مدل‏سازی شکست سد در دیدگاه لاگرانژی با استفاده از روش بدون‌شبکه محلی پترو گلرگین بر پایه تابع شعاعی

در این مطالعه سعی شده است، معادلات حاکم بر جریان سیال در مساله شکست سد با استفاده از مبانی ریاضی روش بدون‌شبکه، رابطه‏سازی شود. در این راستا، ابتدا به معرفی کامل روش بدون‌شبکه محلی پتروگلرکین بر پایه تابع شعاعی پرداخته می‌شود. همچنین به منظور صحت سنجی، یک مثال عددی که دارای پاسخ دقیق می‌باشد، به کمک این روش حل و با پاسخ های دقیق مقایسه گردیده است. نتایج نشان می‏دهد روش بدون‌شبکه به عنوان یک روش‌ کارآمد و دقیق برای دست‏یابی به پاسخ‌های تقریبی معادله‌های‌ دیفرانسیل در روش‌های بدون‌شبکه‌بندی مورد توجه قرار می‌گیرد. در نهایت مساله‏ی شکست سد، با استفاده از تابع شکل شعاعی که در محیط متلب نوشته شده است، به شکل لاگرانژی و با استفاده از روش تصحیح فشار حل و با نتایج دیگر روش‏ها مقایسه شده‏اند. لازم به ذکر است، معادلات حاکم بر جریان شامل معادلات بقای جرم و بقای ممنتوم می‏باشند. نتایج حاصل نشان‌دهنده‌ی دقت بالای روش بدون‌شبکه محلی پتروگلرکین بر پایه تابع شعاعی در مدلسازی مساله شکست سد و نیز عدم نیاز به هیچگونه شبکه‏ی پیش زمینه و تطابق مناسب با شرایط مرزی است.

Lagrangian Approach in Simulating Dam Break Using Meshless Local Petrov-Galerkin (MLPG) Method by Radial Basis Function

Introduction Due to the increasing need water resources, analysis, design and construction of dams is one of the most widely used fields in engineering sciences. In general, dams with their special characteristics have been able use to another of types of the hydraulic structures. One of the most popular numerical methods proposed for the analysis of hydraulic problems is the meshless methods. In the meshless method, the amplitude and boundaries of the structure are created by nodes. The shape function is used to communicate between nodes. In this study first the meshless local PetrovGalerkin (MLPG) method by Radial Basis Function (RBF) has been explained entirely. In the following, MLPG method is verified by exact solution in a numerical example. The Results show that MLPG method presented high accuracy and capability for solving the governing equation of differential equations problem in meshless methods. Finaly, using RBF (MatLab code was adopted) in the fluid flow in dam breaking problem. Methodology Several numerical methods, such as the finite element (FE) and meshless methods, have been developed in the last few decades for solving governing partial differential equations of engineering problems. Approximation in geometry and imposition of boundary conditions in meshless approaches can be mentioned as the drawbacks of the methods. Furthermore, in some engineering problems such as those which are solved in a Lagrangian framework, geometry and boundaries change and, therefore, discretization of the domain should be modified in case of using the FE method, which is quite costly In the meshless methods, the calculation of the integration is based on the Gaussian integration method in the general and the local forms. In the general method, in order to integrate, it is necessary to create meshes in the background of the problem domain; therefore, this method is not a true meshless method. But meshless methods based on the local integration method, such as the meshless local PetrovGalerkin (MLPG) method, have been proposed. In this way, the governing fluid flow in dam breaking problem is expanded using MLPG method. Radial Basis Functions (RBF) is used to communicate between nodes. In order to discretize the derived equations in time domains, Zienkiewicz and Codina (1995) scheme with suitable time step is used. The Mass and momentum conservation laws are governing equations of flow, which are solved by pressure correction in Lagrangian approach. Then these results are compared with another method results. The results showed high accuracy and good conformity compared to available another solutions and the ability of the proposed method in solution of moving fluid with moving boundaries. Results and Discussion In order to demonstrate the accuracy of the present method for dam breaking, at the first a problem verifying with analytical solution. Table 2 shows the analytical, numerical and error values obtained. The comparison shows the high capacity and accuracy of the present method. After verification, the dam breaking problem is investigated. The geometry of the dam breaking problem is shown in Figure 4 and then the present method compare with isogeometric and the least squares method. By comparing the free surface profile between the three methods, it can be easily understood that the meshless local PetrovGalerkin (MLPG) method based on Radial Basis Functions (RBF) has the high accuracy. On the other hand, the close nature of the meshless local PetrovGalerkin (MLPG) method with the least squares method, it is quite clear that the results are in good agreement. The following results are shown in Figures 7 to 9. the water flow velocity resulting from the present method results with the base function of the radial function in 0.15 seconds in the problem compared to the least squares method (Shobeyri and Afshar 2010) and isogeometric (Amini, Maghsoodi et al. 2016). Finally, the pressure obtained from the MLPG method in 0.15 seconds compared to the least squares method (Shobeyri and Afshar 2010) and isogeometric (Amini, Maghsoodi et al. 2016) in Figures 10 to 12 are showed. Conclusion By considering the dam breaking problem, it was found, the Meshless Local PetrovGalerkin (MLPG) method is useful in modeling problems with variable boundary conditions, because only by producing nodes at each stage of analysis can define a new boundary conditions and then in the shortest possible time modeling is done. It is clear the modeling this problem with the other methods such as finite element method is complex, because by changing the boundary conditions, produced the new elements becomes a timeconsuming and complex matter. The Meshless Local PetrovGalerkin (MLPG) Method is an intelligent design for solving problems of variable geometric conditions.