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.     

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.     

Numerical modelling and passive flow control using porous media

The whole flow over a solid body covered by a porous layer is presented. The three main models used in the literature to compute efficiently the fluid flow are given: the reduction of the porous layer to a boundary condition, the coupling of Darcy equation with Navier-Stokes equations and the Brinkman-Navier-Stokes equations or the penalisation method. Numerical simulations on Cartesian grids using the latest model give easily accurate solutions of the flow around solid bodies with or without porous layers. Adding appropriate porous devices to the solid bodies, an efficient passive control of the two-dimensional incompressible flow is achieved. A strong regularisation of the flow is observed and a significant reduction of the vortex induced vibrations or the drag coefficient is obtained.     

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 vibration analysis of laminated composite plates based on FSDT using one-dimensional IRBFN method

This paper presents a new effective radial basis function (RBF) collocation technique for the free vibration analysis of laminated composite plates using the first order shear deformation theory (FSDT). The plates, which can be rectangular or non-rectangular, are simply discretised by means of Cartesian grids. Instead of using conventional differentiated RBF networks, one-dimensional integrated RBF networks (1D-IRBFN) are employed on grid lines to approximate the field variables. A number of examples concerning various thickness-to-span ratios, material properties and boundary conditions are considered. Results obtained are compared with the exact solutions and numerical results by other techniques in the literature to investigate the performance of the proposed method.     

Interface tracking with dynamically-redistributed surface markers in unstructured quadrangular grids

In this work we present an extension of a front-tracking algorithm with a dynamical redistribution of surface markers for interface advection of a fluid body in unstructured quadrangular grids. The interface is described by an ordered list of markers connected by segments which are first advected along the streamlines and then redistributed uniformly along the interface while conserving the spanned area. In this article we perform two-dimensional tests over unstructured meshes and compare the results with those obtained in Cartesian grids. Solid body motions, such as translations and rotations, and more complex vortical flows which progressively deform and stretch the interface line are considered over different geometries. The results show that the interface evolution can be tracked successfully in challenging situations with a very good accuracy in terms of area conservation and a rather moderate decrease in the performance of the method with respect to the structured Cartesian mesh.     

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。     

Solving a Nonlinear Problem in Magneto-Rheological Fluids Using the Immersed Interface Method

Many applications lead to a nonlinear elliptic interface problem in which the discontinuous coefficient depends on the solution and the material properties. A finite difference method based on Cartesian grids and the maximum principle preserving immersed interface method is proposed for the nonlinear elliptic interface problems discussed in this paper. Numerical experiments against the exact solutions reveal that our method is nearly second order accurate in the infinity norm. The method is applied to study the magneto-rheological field-responsive fluids that contain iron particles. Numerical experiments are performed against the results from the literature.     

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.     

Formulation and multigrid solution of Cauchy-Riemann optimal control problems

The formulation of optimal control problems governed by Cauchy-Riemann equations is presented. A distributed control mechanism through divergence and curl sources is considered with the boundary conditions of mixed type. A Lagrange multiplier framework is introduced to characterize the solution to Cauchy-Riemann optimal control problems as the solution of an optimality system of four first-order partial differential equations and two optimality conditions. To solve the optimality system, staggered grids and multigrid methods are investigated. It results that staggered grids provide a natural collocation of the optimization variables and second-order accurate solutions are obtained. The proposed multigrid scheme is based on a coarsening by a factor of three that results in a nested hierarchy of staggered grids. On these grids a distributed-Gauss-Seidel and gradient-based smoothing scheme is employed. Results of numerical experiments validate the proposed optimal control formulation and demonstrate the effectiveness of the staggered-grids multigrid solution procedure.     

Method of adaptive artificial viscosity for gas dynamics equations on triangular and tetrahedral grids

In studies [1–7] a method of adaptive artificial viscosity (AAV) was proposed for the solution of gas dynamics equations. In this paper, this method is extended to the case of triangular grids for two-dimensional (2D) equations in the variables x, y, and r, z, and of tetrahedral grids for equations in Cartesian variables x, y, and z. The calculation results for the test problems are presented.     

