The possibility of using a nodal method allowing irregular distribution of
nodes in a natural way is one of the main advantages of the generalized finite
difference method (GFDM) with regard to the classical finite difference method.
Moreover, this feature has made it one of the most-promising meshless
methods because it also allows us to reduce the time-consuming task of mesh
generation and the numerical solution of integrals. This characteristic allows
us to shape geological features easily whilst maintaining accuracy in the results,
which can be a source of great interest when dealing with this kind of
problems. Two widespread geophysical investigation methods in civil engineering
are the cross-hole method and the seismic refraction method. This
paper shows the use of the GFDM to model the aforementioned geophysical
investigation tests showing precision in the obtained results when comparing
them with experimental data.
References
[1]
Benito, J. J., Ureña, F., & Gavete, L. (2001). Influence Several Factors in the Generalized Finite Difference Method. Applied Mathematical Modeling, 25, 1039-1053. https://doi.org/10.1016/S0307-904X(01)00029-4
[2]
Benito, J. J., Ureña, F., & Gavete, L. (2007). Solving Parabolic and Hyperbolic Equations by the Generalized Finite Difference Method. Journal of Computational and Applied Mathematics, 209, 208-233. https://doi.org/10.1016/j.cam.2006.10.090
[3]
Benito, J. J., Ureña, F., Gavete, L., Salete, E., & Muelas, A. (2013). A GFDM with PML for Seismic Wave Equations in Heterogeneous Media. Journal of Computational and Applied Mathematics, 252, 40-51. https://doi.org/10.1016/j.cam.2012.08.007
[4]
Benito, J. J., Ureña, F., Gavete, L., Salete, E., & Ureña, M. (2017). Implementations with Generalized Finite Differences of the Displacements and Velocity-Stress Formulation of Seismic Wave Propagation Problem. Applied Mathematical Modelling, 52, 1-14. https://doi.org/10.1016/j.apm.2017.07.017
[5]
Benito, J. J., Ureña, F., Salete, E., Muelas, A., Gavete, L., & Galindo, R. (2015). Wave Propagation in Soils Problems Using the Generalized Finite Difference Method. Soil Dynamics and Earthquake Engineering, 79, 190-198. https://doi.org/10.1016/j.soildyn.2015.09.012
[6]
Benito, J. J., Ureña, F., Ureña, M., Salete, E., & Gavete, L. (2018). Schemes in Generalized Finite Differences for Seismic Wave Propagation in Kelvin-Voight Viscoelastic Media. Engineering Analysis with Boundary Elements, 95, 25-32. https://doi.org/10.1016/j.enganabound.2018.06.017
[7]
Benito, J. J., Ureña, F., Ureña, M., Salete, E., & Gavete, L. (2018). A New Meshless Approach to Deal with Interfaces in Seismic Problems. Applied Mathematical Modelling, 58, 447-458. https://doi.org/10.1016/j.apm.2018.02.014
[8]
Gavete, L., Ureña, F., Benito, J. J., Garcia, A., Ureña, M., & Salete, E. (2017). Solving SECOND Order Non-Linear Elliptic Partial Differential Equations Using Generalized Finite Difference Method. Journal of Computational and Applied Mathematics, 318, 378-387. https://doi.org/10.1016/j.cam.2016.07.025
[9]
Jensen, P. S. (1972). Finite Difference Technique for Variable Grids. Computer& Structures, 2, 17-29. https://doi.org/10.1016/0045-7949(72)90020-X
[10]
Lancaster, P., & Salkauskas, K. (1986). Curve and Surface Fitting. Cambridge, MA: Academic Press.
[11]
Levin, D. (1998). The Approximation Power of Moving Least Squares. Mathematics of Computation, 67, 1517-1531. https://doi.org/10.1090/S0025-5718-98-00974-0
[12]
Liszka, T., & Orkisz, J. (1980). The Finite Difference Method at Arbitrary Irregular Grids and Its Application in Applied Mechanics. Computer & Structures, 11, 83-95. https://doi.org/10.1016/0045-7949(80)90149-2
[13]
Moczo, P. (1998). Introduction to Modelling Seismic Wave Propagation by Finite Difference Method, Lectures Notes, Kyoto.
[14]
Orkisz, J. (1998). Finite Difference Method. In M. Kleiber (Ed.), Handbook of Computational Solid Mechanics (Part III). Berlin: Spriger-Verlag.
[15]
Perrone, N., & Kao, R. (1975). A General Finite Difference Method for Arbitrary Meshes, Computer& Structures, 5, 45-58.
[16]
Salete, E., Benito, J. J., Ureña, F., Gavete, L., Ureña, M., & Garcia, A. (2017). Stability of Perfectly Matched Layer Regions in Generalized Finite Difference Method for Wave Problems. Journal of Computational and Applied Mathematics, 312, 231-239. https://doi.org/10.1016/j.cam.2016.05.027
[17]
Salete, E., Ureña, F., Gavete, L., & Benito, J. J. (2011). A Note in the Application of the Generalized Finite Difference Method to Seismic Wave Propagation in 2-D. Journal of Computational and Applied Mathematics, 236, 3016-3025.
[18]
Ureña, F., Benito, J. J., & Gavete, L. (2011). Application of the Generalized Finite Difference Method to Solve the Advection-Diffusion Equation. Journal of Computational and Applied Mathematics, 235, 1849-1855. https://doi.org/10.1016/j.cam.2010.05.026
