Finite-volume algorithm for solving the time-dependent Maxwell equations on unstructured meshes

A finite-volume method is proposed for solving the time-dependent Maxwell equations on unstructured triangular meshes. The results of test computations show that the method has a second-order convergence rate for homogeneous media and a close-to-second-order convergence rate for media with spatially discontinuous permittivity.     

Numerical method for computing two-dimensional unsteady rarefied gas flows in arbitrarily shaped domains

A high-order accurate method for analyzing two-dimensional rarefied gas flows is proposed on the basis of a nonstationary kinetic equation in arbitrarily shaped regions. The basic idea behind the method is the use of hybrid unstructured meshes in physical space. Special attention is given to the performance of the method in a wide range of Knudsen numbers and to accurate approximations of boundary conditions. Examples calculations are provided.     

A finite-element coarse-grid projection method for incompressible flow simulations

Coarse grid projection (CGP) methodology is a novel multigrid method for systems involving decoupled nonlinear evolution equations and linear elliptic Poisson equations. The nonlinear equations are solved on a fine grid and the linear equations are solved on a corresponding coarsened grid. Mapping operators execute data transfer between the grids. The CGP framework is constructed upon spatial and temporal discretization schemes. This framework has been established for finite volume/difference discretizations as well as explicit time integration methods. In this article we present for the first time a version of CGP for finite element discretizations, which uses a semi-implicit time integration scheme. The mapping functions correspond to the finite-element shape functions. With the novel data structure introduced, the mapping computational cost becomes insignificant. We apply CGP to pressure-correction schemes used for the incompressible Navier-Stokes flow computations. This version is validated on standard test cases with realistic boundary conditions using unstructured triangular meshes. We also pioneer investigations of the effects of CGP on the accuracy of the pressure field. It is found that although CGP reduces the pressure field accuracy, it preserves the accuracy of the pressure gradient and thus the velocity field, while achieving speedup factors ranging from approximately 2 to 30. The minimum speedup occurs for velocity Dirichlet boundary conditions, while the maximum speedup occurs for open boundary conditions.     

A new method based on SOM network to generate coarse meshes for overlapping unstructured multigrid algorithm

A new method to generate coarse meshes for overlapping unstructured multigrid algorithm based on self-organizing map (SOM) neural network is presented in this paper. The application of SOM neural network can overcome some limitations of conventional methods and which is designed to pursuit the best structure relation between fine and coarse unstructured meshes with the object to ensure robust convergence for overlapping unstructured multigrid algorithm. Besides, this method can automate the generation of unstructured meshes and is suitable for both two and three dimensions conditions.     

Robin-type wall functions and their numerical implementation

The paper is devoted to numerical implementation of the wall functions of Robin-type for modeling near-wall turbulent flows. The wall functions are based on the transfer of a boundary condition from a wall to some intermediate boundary near the wall. The boundary conditions on the intermediate boundary are of Robin-type and represented in a differential form. The wall functions are formulated in an analytical easy-to-implement form, can take into account the source terms, and do not include free parameters. The relation between the wall functions of Robin type and the theory of Calderon–Ryaben'kii's potentials is demonstrated. A universal robust approach to the implementation of the Robin-type wall functions in finite-volume codes is provided. The example of an impinging jet is considered.     

Implementation of near-wall boundary conditions for modeling boundary layers with free-stream turbulence

The paper is devoted to the extension of the near-wall domain decomposition, earlier developed in some previous works by the authors, to modeling flat-plate boundary layers undergoing laminar-to-turbulent bypass transition. The steady-state wall boundary layers at high-intensity free-stream turbulence are studied on the basis of differential turbulence models with the use of non-overlapping domain decomposition. In the approach the near-wall resolution is replaced by the interface boundary conditions of Robin type. In contrast to the previous studies, the main attention is paid to the laminar–turbulent transitional regime. With the use of modified turbulence models we study an effect of free-stream parameters on the development of dynamic processes in the boundary layer including a transitional regime and fully developed turbulent flow. In addition, for the first time a full scale domain decomposition is realized via iterations between the inner and outer subregions until a convergence. The computational profiles of the velocity and intensity of the turbulence kinetic energy are compared with experimental data. A possible range of location of the near-wall interface boundary is found.     

Immersed Boundary Method on Deformable Unstructured Meshes for Airfoil Aeroacoustic Simulation

Computational Mathematics and Mathematical Physics - An immersed boundary method on unstructured meshes is used in serial eddy-resolving simulations of turbulent flows over individual bodies with a...     