Developments in Cartesian cut cell methods

This paper describes the Cartesian cut cell method, which provides a flexible and efficient alternative to traditional boundary fitted grid methods. The Cartesian cut cell approach uses a background Cartesian grid for the majority of the flow domain with special treatments being applied to cells which are cut by solid bodies, thus retaining a boundary conforming grid. The development of the method is described with applications to problems involving both moving bodies and moving material interfaces.     

An improved immersed boundary method for curvilinear grids

In the present paper we propose an extension of the direct-forcing immersed boundary technique, recently developed and employed by Verzicco and co-authors [Fadlun EA, Verzicco R, Orlandi P, Mohd-Yusof J. Combined immersed-boundary finite-difference methods for three-dimensional complex flow simulations. J Comput Phys 2000;161:35–60; Verzicco R, Fatica M, Iaccarino G, Moin P, Khalighi B. Large eddy simulation of a road vehicle with drag-reduction devices. AIAA J 2002;40(12):2447–55; Cristallo A, Verzicco R. Combined immersed boundary/large-eddy-simulations of incompressible three-dimensional complex flows. Flow Turbul Combust 2006;77(1–4):3–26.] and successively improved by Balaras and co-authors [Gilmanov A, Sotiropoulos F, Balaras E. A general reconstruction algorithm for simulating flows with complex 3D immersed boundaries on Cartesian grids. J Comput Phys 2003;191:660–9; Balaras E. Modeling complex boundaries using an external force field on fixed Cartesian grids in large-eddy simulations. Comput Fluids 2004;33:375–404]. We extend the aforementioned technique to curvilinear-coordinate, structured grid, Navier–Stokes solvers. This improved technique allows for more flexibility and efficiency when compared to standard methods in which the technique is coupled with orthogonal-grid solvers. Additional modifications are also proposed with respect to the state-of-art, which allow to deal with general shaped, multiple-body immersed surfaces and to make the interpolation of the velocity field off the body suitable for curvilinear grids. Several tests have been carried out to check the reliability of the proposed technique: first we have considered the three-dimensional Stokes flow around a sphere, and compared the numerical results with the analytical ones. Second we have considered the two-dimensional unsteady flow around a circular cylinder placed between two parallel solid walls and compared the results with those of the database of the Priority Research Program ‘Flow Simulation on High Performance Computers’ of the German Research Association (DFG). Third, we have considered the two-dimensional flow within a S-shaped duct containing an elliptical valve. Finally, we have applied the technique to the study of a practical high-Reynolds number industrial problem.The geometrical configuration of the first two test cases is suited for both Cartesian and curvilinear algorithms. The geometry of the third test case is suited for curvilinear meshes and makes the use of Cartesian grids very inefficient and less accurate than the curvilinear ones. In these cases Cartesian – as well as curvilinear – mesh simulations have been carried out. Finally, the geometry of the high-Reynolds industrial problem is suited for curvilinear grids.The proposed technique has shown to preserve at least the same level of accuracy of its Cartesian counterpart allowing to reduce in a considerable way the computational cost of the simulations. When the geometry is better suited for curvilinear meshes, the reduction of the computational cost is accompanied by an increased accuracy with respect to the Cartesian counterpart.We also propose a simplified direct-forcing, semi-implicit method, allowing reduced computational cost with respect to the literature techniques. We have checked the accuracy of the technique and shown that when the Reynolds number is large enough, the present simplified technique allows the use of time steps much larger than those allowed by the explicit time-advancement scheme, preserving the accuracy of the results.     

