基于笛卡尔切割单元法的复杂河道地理信息系统环评可视化

复杂河道中污染物扩散计算及其在地理信息系统(GIS)上的可视化,对于地表水环境影响评价(EIA)具有非常重要的意义,但在网格生成、污染物计算模型及计算结果可视化方面存在着诸多困难。针对点源岸边排放河流污染物计算及基于GIS可视化中的难点问题,提出了基于切割单元法的地面水环境影响评价可视化方法。将切割单元法应用于网格剖分,通过切割单元交点追踪算法及河道轮廓线内背景网格筛取算法,生成了复杂河道笛卡尔网格。提出了基于污染物二维稳态衰减模式的网格自适应加密与稀疏算法,在非结构化笛卡尔网格基础上采用了基于河流几何信息判断的点源岸边排放河流污染预测算法与区域填充算法,实现了环境影响评价计算结果的可视化显示。通过一个河流污染环境影响评价可视化的实例,验证了所提方法的可行性与有效性。     

A block LU-SGS implicit dual time-stepping algorithm for hybrid dynamic meshes

A block lower-upper symmetric Gauss-Seidel (BLU-SGS) implicit dual time-stepping method is developed for moving body problems with hybrid dynamic grids. To simulate flows over complex configurations, a hybrid grid method is adopted in this paper. Body-fitted quadrilateral (quad) grids are generated first near solid bodies. An adaptive Cartesian mesh is then generated to cover the entire computational domain. Cartesian cells which overlap the quad grids are removed from the computational domain, and a gap is produced between the quad grids and the adaptive Cartesian grid. Finally triangular grids are used to fill this gap. With the motion of moving bodies, the quad grids move with the bodies, while the adaptive Cartesian grid remains stationary. Meanwhile, the triangular grids are deformed according to the motion of solid bodies with a ‘spring’ analogy approach. If the triangular grids become too skewed, or the adaptive Cartesian grid crosses into the quad grids, the triangular grids are regenerated. Then the flow solution is interpolated from the old to the new grid. The fully implicit equation is solved using a dual time-stepping solver. A Godunov-type scheme with Roe’s flux splitting is used to compute the inviscid flux. Several sub-iteration schemes are investigated in this study. Both supersonic and transonic unsteady cases are tested to demonstrate the accuracy and efficiency of the method.     

Ghost-Cell Method with far-field coarsening and mesh adaptation for Cartesian grids

Cartesian grid methods for inviscid computational fluid dynamics offer great promise for the development of very rapid conceptual design tools. The present paper deals with a number of new features for Cartesian grid methods which appear to be particularly well suited for this application. A key ingredient is the implementation of non-penetration boundary conditions at solid walls which is based upon the curvature-corrected symmetry technique (CCST) developed by the present authors for body-fitted grids. The method introduces ghost cells near the boundaries whose values are developed from an assumed flow-field model in vicinity of the wall. This method was shown to be substantially more accurate than traditional surface boundary condition approaches. This improved boundary condition has been adapted to a Cartesian mesh formulation, which we termed the Ghost-Cell Method. In this approach, all cell centers exterior to the body are computed with fluxes at the four surrounding cell edges, without any cut cells which complicate other Cartesian mesh methods. Another typical drawback of non-adaptive Cartesian grid methods is that any Cartesian grid clustering near the body must be maintained to the far field boundary. To address this issue, we have introduced a far-field grid coarsening, based on the iblanking approach, which maintains the structured nature of the grid while computing only the active cell centers. In addition, to highlight the advantages of grid adaptation in connection with Cartesian mesh methods, we have introduced a rudimentary procedure which detects the shock position and automatically refines the mesh by locally reducing the grid dimension to one fourth of its original dimension. The merits of the Ghost-Cell Method are established by the computation of the compressible flow about circular cylinders. The results show the surface non-penetration condition to be satisfied in the limit of vanishing cell size and the method to be second-order accurate in space. The results of the far-field grid coarsening indicate that the numerical solutions are unaffected by the coarsening, while the number of the computed cells and the CPU time are reduced to less than 50% of the uncoarsened solution values. The results computed using the mesh adaptation at the shock indicate that the method is effective in reducing the shock transition region thickness, without modifying the flow solution away from the shock. Finally, some test cases with airfoils located at different positions have been considered and the results are proven to be practically independent of the position of the body with respect to the grid. Although this paper is limited to two-dimensional applications, the methodology is well-suited for general three-dimensional geometries, which will be the subject of future research.     

