به‌کارگیری و مقایسه روش‌های عددی FDM، DQ وRBF-DQ در مدلسازی جریان آب زیرزمینی در سفره های محصور
محورهای موضوعی : مدیریت منابع آبآتنا نقی پور کادر 1 , علی خوش فطرت 2
1 - دانش آموخته گروه مهندسی عمران، دانشگاه شهید اشرفی اصفهانی، اصفهان، ایران
2 - گروه مهندسی عمران، واحد اصفهان (خوراسگان)، دانشگاه آزاد اسلامی، اصفهان، ایران
کلید واژه: روش DQ, روش RBF-DQ, جریان غیرماندگار, جریان ماندگار, معادلات آب‌های زیرزمینی, سفره‌های محصور,
چکیده مقاله :
روشDQ ((Differential Quadrature یکی از روش‌های عددی جدید مرتبه بالا با دقت زیاد می‌باشد که هزینه محاسباتی بسیار پایین از مزایای این روش است اما ایراد این روش، فقدان انعطاف‌پذیری هندسی در دامنه مدلسازی است. در روش RBF-DQ (Radial Basis Function-based Differential Quadrature) علاوه بر بهرهبردن از ویژگیهای روش DQ در تخمین مستقیم مشتق، با بکارگیری توابع پایهی شعاعی، از مزایای روش‌های عددی بدون شبکه نیز می‌توان بهره برد ضمن آنکه می‌توان این روش را در مسائل با دامنه منظم و نامنظم به‌کارگرفت. در اين تحقيق برای اولین بار از این دو روش برای حل معادلات دیفرانسیل حاکم بر جریان آب‌های زیرزمینی در سفره‌های تحت‌فشار در دو حالت دائمی و غیردائمی استفاده شده و کارآیی آنها در حل این معادلات از طریق مقایسه با حل دقیق به دست آمده از روش‌های تیم و تیس با روش تفاضل محدود که یک روش سنتی می‌باشد، مقایسه شده است. نتایج این تحقیق حاکی از دقت بالای روش‌های DQ و RBF-DQ در مدلسازی عددی جریان آب زیرزمینی در سفره‌های محصور است و روش DQ از نظر دقت و زمان محاسبات بر روش RBF-DQ برتری دارد.
The Differential Quadrature (DQ) method is a high-order numerical approach known for its remarkable accuracy and low computational cost, making it an attractive option for numerical modeling. However, a notable limitation of this method is its lack of geometric flexibility in modeling domains. The Radial Basis Function-based Differential Quadrature (RBF-DQ) method addresses this limitation by combining the DQ method's direct derivative estimation with the flexibility of mesh-free numerical techniques, making it suitable for both regular and irregular domains. This study compares the performance of the DQ, RBF-DQ, and Finite Difference (FD) methods — an established numerical technique in solving groundwater flow equations in confined aquifers for both steady-state and unsteady-state conditions. Exact solutions for these problems are derived using the Thiem and Theis methods. The results demonstrate the high accuracy of both the DQ and RBF-DQ methods in modeling groundwater flow in confined aquifers. Additionally, the DQ method outperforms the RBF-DQ method in terms of both accuracy and computational efficiency.
Abdollahian, M., GhorbanpourArani, A., MosallaieBarzoki, A A., Kolahchi, R., & Loghman, A. )2014(. Non-local wave propagation in embedded armchair TWBNNTs conveying viscous fluid using DQM. Physica B, 1-15. https://doi.org/10.1016/j.physb.2013.02.037. (In Persian)
Behroozi, A M., & Vaghefi, M. (2022). Thin plates spline based differential quadrature for numerical solution of groundwater flow. Engineering Computation, 3(6). https://doi.org/10.1108/EC-06-2021-0331 2194-2208. (In Persian)
Bellman, R., & Casti, J. (1971). Differential quadrature and long-term integration. Journal of Mathematical Analysis Applications, 34, 235-238. https://doi.org/10.1016/0022-247X (71)90110-7
Boujoudar, M., Beljadid, A., & Taik, A. (2024). LRBF meshless methods for predicting soil moisture distribution in root zone. Preprint submitted to Elsevier, 1-27. https://doi.org/10.13140/RG.2.2.33221.87523
Chaabelasri, E., Jeyar, M., & Borthwick, A G L. (2019). Explicit radial basis function collocation method for computing shallow water flows. Procedia Computer Science, 148, 361-370. https://doi.org/10.1016/j.procs.2019.01.044
Dehghan, M., & Mohammadi, V. (2015). The numerical solution of Cahn–Hilliard (CH) equation in one, two and three dimensions via globally radial basis functions (GRBFs) and RBFs-differential quadrature (RBFs-DQ) methods. Engineering Analysis with Boundary Elements, 51, 74-100. https://doi.org/10.1016/j.enganabound.2014.10.008. (In Persian)
Eldho, T I., & Boddula S. (2016). Simulation-optimization models for the remediation of groundwater contamination. American Society of Civil Engineers, 381-391. https://doi.org/10.1061/9780784480168.038