On Cartesian stiffness matrices in rigid body dynamics: an energetic perspective

Several Cartesian stiffness matrices for a single rigid body subject to a conservative force field are developed in this paper. The treatment is based on energetic arguments and an Euler angle parameterization of the rotation of the rigid body is employed. Several new representations for the stiffness matrix are obtained and the relation to other works on Cartesian stiffness matrices and Hessians is illuminated. Additional details are presented with respect to determining the Cartesian stiffness matrix for a pair of rigid bodies, as well as for a system of rigid bodies constrained to a plane.     

Discontinuous Galerkin method on three-dimensional tetrahedral grids: Using the operator programming method

In the numerical simulation of gas-dynamic flows in domains with a complex geometry, it is necessary to use detailed unstructured grids and highly accurate numerical methods. The Galerkin method with discontinuous base functions (or the discontinuous Galerkin method) works well in dealing with such problems. This technique has several advantages inherent both in finite-element and in finite-difference approximations. At the same time, the discontinuous Galerkin method is computationally complex; therefore, the question arises about the most efficient use of the full potential of computers. In order to speed up the computations, we applied the operator programming method to develop the computational module. It allows presenting mathematical formulas in programs in compact form and helps to port programs to parallel architectures such as NVidia CUDA and Intel Xeon Phi. Earlier the operator programming method was implemented for regular three-dimensional Cartesian grids and three-dimensional locally adaptive grids. In this work, this method is applied to threedimensional tetrahedral grids. This example demonstrates that the method in question can be efficiently implemented on arbitrary three-dimensional grids. Besides, we demonstrate the use of the template metaprogramming methods of the C++ programming language in order to speed up computations.     

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.     

A second-order curvilinear to Cartesian transformation of immersed interfaces and boundaries. Application to fictitious domains and multiphase flows

A global methodology dealing with fictitious domains of all kinds on curvilinear grids is presented. The main idea is to transform the curvilinear framework and its associated elements (velocity, immersed interfaces…) into a Cartesian grid. On such grids, many operations can be performed much faster than on curvilinear grids. The method is coupled with a Thread Ray-casting algorithm which works on Cartesian grids only. This algorithm computes quickly the Heaviside function related to the interior of an object on an Eulerian grid. The approach is also coupled with an immersed boundary method (L²-penalty) or with phase advection methods such as VOF–PLIC, VOF–TVD, Front-tracking or Level-set approaches. Applications, convergence and speed tests are performed for shape initializations, immersed boundary methods, and interface tracking.     

A unifying grid approach for solving potential flows applicable to structured and unstructured grid configurations

In this study, an efficient numerical method is proposed for unifying the structured and unstructured grid approaches for solving the potential flows. The new method, named as the “alternating cell directions implicit - ACDI”, solves for the structured and unstructured grid configurations equally well. The new method in effect applies a line implicit method similar to the Line Gauss Seidel scheme for complex unstructured grids including mixed type quadrilateral and triangle cells. To this end, designated alternating directions are taken along chains of contiguous cells, i.e. ‘cell directions’, and an ADI-like sweeping is made to update these cells using a Line Gauss Seidel like scheme. The algorithm makes sure that the entire flow field is updated by traversing each cell twice at each time step for unstructured quadrilateral grids that may contain triangular cells. In this study, a cell-centered finite volume formulation of the ACDI method is demonstrated. The solutions are obtained for incompressible potential flows around a circular cylinder and a forward step. The results are compared with the analytical solutions and numerical solutions using the implicit ADI and the explicit Runge-Kutta methods on single-and multi-block structured and unstructured grids. The results demonstrate that the present ACDI method is unconditionally stable, easy to use and has the same computational performance in terms of convergence, accuracy and run times for both the structured and unstructured grids.     

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.     

复杂外形三维非结构粘性直角网格的生成

在保证非结构网格逻辑关系不变的条件下,采用光顺、投影、分层加密的方法,解决直角网格物面不贴体的难点,建立三维非结构贴体直角网格生成系统,以适应粘性流场和湍流的计算,同时对自适应网格的流场计算进行研究。对航空航天领域几类复杂外形飞行器进行了三维非结构贴体直角网格的生成。