Interior Nodes Research Articles

A fourth-order seven-point cubature on regular hexagons

We investigate the central moments of (regular) hexagons and derive accordingly a discrete approximation to definite integrals on hexagons. The seven-point cubature rule makes use of interior and neighbor center nodes, and is of fourth order by construction. The result is expected to be useful in two-dimensional (open-field) applications of integral equations or image processing.

Local integration of population dynamics via moving least squares approximation

This paper applies an approach based on the Galerkin and collocation methods so-called meshless local Petrov---Galerkin (MLPG) method to treat a nonlinear partial integro-differential equation arising in population dynamics. In the proposed method, the MLPG method is applied to the interior nodes while the meshless collocation method is used for the nodes on the boundary, so the Dirichlet boundary condition is imposed directly. In MLPG method, it does not require any background integration cells so that all integrations are carried out locally over small quadrature domains of regular shapes, such as circles or squares in two dimensions and spheres or cubes in three dimensions. The moving least squares approximation is proposed to construct shape functions. A one-step time discretization method is employed to approximate the time derivative. To treat the nonlinearity, a simple predictor---corrector scheme is performed. Also the integral term, which is a kind of convolution, is treated by the cubic spline interpolation. Convergence in both time and spatial discretizations is shown and more, stability of the method is illustrated.

Justification of the saturation assumption

The saturation assumption is widely used in computational science and engineering, usually without any rigorous theoretical justification and even despite of counterexamples for some coarse meshes known in the mathematical literature. On the other hand, there is overwhelming numerical evidence at least in an asymptotic regime for the validity of the saturation. In the generalized form, the assumption states, for any $$0<\varepsilon \le 1$$0<?≤1, that $$\begin{aligned} ||| u - {\hat{U}} |||^2 \le (1- \varepsilon /C) ||| u - U |||^2+ \varepsilon \mathrm{osc}^2(f,\mathcal {N}) \end{aligned}$$|||u-U^|||2≤(1-?/C)|||u-U|||2+?osc2(f,N)for the exact solution u and the first-order conforming finite element solution U (resp. $${\hat{U}}$$U^) of the Poisson model problem with respect to a regular triangulation $$\mathcal {T}$$T (resp. $${\hat{\mathcal {T}}}$$T^) and its uniform refinement $${\hat{\mathcal {T}}}$$T^ within the class $$\mathbb {T}$$T of admissible triangulations. The point is that the patch-oriented oscillations $$\mathrm{osc}(f,\mathcal {N})$$osc(f,N) vanish for constant right-hand sides $$f\equiv 1$$f?1 and may be of higher order for smooth f, while the strong reduction factor $$(1- \varepsilon /C)<1$$(1-?/C)<1 involves some universal constant C which exclusively depends on the set of admissible triangulations and so on the initial triangulation only. This paper proves the inequality (SA) for the energy norms of the errors for any admissible triangulation $$\mathcal {T}$$T in $$\mathbb {T}$$T up to computable pathological situations characterized by failing the weak saturation test (WS). This computational test (WS) for some triangulation $$\mathcal {T}$$T states that the solutions U and $${\hat{U}}$$U^ do not coincide for the constant right-hand side $$f\equiv 1$$f?1. The set of possible counterexamples is characterized as $$\mathcal {T}$$T with no interior node or exactly one interior node which is the vertex of all triangles and $${\hat{\mathcal {T}}}$$T^ is a particular uniform bisec3 refinement. In particular, the strong saturation assumption holds for all triangulations with more than one degree of freedom. The weak saturation test (WS) is only required for zero or one degree of freedom and gives a definite outcome with O(1) operations. The only counterexamples known so far are regular n-polygons. The paper also discusses a generalization to linear elliptic second-order PDEs with small convection to prove that saturation is somehow generic and fails only in very particular situations characterised by (WS).

High-order curvilinear meshing using a thermo-elastic analogy

With high-order methods becoming increasingly popular in both academia and industry, generating curvilinear meshes that align with the boundaries of complex geometries continues to present a significant challenge. Whereas traditional low-order methods use planar-faced elements, high-order methods introduce curvature into elements that may, if added naively, cause the element to self-intersect. Over the last few years, several curvilinear mesh generation techniques have been designed to tackle this issue, utilizing mesh deformation to move the interior nodes of the mesh in order to accommodate curvature at the boundary. Many of these are based on elastic models, where the mesh is treated as a solid body and deformed according to a linear or non-linear stress tensor. However, such methods typically have no explicit control over the validity of the elements in the resulting mesh. In this article, we present an extension of this elastic formulation, whereby a thermal stress term is introduced to ‘heat’ or ‘cool’ elements as they deform. We outline a proof-of-concept implementation and show that the adoption of a thermo-elastic analogy leads to an additional degree of robustness, by considering examples in both two and three dimensions.