A strictly conservative Cartesian cut-cell method for compressible viscous flows on adaptive grids

A Cartesian cut-cell method which allows the solution of two- and three-dimensional viscous, compressible flow problems on arbitrarily refined graded meshes is presented. The finite-volume method uses cut cells at the boundaries rendering the method strictly conservative in terms of mass, momentum, and energy. For three-dimensional compressible flows, such a method has not been presented in the literature, yet. Since ghost cells can be arbitrarily positioned in space the proposed method is flexible in terms of shape and size of embedded boundaries. A key issue for Cartesian grid methods is the discretization at mesh interfaces and boundaries and the specification of boundary conditions. A linear least-squares method is used to reconstruct the cell center gradients in irregular regions of the mesh, which are used to formulate the surface flux. Expressions to impose boundary conditions and to compute the viscous terms on the boundary are derived. The overall discretization is shown to be second-order accurate in L¹. The accuracy of the method and the quality of the solutions are demonstrated in several two- and three-dimensional test cases of steady and unsteady flows.     

Progress of current seamless virtual boundary method

In this paper, a current virtual boundary method, i.e., a seamless virtual boundary method (VBM) is presented. In the seamless VBM, the forcing term is added not only to the grid points near the boundary but also to the grid points inside the boundary, in order to remove unphysical oscillations near the boundary. The development of seamless VBM can be applied to solve for heat transfer and moving boundary problems in both the Cartesian and the curvilinear coordinates, and the lattice Boltzmann equation. A series of the method is validated in the typical test problems. Therefore, it is concluded that the present method is a very versatile numerical approach for solving the incompressible Navier–Stokes equations.     

Cartesian cut cell approach for simulating incompressible flows with rigid bodies of arbitrary shape

In this paper, a Cartesian grid method with cut cell approach has been developed to simulate two dimensional unsteady viscous incompressible flows with rigid bodies of arbitrary shape. A collocated finite volume method with nominally second-order accurate schemes in space is used for discretization. A pressure-free projection method is used to solve the equations governing incompressible flows. For fixed-body problems, the Adams-Bashforth scheme is employed for the advection terms and the Crank-Nicholson scheme for the diffusion terms. For moving-body problems, the fully implicit scheme is employed for both terms. The present cut cell approach with cell merging process ensures global mass/momentum conservation and avoid exceptionally small size of control volume which causes impractical time step size. The cell merging process not only keeps the shape resolution as good as before merging, but also makes both the location of cut face center and the construction of interpolation stencil easy and systematic, hence enables the straightforward extension to three dimensional space in the future. Various test examples, including a moving-body problem, were computed and validated against previous simulations or experiments to prove the accuracy and effectiveness of the present method. The observed order of accuracy in the spatial discretization is superlinear.     

A block LU-SGS implicit unsteady incompressible flow solver on hybrid dynamic grids for 2D external bio-fluid simulations

A hybrid dynamic grid generation technique for two-dimensional (2D) morphing bodies and a block lower-upper symmetric Gauss-Seidel (BLU-SGS) implicit dual-time-stepping method for unsteady incompressible flows are presented for external bio-fluid simulations. To discretize the complicated computational domain around 2D morphing configurations such as fishes and insect/bird wings, the initial grids are generated by a hybrid grid strategy firstly. Body-fitted quadrilateral (quad) grids are generated first near solid bodies. An adaptive Cartesian mesh is then generated to cover the entire computational domain. Cartesian cells which overlap the quad grids are removed from the computational domain, and a gap is produced between the quad grids and the adaptive Cartesian grid. Finally triangular grids are used to fill this gap. During the unsteady movement of morphing bodies, the dynamic grids are generated by a coupling strategy of the interpolation method based on ‘Delaunay graph’ and local remeshing technique. With the motion of moving/morphing bodies, the grids are deformed according to the motion of morphing body boundaries firstly with the interpolation strategy based on ‘Delaunay graph’ proposed by Liu and Qin. Then the quality of deformed grids is checked. If the grids become too skewed, or even intersect each other, the grids are regenerated locally. After the local remeshing, the flow solution is interpolated from the old to the new grid. Based on the hybrid dynamic grid technique, an efficient implicit finite volume solver is set up also to solve the unsteady incompressible flows for external bio-fluid dynamics. The fully implicit equation is solved using a dual-time-stepping approach, coupling with the artificial compressibility method (ACM) for incompressible flows. In order to accelerate the convergence history in each sub-iteration, a block lower-upper symmetric Gauss-Seidel implicit method is introduced also into the solver. The hybrid dynamic grid generator is tested by a group of cases of morphing bodies, while the implicit unsteady solver is validated by typical unsteady incompressible flow case, and the results demonstrate the accuracy and efficiency of present solver. Finally, some applications for fish swimming and insect wing flapping are carried out to demonstrate the ability for 2D external bio-fluid simulations.     

