshes generated by NETGEN and PDE to meet these requirements. Therefore, this section provides a comparative analysis among four approaches: NETGEN, PDE, NETGEN with INRCDTS, and PDE with INRCDTS. The problem formulation specifies that the goal of the algorithmic approach is to determine an optimal solution that satisfies two requirements: 1) complete coverage and 2) maximum formation. These requirements must be met with a fixed number of nodes and specific internodal spacing. Other quality criteria, such as an internodal spacing ranging between and may be used to simplify the mesh network. However economic decisions, such as the number of nodes, are typically the determining factor. Sixty nodes are desired in the final mesh network configuration. This number was chosen heuristically based on two boundary cases for the problem illustrated by Fig. 1 (the domain with an area of 60 450 square meters). The two boundary cases were chosen for an NSL of 30 and 40 m. These lengths were reasonable distances for wireless distance communications in everyday use. Each sensor covers an area of . An NSL of 40 m is equivalent to an of 23 m requiring 36 nodes to cover the polygonal area. An NSL of 30 m is equivalent to an of 17.3 m requiring 64 nodes to cover the polygonal area. Therefore, an NSL of 30 (or 40) m would imply usage

Fig. 8. NETGEN mesh simplification with INRCDTS algorithm.

of 36 (or 64) nodes. Sixty nodes provide some flexibility in internodal spacing to allow for the geometric irregularities of the polygonal area of Fig. 1 as well as for placement of nodes on the boundary of the domain. A. INRCDTS Mesh Simplification Results Figs. 7 and 8 depict the application of the INRCDTS mesh simplification algorithm to the PDE and NETGEN meshes of Figs. 2 and 4, respectively. The PDE mesh and NETGEN meshes have been reduced from 204 and 126 nodes, respectively, to 60 nodes each. B. K-Connectivity Fig. 9 shows the percentage of nodes with a K-Connectivity of 1 through 7 for each mesh generation technique. PDE and NETGEN meshing techniques with the novel INRCDTS mesh simplification algorithm clearly provide the highest degrees of connectivity resulting in multiple communications paths


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TABLE III BASIC METRICS SUMMARY FOR SIMULATION APPROACHES

TABLE IV MESH EVALUATION METRICS SUMMARY FOR SIMULATION APPROACHES

Fig. 9. Percent of node connectivity by K-Connectivity and mesh technique.

through the mesh. The greater the K-Connectivity, the more resilient the mesh network is to failure with multiple network paths to transmit acquired sensing data to base stations outside the domain. C. Description of Each Metric Average triangle size (ATS) is the average triangle size in the generated mesh. (formation effectiveness) is a measure of how well the mesh triangles conform to an equilateral triangle with the desired side lengths. The ideal is 1. An indicates that some of the triangles are larger than the ideal triangle size. (coverage effectiveness) provides a measure of how much of the domain is covered by the mesh network. An ideal is 1. Average edge length (AEL) is the average distance between nodes in the network. Sensing coverage overlap (SCO) is the fraction of the area that has overlap in sensor coverage. Average degree is the average number of edges incident to all nodes in the network. q (triangle quality metric) is the average triangle quality level for all triangles in the mesh network. (average equilateral angle skewness) is a measure of the equilateral angle skew with 0 meaning that the triangles are as well formed as possible. (average equilateral volume skewness) is the average difference between average and ideal triangle sizes in the mesh. Skewness values range from 0 (best) to 1 (worst). (average aspect ratio) is the average ratio of the length of the longest side of a triangle to the length of the shortest side of the triangle. The ideal value is 1. # of Nodes generated in the mesh network is variable using MATLAB PDE and NETGEN, while the INRCDTS mesh simplification algorithm tailors the mesh to the specific number of desired nodes.