Error analysis of staggered finite difference finite volume schemes on unstructured meshes

This work combines the consistency in lower‐order differential operators with external approximations of functional spaces to obtain error estimates for finite difference finite volume schemes on unstructured nonuniform meshes. This combined approach is first applied to a one‐dimensional elliptic boundary value problem on nonuniform meshes, and a first‐order convergence rate is obtained, which agrees with the results previously reported. The approach is also applied to the staggered Marker‐and‐Cell scheme for the two‐dimensional incompressible Stokes problem on unstructured meshes. A first‐order convergence rate is obtained, which improves over a previously reported result in that it also holds on unstructured meshes. For both problems considered in this work, the convergence rate is one order higher on meshes satisfying special requirements. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1159–1182, 2017     

Automatic parallel generation of tetrahedral grids by using a domain decomposition approach

An algorithm for the automatic parallel generation of three-dimensional unstructured grids based on geometric domain decomposition is proposed. A software package based on this algorithm is described. Examples of generating meshes for some application problems on a multiprocessor computer are presented. It is shown that the parallel algorithm can significantly (by a factor of several tens) reduce the mesh generation time. Moreover, it can easily generate meshes with as many as 5 × 10⁷ elements, which can hardly be generated sequentially. Issues concerning the speedup and the improvement of the efficiency of the computations and of the quality of the resulting meshes are discussed.     

Construction of edge-based 1-exact schemes for solving the Euler equations on hybrid unstructured meshes

In this paper, 1-exact vertex-centered finite-volume schemes with an edge-based approximation of fluxes are constructed for numerically solving hyperbolic problems on hybrid unstructured meshes. The 1-exactness property is ensured by introducing a new type of control volumes, which are called semitransparent cells. The features of a parallel algorithm implementing the computations using semitransparent cells on modern supercomputers are described. The results of solving linear and nonlinear test problems are given.     

Application of numerical schemes with singling out the boundary layer for the computation of turbulent flows using eddy-resolving approaches on unstructured grids

The use of eddy-resolving approaches to solving problems on arbitrary unstructured grids is investigated. The applications of such approaches requires the use of low dissipation numerical schemes, which can lead to numerical oscillations of the solution on unstructured grids. Numerical oscillations typically occur in domains with large gradients of velocities, in particular, in the near-wall layer. It is proposed to single out the boundary layer and use a numerical scheme with increased numerical dissipation in it. The algorithm for singling out the boundary layer uses a switching function to change the parameters of the numerical scheme. This algorithm is formulated based on the BCD scheme from the family NVD. Its validity and advantages are investigated using the zonal RANS–LES approach for solving some problems of turbulent flow of incompressible fluids.     

A discrete duality finite volume discretization of the vorticity‐velocity‐pressure stokes problem on almost arbitrary two‐dimensional grids

We present an application of the discrete duality finite volume method to the numerical approximation of the vorticity‐velocity‐pressure formulation of the two‐dimensional Stokes equations, associated to various nonstandard boundary conditions. The finite volume method is based on the use of discrete differential operators obeying some discrete duality principles. The scheme may be seen as an extension of the classical Marker and Cell scheme to almost arbitrary meshes, thanks to an appropriate choice of degrees of freedom. The efficiency of the scheme is illustrated by numerical examples over unstructured triangular and locally refined nonconforming meshes, which confirm the theoretical convergence analysis led in the article. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1–30, 2015     

Simulation of elastic wave propagation in geological media: Intercomparison of three numerical methods

For wave propagation in heterogeneous media, we compare numerical results produced by grid-characteristic methods on structured rectangular and unstructured triangular meshes and by a discontinuous Galerkin method on unstructured triangular meshes as applied to the linear system of elasticity equations in the context of direct seismic exploration with an anticlinal trap model. It is shown that the resulting synthetic seismograms are in reasonable quantitative agreement. The grid-characteristic method on structured meshes requires more nodes for approximating curved boundaries, but it has a higher computation speed, which makes it preferable for the given class of problems.     

Laplace–Beltrami enhancement for unstructured two-dimensional meshes having dendritic elements and boundary node movement

The Laplace–Beltrami mesh enhancement algorithm of Hansen et al. ,  and  has been implemented and broadened to include meshes containing dendritic elements and allowing for boundary node movement. This implementation operates on an unstructured two-dimensional mesh by forming an equivalent weak statement using finite element interpolation, assembly, and solution ideas to iteratively place those nodes allowed to move. Moving boundary nodes are constrained to follow the boundary geometry described as a Wilson–Fowler spline (e.g., [3, Section 2.1.3.1]). Implementation details concerning the element basis set modifications, the metric tensor for dendritic element treatment and boundary node movement are presented. Laplacian (e.g., [6]) enhancement is included as a special case. Results are presented which illustrate the algorithm for three test problems.     