Aeroelastic simulations using gridless boundary condition and small perturbation techniques

This paper examines the use of stationary Cartesian mesh for non-linear flutter computations involving complex geometries. The surface boundary conditions are implemented using reflected points which are determined via a gridless approach. The method uses a cloud of nodes in the vicinity of the surface to get a weighted-average of the flow properties using radial basis functions. To ensure computational efficiency and for local grid refinements, multigrid computations within an embedded grids framework are used. As the displacements of moving surfaces from their original position are typically small for flutter problems, a small perturbation boundary condition method is used to account for the moving surfaces. The method therefore does not require repeated grid re-generation for the deforming surfaces. The overall method is both accurate and robust. Computations of the well-known Onera M6 wing, RAE wing-body configuration, the AGARD 445.6 wing flutter test case show good accuracy and efficiencies. Simulations of the aeroelastic behavior of a complete fighter-type aircraft with wing tip missiles at high transonic speeds further demonstrate the practical usefulness of the present boundary conditions technique.     

Ghost-cell method for analysis of inviscid three-dimensional flows on Cartesian-grids

The present paper deals with the implementation of non-penetration boundary conditions at solid walls for three-dimensional inviscid flow computations on Cartesian grids. The crux of the method is the curvature-corrected symmetry technique (CCST) developed by the present authors for body-fitted grids. The method introduces ghost cells near the boundaries whose values are developed from an assumed flow-field model in vicinity of the wall consisting of a vortex flow, with locally symmetric distribution of entropy and total enthalpy. In three dimensions this procedure is implemented in the so-called “osculating plane”. This method was shown to be substantially more accurate than traditional surface boundary condition approaches. This improved boundary condition is adapted to a Cartesian mesh formulation, which we have termed the “ghost-cell method”. In this approach, all cell centers exterior to the body are computed with fluxes at the six surrounding cell faces, without any cut cell. A multiple-valued point technique is used to compute sharp edges. The merits of the ghost-cell method for three-dimensional inviscid flow computations are established by computing compressible and transonic flows about a sphere, an oblate and a prolate spheroid, a cylindrical wing with an end-plate, the ONERA M6 wing and detailed comparison to body-fitted grid computations and to published data. The computed results show the surface non-penetration condition to be satisfied in the limit of vanishing cell size and the method to be second-order accurate in space. The comparison with body-fitted results proves that the accuracy is comparable to the accuracy of CCST computations on body-fitted grids and remarkably superior to body-fitted computations based on traditional pressure extrapolation, non-penetration boundary conditions. In addition, we prove that the results are independent of the position of the body with respect to the grid. Finally, we show that the ONERA M6 wing results compare very well with published data.     

Flow simulations in arbitrarily complex cardiovascular anatomies – An unstructured Cartesian grid approach

Image guided computational fluid dynamics is attracting increasing attention as a tool for refining in vivo flow measurements or predicting the outcome of different surgical scenarios. Sharp interface Cartesian/Immersed-Boundary methods constitute an attractive option for handling complex in vivo geometries but their capability to carry out fine-mesh simulations in the branching, multi-vessel configurations typically encountered in cardiovascular anatomies or pulmonary airways has yet to be demonstrated. A major computational challenge stems from the fact that when such a complex geometry is immersed in a rectangular Cartesian box the excessively large number of grid nodes in the exterior of the flow domain imposes an unnecessary burden on both memory and computational overhead of the Cartesian solver without enhancing the numerical resolution in the region of interest. For many anatomies, this added burden could be large enough to render comprehensive mesh refinement studies impossible. To remedy this situation, we recast the original structured Cartesian formulation of Gilmanov and Sotiropoulos [Gilmanov A, Sotiropoulos F. A hybrid Cartesian/immersed boundary method for simulating flows with 3D, geometrically complex, moving bodies. J Comput Phys 2005;207(2):457–92] into an unstructured Cartesian grid layout. This simple yet powerful approach retains the simplicity and computational efficiency of a Cartesian grid solver, while drastically reducing its memory footprint. The method is applied to carry out systematic mesh refinement studies for several internal flow problems ranging in complexity from flow in a 90° pipe bend to flow in an actual, patient-specific anatomy reconstructed from magnetic resonance images. Finally, we tackle the challenging clinical scenario of a single-ventricle patient with severe arterio-venous malformations, seeking to provide a fluid dynamics prospective on a clinical problem and suggestions for procedure improvements. Results from these simulations demonstrate very complex cardiovascular flow dynamics and underscore the need for high-resolution simulations prior to drawing any clinical recommendations.     

