Nikolay I. Khokhlov

Numerical modeling of elastic wave propagation in heterogeneous media is an effective tool for solving a wide range of applied problems. A significant advantage of the proposed approach is the ability to explicitly delineate the boundaries of heterogeneities, which improves the accuracy of the simulation. One of the key applications of the methods under consideration is the exploration and characterization of oil and gas reservoirs using seismic surveys. In this paper, we investigate the application of the grid-characteristic method combined with hierarchically refined meshes for the numerical modeling of elastic media containing fluid-saturated inclusions. The efficiency of using hierarchical meshes was assessed in a series of numerical experiments that measured wave propagation velocity in fractured media and compared the resulting velocities, wavefield patterns, and computation times for different levels of mesh hierarchy. The experimental results showed that this approach significantly reduced computational costs without compromising modeling accuracy. As a demonstration of the technique, a series of simulations was performed to numerically study the dependence of P-wave velocity on the orientation angle of the fractures. The results illustrate the potential of the developed approach for modeling wave processes in heterogeneous geophysical media.

Full-waveform inversion (FWI) is a powerful interpretation method in geophysics for inferring high-resolution subsurface models by minimizing the difference between observed and simulated seismic data. In mineral exploration, FWI has shown particular promise for delineating complex ore bodies in hard-rock environments where conventional reflection seismic methods often fail. However, traditional FWI remains computationally expensive due to the iterative solution of forward and adjoint problems. The integration of deep learning, particularly the U-Net architecture, has recently emerged as a promising approach to address these computational challenges. Originally developed for biomedical image segmentation, U-Net employs a symmetric encoder–decoder structure with skip connections, enabling precise localization and efficient feature extraction from complex data. This paper proposes a modified dual-path architecture, termed DU-Net, specifically designed for the simultaneous detection and extraction of high-contrast velocity anomalies (representing potential ore bodies) and reconstruction of the background velocity model. The key innovation lies in parallel processing branches—one dedicated to anomaly segmentation and another to background reconstruction—combined with a specialized composite loss function, SeismoLoss, that independently supervises each component. This design allows the network to focus on the distinctive features of the anomaly while filtering out background complexity that typically degrades prediction quality in single-path approaches. We provide a detailed description of the DU-Net architecture and evaluate its performance on two synthetic datasets representing different styles of mineralization and host-rock complexity. Experimental results demonstrate that DU-Net achieves superior accuracy in localizing anomalous bodies and reconstructing background geology compared to the standard U-Net baseline, with a substantial reduction in boundary blurring artifacts.

We propose a second-order accurate grid-characteristic numerical scheme for one-dimensional linear hyperbolic systems with piecewise-constant coefficients and discontinuous solutions at material interfaces. The method assumes that all coefficient jumps are aligned with the computational grid nodes. To achieve high-order accuracy without solving auxiliary linear systems or introducing additional degrees of freedom, we construct modified interpolation stencils that account for both solution jumps and kinks in the derivatives across interfaces. The approach relies on locally transforming the solution into a smooth auxiliary function, enabling standard polynomial interpolation while preserving physical interface conditions. The scheme is explicit, computationally efficient, and naturally extends to arbitrary order via wider stencils. Numerical experiments for a one-dimensional Maxwell system with discontinuous impedance and wave speed confirm second- and third-order convergence. The method is particularly suited for geophysical and acoustic applications involving multiple contrasting layers.

Nowadays, the solution of applied problems of seismic exploration and ultrasonic nondestructive testing is accompanied by the use of computer simulation. This poses a challenge for scientists to develop new modifications of numerical methods that can increase calculation accuracy while minimizing the cost of computing resources. In contrast to numerical methods on unstructured computational meshes, the use of Chimera computational meshes (or superimposed or adaptive meshes) makes it possible to describe boundaries and contact boundaries of arbitrary shape, but at the same time spending less RAM and time on calculations. This is especially important for the active use of neural networks in solving inverse problems, since when training samples are generated, the accuracy of modeling, speed of calculations, and amount of RAM spent are crucial. The paper considers and compares various modifications of the grid-characteristic method on Chimera computational meshes. Examples of test calculations are given.

Differential equations describing the behavior of continuous media with creep involve integral type operators, in accordance with Volterra’s linear theory, which is applicable to a wide range of materials with amorphous and heterogeneous structures. In these equations, the kernel of the integral operator is represented as a sum of exponentials or as a weakly singular kernel (Rabotnov function). Obtaining an analytical solution for the equations in question is problematic in some cases, so it is necessary to develop a numerical method and algorithm, taking into account the memory of the considered medium. In this paper, the equations are solved using the grid-characteristic method and dimensional splitting (for multidimensional problems). The approximation and stability of the proposed method are numerically investigated.

The process of fatigue fracturing is very important in aircraft manufacturing. The turbine blades are loaded during each flight with tiny amplitudes, but it still leads to their damage. The elaboration of modern laboratory devices for very high cycling loading allows us to test the materials under high-frequency loading. The computer simulation in this domain can give us the data, that is necessary to construct reliable mechanical models. In this work, the problem of the fatigue fracturing of the plate with the hole is considered. The full-wave simulation of each cycle of the loading was carried out using the grid-characteristic method. The usage of the overlapping grids allows us to describe the complex form of the hole without significant increases in the computational complexity. The multi-modal continuum model of destruction was used to describe the plate damage.

When modeling seismic processes in heterogeneous media, problems associated with the multi scales of the effects under consideration often arise. Thus, the sizes of geological heterogeneities can be units and tens of meters, while the size of the computational domain can reach units and tens of kilometers. When discretizing the computational domain with a grid with a constant step, problems associated with the size of the occupied memory and the calculation time may arise. In this paper, an algorithm is proposed that uses hierarchical computational grids and a grid-characteristic method for carrying out such calculations. Using the example of a response from a fractured layer, a significant reduction in the calculation time is shown when using the proposed approach.

This paper presents a new algorithm for parallelizing the grid-characteristic method in sharedmemory systems. The OpenMP task parallelism mechanism is used for parallelization. A modification of the grid-characteristic method is considered that uses a set of overlapped grids to determine a complex heterogeneous structure of the computational domain. The complexity of parallelizing the algorithm is represented by the presence of many different-sized grids. The proposed algorithm is described and compared with basic parallelization algorithms. Basic algorithms mean separate parallelization within each computational grid using the loop parallelization mechanism. An analysis of the efficiency of the post-doubling and parallel algorithms is performed. The advantage of the proposed algorithm for a number of problems is demonstrated. The results of testing and calculating the propagation of wave disturbances in a fractured layer are presented. Each crack in the example is specified by a separate computational grid, which significantly increases the multi-scale problem and the number of computational grids. Work is underway to transfer the algorithm to the three-dimensional case.