Heterogeneous Computing in Resource-Intensive CFD Simulations

A software package implementing a fully heterogeneous mode of computations on CPUs and GPU accelerators for efficient use of hybrid supercomputers has been developed at the Keldysh Institute of Applied Mathematics of the Russian Academy of Sciences. The package involves a distributed preprocessor ensuring work with fine unstructured meshes. Combined compression of the grid topology is used to reduce the amount of storage required for superlarge grid data. The study involves petascale computational resources.     

A unified framework of multi-objective cost functions for partitioning unstructured finite element meshes

Good performance of parallel finite element computations on unstructured meshes requires high-quality mesh partitioning. Such a decomposition task is normally done by a graph-based partitioning approach. However, the main shortcoming of graph partitioning algorithms is that minimizing the so-called edge cut is not entirely the same as minimizing the communication overhead. This paper thus proposes a unified framework of multi-objective cost functions, which take into account several factors that are not captured by the graph-based partitioning approach. Freely adjustable weighting parameters in the framework also promote a flexible treatment of different optimization objectives. A greedy-style post-improvement procedure is designed to use these cost functions to improve the quality of subdomain meshes arising from the graph-based partitioning approach. Both serial and parallel implementation of the post-improvement procedure have been done. Numerical experiments show that communication overhead can indeed be reduced by this improvement procedure, thereby increasing the performance of parallel finite element computations.     

无结构网格有限体积方法及在热对流中的应用研究

首次将无结构三角网格的有限体积方法和压强连接半隐式算法相结合 ,用于求解非平行壁管道中的热对流问题 .并由此分析了化学汽相淀积薄膜生长的均匀性问题 .计算结果对于分析一类管道中热和动量输运现象均有普遍指导意义     

Adaptation and optimization of basic operations for an unstructured mesh CFD algorithm for computation on massively parallel accelerators

The design of efficient algorithms for large-scale gas dynamics computations with hybrid (heterogeneous) computing systems whose high performance relies on massively parallel accelerators is addressed. A high-order accurate finite volume algorithm with polynomial reconstruction on unstructured hybrid meshes is used to compute compressible gas flows in domains of complex geometry. The basic operations of the algorithm are implemented in detail for massively parallel accelerators, including AMD and NVIDIA graphics processing units (GPUs). Major optimization approaches and a computation transfer technique are covered. The underlying programming tool is the Open Computing Language (OpenCL) standard, which performs on accelerators of various architectures, both existing and emerging.     

Third-order accurate conservative method on unstructured meshes for gasdynamic simulations

A third-order accurate finite-volume method on unstructured meshes is proposed for solving viscous gasdynamic problems. The method is described as applied to the advection equation. The accuracy of the method is verified by computing the evolution of a vortex on meshes of various degrees of detail with variously shaped cells. Additionally, unsteady flows around a cylinder and a symmetric airfoil are computed. The numerical results are presented in the form of plots and tables.     

Computation of dynamical phase transitions in solids

This paper deals with the dynamics of phase boundaries in a nonlinear elastic two-phase material. We consider the elasticity system in 1D and the equations of anti-plane shear motion in 2D, where effects of viscosity and capillarity are neglected. These first-order conservation laws allow to represent phase boundaries as shock-like sharp interfaces. However, in contrast to what is known for homogeneous materials, the entropy inequality does not select a unique solution, and an additional criterion, the so-called kinetic relation, is required.Based on a scheme introduced by Hou, Rosakis and LeFloch [T. Hou, Ph. Rosakis, P.G. LeFloch, A level-set approach to the computation of twinning and phase-transition dynamics, J. Comput. Phys. 150 (1999) 302–331] we focus on the numerical solution of a specific model system. Using a level-set technique to enforce the kinetic relation on the discrete level leads to a reformulation of the original system in the form of a system of conservation laws coupled to a Hamilton–Jacobi equation for each phase boundary. The numerical method for the reformulated system is constructed for unstructured meshes (in 2D), and a self-adaptive algorithm is introduced.In the 1D-case we show that the reformulated system has a solution that corresponds to exact dynamical phase boundaries of the elasticity system which obey the kinetic relation. To validate the method in 2D, we present computations on the interaction of a plane wave with a phase boundary. The efficiency of the adaptation mechanism is demonstrated by an example showing the development of microstructures by twinning.