توسعه روش نیوتنرافسون با جهت‌های متغیر برای تحلیل شبکه‌های تهویه فضاهای زیرزمینی

یکی از روش های طراحی شبکه تهویه فضاهای زیرزمینی، روش رایانه ای است. روش رایانه ای بر پایه روش های ریاضی استوار است. روش هایی همچون روش نیوتنرافسون، روش هاردی کراس و مدل های اصلاح‌شده آن، مسیر بحرانی، تحلیل خطی و برنامه ریزی غیرخطی به‌عنوان برخی از روش های ریاضی محسوب می شوند. روش نیوتن رافسون یکی از روش های حل معادلات غیرخطی در محاسبات عددی به‌حساب می‌آید. این روش بر پایه تعریف مشتق و تصحیح آن استوار است. استفاده از روش نیوتنرافسون در تحلیل شبکه‌های تهویه فضاهای زیرزمینی رایج است ولی بررسی دقیق این روش تاکنون به‌طور کامل انجام نشده است. هدف از ارائه این مقاله، بررسی روش نیوتنرافسون و بهبود این روش در تحلیل شبکه تهویه فضاهای زیرزمینی با استفاده از جهت های متغیر است. تحلیل انجام‌شده بیانگر این مطلب است که استفاده از این روش در شبکه تهویه فضاهای زیرزمینی در بعضی از مدل ها برای رسیدن به جواب نهایی دچار واگرایی می شود و بر این اساس بهبود این روش امری ضروری است.

Extension of NewtonRaphson Method with Variable Directions in Ventilation Network Analysis of Underground Excavations

Summary Computer method is one of the ventilation network design methods in underground excavations. Computer method has been designed based on mathematical approximate methods. Methods such as NewtonRaphson method, Hardy Cross method and its modified versions, critical path, linear analysis, nonlinear programming and optimization techniques are considered as some mathematical approximate methods. NewtonRaphson method is one of the methods of solving nonlinear equations in numerical calculations. This method is designed based on the definition of derivative and its correction. Using NewtonRaphson method for the analysis of ventilation networks in underground excavations has been common. However, so far, perfect investigation of this method has not been carried out. The purpose of this paper is to investigate NewtonRaphson method and its improvement in the analysis of ventilation networks in underground excavations. The analysis indicates that this method in some models is unable to find the final true answer because instead of convergence, it diverges. Thus, the improvement of this method seems to be necessary.   Introduction Ventilation design of underground excavations is done based on preparing underground excavation map, identification of branches and injunctions in ventilation network, calculation of mine resistance for any of branches, calculation of air flow quantity for any of branches, calculation of pressure loss for any of branches, calculation of natural ventilation, network adjustment, selection of regulator doors and selection of main fans. Various methods have been presented such a manual and computer methods for the analysis of ventilation networks of underground excavations. The choice of analysis method depends on the purpose of ventilation network analysis.   Methodology and Approaches One of the methods for solving nonlinear equations in numerical calculations is NewtonRaphson method. This method is designed based on the definition of derivative and its correction. In this method, the error amount of the initial guess is calculated for purpose function, and then, the initial guess is corrected. In this method, based on the equation  and injunction rules, supposition flow quantity for any of branches.is supposed. Based on the ventilation network fans and mathematical approximate equations, the error of any ring in the ventilation network is calculated, and then, supposition flow quantity is corrected. Based on the mathematical equations, correction operations of air flow quantity are repeated until the calculation accuracy becomes equal to or smaller than the calculation error.   Results and Conclusions NewtonRaphson method is one of solution methods in ventilation networks analysis. Convergence to the final solution in this method depends on supposition flow quantity, the direction of supposition flow and the arrangement of ring selection. If the direction of the hypothetical flow in the network is in accordance with the actual flow direction, the fastest method for reaching the final solution in the analysis of underground ventilation networks will be the use of NewtonRaphson method. However, if the direction of the hypothetical flow in the network is not in accordance with the actual flow direction, the validation of this method will indicate that this method in some models becomes divergent. Therefore, improvement of NewtonRaphson method is necessary. The improvement of NewtonRaphson method in this paper is made in two stages.  The first stage is the identification of negative flow intensity, and then, the flow direction is reversed. In the second stage, the correction of the equation  is performed, and then, the calculations are repeated.