Efficient Interlayer Network Codes for Fair Layered Multicast Streaming

Multilayer video streaming allows to provide different video qualities to a group of multicast receivers with heterogeneous receive rates. The number of layers received (and thus the receive rate) determines the quality of the decoded video stream. For such layered multicast streaming, network coding provides higher capacity than multicast routing. Network coding can be performed within a layer or across layers, and in general, interlayer coding outperforms intralayer coding. An optimal solution to a network-coded layered multicast problem may require decoding of the network code at interior nodes to extract information to be forwarded. However, decoding consumes resources and introduces delay, which is particularly undesirable at interior nodes (the routers) of the network. In this paper, we thus focus on the interlayer network coding problem without decoding at interior nodes. We show that the problem is NP-hard and propose a heuristic algorithm for rate allocation and coding based on the Edmonds-Karp maximum flow algorithm. We prove that our algorithm ensures decodability of the information received and provides some fairness properties. Finally, we perform extensive simulations and show that our algorithm may even outperform other heuristics that do require decoding at interior nodes.

An Adaptive Spanning Tree-Based Data Collection Scheme in Wireless Sensor Networks

In-network data aggregation is a widely used method for collecting data efficiently in wireless sensor networks (WSNs). The authors focus on how to achieve high aggregation efficiency and prolonging networks’ lifetime. Firstly, this paper proposes an adaptive spanning tree algorithm (AST), which can adaptively build and adjust an aggregation spanning tree. Owing to the strategies of random waiting and alternative father nodes, AST can achieve a relatively balanced spanning tree and flexible tree adjustment. Then a redundant aggregation scheme (RAG) is illustrated. In RAG, interior nodes help to forward data for their sibling nodes and thus provide reliable data transmission for WSN. Finally, the simulations demonstrate that (1) AST can prolong the lifetime and (2) RAG makes a better trade-off between storage and aggregation ratio, comparing to other aggregation schemes.

TH-CD-303-01: On Characterizing 4D Lung Motion Using Coupled Multi-Physics Framework of Computational Fluid Dynamics, Linear Elasticity Estimation and 4D Lung Deformable Image Registration

Purpose: Modeling human lung dynamics is critical in quantitatively representing the subject specific 4D tumor motion. However, the process is numerically complicated by fluid-structure interactions (FSI) between air and the visco-elastic lung tissue and the lack of information on the anisotropic lung elasticity. Methods: We present a mathematical method to couple image registration and physics-based modeling approaches in order to represent a consistent volumetric lung dynamics. The Computational Fluid Dynamics (CFD) simulation assumes the lung to be a poro-elastic medium with spatially dependent elastic property. The simulation is performed on a 3D lung geometry reconstructed from four-dimensional computed tomography (4DCT) scan dataset of a human subject. The heterogeneous linear elastic Young’s Modulus (YM) values are estimated from a linear elastic deformation model with the same lung geometry and 4D lung Deformable Image Registration (DIR). The deformation obtained from the CFD simulation is then coupled with the displacement obtained from the DIR by means of the Tikhonov regularization (TR) algorithm. Numerical results present an analytic inter-model consistent estimation of the volumetric lung dynamics. The fusion method was validated by comparing the optimal displacement with 5D model-generated landmarks. Results: A set of five 4DCT datasets was employed for this study. For the landmarks inside the lung, the optimization results in a significant reduction in error. Specifically, in the left lung, the average error for the landmarks on the surface and edges is reduced by 1.11 mm while this value is 2.96 mm for the nodes, while, in the right lung, the average displacement errors after optimization are reduced by 1.34 mm and 0.86 mm for the interior and surface nodes respectively. Conclusion: These results shows that although optimization reduces the displacement error both at the surface and the interior nodes, it is more effective for the error reduction of interior nodes.

