کاربرد روش تبدیل دیفرانسیلی در یافتن سطح آزاد نشت محیط متخلخل یک بعدی

نوع مقاله : مقاله پژوهشی

نویسندگان

• محمدصالح برادران ¹
• احمد آفتابی ثانی ²
• سعید ابریشمی ³

¹ دانشکده مهندسی، دانشگاه آزاد اسلامی مشهد

² دانشکده مهندسی عمران، دانشگاه فردوسی مشهد.

³ دانشکده مهندسی عمران، دانشگاه فردوسی مشهد

10.22067/jfcei.2024.82170.1226

چکیده

تاکنون روش‌های عددی متنوعی مانند تفاضلات محدود، حجم محدود و اجزای محدود برای حل مسئله‌ی سطح آزادِ نشت، در محیط یک بُعدی و دو بُعدی، بکار رفته است. معادله‌ی ریاضی حاکم بر مسئله‌ی نشت در محیط متخلخل، معادله‌ی لاپلاس می‌باشد. روش‌های تفاضلات محدود و اجزای محدود، بر پایه شبکه‌بندی سطح مورد نظر و بدست آوردن مقادیر هد در این سطح و تکرار این روند تا رسیدن به همگرایی موردنظر، مبتنی هستند. در این مقاله، مسأله‌ی سطح آزاد نشت در ستون آب یک‌بعدی، با استفاده از روش تبدیل دیفرانسیلی که روشی است مبتنی بر سری تیلور، با درنظر گرفتن شرایط مرزی موجود، حل شده و سپس، جواب حاصل با حل تحلیلی مسأله که در بخش پیوست مقاله درج شده است، مقایسه می‌گردد. خوشبختانه، نتایج حاصل نشان‌دهنده این است که روش تبدیل دیفرانسیلی پاسخ بسیار دقیق و خوبی ارائه می‌دهد. ناگفته نماند، روشِ تبدیل دیفرانسیلی یک روش نیمه ـ تحلیلی است که تاکنون، برای حل انواع معادلات دیفرانسیل مورد استفاده قرار گرفته است، اما هنوز در حل مسأله‌ی سطح آزاد نشت بکار گرفته نشده است.

کلیدواژه‌ها

• روش تبدیل دیفرانسیلی
• محیط متخلخل
• معادله لاپلاس
• سطح آزاد نشت
• روش فشار گسترده

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عنوان مقاله [English]

Application of Differential Transform Method for Solving Free-Surface Seepage Problem of One-Dimensional Porous Media

نویسندگان [English]

• Mohammad Saleh Baradaran ¹
• Ahmad Aftabi Sani ²
• Saeed abrishami ³

¹ Department of Civil Engineering, Mashhad Branch, Islamic Azad University, Mashhad, Iran

² Department of Civil Engineering, Faculty of Engineering, Ferdowsi University of Mashhad, Mashhad, Iran

³ Department of Civil Engineering, Faculty of Engineering, Ferdowsi University of Mashhad, Mashhad, Iran

چکیده [English]

Several numerical methods, such as finite difference, finite volume, finite element, and so on, have been utilized to solve the problem of the free surface seepage in one-dimensional and two-dimensional domains. The governing equation of the seepage problem in porous media is the well-known Laplace equation. The finite difference and finite element methods are based on meshing the desired surface and obtaining the head values at this level; and repeating this process until convergence is reached. In this paper, a new method is proposed to solve the one-dimensional free-surface seepage problem in one-dimensional water column based on the Taylor-series expansion, entitled Differential Transform Method (DTM). It should be mentioned, up to now, DTM is employed to solve the various form of differential equations, but it has not been used in seepage problems. For some verification purposes, the obtained results are compared with the well-known finite difference method, and the good agreement is observed. Moreover, the analytical solution of the problem is attained by the use of mathematical techniques and it is reported in the paper as an appendix.

کلیدواژه‌ها [English]

• Differential transform method
• porous media
• Laplace equation
• free surface seepage
• extended pressure method

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