Modeling 1-D elastic P-waves in a fractured rock with hyperbolic jump conditions

The propagation of elastic waves in a fractured rock is investigated, both theoretically and numerically. Outside the fractures, the propagation of compressional waves is described in the simple framework of one-dimensional linear elastodynamics. The focus here is on the interactions between the waves and fractures: for this purpose, the mechanical behavior of the fractures is modeled using nonlinear jump conditions deduced from the Bandis-Barton model classicaly used in geomechanics. Well-posedness of the initial-boundary value problem thus obtained is proved. Numerical modeling is performed by coupling a time-domain finite-difference scheme with an interface method accounting for the jump conditions. The numerical experiments show the effects of contact nonlinearities. The harmonics generated may provide a non-destructive means of evaluating the mechanical properties of fractures.Comment: accepted and to be published in the Journal of Computational and Applied Mathematic

Wave propagation across acoustic / Biot's media: a finite-difference method

Numerical methods are developed to simulate the wave propagation in heterogeneous 2D fluid / poroelastic media. Wave propagation is described by the usual acoustics equations (in the fluid medium) and by the low-frequency Biot's equations (in the porous medium). Interface conditions are introduced to model various hydraulic contacts between the two media: open pores, sealed pores, and imperfect pores. Well-possedness of the initial-boundary value problem is proven. Cartesian grid numerical methods previously developed in porous heterogeneous media are adapted to the present context: a fourth-order ADER scheme with Strang splitting for time-marching; a space-time mesh-refinement to capture the slow compressional wave predicted by Biot's theory; and an immersed interface method to discretize the interface conditions and to introduce a subcell resolution. Numerical experiments and comparisons with exact solutions are proposed for the three types of interface conditions, demonstrating the accuracy of the approach.Comment: Communications in Computational Physics (2012) XX

Free and smooth boundaries in 2-D finite-difference schemes for transient elastic waves

A method is proposed for accurately describing arbitrary-shaped free boundaries in single-grid finite-difference schemes for elastodynamics, in a time-domain velocity-stress framework. The basic idea is as follows: fictitious values of the solution are built in vacuum, and injected into the numerical integration scheme near boundaries. The most original feature of this method is the way in which these fictitious values are calculated. They are based on boundary conditions and compatibility conditions satisfied by the successive spatial derivatives of the solution, up to a given order that depends on the spatial accuracy of the integration scheme adopted. Since the work is mostly done during the preprocessing step, the extra computational cost is negligible. Stress-free conditions can be designed at any arbitrary order without any numerical instability, as numerically checked. Using 10 grid nodes per minimal S-wavelength with a propagation distance of 50 wavelengths yields highly accurate results. With 5 grid nodes per minimal S-wavelength, the solution is less accurate but still acceptable. A subcell resolution of the boundary inside the Cartesian meshing is obtained, and the spurious diffractions induced by staircase descriptions of boundaries are avoided. Contrary to what occurs with the vacuum method, the quality of the numerical solution obtained with this method is almost independent of the angle between the free boundary and the Cartesian meshing.Comment: accepted and to be published in Geophys. J. In

Modeling Of Dynamic Behavior In Closed Crack And Nonlinear Ultrasonic Array Imaging

Ultrasonic testing (UT) utilizes the traveling time and amplitude of a scattered wave from cracks in a material. A distinct scattered wave can be obtained from a crack with opening faces. It is difficult, by contrast, to detect signals from closed cracks such as stress corrosion and fatigue cracks using the conventional UT. Since the crack faces are in contact due to a residual stress, most of the incident wave penetrates the crack faces and a little scattered wave will be generated. A nonlinear ultrasonic method based on contact acoustic nonlinearity (CAN) which utilizes the dynamic behaviors of the contact and separation states of the crack faces is a promising method. The clapping motion of the crack faces generates harmonics in the frequency spectrum. However, the generation of the harmonics from the crack faces is so sensitive that the voltage, angle, cycle, and frequency of the incident wave should be set in a well-chosen method.In this thesis, a modeling of the generation of the harmonics wave from the closed crack was performed to enhance the reliability of the nonlinear ultrasonic method. Here, an elastodynamic finite integration technique (EFIT) was introduced to simulate a transient motion of the scattered wave from the closed crack. The EFIT adopted a set of split computational nodes at the interface of the closed crack to show the contact and separation depending on the stress and opening displacement of the interface. The numerical results for one-dimensional wave field showed good agreement with the analytical solutions. The simulation results revealed that a closing velocity of the interface was determined by the compressive pressure of the material and was validated by the experimental measurement with polymethylmethacrylate (PMMA) specimens. The appropriate conditions to obtain the nonlinear ultrasonic wave in the case of ultrasonic array testing were determined by performing two-dimensional simulations.An imaging method of the closed crack using an array transducer was investigated using the EFIT simulation. The full waveforms sampling and processing (FSAP) was applied as the imaging technique. For the generation of the nonlinear ultrasonic wave from the closed crack, the FSAP was modified to an algorithm which can transmit a strong beam from the array transducer by setting the delay for all elements electronically. The second harmonic component which extracted from the scattered wave using a band-pass filter was used as the input to the FSAP imaging technique. From the results, it was found that the shape and the location of the closed crack can be reconstructed when the amplitude, frequency, cycle, and angle of the incident wave are set at appropriate values