D. Comparative Analysis Tables III and IV list the comparative analysis of the INRCDTS enhanced approaches developed in this paper (PDE Toolbox with INRCDTS and NETGEN with INRCDTS) versus the same approaches without INRCDTS. An explanation of each metric value for the MATLAB PDE, NETGEN, and INRCDTS simulations follows. Analysis of ATS. NETGEN generates an average triangle size that is closest to the ideal triangle size. However, NETGEN uses 126 nodes rather than the desired node count of 60. have an average triangle size closest to the ideal while using the desired number of nodes. Analysis of . PDE and NETGEN produce meshes with variable triangle sizes resulting in low values. An indicates that some of the triangles are larger than the ideal triangle size. Analysis of . is calculated on the triangles formed in the mesh. All of the values in Table III have a very high value. Analysis of AEL. The PDE and NETGEN algorithms with INRCDTS have an edge length very close to the desired edge length of 40 m. This is entirely due to the INRCDTS algorithm. A low AEL is an indication of a wide variation in triangles sizes as is evident from the PDE deployment in Fig. 2 and the NETGEN deployment in Fig. 4. Analysis of SCO. The PDE algorithm generates a large number of nodes resulting in the highest SCO value. Our design goals, as noted in Section VI-A, were to minimize both gaps and overlap in coverage. have the lowest SCO value. INRCDTS will generate a more optimal mesh if the initial mesh is of high quality. Analysis of . NETGEN produces a more uniform mesh than PDE, as previously noted, resulting in a lower average degree per node. A high average node degree for node is the average number of paths out of for moving data through the network. Analysis of q. The average triangle quality level is approximately the same with this metric for all algorithmic

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approaches in Table III. Averaging tends to level out the triangle quality values. Even though there is a wide variation in triangle sizes for each algorithmic approach, there appears to be a Gaussian distribution of triangle sizes per each approach. The mesh simplification algorithm produces meshes of high quality, , in both cases. Analysis of . All of the algorithmic approaches in Table III produce approximately the same degree of skew averaged over all of the triangles in the mesh. Analysis of . PDE and NETGEN produce the greatest number of nodes with varying sizes, which result in poor values. INRCDTS provides dramatic improvement in as noted in Table III. Analysis of . Table III indicates that, on average, the longest side of a triangle is approximately 1/3 longer than the shortest side. Analysis of # Nodes. The desired number of nodes is 60. INRCDTS will simplify a mesh to meet the exact required number of nodes. The PDE and NETGEN algorithm generate a mesh of an area based on a desired maximum edge length resulting in a variable number of nodes. NETGEN is based on the advancing front mesh generation algorithm which is known to produce a more uniform mesh than PDE with Delaunay triangulation. IX. CONCLUSION This paper presented centralized mesh generation techniques which are based on Delaunay triangulation and advancing front algorithms. A newly developed mesh simplification algorithm INRCDTS was then applied to optimize the generated meshes (from Delaunay triangulation and advancing front algorithms) based on user-defined criteria (sensor node spacing and number of sensor nodes). The major contributions of this paper are: 1) a novel mesh simplification algorithm INRCDTS for improving the quality of meshes generated by a centralized algorithm and 2) a centralized mesh generation approach with INRCDTS that may be used for any nonintersecting closed polygonal area. These were elaborated upon by providing a comparison of two centralized mesh generation techniques for configuration of an SN deployment in an irregular shaped polygonal area. In addition, the paper provides the validation of the INRCDTS algorithm on PDE and NETGEN generated meshes and a proof of the geometric analysis of the relationship between coverage and connectivity. The second part of this investigation [33] will compare and contrast the centralized algorithms with decentralized algorithms for mobile sensor nodes. Decentralized algorithms require no a priori information but are capable of automatically deploying mobile sensor nodes to any shape of an environment using only local information. REFERENCES [1] J. O’Rourke, Art Gallery Theorems and Algorithms. New York, NY, USA: Oxford Univ., 1987. [2] B. Wang, “Coverage problems in sensor networks: A survey,” ACM Computing Surveys, vol. 43, no. 4, Oct. 2011. [3] Y. Wang, C. Hu, and Y. Tseng, “Efficient deployment algorithms for ensuring coverage and connectivity of wireless sensor networks,” in Proc. 1st Int. Conf .Wireless Internet, Sep. 2005, pp. 114–121.

