Numerical modeling of mechanical wave propagation
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Numerical modeling of mechanical wave propagation G. Seriani1 · S. P. Oliveira2 Received: 22 June 2020 / Accepted: 12 August 2020 / Published online: 12 October 2020 © The Author(s) 2020
Abstract The numerical modeling of mechanical waves is currently a fundamental tool for the study and investigation of their propagation in media with heterogeneous physical properties and/or complex geometry, as, in these cases, analytical methods are usually not applicable. These techniques are used in geophysics (geophysical interpretation, subsoil imaging, development of new methods of exploration), seismology (study of earthquakes, regional and global seismology, accurate calculation of synthetic seismograms), in the development of new methods for ultrasonic diagnostics in materials science (non-destructive methods) and medicine (acoustic tomography). In this paper we present a review of numerical methods that have been developed and are currently used. In particular we review the key concepts and pioneering ideas behind finite-difference methods, pseudospectral methods, finite-volume methods, Galerkin continuous and discontinuous finite-element methods (classical or based on spectral interpolation), and still others such as physics-compatible, and multiscale methods. We focus on their formulations in time domain along with the main temporal discretization schemes. We present the theory and implementation for some of these methods. Moreover, their computational characteristics are evaluated in order to aid the choice of the method for each practical situation.
Contents 1 Introduction . . . . . . . 2 Governing equations . . . 2.1 Scalar wave equation 2.2 Elastic wave equation 2.2.1 Viscoelasticity 2.2.2 Poroelasticity .
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G. Seriani [email protected]
1
National Institute of Oceanography and Applied Geophysics-OGS, Borgo Grotta Gigante, 42/c, 34010 Sgonico, TS, Italy
2
Department of Mathematics and Graduate Program in Geology, Federal University of Paraná, Caixa Postal 19096, Curitiba, PR 81531-980, Brazil
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460 2.3 Velocity-stress formulation . . . . . . . . . . . 2.4 Boundary conditions . . . . . . . . . . . . . . 2.5 Variational formulation . . . . . . . . . . . . . 3 Model discretization . . . . . . . . . . . . . . . . . 3.1 Spatial discretization . . . . . . . . . . . . . . 3.2 Temporal discretization . . . . . . . . . . . . . 4 Temporal discretization methods . . . . . . . . . . 4.1 Newmark methods . . . . . . . . . . . . . . . 4.2 Lax–Wendroff methods . . . . . . . . . . . . . 4.3 Runge–Kutta and sympl
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