混合重叠网格通信优化算法

重叠网格技术广泛应用在复杂外型和运动边界问题的流场数值模拟中.本文在并行重叠网格隐式挖洞算法实现的基础上,提出了笛卡尔辅助网格和多块结构网格的混合重叠网格方法.通过笛卡尔辅助网格实现重叠网格洞边界和网格插值关系的快速建立.通过定义重叠区域网格权重、部件网格与背景网格绑定的方法,建立了混合网格的并行分配模式,有效减少重叠插值信息在各进程间的通信,实现计算负载和通信负载在各个进程的均匀分配.测试表明该方法可应用于数千万量级的重叠网格系统,可扩展至千核规模,高效的实现多个物体构成的复杂网格系统的重叠关系建立.     

Numerical simulation of the unsteady three-dimensional flow in confined domains crossed by moving bodies

A numerical method devoted to the prediction of unsteady flows in complex domains with moving boundaries is presented. Based on the unsteady Euler equations with source terms to take diffusive effects into account as well as additional mass, momentum or enthalpy sources, it has been specially developed to model the thermal and dynamic behavior of the ambient air inside underground stations in the presence of moving trains. The numerical solution method is a unstructured finite-volume cell-centered scheme using the SIMPLE algorithm coupled with a second-order intermediate time stepping scheme. The spatial discretization is realized with an automatic Cartesian grid generator, complemented by a technique of sliding grids to handle straight moving bodies inside the domain.     

A FV-TD electromagnetic solver using adaptive Cartesian grids

A second-order finite-volume (FV) method has been developed to solve the time-domain (TD) Maxwell equations, which govern the dynamics of electromagnetic waves. The computational electromagnetic (CEM) solver is capable of handling arbitrary grids, including structured, unstructured, and adaptive Cartesian grids, which are topologically arbitrary. It is argued in this paper that the adaptive Cartesian grid is better than a tetrahedral grid for complex geometries considering both efficiency and accuracy. A cell-wise linear reconstruction scheme is employed to achieve second-order spatial accuracy. Second-order time accuracy is obtained through a two-step Runge-Kutta scheme. Issues on automatic adaptive Cartesian grid generation such as cell-cutting and cell-merging are discussed. A multi-dimensional characteristic absorbing boundary condition (MDC-ABC) is developed at the truncated far-field boundary to reduce reflected waves from this artificial boundary. The CEM solver is demonstrated with several test cases with analytical solutions.     

Free-boundary method for the numerical solution of gas-dynamic equations in domains with varying geometry

A novel method is proposed for numerical solution of gas-dynamic equations on stationary Cartesian grids in domains containing solid impermeable and, in the general case, moving inclusions (objects). The suggested technique is based on the immersed boundary method, in which the computational domain (including solid objects) is covered by a single Cartesian grid and the calculation is carried out by the shock-capturing method over all cells. Under this approach, the influence of the solid inclusions on the flow of the gas medium is simulated by the introduction of specially selected mass, momentum, and energy fluxes into the right-hand side of the equations. The currently developed methods for the solution of this class of problems are surveyed and the advantages of the proposed approach are discussed. The method is verified by calculating some test problems that admit analytical solutions and it is used to solve the problem of supersonic flow around a blunt body. The results are compared with the calculation findings based on the standard curvilinear grid tied to the geometry of the body.     

Hybrid Spectral Difference Methods for Elliptic Equations on Exterior Domains with the Discrete Radial Absorbing Boundary Condition

The hybrid spectral difference methods (HSD) for the Laplace and Helmholtz equations in exterior domains are proposed. We consider the fictitious domain method with the absorbing boundary conditions (ABCs). The HSD method is a finite difference version of the hybridized Galerkin method, and it consists of two types of finite difference approximations; the cell finite difference and the interface finite difference. The fictitious domain is composed of two subregions; the Cartesian grid region and the boundary layer region in which the radial grid is imposed. The boundary layer region with the radial grid makes it easy to implement the discrete radial ABC. The discrete radial ABC is a discrete version of the Bayliss–Gunzburger–Turkel ABC without pertaining any radial derivatives. Numerical experiments confirming efficiency of our numerical scheme are provided.     