A finite element method for prediction of unknown boundary conditions in two-dimensional steady-state heat conduction problems

This paper presents a finite element method in predicting unknown boundary conditions of homogeneous and two-layered materials subjected to steady-state heat conduction. Firstly, a finite element formulation is introduced to solve a steady-state boundary inverse heat conduction problem of homogeneous material. Effects of bias error of temperature specified on interior nodes and locations of such nodes on accuracy of predicted temperature distribution are examined. Then, modified cubic spline is specified on material boundary to stabilize predicted temperature distribution. Cubic spline functions using different numbers of control points are used in examining their effects on accuracy of predicted temperature distribution and computing time when specifying no bias temperatures. Finally, the formulation with cubic spline function specification is employed in predicting unknown boundary conditions of two-layered materials with thermal conductivity ratio of 0.1, 1, and 10. Concept of coincident nodes is applied in handling physical condition characterized by thermal contact resistance and heat source strength at layer interface. Effect of bias error of temperatures specified on nodes within thicker layer is examined under three interface conditions. Cubic spline function with five control points can predict temperature distributions accurately for all interface conditions when specifying no bias temperatures. RMS errors vary linearly with bias errors for interface conditions with no heat source but are drastically affected by bias error when heat source exists at the interface.

Strong sum distance in fuzzy graphs.

In this paper the idea of strong sum distance which is a metric, in a fuzzy graph is introduced. Based on this metric the concepts of eccentricity, radius, diameter, center and self centered fuzzy graphs are studied. Some properties of eccentric nodes, peripheral nodes and central nodes are obtained. A characterisation of self centered complete fuzzy graph is obtained and conditions under which a fuzzy cycle is self centered are established. We have proved that based on this metric, an eccentric node of a fuzzy tree G is a fuzzy end node of G and a node is an eccentric node of a fuzzy tree if and only if it is a peripheral node of G and the center of a fuzzy tree consists of either one or two neighboring nodes. The concepts of boundary nodes and interior nodes in a fuzzy graph based on strong sum distance are introduced. Some properties of boundary nodes, interior nodes and complete nodes are studied.

Automatic quadrilateral mesh generation and quality improvement techniques for an improved combination method

This paper developed an automatic quadrilateral meshing system to resolve the numerical simulation problems in the fields of geological mechanics, hydraulics, hydrology, and water resources. An improved combination method was proposed to automatically convert Delaunay triangular meshes into quadrilateral meshes by combining adjacent pairs of triangles through searching advancing-front edges. In order to implement the full conversion, seven combination patterns were established to select appropriate pairs of adjacent triangles. To handle the residual triangles, each nearest pair of residual triangles were moved to each other along a proper path and recombined into a quadrilateral. The movement of residual triangles was implemented through the destruction and reconstruction of neighboring quadrilaterals. A transformation template was established to convert the only residual triangle in each domain into quadrilaterals. To ensure the geometric topology of the resulting mesh, nine topological optimization modes were proposed to improve the topological connections of the degenerated quadrilaterals especially those on the boundaries of the mesh. An area-weighted Laplacian method was used to smooth the interior nodes, and an objective function method was employed to adjust the positions of special nodes. Finally, practical examples of two treatment strategies for river and overland flows were provided to demonstrate the accuracy and reliability of the meshing and optimization algorithms proposed in this paper.

Local integration of 2-D fractional telegraph equation via moving least squares approximation

In this paper, a classical type of two-dimensional time-fractional telegraph equation defined by Caputo sense for (1<α<2) is analyzed by an approach based on the Galerkin weak form and moving least squares (MLS) approximation subject to given appropriate initial and Dirichlet boundary conditions. In the proposed method, which is a kind of the Meshless local Petrov–Galerkin (MLPG) method, meshless Galerkin weak form is applied to the interior nodes while the meshless collocation method is used for the nodes on the boundary, so the Dirichlet boundary condition is imposed directly. In MLPG method, it does not require any background integration cells so that all integrations are carried out locally over small quadrature domains of regular shapes, such as circles or squares. The moving least squares approximation is proposed to construct shape functions. Two numerical examples are presented and satisfactory agreements are achieved.

A Boundary Enhancement for the Stabilized Conforming Nodal Integration of Galerkin Meshfree Methods

The stabilized conforming nodal integration (SCNI) has been successfully developed for Galerkin meshfree methods based upon the linear exactness requirement. In this study, it is shown that for a given problem domain, when the support of the meshfree shape functions associated with the interior nodes do not cover the essential boundary, the linear exactness can be perfectly achieved by the standard SCNI formulation. On the other hand, when the essential boundary lies in the support of the meshfree shape functions of the interior nodes, a linear field may not be exactly obtained with the original SCNI formulation where the essential boundary conditions are enforced via the nodally exact transformation method, and the error even becomes more pronounced with the increase of support size. To resolve this issue, a flux term associated with the essential boundary is recovered in the variational formulation and it turns out to be proper to keep this term since the meshfree shape functions of interior nodes usually do not vanish on the boundary. Consequently the original SCNI integration constraint is revised and the stiffness matrix is enhanced by an additional stiffness contribution from the flux integration along the essential boundary. It is demonstrated that the proposed enhanced formulation is capable of exactly reproducing linear fields regardless of the support sizes. Moreover, several benchmark examples reveal that the present SCNI formulation with boundary enhancement yields better accuracy compared with the original SCNI approach, particularly for meshfree discretizations with larger support sizes.

