Numerical Simulation of 2D Sediment Transport Equations via Element Free Galerkin
Author
Abstract
In this research, the element free Galerkin is implemented to simulate the bed-load sediment transport equations in two dimensions. In this method, which is a meshless method, the computational domain is discretized by a set of arbitrarily scattered nodes and there is no need to use meshes, elements or any other connectivity information in nodes. The hydrodynamical part of sediment transport equations is modeled using 2D shallow water equations; and the Exner equation describes the sediment continuity. Eventually, to appraise the ability of considered method, several benchmark examples are solved and then, the obtained results are compared with previously published works
Keywords
2. Noroozi, S., Kolahdoozan, M., and Zarrati, A. R., “Study on Sediment Exchange Between Bed and Flow with a 2D Shallow Water Model”, 7th International River Engineering Conference, Shahid Chamran University, Ahwaz, 2007.
3. Tassi, P. A., Rhebergen, S., Vionnet, C. A., and Bokhove, O., “A Discontinuous Galerkin Finite Element Model for River Bed Evolution under Shallow Flows”, Computer Methods in Applied Mechanics and Engineering, Vol. 197, pp. 2930-2947, 2008.
4. Izem, N., Seaid, M., and Wakrim, M., “A High-Order Nodal Discontinuous Galerkin Method for 1D Morphodynamic Modelling”, International Journal of Computer Applications, Vol. 41, pp. 19-27, 2012.
5. Hudson, J., and Sweby, P. K., “Formulations for Numerically Approximating Hyperbolic Systems Governing Sediment Transport”, Journal of Scientific Computing, Vol. 19, pp. 225-252, 2003.
6. Hudson, J., “Numerical Techniques for Morphodynamic Modelling”, Ph.D. Thesis, University of Reading, Department of Mathematics, 2001.
7. Črnjarić-Žic, N., Vuković, S., and Sopta, L., “Extension of ENO and WENO Schemes to One-DImensional Sediment Transport Equations”, Computers & Fluids, Vol. 31, pp. 31-56, 2004.
8. Caleffi, V., Valiani, A., and Bernini, A., “High-Order Balanced CWENO Scheme for Movable Bed Shallow Water Equations”, Advances in Water Resources, Vol. 30, pp. 730-741, 2007.
9. Sportiello, V., Implicit Simulations for Morphodynamic Shallow Water Flows, in Department of Civil and Industrial Engineering, University of Pisa, 2012.
10. Delis, A. I. and Papoglou, I., “Relaxation Approximation to Bed-Load Sediment Transport”, Journal of Computational and Applied Mathematics, Vol. 231, pp. 521-546, 2008.
11. Benkhaldoun, F., and Seaïd, M., “Combined Characteristics and Finite Volume Methods for Sediment Transport and Bed Morphology in Surface Water Flows”, Mathematics and Computers in Simulation, Vol. 81, pp. 2073-2086, 2011.
12. Benkhaldoun, F., Seaïd, M., and Sahmim, S., “Mathematical Development and Verification of a Finite Volume Model for Morphodynamic Flow Applications”, Advances in Applied Mathematics and Mechanics, Vol 3, pp. 470-492, 2011.
13. Benkhaldoun, F., Sahmim, S., and Seaïd, M., “A Two-Dimensional Finite Volume Morphodynamic Model on Unstructured Triangular Grids,” International Journal for Numerical Methods in Fluids, Vol. 63, pp. 1296-1327, 2009.
14. Bilanceri, M., Beux, F., Elmahi, I., Guillard, H., and Salvetti, M. V., “Implicit Time Advancing Combined with Two Finite-Volume Methods in the Simulation of Morphodynamic Flows”, Mathematics and Computers in Simulation, Vol. 99, pp. 153-169, 2014.
15. Bilanceri, M., Beux, F., Elmahi, I., Guillard, H., and Salvettia, M. V., “Linearized Implicit Time Advancing and Defect Correction Applied to Sediment Transport Simulations”, Computers & Fluids, Vol. 63, pp. 82-104, 2012.
16. Castro Díaz, M. J., Fernández-Nieto, E. D., Ferreiro, A. M., and Parés, C., “Two-Dimensional Sediment Transport Models in Shallow Water Equations. A Second order Finite Volume Approach on Unstructured Meshes”, Computer Methods in Applied Mechanics and Engineering, Vol. 198, pp. 2520-2538, 2009.
17. Idelsohn, S. R., Oñate, E., Calvo, E., and Del Pin, F. “Meshless Finite Element Ideas”, Proceeding in Fifth World Congress on Computational Mechanics, Vienna, Austria, 2002.
18. Belytschko, T., Lu, Y. Y. and Gu, L., “Element Free Galerkin Methods”, International Journal for Numerical Methods in Engineering, Vol. 37, pp. 229-256, 1994.
19. Grass, A. J., Sediment Transport by Wave and Currents, SERC London Cent. Mar. Technol, Report No, FL29, 1981.
20. Liu, G. R., Mesh-Free Methods: Moving beyond the Finite Element Methods, CRC Press, 2003.
21. Hudson, J., and Sweby, P. K, “A High‐Resolution Scheme for the Equations Governing 2D Bed‐Load Sediment Transport”, International Journal for Numerical Methods in Fluids, Vol. 47, pp. 1085-1091, 2005.
