Rotated Staggered Finite-difference Grid Research Articles

Numerical Simulation-Based Damage Identification in Concrete

High costs for the repair of concrete structures can be prevented if damage at an early stage of degradation is detected and precautionary maintenance measures are applied. To this end, we use numerical wave propagation simulations to identify simulated damage in concrete using convolutional neural networks. Damage in concrete subjected to compression is modeled at the mesoscale using the discrete element method. Ultrasonic wave propagation simulation on the damaged concrete specimens is performed using the rotated staggered finite-difference grid method. The simulated ultrasonic signals are used to train a CNN-based classifier capable of classifying three different damage stages (microcrack initiation, microcrack growth and microcrack coalescence leading to macrocracks) with an overall accuracy of 77%. The performance of the classifier is improved by refining the dataset via an analysis of the averaged envelope of the signal. The classifier using the refined dataset has an overall accuracy of 90%.

Ultrasonic prediction of crack density using machine learning: A numerical investigation

Cracks are accounted as the most destructive discontinuity in rock, soil, and concrete. Enhancing our knowledge from their properties such as crack distribution, density, and/or aspect ratio is crucial in geo-systems. The most well-known mechanical parameter for such an evaluation is wave velocity through which one can qualitatively or quantitatively characterize the porous media. In small scales, such information is obtained using the ultrasonic pulse velocity (UPV) technique as a non-destructive test. In large-scale geo-systems, however, it is inverted from seismic data. In this paper, we take advantage of the recent advancements in machine learning (ML) for analyzing wave signals and predict rock properties such as crack density (CD) – the number of cracks per unit volume. To this end, we designed numerical models with different CDs and, using the rotated staggered finite-difference grid (RSG) technique, simulated wave propagation. Two ML networks, namely Convolutional Neural Networks (CNN) and Long Short-Term Memory (LSTM), are then used to predict CD values. Results show that, by selecting an optimum value for wavelength to crack length ratio, the accuracy of predictions of test data can reach R2 > 96% with mean square error (MSE) < 25e-4 (normalized values). Overall, we found that: (i) performance of both CNN and LSTM is highly promising, (ii) accuracy of the transmitted signals is slightly higher than the reflected signals, (iii) accuracy of 2D signals is marginally higher than 1D signals, (iv) accuracy of horizontal and vertical component signals are comparable, (v) accuracy of coda signals is less when the whole signals are used. Our results, thus, reveal that the ML methods can provide rapid solutions and estimations for crack density, without the necessity of further modeling.

Digital material laboratory: Wave propagation effects in open-cell aluminium foams

This paper is concerned with numerical wave propagation effects in highly porous media using digitized images of aluminium foam. Starting point is a virtual material laboratory approach. The aluminium foam microstructure is imaged by 3D X-ray tomography. Effective velocities for the fluid-saturated media are derived by dynamic wave propagation simulations. We apply a displacement-stress rotated staggered finite-difference grid technique to solve the elastodynamic wave equation. The used setup is similar to laboratory ultrasound measurements and computed results are in agreement with our experimental data. Theoretical investigations allow to quantify the influence of the interaction of foam and fluid during wave propagation. Together with simulations using an artificial dense foam we are able to determine the tortuosity of aluminium foam.

Digital rock physics: Effect of fluid viscosity on effective elastic properties

Abstract This paper is concerned with the effect of pore fluid viscosity on effective elastic properties using digitized rocks. We determine a significant velocity dispersion in wave propagation simulations by the variation of the pore fluid viscosity. Several attenuation regimes are considered which may contribute to this observation. Starting point is a virtual rock physics approach. Numerical simulations of effective transport and effective mechanical properties are applied to statistically representative rock samples. The rock microstructure is imaged by 3D X-ray tomography. Permeability values were estimated through Lattice-Boltzmann flow simulations. The dry rock moduli and the tortuosity are derived by dynamic wave propagation simulations. We apply a displacement-stress rotated staggered finite-difference grid technique to solve the elastodynamic wave equation. An accurate approximation of a Newtonian fluid is implemented in this technique by using a generalized Maxwell body. We give a practical description of how to use this approach and discuss the application limits. Additionally, we show the simulated signature of a theoretically predicted slow S-wave.

Numerical Rock Physics: Fluid Effects on Wave Propagation

This paper is concerned with numerical considerations of fluid effects on wave propagation. The focus is on effective elastic properties (i.e. velocities) in different kinds of dry and fluid-saturated fractured media. We apply the so-called rotated staggered finite-difference grid (RSG) technique. Using this modified grid it is possible to simulate the propagation of elastic waves in a 2D or 3D medium containing cracks, pores or free surfaces without explicit boundary conditions and without averaging elastic moduli. Therefore the RSG allows an efficient and precise numerical study of effective velocities in fractured structures. This is also true for structures where theoretically it is only possible to predict upper and lower bounds. We simulate the propagation of plane P- and S-waves through three kinds of randomly cracked 3D media. Each model realization differs in the porosity of the medium and is performed for dry and fluid-saturated pores. The synthetic results are compared with the predictions of the well known Gassmann equation and the Biot velocity relations. Although we have a very low porosity in our models, the numerical calculations showed that the Gassmann equation cannot be applied for isolated pores (thin penny-shaped cracks). For Fontainebleau sandstone we observe with our dynamic finite-difference approach the exact same elastic properties as with a static finite-element approach. For this case the Gassmann equation can be checked successfully. Additionally, we show that so-called open-cell Gaussian random field models are an useful tool to study wave propagation in fluid-saturated fractured media. For all synthetic models considered in this study the high-frequency limit of the Biot velocity relations is very close to the predictions of the Gassmann equation. However, using synthetic rock models saturated with artificial ``heavy' water we can roughly estimate the corresponding tortuosity parameter.

Effective elastic properties of randomly fractured soils: 3D numerical experiments

ABSTRACTThis paper is concerned with numerical tests of several rock physical relationships. The focus is on effective velocities and scattering attenuation in 3D fractured media. We apply the so‐called rotated staggered finite‐difference grid (RSG) technique for numerical experiments. Using this modified grid, it is possible to simulate the propagation of elastic waves in a 3D medium containing cracks, pores or free surfaces without applying explicit boundary conditions and without averaging the elastic moduli. We simulate the propagation of plane waves through a set of randomly cracked 3D media. In these numerical experiments we vary the number and the distribution of cracks. The synthetic results are compared with several (most popular) theories predicting the effective elastic properties of fractured materials. We find that, for randomly distributed and randomly orientated non‐intersecting thin penny‐shaped dry cracks, the numerical simulations of P‐ and S‐wave velocities are in good agreement with the predictions of the self‐consistent approximation. We observe similar results for fluid‐filled cracks. The standard Gassmann equation cannot be applied to our 3D fractured media, although we have very low porosity in our models. This is explained by the absence of a connected porosity. There is only a slight difference in effective velocities between the cases of intersecting and non‐intersecting cracks. This can be clearly demonstrated up to a crack density that is close to the connectivity percolation threshold. For crack densities beyond this threshold, we observe that the differential effective‐medium (DEM) theory gives the best fit with numerical results for intersecting cracks. Additionally, it is shown that the scattering attenuation coefficient (of the mean field) predicted by the classical Hudson approach is in excellent agreement with our numerical results.