Local integration of 2-D fractional telegraph equation via local radial point interpolant approximation

In this article, a general type of two-dimensional time-fractional telegraph equation explained by the Caputo derivative sense for (1 < α ≤ 2) is considered and analyzed by a method based on the Galerkin weak form and local radial point interpolant (LRPI) approximation subject to given appropriate initial and Dirichlet boundary conditions. In the proposed method, so-called meshless local radial point interpolation (MLRPI) method, a meshless Galerkin weak form is applied to the interior nodes while the meshless collocation method is used for the nodes on the boundary, so the Dirichlet boundary condition is imposed directly. The point interpolation method is proposed to construct shape functions using the radial basis functions. In the MLRPI method, it does not require any background integration cells so that all integrations are carried out locally over small quadrature domains of regular shapes, such as circles or squares. Two numerical examples are presented and satisfactory agreements are achieved.

Optimizing R-tree for flash memory

R-tree has been widely used in spatial data management and data analysis to improve the performance of spatial data retrieval. However, the original R-tree is designed for magnetic disks, and has poor performance on flash memory, due to the special features of flash memory such as asymmetric read/write speeds (fast read, slow write) and the erase-before-write feature. Particularly, the original updating mechanism of R-tree usually has to update a few interior nodes when inserting an indexing item into or deleting an item from a leaf node, yielding many slow writes to flash memory. With the wide use of flash memory in many location-based fields, e.g., to store moving trajectories in intelligent transportation systems, how to optimize R-tree for flash memory has become a critical issue. In this paper, we propose a novel spatial index named Flash-Optimized R-tree that is optimized for flash memory. In particular, we propose to defer the node-splitting operations on R-tree by introducing overflow nodes, which results in an unbalanced tree structure. With this mechanism, we can reduce random writes to flash memory and improve the overall performance of R-tree. In addition, we present a new buffering scheme to efficiently cache the updates to the tree, which can further reduce random writes to flash memory. We conduct extensive experiments on real flash-memory storage devices as well as a flash memory simulation platform to evaluate the performance of our proposal, and the results suggest the efficiency of our proposal with respect to different metrics.

A new directional query method for polygon dataset in spatial database

Directional query is critical in many domains, especially in geographic information systems (GIS), which are also frequently used as selection conditions in spatial queries. In this paper, we explore the processing of directional queries based on MRD-tree and open shape model. The interior node of MRD-tree contains not only index entries but also data entries, where the data entry contains the pointer of actual spatial object in disk, the maximum boundary rectangle (MBR) and the maximum enclosed circle (MEC) of the spatial data object. And open shape model converts the processing of the direction predicates into the processing of topological operations between open shapes and closed geometry objects. So a number of spatial objects were excluded immensely to compute during the filter step, and it is unnecessary to know the boundary of the embedding world and also eliminates the computation related to the world boundary. Our experimental evaluation shows that directional query based on MRD-tree Index and open shape model consistently outperforms classical directional queries based on R-tree and range model while the degree of performance improvement varies by several parameters.

Soft-Split Random Forest for Anatomy Labeling.

Random Forest (RF) has been widely used in the learning-based labeling. In RF, each sample is directed from the root to each leaf based on the decisions made in the interior nodes, also called splitting nodes. The splitting nodes assign a testing sample to either left or right child based on the learned splitting function. The final prediction is determined as the average of label probability distributions stored in all arrived leaf nodes. For ambiguous testing samples, which often lie near the splitting boundaries, the conventional splitting function, also referred to as hard split function, tends to make wrong assignments, hence leading to wrong predictions. To overcome this limitation, we propose a novel soft-split random forest (SSRF) framework to improve the reliability of node splitting and finally the accuracy of classification. Specifically, a soft split function is employed to assign a testing sample into both left and right child nodes with their certain probabilities, which can effectively reduce influence of the wrong node assignment on the prediction accuracy. As a result, each testing sample can arrive at multiple leaf nodes, and their respective results can be fused to obtain the final prediction according to the weights accumulated along the path from the root node to each leaf node. Besides, considering the importance of context information, we also adopt a Haar-features based context model to iteratively refine the classification map. We have comprehensively evaluated our method on two public datasets, respectively, for labeling hippocampus in MR images and also labeling three organs in Head & Neck CT images. Compared with the hard-split RF (HSRF), our method achieved a notable improvement in labeling accuracy.