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[29] A. Bakker, “Applied computational fluid dynamics,” Fluent Inc., Computational Fluid Dynamics Class Lecture 7, 2006. [30] B. Shucker and J. Bennett, “Scalable control of distributed robotic macrosensors,” in Proc. 7th Int. Symp. Distrib. Auton. Robotic Syst., Jun. 23–25, 2004, pp. 379–388. [31] W. Spears, D. Spears, R. Heil, and D. Zarzhitsky, Physicomimetics for Mobile Robot Formations. New York, NY, USA: ACM, 2004. [32] P. Lindstrom, “Model simplification using image and geometry-based metrics,” Ph.D. dissertation, Comput. Sci. Dept., Georgia Inst. Technol., Atlanta, GA, USA, 2000. [33] K. Derr and M. Manic, “Wireless sensor network configuration, part II: Adaptive coverage for decentralized algorithms,” IEEE Trans. Ind. Inf., vol. PP, no. 99, DOI: 10.1109/TII.2013.2245907. [34] O. Hjelle and M. Daehlen, Triangles and Applications. Berlin, Germany: Springer-Verlag, 2010, ch. 1.

Kurt Derr (M’04) received the B.S. degree in electrical engineering from the Florida Institute of Technology, Melbourne, FL, USA, and the M.S. degree in computer science from the University of Idaho, Idaho Falls, ID, USA, where he is currently working toward the Ph.D. degree in computer science. He has worked as a Computer Scientist with the Idaho National Laboratory, Idaho Falls, ID, USA, since 1986 and began teaching courses as Affiliate Faculty with the University of Idaho in 1987. He and has coauthored four patents and authored a book on object-oriented technology, Applying OMT (SIGS Publications, 1995). He has authored and coauthored several recent journal and conference papers on computational intelligence in wireless networking environments. Mr. Derr was the recipient of an R&D 100 Award in 2009 for RFinity, Mobile Open-Encryption Platform

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Milos Mani (S’95–M’05–SM’06) received the Dipl. Ing. and M.S. degrees in electrical engineering and computer science from the University of Nis, Nis, Serbia, in 1991 and 1997, respectively, and the Ph.D. degree in computer science from the University of Idaho, Idaho Falls, ID, USA, in 2003. He is an Associate Professor with the Computer Science Department, Idaho Falls, ID, USA, where he has led the Computer Science Program and is a Director of the Modern Heuristics Group. He has over 20 years of academic and industrial experience, including an appointment with the Electrical and Computer Engineering Department and Neuroscience program at the University of Idaho. As a university collaborator or principal investigator he lead number of research grants with the National Science Foundation, Idaho National Laboratory, EPSCoR, Department of Air Force, and Hewlett-Packard, in the area of data mining and computational intelligence applications in process control, network security and infrastructure protection. He has published over 100 refereed articles in international journals, books, and conferences. Dr. Manic is a Secretary of IEEE Industrial Electronics Society and is a member of several technical committees and boards of this Society such as IES Committees (previously the Boards) for Conferences and for Publications. Also involved in various capacities in Technical Committees on Education, Industrial Informatics, Factory Automation, Smart Grids, Standards, and Web and Information Committee (WIC), and is a co-founder and chair of Technical Committee on Resilience and Security in Industry. Dr. Manic is an associate editor of the IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, the IEEE TRANSACTIONS ON INDUSTRIAL INFORMATICS, and The International Journal of Engineering Education, and has completed over six special sections in the IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS and the IEEE TRANSACTIONS ON INDUSTRIAL INFORMATICS as a lead or co-editor. He has served conferences in various capacities as a General Co-Chair, Technical Program Chair, Track Chair, and Special Session Chair at various IES conferences (IESRCS, IECON, ICELIE, HSI, INDIN, ICIEA, ISIE).