The FMM accelerated PIES with the modified binary tree in solving potential problems for the domains with curvilinear boundaries

The paper presents an accelerating of solving potential boundary value problems (BVPs) with curvilinear boundaries by modified parametric integral equations system (PIES). The fast multipole method (FMM) known from the literature was included into modified PIES. To consider complex curvilinear shapes of a boundary, the modification of a binary tree used by the FMM is proposed. The FMM combined with the PIES, called the fast PIES, also allows a significant reduction of random access memory (RAM) utilization. Therefore, it is possible to solve complex engineering problems on a standard personal computer (PC). The proposed algorithm is based on the modified PIES and allows for obtaining accurate solutions of complex BVPs described by the curvilinear boundary at a reasonable time on the PC.

Boundary Conditions for the Simulation of Wave Breaking

In this paper we propose a new numerical model for the simulation of the wave breaking. The three-dimensional equations of motion are expressed in integral contravariant form and are solved on a curvilinear boundary conforming grid that is able to represent the complex geometry of coastal regions. A time-dependent transformation of the vertical coordinate that is a function of the oscillation of the turbulent wave boundary layer is proposed. A new numerical scheme for the simulation of the resulting equations is proposed. New boundary conditions at the free surface and bottom for the equations of motion expressed in contravariant form are proposed. We present an analysis of the importance of the correct positioning, inside the oscillating turbulent boundary layer, of the centre of the calculation grid cell closest to the bottom, in order to correctly simulate the height of the breaking waves.

Filtration in porous rock with initial deposit

The problems of underground fluid mechanics play an important role in the design and preparation for the construction of tunnels and underground structures. To strengthen the insecure soil a grout solution is pumped under pressure in the porous rock. The liquid solution filters in the pores of the rock and strengthens the soil after hardening. A macroscopic model of deep bed filtration of a monodisperse suspension in a porous medium with a size-exclusion mechanism for the suspended particles capture in the absence of mobilization of retained particles is considered. The solids are transported by the carrier fluid through large pores and get stuck at the inlet of small pores. It is assumed that the accessibility factor of pores and the fractional flow of particles depend on the concentration of the retained particles, and at the initial moment the porous medium contains an unevenly distributed deposit. The latter assumption leads to inhomogeneity of the porous medium. A quasilinear hyperbolic system of two first-order equations serves as a mathematical model of the problem. The aim of the work is to obtain the asymptotic solution near the moving curvilinear boundary - the concentration front of suspended particles of the suspension. To obtain a solution to the problem, methods of nonlinear asymptotic analysis are used. The asymptotic solution is based on a small-time parameter, measured from the moment of the concentration front passage at each point of the porous medium. The terms of the asymptotics are determined explicitly from a recurrent system of ordinary differential and algebraic equations. The numerical calculation is performed by the finite difference method using an explicit TVD scheme. Calculations for three types of microscopic suspended particles show that the asymptotics is close to the solution of the problem. The time interval of applicability of the asymptotic solution is determined on the basis of numerical calculation. The constructed asymptotics, which explicitly determines the dependence on the parameters of the system, allows to plan experiments and reduce the amount of laboratory research.

Mathematical modelling of soil massifs strained-deformed state under soil water level decreasing

The analytical solutions for the determination of vertical displacements at any point of single-layer and multilayer soil compositions under filtration water flow influence, saline solutions presence and filtration considering soil changing filtration and deformation characteristics have been obtained. The mathematical models of soil filtration and the stress-deformed state from water-saturated ground massifs and bases deformations forecast under internal volumetric forces influence (hydrodynamic forces of the filtration flow, changes in the soils own weight) have been developed and substantiated. Numerical solutions of the corresponding boundary filtration problems and SDS of soil in regions with time-varying curvilinear boundary have been obtained for these mathematical models. They have enabled to perform water-saturated soils and bases deformations forecast under the change of hydrogeological conditions and man-made factors effect.

Stabilized numerical method for solving the Oseen type problem with a singularity

На основе метода декомпозиции области построен стабилизационный неконформный метод конечных элементов для решения задачи типа Озеена. Для конвективно доминирующих течений с разрывным коэффициентом вязкости определена шкала оптимального выбора стабилизирующего параметра. Результаты численных экспериментов согласуются с теоретической оценкой сходимости. Purpose. To construct modified approximation approach using the finite element method and to perform numerical analysis for a two dimensional problem on the flow of a viscous inhomogeneous fluids — the Oseen type problem, that is obtained by sampling in time and linearizing the incompressible Navier—Stokes equations. To consider the convection dominated flow case. Methodology. Based on the domain decomposition method with a smooth curvilinear boundary between subdomains, a stabilization nonconformal finite element method is constructed that satisfies the inf-sup-stability condition. To solve the resulting system of linear algebraic equations, an iterative process is considered that uses the decomposition of the vector in the Krylov subspace with minimal inviscidity, with a block preconditioning of the matrix. Findings. The results of the numerical experiments demonstrate the robustness of the considered method for different (even small) discontinuous values of viscosity. The differences between finite element and exact solutions for the velocity field and pressure in the norms of the grid spaces decrease as

Formation of Surface Plasmon-Polariton Vortices at Reflection from Curvilinear Boundary

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