Ghosh, A., & Chakraborty, R. (2011). Finite difference method for computation of 1d pollutant migration through saturated homogeneous soil media. International Journal of Geomechanics, 10, 12-22. https://doi.org/10.1061/(ASCE)GM. 1943-5622.0000068
Hashemi, M R., & Hatam, F. (2011). Unsteady seepage analysis using local radial basis function-based differential quadrature method. Applied Mathematical Modeling, 35, 4934-4950. https://doi.org/10.1016/j.apm.2011.04.002. (In Persian)
Hatami, M., & Ganji, D D. (2014). Motion of a spherical particle in a fluid forced vortex by DQM and DTM. Particuology, 16, 206-212. https://doi.org/10.1016/j.partic.2014.01.001. (In Persian)
Hardy, RL. (1971). Multiquadric equations of topography and other irregular surfaces. Journal Geophysical Research, 76, 1905-1915. https://doi.org/10.1029/JB076i008p01905
Hung, C S., Lee, C F., & Cheng, A H D. (2007). Error estimate, optimal shape factor, and high precision computation of multiquadric collocation method. Engineering Analysis with Boundary Elements, 31, 614-623. https://doi.org/10.1016/j.enganabound.2006.11.011
Kaya, B. (2010). Investigation of gradually varied flows using differential quadrature method. Scientific Research and Essays, 13, 2630-2638. https://doi.org/10.1061/(ASCE)HE.1943-5584.0000509
Khoshfetrat, A., & Abedini, M J. (2011). A hybrid DQ/LMQRBF-DQ approach for numerical solution of Poisson-type and Burger’s equations in irregular domain. Applied Mathematical Modelling, 36, 1885-1901. https://doi.org/10.1016/j.apm.2011.07.079
Meenal, M., & Eldho, T I. (2012). Simulation–optimization model for groundwater contamination remediation using meshfree point collocation method and particle swarm optimization. Sadhana, 37, 351-369. https://doi.org/10.1007/s12046-012-0086-0
Rahman, S., & Bhuiyan, M. (2012). Simulation of subsurface water flow by galerkin finite element method in dhaka city aquifer. Journal of Hydrolic American Society of Civil Engineers, 1-10. https://doi.org/10.1061/9780784410363
Raj, S., & Pradhan, V H. (2013). Numerical simulation of one - dimensional solute transport equation in an adsorbing medium by using differential quadrature method. International Journal of Mathematics and
Computer Applications Research (IJMCAR), 3 (3), 23-36. https://doi.org/10.48550/arXiv.1510.08011
Safavi, H R. (2006). Engineering hydrology. Arkan Publications, Isfahan. (In Persian)
Shen, Q. (2010). Local RBF-based differential quadrature collocation method for the boundary layer problems. Engineering Analysis with Boundary Elements, 34, 213-228. https://doi.org/10.1016/j.enganabound.2009.10.004
Shu, C., & Wu, YL. (2002). Development of RBF-DQ method for derivative approximation and its application to simulate natural convection in concentric annuli. Springer-Verlag, 8, 477-485. https://doi.org/10.1007/s00466-002-0357-4
Soleimani, S., Qajarjazi, A., Bararnia, H., Barari, A., & Domairry, G., (2011). Entropy generation due to natural convection in a partially heated cavity by local RBF-DQ method. Meccanica, 46, 1023-1033. https://doi.org/10.1007/s11012-010-9358-0
Xie, Y., Wu, J., & Xie, C. (2015). Cubic-spline multiscale finite element method for solving nodal darcian velocities in porous media, Journal of Hydraulic, 20 (11), 1-10. https://doi.org/10.1061/(ASCE)HE.1943-5584.0001222
Zhang, J., Ross, M., Fu, C., & Trout, K. (2014). Certification tests of MODFLOW implementation in the integrated hydrologic model. Journal Hydraulic, 19(3), 643-648. https://doi.org/10.1061/(ASCE)HE.1943-5584.0000822
Zhang, J., Bai, SH., Ma, ZH., An, D., Jiang, Y., Jiang, L., Xi, B., Yang, Y., & Li, M. (2013). Analysis for remedial alternatives of unregulated municipal solid waste landfills leachate-contaminated groundwater. Higher Education Press and Springer, 1-10. https://doi.org/10.1007/s11707-013-0374-y
Zhou, F., Xu, J., & Wang, X. (2017). Finite layer formulations for land subsidence due to groundwater withdrawal. Journal of Performance of Constructed Facilities, 17 (11). https://doi.org/10.1061/(ASCE)GM.1943-5622.0000996