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Connected dominating set (CDS) problem is the most widely used method for backbone formation ‎in wireless sensor networks. To date, numerous algorithms have been proposed for backbone ‎construction on minimum CDS (MCDS) problem in unit disk graphs (UDG); however, only a few ‎algorithms have been proposed on MCDS problem in disk graphs with bidirectional links (DGB) ‎and on degree-constrained minimum-weight CDS (DC-MWCDS) problem in UDG. To the best of ‎our knowledge, no work has been done on DC-MWCDS problem in DGB. In this paper, we present ‎OEDC-MWCDS problem (optimal energy and degree constrained minimum-weight connected ‎dominating set) for constructing energy efficient backbone in wireless sensor networks. Then, we ‎model a wireless sensor network as a disk graph with bidirectional links and propose a backbone ‎construction algorithm called EBC-PSO (Energy efficient Backbone Construction utilizing Particle ‎Swarm Optimization algorithm) to obtain a CDS with the minimum weight subject to the optimal ‎energy and degree constraint. The main objective of the proposed algorithm is to find the optimal ‎values of energy and degree of constraint to maximize network lifetime. In the proposed algorithm, ‎optimal coefficients of minimum remaining energy and maximum degree of nodes are determined ‎utilizing PSO algorithm. Then, in the selection of DS nodes, these coefficients are used. Simulation ‎results verify the performance of the proposed algorithm in terms of network lifetime and ‎backbone size.‎

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The aim of this paper is to analyze the strain components in the direct extrusion process to predict the curvature of the exit product. For this purpose, Riemann mapping theory is used to model the deformation zone and create a one-to-one correspondence between the input and output cross sections of the die. With the help of the Bezier curves, flow lines are created between these points and then an upper bound solution is obtained for the velocity field. The process pressure and the distribution of the strain components are determined for the square, hexagonal, and rectangular sections using the obtained velocity field. A theoretical method based on the elastic-plastic bending of beams is presented for calculating the curvature of the exit product for the eccentric dies. In this theoretical method, the distribution of stress components and the bending moments is calculated using the specified strain components. In fact, the amount of bending moments indicates the curvature of the exit product. Finally, the presented theoretical model is validated through comparison with the results of the finite element simulation and the previous studies. The results show that Riemann conformal mapping theory and upper bound method can be used to determine the distribution of strain components and predict the curvature of the output product, in addition to estimate the process pressure.