22. Belytschko, T., Organ, D., and Krongauz, Y., “A Coupled Finite Element-Element-Free Galerkin Method”, Computational Mechanics, Vol. 17, pp.186-195, 1995.
3. Tassi, P. A., Rhebergen, S., Vionnet, C. A., and Bokhove, O., “A Discontinuous Galerkin Finite Element Model for River Bed Evolution under Shallow Flows”, Computer Methods in Applied Mechanics and Engineering, Vol. 197, pp. 2930-2947, 2008.
4. Izem, N., Seaid, M., and Wakrim, M., “A High-Order Nodal Discontinuous Galerkin Method for 1D Morphodynamic Modelling”, International Journal of Computer Applications, Vol. 41, pp. 19-27, 2012.
5. Hudson, J., and Sweby, P. K., “Formulations for Numerically Approximating Hyperbolic Systems Governing Sediment Transport”, Journal of Scientific Computing, Vol. 19, pp. 225-252, 2003.
6. Hudson, J., “Numerical Techniques for Morphodynamic Modelling”, Ph.D. Thesis, University of Reading, Department of Mathematics, 2001.
7. Črnjarić-Žic, N., Vuković, S., and Sopta, L., “Extension of ENO and WENO Schemes to One-DImensional Sediment Transport Equations”, Computers & Fluids, Vol. 31, pp. 31-56, 2004.
8. Caleffi, V., Valiani, A., and Bernini, A., “High-Order Balanced CWENO Scheme for Movable Bed Shallow Water Equations”, Advances in Water Resources, Vol. 30, pp. 730-741, 2007.
9. Sportiello, V., Implicit Simulations for Morphodynamic Shallow Water Flows, in Department of Civil and Industrial Engineering, University of Pisa, 2012.
10. Delis, A. I. and Papoglou, I., “Relaxation Approximation to Bed-Load Sediment Transport”, Journal of Computational and Applied Mathematics, Vol. 231, pp. 521-546, 2008.
11. Benkhaldoun, F., and Seaïd, M., “Combined Characteristics and Finite Volume Methods for Sediment Transport and Bed Morphology in Surface Water Flows”, Mathematics and Computers in Simulation, Vol. 81, pp. 2073-2086, 2011.
12. Benkhaldoun, F., Seaïd, M., and Sahmim, S., “Mathematical Development and Verification of a Finite Volume Model for Morphodynamic Flow Applications”, Advances in Applied Mathematics and Mechanics, Vol 3, pp. 470-492, 2011.
13. Benkhaldoun, F., Sahmim, S., and Seaïd, M., “A Two-Dimensional Finite Volume Morphodynamic Model on Unstructured Triangular Grids,” International Journal for Numerical Methods in Fluids, Vol. 63, pp. 1296-1327, 2009.
14. Bilanceri, M., Beux, F., Elmahi, I., Guillard, H., and Salvetti, M. V., “Implicit Time Advancing Combined with Two Finite-Volume Methods in the Simulation of Morphodynamic Flows”, Mathematics and Computers in Simulation, Vol. 99, pp. 153-169, 2014.
15. Bilanceri, M., Beux, F., Elmahi, I., Guillard, H., and Salvettia, M. V., “Linearized Implicit Time Advancing and Defect Correction Applied to Sediment Transport Simulations”, Computers & Fluids, Vol. 63, pp. 82-104, 2012.
16. Castro Díaz, M. J., Fernández-Nieto, E. D., Ferreiro, A. M., and Parés, C., “Two-Dimensional Sediment Transport Models in Shallow Water Equations. A Second order Finite Volume Approach on Unstructured Meshes”, Computer Methods in Applied Mechanics and Engineering, Vol. 198, pp. 2520-2538, 2009.
17. Idelsohn, S. R., Oñate, E., Calvo, E., and Del Pin, F. “Meshless Finite Element Ideas”, Proceeding in Fifth World Congress on Computational Mechanics, Vienna, Austria, 2002.
18. Belytschko, T., Lu, Y. Y. and Gu, L., “Element Free Galerkin Methods”, International Journal for Numerical Methods in Engineering, Vol. 37, pp. 229-256, 1994.
19. Grass, A. J., Sediment Transport by Wave and Currents, SERC London Cent. Mar. Technol, Report No, FL29, 1981.
20. Liu, G. R., Mesh-Free Methods: Moving beyond the Finite Element Methods, CRC Press, 2003.
21. Hudson, J., and Sweby, P. K, “A High‐Resolution Scheme for the Equations Governing 2D Bed‐Load Sediment Transport”, International Journal for Numerical Methods in Fluids, Vol. 47, pp. 1085-1091, 2005.
22. Belytschko, T., Organ, D., and Krongauz, Y., “A Coupled Finite Element-Element-Free Galerkin Method”, Computational Mechanics, Vol. 17, pp.186-195, 1995.
)