انجمن هیدرولیک ایران نشریه علمی هیدرولیک 2345-4237 20 3 2025 09 23 Numerical simulation of dam-break problem including channel slope and friction using MQ method حل مسئله شکست سد با استفاده از روش بدون شبکه چند ربعی با وجود اصطکاک و شیب بستر 1 15 218263 10.30482/jhyd.2024.462205.1706 FA سیما شوکت دانشگاه قم علیرضا فلاح دانشگاه قم رضا بابایی دانشگاه قم 0000-0002-3068-3602 احسان جباری دانشگاه قم 0000-0002-6345-8567 Journal Article 2024 06 13 Introduction<br>In this research, the shallow water equations of dam-break flow have been rewritten in the semi-implicit form to create a system of linear equations. The existing methods used for solving the dam break problem have their pros and cons. For example, the finite volume method requires meshing of the entire problem domain; consequently, in large-scale domains such as the dam break problem, due to using large finite volumes, accuracy will be reduced. The finite difference method also requires points on a regular grid to define the geometry of the problem. Furthermore, such methods obtain the solutions in specific computational nodes or elements of the domain while other points require the interpolation procedure, in which, more errors may occur. Besides, most of the numerical methods for modeling discontinuities, which essentially occur in the dam-break problem, have a significant error and computational cost. In this research, the Multiquadric (MQ) method was employed to resolve some of the mentioned weaknesses in solving the problem. The development is carried out in solving systems of unsteady non-linear differential equations by making more convenient the implementation of Radial Basis Functions (RBFs). Since the accuracy of the MQ method drastically depends on its optimal shape parameter, an efficient optimization approach is implemented (Kahid et al., 2020).<br>Methodology<br>The Partial Differential Equations (PDE) of the dam break problem include the continuity and momentum equations in the x and y directions. In this research, the system of nonlinear equations is reproduced to solve the dam break problem due to the presence of slope and friction of the bed in the flow. Also, the forward finite difference method is used to discretize the time derivatives. Besides, a previously experienced optimization scheme (Kahid et al., 2020), is used for finding the optimal shape parameter as the critical factor in improving the accuracy of the method. In addition, in this approach, the initial conditions of the problem are estimated using the MQ function (for finding the optimized shape parameters), and it may be shown that the obtained optimal shape parameters are also the optimal values for the next time steps. Therefore, there is no need for the shape parameter to be optimized for all subsequent time steps. As a result, the computational cost was significantly reduced. Furthermore, to reduce the non-physical or numerical oscillations, viscous terms were added to the governing equations, which are coefficients of the second derivatives in the x and y directions. In this regard, the optimal values of the unknown coefficients were found as -0.25 and -0.25 in the x and y directions, respectively.<br>Results and Discussion<br>MQ-RBF resolves the disadvantages of mesh-based methods, such as the high cost of meshing, the need for interpolation between the grid nodes, etc. In this research, the MQ method has been successfully used to model the dam break problem by considering the friction and slope of the bed. An optimization scheme is used to find the optimal shape parameter as the main critical factor in increasing the accuracy of the MQ method. For instance, six numerical examples are presented to validate the reliability of the proposed approach, for 1D and 2D domains. The examples highlight the capabilities and demonstrate the long-term stability of the proposed approach. In addition, the results of the proposed approach were compared with other numerical and analytical methods (Stoker, 1992), which showed that the solution obtained by the present approach has an acceptable precision compared to the other solutions. Also, this approach may be developed to be used in solving partial and gradual dam break problems.<br>Conclusion<br>In this research, the dam break problem has been solved using the meshless MQ method in the presence of the bed and friction slopes. Besides, using viscous terms the numerical oscillations in the discontinuities were successfully captured. In this regard, the optimal values of the coefficients of the viscous terms were found. The capability and reliability of the numerical model were examined through some 1D and 2D examples. The results showed that the proposed method has acceptable accuracy in simulating the flood flow caused by the dam failure including the friction and slope of the bed, as well as its shock front, with less computational costs compared to other numerical methods. شکست سد پدیده‌ای است که در پی آن سیلاب عظیمی به سمت پایین‌دست، جریان می‌یابد. حل معادلات حاکم بر پدیده آب‌های کم‌عمق به دلیل غیرخطی بودن نیازمند ملاحظات خاص است. در این تحقیق، معادلات برای استفاده از روش حل شبه‌خطی دستگاه معادلات غیرخطی، بازنویسی گردید. روش بدون شبکه پایه شعاعی با مزایایی همچون ایجاد یک تابع پیوسته پاسخ در سراسر میدان محاسباتی، قابلیت بالا در شبیه‌سازی هندسه‌های نامنظم و پیچیده، توانایی شبیه‌سازی ناپیوستگی‌های موجود در پاسخ‌ها، تعمیم آسان به مسائل سه‌بعدی و غیره شناخته می‌شود. این روش برای احتراز از برخی از مشکلات روش‌های باشبکه، شامل تعریف شبکه و نیاز به شبکه منظم از خطوط متقاطع، بعضا نیاز به حل اساسی وابسته به شرایط مسئله، تکینگی و گسسته‌سازی کل میدان و نیز برخورداری از مزایای روش های بدون شبکه مانند پیوستگی تابع پاسخ در میدان و آسانی تغییر در آرایش نقاط توسعه داده شد. به این منظور، دستگاه معادلات دیفرانسیل حاکم، برای حل مسئله شکست سد با وجود شیب و اصطکاک بستر بازتولید گردید. علاوه براین برای گسسته‌سازی جمله‌های زمانی از روش اختلاف ‌محدود پیشرو و برای کاهش نوسانات غیرفیزیکی از جملات مستهلک کننده در معادلات حاکم استفاده گردید که این جملات به صورت ضرایبی از مشتقات مرتبه دو در جهات x و y در معادلات اعمال می‌شوند که مقادیر بهینه این ضرائب در جهت هر دو محور x و y برابر با 25/0- بدست آمد. در نهایت با حل چند مثال، کارآمدی و دقت قابل قبول این روش در مقایسه با روش‌های عددی و تحلیلی موجود نشان داده شد. https://jhyd.iha.ir/article_218263_b8bbe602ef97869dfb0c159f42d31998.pdf