An augmented method for free boundary problems with moving contact lines

An augmented immersed interface method (IIM) is proposed for simulating one-phase moving contact line problems in which a liquid drop spreads or recoils on a solid substrate. While the present two-dimensional mathematical model is a free boundary problem, in our new numerical method, the fluid domain enclosed by the free boundary is embedded into a rectangular one so that the problem can be solved by a regular Cartesian grid method. We introduce an augmented variable along the free boundary so that the stress balancing boundary condition is satisfied. A hybrid time discretization is used in the projection method for better stability. The resultant Helmholtz/Poisson equations with interfaces then are solved by the IIM in an efficient way. Several numerical tests including an accuracy check, and the spreading and recoiling processes of a liquid drop are presented in detail.     

The direct matrix imbedding technique for computing three-dimensional potential flow about arbitrarily shaped bodies

The direct matrix imbedding technique is used to solve Laplace's equation for the velocity potential numerically about arbitrarily shaped bodies with normal gradient boundary conditions in two and three dimensions. The bodies are imbedded in Cartesian grids overlaying relatively large rectangular and box regions. Solutions are obtained only in those parts of the grid necessary for constructing solutions to potential flow problems. An important subclass of these problems, considered in this paper, is ship wave problems in channels. Uniform and stretched Cartesian grids are considered, and solutions are obtained very quickly. Results are presented.     

An adaptive multilevel multigrid formulation for Cartesian hierarchical grid methods

A Cartesian grid method with adaptive mesh refinement and multigrid acceleration is presented for the compressible Navier-Stokes equations. Cut cells are used to represent boundaries on the Cartesian grid, while ghost cells are introduced to facilitate the implementation of boundary conditions. A cell-tree data structure is used to organize the grid cells in a hierarchical manner. Cells of all refinement levels are present in this data structure such that grid level changes as they are required in a multigrid context do not have to be carried out explicitly. Adaptive mesh refinement is introduced using phenomenon-based sensors. The application of the multilevel method in conjunction with the Cartesian cut-cell method to problems with curved boundaries is described in detail. A 5-step Runge-Kutta multigrid scheme with local time stepping is used for steady problems and also for the inner integration within a dual time-stepping method for unsteady problems. The inefficiency of customary multigrid methods on Cartesian grids with embedded boundaries requires a new multilevel concept for this application, which is introduced in this paper. This new concept is based on the following novelties: a formulation of a multigrid method for Cartesian hierarchical grid methods, the concept of averaged control volumes, and a mesh adaptation strategy allowing to directly control the number of refined and coarsened cells.     

Efficient computation of volume fractions for multi-material cell complexes in a grid by slicing

The paper presents a novel slicing based method for computation of volume fractions in multi-material solids given as a B-rep whose faces are triangulated and shared by either one or two materials. Such objects occur naturally in geoscience applications and the said computation is necessary for property estimation problems and iterative forward modeling. Each facet in the model is cut by the planes delineating the given grid structure or grid cells. The method, instead of classifying the points or cells with respect to the solid, exploits the convexity of triangles and the simple axis-oriented disposition of the cutting surfaces to construct a novel intermediate space enumeration representation called slice-representation, from which both the cell containment test and the volume-fraction computation are done easily. Cartesian and cylindrical grids with uniform and non-uniform spacings have been dealt with in this paper. After slicing, each triangle contributes polygonal facets, with potential elliptical edges, to the grid cells through which it passes. The volume fractions of different materials in a grid cell that is in interaction with the material interfaces are obtained by accumulating the volume contributions computed from each facet in the grid cell. The method is fast, accurate, robust and memory efficient. Examples illustrating the method and performance are included in the paper.     

Modeling complex boundaries using an external force field on fixed Cartesian grids in large-eddy simulations

In the present study a methodology to perform large-eddy simulations around complex boundaries on fixed Cartesian grids is presented. A novel interpolation scheme which is applicable to boundaries of arbitrary shape, does not involve special treatments, and allows the accurate imposition of the desired boundary conditions is introduced. A method to overcome the problems associated with the computation of the subgrid scale terms near solid boundaries is also discussed. A detailed study on the accuracy and efficiency of the method is carried out for the cases of Stokes flow around a cylinder in the vicinity of a moving plate, the three-dimensional flow around a circular cylinder, and fully developed turbulent flow in a plane channel with a wavy wall. It is demonstrated that the method is second-order accurate, and that the solid boundaries are mimicked “exactly” on the Cartesian grid within the overall accuracy of the scheme. For all cases under consideration the results obtained are in very good agreement with analytical and numerical data.