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

Document Type : Research Article

Authors

Department of Civil Engineering, Shahrood University of technology, Shahrood, Iran

Abstract

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 Petrov-Galerkin (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 Petrov-Galerkin (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 Petrov-Galerkin (MLPG) method based on Radial Basis Functions (RBF) has the high accuracy. On the other hand, the close nature of the meshless local Petrov-Galerkin (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 Petrov-Galerkin (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 time-consuming and complex matter. The Meshless Local Petrov-Galerkin (MLPG) Method is an intelligent design for solving problems of variable geometric conditions.

Keywords


Akbarimakoui, M., Amini, R. and Mosavi Nezhad, S.M. (2018). Fluid flow modeling using meshless local Petrov-Galerkin (MLPG) method by Radial Basis Function. Journal of Hydraulics, 13(3), 95-106. (in Persian).
Amini, R., Maghsoodi, R. and Moghaddam, N. (2016). Simulating free surface problem using isogeometric analysis. Journal of the Brazilian Society of Mechanical Sciences and Engineering, 38(2), 413-421.
Amini, R., Akbarimakoui, M. and Mosavi Nezhad, S.M. (2018). Fluid flow modeling in channel using meshless local Petrov-Galerkin (MLPG) method by Radial Basis Function. Modares Mechanical Engineering, 18(8), 241-249. (in Persian).
Arami Fadafan, M. and Hessami Kermani, M.-R. (2018). Moving particle semi-implicit method with improved pressures stability properties. Journal of Hydroinformatics, 20(6), 1268-1285.
Atluri, S.N., Kim, H.-G. and Cho, J.Y. (1999). A critical assessment of the truly meshless local Petrov-Galerkin (MLPG), and local boundary integral equation (LBIE) methods. Computational mechanics 24(5), 348-372.
Belytschko, T., Lu, Y.Y. and Gu, L. (1994). Element‐free Galerkin methods. International journal for numerical methods in engineering, 37(2), 229-256.
Eslamlooian, A. and Amiri, S.M. (2020). Evaluation of well-balanced form of Weighted Average Flux scheme for simulation of flow in open channels. Journal of Hydraulics, 15(1), 143-155. (in Persian).
Farzin, S., Hassanzadeh, Y., Alami, M.T. and Fatehi, R. (2014). An Implicit Incompressible SPH Method for Free Surface Flow Problems. Modares Mechanical Engineering, 14(4), 99-110. (in Persian).
Gingold, R.A. and Monaghan, J.J. (1977). Smoothed particle hydrodynamics: theory and application to non-spherical stars. Monthly notices of the royal astronomical society, 181(3), 375-389.
Hardy, R.L. (1990). Theory and applications of the multiquadric-biharmonic method 20 years of discovery 1968–1988. Computers & Mathematics with Applications, 19(8-9), 163-208.
Kahid Basiri, H., Babaee, R., Fallah, A. and Jabbari, E. (2020). Development of multiquadric method for solving dam break problem. Journal of Hydraulics, 14(4), 83-98. (in Persian).
Liu, G.-R. and Gu, Y.-T. (2005). An introduction to meshfree methods and their programming, Springer Science & Business Media.
Liu, G.-R. and Gu, Y. (2001). A point interpolation method for two-dimensional solids. International Journal for Numerical Methods in Engineering, 50(4), 937-951.
Liu, G. and Gu, Y. (2001). A local radial point interpolation method (LRPIM) for free vibration analyses of 2-D solids. Journal of Sound and vibration, 246(1), 29-46.
Liu, W.K., Jun, S. and Zhang, Y.F. (1995). Reproducing kernel particle methods. International journal for numerical methods in fluids 20(8‐9), 1081-1106.
Melenk, J. and Babuska, I. (1997). Approximation with harmonic and generalized harmonic polynomials in the partition of unity method. Computer Assisted Mechanics and Engineering Sciences, 4, 607-632.
Moussavinezhad, S., Shahabian, F. and Hosseini, S. M. (2013a). Two-dimensional elastic wave propagation analysis in finite length FG thick hollow cylinders with 2D nonlinear grading patterns using MLPG method. CMES Comput. Model. Eng. Sci, 91, 177-204.
Moussavinezhad, S., Shahabian, F. and Hosseini, S. M. (2013b). Two-dimensional stress-wave propagation in finite-length FG cylinders with two-directional nonlinear grading patterns using the MLPG method. Journal of Engineering Mechanics,  140(3), 575-592.
Nayroles, B., Touzot, G. and Villon, P. (1992). Generalizing the finite element method: diffuse approximation and diffuse elements. Computational mechanics, 10(5), 307-318.
Nithiarasu, P. (2005). An arbitrary Lagrangian Eulerian (ALE) formulation for free surface flows using the characteristic‐based split (CBS) scheme. International Journal for Numerical Methods in Fluids, 48(12), 1415-1428.
Onate, E., Idelsohn, S., Zienkiewicz, O. and Taylor, R. (1996). A finite point method in computational mechanics. Applications to convective transport and fluid flow. International journal for numerical methods in engineering, 39(22), 3839-3866.
Pahange, H. and Abolbashiri, M. (2016). Simulation, analysis and optimization of airplane wing leading edge structure against bird strike.
Shao, S. and Lo, E.Y.  (2003). Incompressible SPH method for simulating Newtonian and non-Newtonian flows with a free surface. Advances in water resources, 26(7), 787-800.
Shobeyri, G. and Afshar, M. (2010). Simulating free surface problems using discrete least squares meshless method. Computers & Fluids, 39(3), 461-470.
Sukumar, N., Moran, B. and Belytschko, T. (1998). The natural element method in solid mechanics. International journal for numerical methods in engineering, 43(5), 839-887.
Valette, R., Pereira, A., Riber, S., Sardo, L., Larcher, A. and Hachem, E. (2021). Viscoplastic dam-breaks. Journal of Non-Newtonian Fluid Mechanics, 287, 104447.
Wendland, H. (1999). Meshless Galerkin methods using radial basis functions. Mathematics of Computation of the American Mathematical Society, 68(228), 1521-1531.
Xu, T. and Jin, Y.C. (2019). Improvement of a projection-based particle method in free-surface flows by improved Laplacian model and stabilization techniques. Comput. Fluids, 191, 104235.
Yang, S., Yang, W., Qin, S., Li, Q. and Yang, B. (2018). Numerical study on characteristics of dam-break wave. Ocean Eng., 159, 358–371.
Ye, Y., Xu, T. and Zhu, D. (2020). Numerical analysis of dam-break waves propagating over dry and wet beds by the mesh-free method. Ocean Engineering, 217, 107-118.
Zhu, T., Zhang, J.-D. and Atluri, S. (1998). A local boundary integral equation (LBIE) method in computational mechanics, and a meshless discretization approach. Computational mechanics,  21(3), 223-235.
Zienkiewicz, O.C. and Codina, R. (1995). A general algorithm for compressible and incompressible flow—Part I. the split, characteristic based scheme. International Journal for Numerical Methods in Fluids, 20(89), 869-885.
Zounemat-Kermani, M. and Ghiasi-Tarzi, O. (2017). Using natural element mesh-free numerical method in solving shallow water equations. European Journal of Environmental and Civil Engineering, 21(6), 753-767
  • Receive Date: 14 December 2020
  • Revise Date: 19 January 2021
  • Accept Date: 24 January 2021
  • First Publish Date: 24 January 2021