A probabilistic analysis of border node-based MFR routing protocol for vehicular ad hoc networks

In this paper, we examined the significance of position-based routing with border-node-based forwarding for vehicular ad hoc network VANET to optimise path length between vehicles. We have mathematically evaluated the performance of border node-based forwarding method such as Border node-based MFR B-MFR. B-MFR strongly avoids the interior nodes within the transmission range for further transmitting the packets. Furthermore, we have developed a mathematical model for B-MFR to determine average number of hop counts, expected distance between source and destination, and maximum progress of the next-hop node towards the destination node using Poisson distribution for node deployment. Analytical and simulation results indicate that B-MFR can provide efficient routing in VANETs. In our work, results clearly show that using the border-nodes is an advantage to maximise the performance of routing protocol in terms of average number of hops with minimum delay. We have mathematically simulated the proposed protocol in MATLAB and compared the result with existing routing protocol MFR.

An Investigation of Starting Point Preferences in Human Performance on Traveling Salesman Problems

Previous studies have shown that people start traveling sales problem tours significantly more often from boundary than from interior nodes. There are a number of possible reasons for such a tendency: first, it may arise as a direct result of the processes involved in tour construction; second, boundary points may be perceptually more salient than interior points, and selected for that reason; third, starting from the boundary may make the task easier or be more likely to result in a better tour than starting from the interior. The present research investigated each of these possibilities by analyzing start point frequencies in previously unpublished data and by conducting an experiment. The analysis of start points provided some slight but contradictory support for the hypothesis that start selections result from the process of tour construction, but no evidence for the perceptual salience explanation. The experiment required participants to start tours either from a boundary or from an interior point, to test whether there was an effect on the quality of tour construction. No evidence was found that starting point affected either the length of tours or the time required to produce them. However, there was some indication that starting from a central location may be more likely to result in crossed arcs.

Network Codes Resilient to Jamming and Eavesdropping

We consider the problem of communicating information over a network secretly and reliably in the presence of a hidden adversary who can eavesdrop and inject malicious errors. We provide polynomial-time distributed network codes that are information-theoretically rate-optimal for this scenario, improving on the rates achievable in prior work by Ngai Our main contribution shows that as long as the sum of the number of links the adversary can jam (denoted by ZO) and the number of links he can eavesdrop on (denoted by ZI) is less than the network capacity (denoted by C) (i.e., ), our codes can communicate (with vanishingly small error probability) a single bit correctly and without leaking any information to the adversary. We then use this scheme as a module to design codes that allow communication at the source rate of C- ZO when there are no security requirements, and codes that allow communication at the source rate of C- ZO- ZI while keeping the communicated message provably secret from the adversary. Interior nodes are oblivious to the presence of adversaries and perform random linear network coding; only the source and destination need to be tweaked. We also prove that the rate-region obtained is information-theoretically optimal. In proving our results, we correct an error in prior work by a subset of the authors in this paper.

Meshless local Petrov–Galerkin (MLPG) method for three-dimensional nonlinear wave equations via moving least squares approximation

This paper proposes an approach based on the Galerkin weak form and moving least squares (MLS) approximation to simulate three space dimensional nonlinear wave equation of the form utt+αut+βu=uxx+uyy+uzz+δg(u)ut+f(x,y,z,t),0<x,y,z<1,t>0 subject to given appropriate initial and Dirichlet boundary conditions. The main difficulty of methods in fully three-dimensional problems is the large computational costs. In the proposed method, which is a kind of Meshless local Petrov–Galerkin (MLPG) method, meshless Galerkin weak form is applied to the interior nodes while the meshless collocation method is used for the nodes on the boundary, so the Dirichlet boundary condition is imposed directly. In MLPG method, it does not require any background integration cells so that all integrations are carried out locally over small quadrature domains of regular shapes, such as circles or squares in two dimensions and spheres or cubes in three dimensions. The moving least squares approximation is proposed to construct shape functions. A two-step time discretization method is employed to approximate the time derivatives. To treat the nonlinearity, a kind of predictor–corrector scheme combined with one-step time discretization and Crank–Nicolson technique is adopted. Several numerical examples are presented and satisfactory agreements are achieved.