Ilia Marchevsky

In recent years, nonlinear random vibration analysis of engineering structures subjected to complex environmental excitations has attracted increasing attention from researchers. However, the inherent coupling of nonlinearity and randomness in multi-degree-of-freedom systems within these complex environments presents a significant challenge. Historically, random excitations have often been simplified to white noise, thereby transforming stochastic dynamical systems into Markov systems whose transition probability density function satisfies the well-known Fokker–Planck–Kolmogorov (FPK) equation and its equivalent Chapman–Kolmogorov–Smoluchowski equation. Building upon this framework, researchers have developed a series of theoretical and analytical methods for nonlinear stochastic systems. Nevertheless, the computational complexity of solving the FPK equation restricts the dimensionality of solvable systems. To overcome this limitation, novel equations characterizing probability density propagation and evolution have been derived on the basis of the principle of probability conservation, namely, the generalized density evolution equation. Also, the probability density integral equation is further developed. Corresponding random vibration analysis methods have also been developed. This paper provides a comprehensive overview of the fundamental principles of these methods and the associated recent advances in nonlinear random vibration analysis.

This paper presents a virtual boundary integral neural network (VBINN) for three-dimensional exterior acoustic problems. The method introduces a virtual boundary within the scatterer or vibrating body and represents the associated source density using a neural network. Based on the acoustic Green’s function, this representation satisfies the Sommerfeld radiation condition and enables direct evaluation of the acoustic pressure and its normal derivative at arbitrary field points. Because the integration surface is separated from the physical boundary, the formulation avoids the singular and near singular kernel evaluations in conventional boundary integral learning methods. To reduce sensitivity to boundary placement, the geometric parameters of the virtual boundary are optimized jointly with the source density during training. Numerical examples for acoustic scattering, multiple body interaction, and underwater acoustic propagation show close agreement with analytical solutions and COMSOL results, and the Burton-Miller extension further improves stability near characteristic frequencies. These results demonstrate the potential of VBINN for three-dimensional exterior acoustic analysis.

Some types of integrals are considered that arise in the numerical solution of BIEs that corresponds to the Laplace equation. The solution of the original boundary value problem is sought in the form of a single- or double-layer potential, and the solution of the BIE itself represents the density of the potential, often assumed to be piecewise constant. In vortex particle methods of CFD, the solution of the BIE is often sought as the piecewise-constant intensity of a vortex sheet. If the collocation method or the Galerkin approach is used for the numerical solution of the BIE, then it becomes necessary to compute integrals of the Newtonian potential or its gradient over one or two triangular cells of the surface mesh.Convenient for practical application exact analytical formulae for integrals of the Newtonian potential and its gradient over a triangular mesh cell are presented, and a scheme for singularity extraction for computing double integrals over two neighboring cells sharing a common edge or a common vertex is proposed. Computational formulae for exact evaluation of integrals from the singular parts of the functions are given; all possible cases of mutual arrangement of cell pairs are considered.

The Level-Set Staggered (LS-STAG) immersed boundary method for viscous incompressible flow simulation combines the advantages of immersed boundary methods, the marker and cells method, and level-set method. The general approach to the application of the LS-STAG method for Reynolds-averaged Navier–Stokes, Large Eddy Simulation, and Detached Eddy Simulation simulation is suggested. According to the concept of the LS-STAG method, normal Reynolds stress components are sampled on the base mesh (similar to pressure discretization), while shear ones are sampled at the upper right corners of the base mesh cells. Thus, for shear Reynolds stresses, an additional mesh (xy-mesh) is introduced. In this research, the LS-STAG discretization for convective and diffusive fluxes on the xy-mesh is developed. To validate this approach, the flow past a circular airfoil at the Reynolds number Re=3900 was simulated. Computational results are in good agreement with established results from the literature.

Vortex particle methods of computational hydrodynamics are widely employed by engineers to solve the problems of flow simulation and estimation of unsteady hydrodynamic loads acting on bodies. The main advantage of such methods is a relatively low computational cost, but their applicability is limited to subsonic incompressible single-phase non-heat-conducting flows. If high order discretization is required, the usage of direct algorithms leads to a significant increase in computational complexity and memory demand. To overcome this limitation, approximate fast algorithms of quasilinear computational complexity were developed and implemented for the most time consuming operations, such as the computation of convective velocities and the solution of the boundary integral equation. The general principles of fast algorithms were described. Their modifications for the problems mentioned above were discussed, and their efficiency was evaluated. The results obtained show that the application of fast algorithms enables a computational speedup of up to several hundred times for around a million vortex particles.