Yu. Vassilevski

A monotone nonlinear finite volume method for diffusion equations on conformal polyhedral meshes

Abstract We have developed a new monotone cell-centered finite volume method for the discretization of diffusion equations on conformal polyhedral meshes. The proposed method is based on a nonlinear two-point flux approximation. For problems with smooth diffusion tensors and Dirichlet boundary conditions the method is interpolation-free. An adaptive interpolation is applied on faces where diffusion tensor jumps or Neumann boundary conditions are imposed. The interpolation is based on physical relationships such as continuity of the diffusion flux. The second-order convergence rate is verified with numerical experiments.

Minimal stencil finite volume scheme with the discrete maximum principle

Abstract We propose a cell-centered finite volume (FV) scheme with the minimal stencil formed by the closest neighbouring cells. The discrete solution satisfies the discrete maximum principle and approximates the exact solution with second-order accuracy. The coefficients in the FV stencil depend on the solution; therefore, the FV scheme is nonlinear. The scheme is applied to a steady state advection-diffusion equation discretized on a general polygonal mesh.

Two splitting schemes for nonstationary convection-diffusion problems on tetrahedral meshes

Two splitting schemes are proposed for the numerical solution of three-dimensional nonstationary convection-diffusion problems on unstructured meshes in the case of a full diffusion tensor. An advantage of the first scheme is that splitting is generated by the properties of the approximation spaces and does not reduce the order of accuracy. An advantage of the second scheme is that the resulting numerical solutions are nonnegative. A numerical study is conducted to compare the splitting schemes with classical methods, such as finite elements and mixed finite elements. The numerical results show that the splitting schemes are characterized by low dissipation, high-order accuracy, and versatility.

Numerical issues of modelling blood flow in networks of vessels with pathologies

The synthesis of the blood circulation model and the elastic fiber model of the vessel wall allows us to take into account the influence of possible vessel pathologies on the global blood flow. The interaction is based on the state equation representing the dependence of the transmural pressure on the cross-section of the vessel. Numerical properties of both models are considered in the paper. The mathematical modelling of blood circulation is a fundamental problem lying at the junction of several disciplines, such as differential equations, numerical analysis, elasticity theory, and physiology. Several numerical implementations of blood circu- lation models taking into account elastic properties of blood vessels were created in the last decade (6,9,10,14,21,22). Previously we proposed an approach to synthesis of the blood circulation model and the elastic model of the vessel wall (24) taking into account the influence of possible vessel pathologies on the global blood flow. The distinctive feature of the approach is the use of merely one-dimensional dif- ferential operators, which provided us with an efficient numerical simulation tech- nology. The mathematical blood flow model is a system of differential equations for each vessel linked by boundary conditions at the points of vessel junctions (22). The mathematical model of the elastic vessel wall is based on the fiber approach (17,18) to the calculation of the reaction force as a response to the deformation of a fiber. The representation of an elastic body by sets of fibers of different configurations was successfully used for simulation of cardiac work (13) and collapsed veins (18). In our model we used the same types of fibers as in (18). The synthesis of both models is based on the state equation representing the dependence of the transmural pressure on the cross-section area of the vessel. This

Two-phase water flooding simulations on dynamic adaptive octree grids with two-point nonlinear fluxes

Abstract We present a method for numerical simulation of the two-phase water flooding problem on general polyhedral grids not aligned with permeability tensor K (K-nonorthogonal grids) and dynamic octree grids adapted to the front between the phases. The discretization is based on the cell-centered monotone finite volume (FV) method with the nonlinear two-point flux approximation (TPFA) applicable to general K-non-orthogonal polyhedral grids. We use fully implicit discretization in time to avoid the restriction on the time step caused by the minimal mesh size. In our numerical experiments we demonstrate the superiority of the nonlinear TPFA on a K-non-orthogonal grid over linear TPFA and considerable speed-up of the simulation on a dynamically adapted octree grid with minimal loss in accuracy compared to the simulation on a fine regular grid.

Modelling of bioimpedance measurements: unstructured mesh application to real human anatomy

Abstract A technology for high-resolution efficient numerical modelling of bioimpedance measurements is considered that includes 3D image segmentation, adaptive unstructured tetrahedral mesh generation, finite-element discretization, and analysis of simulation data. The first-order convergence of the proposed numerical methods on a series of unmatched meshes and roughly second-order convergence on a series of nested meshes are shown. The current, potential, and sensitivity field distributions are computed for conventional schemes of bioimpedance measurements using segmented geometrical torso model of the Visible Human Project (VHP) man. Use of the adaptive tetrahedral meshes reduces significantly the number of mesh elements and, hence, the associated computational cost compared to rectangular meshes while keeping the model accuracy.

CFD technology for 3D simulation of large-scale hydrodynamic events and disasters

Abstract In this paper we discuss the basic components of the computational technology for the simulation of complex hydrodynamic events, such as a break of a dam, a wave pileup, a landslide, or a mud flow. The technology uses three-dimensional equations of fluid dynamics with free boundaries. The mathematical model is based on the Navier-Stokes equations with nonlinear defining relations between the stress tensor and the rate of strain tensor. The assignment of a particular defining relation allows one to simulate both Newtonian flows (break of a dam, wave pileup), and non-Newtonian ones (landslide, mud flow, snow avalanche, flood of lava). The numerical model developed in the paper uses the method of the grid level set function for calculation of a free surface flow evolution and adaptively reconstructed three-dimensional grids of the octree type for discretization of the flow equations. The predictive accuracy of this technology is demonstrated in the paper by comparing the results of certain numerical calculations with physical experiments; the efficiency of the technology is illustrated by simulation of the break of a dam and a mud flow using the actual 3D topology of the area around the Sayano-Shushenskaya dam.

Free surface flow modelling on dynamically refined hexahedral meshes

Abstract An efficient method for modelling incompressible free surface flows is presented. The method unites the projection method for solving the Navier–Stokes equations and the particle level set method for free surface evolution. The method uses adaptively refined hexahedral meshes built on an enhanced octree data structure.

Hessian-based anisotropic mesh adaptation in domains with discrete boundaries

Black-box methodology for generating anisotropic adaptive tetrahedral meshes in domains with discrete boundaries is described. A new high-order reconstruction method for triangular surface meshes is proposed. The performance of the method for a model convection–diffusion problem is demonstrated.

Sensitivity field distributions for segmental bioelectrical impedance analysis based on real human anatomy

In this work, an adaptive unstructured tetrahedral mesh generation technology is applied for simulation of segmental bioimpedance measurements using high-resolution whole-body model of the Visible Human Project man. Sensitivity field distributions for a conventional tetrapolar, as well as eight- and ten-electrode measurement configurations are obtained. Based on the ten-electrode configuration, we suggest an algorithm for monitoring changes in the upper lung area.