Parallel adaptive mesh generation

Unstructured meshes have proved to be a powerful tool for adaptive remeshing of finite element idealizations. This paper presents a transputer-based parallel algorithm for two dimensional unstructured mesh generation. A conventional mesh generation algorithm for unstructured meshes is reviewed by the authors, and some program modules of sequential C source code are given. The concept of adaptivity in the finite element method is discussed to establish the connection between unstructured mesh generation and adaptive remeshing.After these primary concepts of unstructured mesh generation and adaptivity have been presented, the scope of the paper is widened to include parallel processing for un-structured mesh generation. The hardware and software used is described and the parallel algorithms are discussed. The Parallel C environment for processor farming is described with reference to the mesh generation problem. The existence of inherent parallelism within the sequential algorithm is identified and a parallel scheme for unstructured mesh generation is formulated. The key parts of the source code for the parallel mesh generation algorithm are given and discussed. Numerical examples giving run times and the consequent “speed-ups” for the parallel code when executed on various numbers of transputers are given. Comparisons between sequential and parallel codes are also given. The “speed-ups” achieved when compared with the sequential code are significant. The “speed-ups” achieved when networking further transputers is not always sustained. It is demonstrated that the consequent “speed-up” depends on parameters relating to the size of the problem.     

Time-accurate Navier-Stokes simulation of vortex convection using an unstructured dynamic mesh procedure

A two-dimensional Navier-Stokes flow solver is developed for the simulation of unsteady flows on unstructured adaptive meshes. The solver is based on a second-order accurate implicit time integration using a point Gauss-Seidel relaxation scheme and a dual time-step subiteration. A vertex-centered, finite-volume discretization is used in conjunction with Roe’s flux-difference splitting. The Spalart-Allmaras one equation model is employed for the simulation of turbulence. An unsteady solution-adaptive dynamic mesh scheme is used by adding and deleting mesh points to take account of spatial and temporal variations of the flowfield. Unsteady viscous flow for a traveling vortex in a free stream is simulated to validate the accuracy of the dynamic mesh adaptation procedure. Flow around a circular cylinder and two blade-vortex interaction problems are investigated for demonstration of the present method. Computed results show good agreement with existing experimental and computational results. It was found that unsteady time-accurate viscous flows can be accurately simulated using the present unstructured dynamic mesh adaptation procedure.     

Parallel three-dimensional direct simulation Monte Carlo method and its applications

The development of a parallel three-dimensional direct simulation Monte Carlo (DSMC) method using unstructured cells is reported. Variable hard sphere molecular model and no time counter method are used for the molecular collision kinetics, while the cell-by-cell ray-tracing technique is implemented for particle movement. Developed serial code has been verified by comparing the results of a supersonic corner flow with those of Bird’s three-dimensional structured DSMC code. In addition, a benchmark test is performed for an orifice expanding flow to verify the parallel implementation of DSMC method by comparing with available experimental data. Static physical domain decomposition is used to distribute the workload among multiple processors by considering the estimated particle weighting distribution. Two-step multi-level graph partitioning technique is used to perform the required domain decomposition. Completed code is then applied to compute a hypersonic flow over a sphere (external flow) and the flow field of a spiral drag pump (internal flow), respectively. Results of the former are in good agreement with previous numerical results using axisymmetric DSMC method and experimental data. Results of the latter also agree well with previous numerical results.     

A Parallel Particle Tracking Framework for Applications in Scientific Computing

Particle tracking methods are a versatile computational technique central to the simulation of a wide range of scientific applications. In this paper, we present a new parallel particle tracking framework for the applications of scientific computing. The framework includes the in-element particle tracking method, which is based on the assumption that particle trajectories are computed by problem data localized to individual elements, as well as the dynamic partitioning of particle-mesh computational systems. The ultimate goal of this research is to develop a parallel in-element particle tracking framework capable of interfacing with a different order of accuracy of ordinary differential equation (ODE) solver. The parallel efficiency of such particle-mesh systems depends on the partitioning of both the mesh elements and the particles; this distribution can change dramatically because of movement of the particles and adaptive refinement of the mesh. To address this problem we introduce a combined load function that is a function of both the particle and mesh element distributions. We present experimental results that detail the performance of this parallel load balancing approach for a three-dimensional particle-mesh test problem on an unstructured, adaptive mesh, and demonstrate the ability of interfacing with different ODE solvers.     

Parallel unstructured tetrahedral mesh adaptation: algorithms,implementation and scalability

The use of unstructured adaptive tetrahedral meshes in the solution of transient flows poses a challenge for parallel computing due to the irregular and frequently changing nature of the data and its distribution. A parallel mesh adaptation algorithm, PTETRAD, for unstructured tetrahedral meshes (based on the serial code TETRAD) is described and analysed. The portable implementation of the parallel code in C with MPI is described and discussed. The scalability of the code is considered, analysed and illustrated by numerical experiments using a shock wave diffraction problem. Copyright © 1999 John Wiley & Sons, Ltd.     

Three dimensional discontinuous Galerkin methods for Euler equations on adaptive conforming meshes

In the numerical simulation of three dimensional fluid dynamical equations, the huge computational quantity is a main challenge. In this paper, the discontinuous Galerkin (DG) finite element method combined with the adaptive mesh refinement (AMR) is studied to solve the three dimensional Euler equations based on conforming unstructured tetrahedron meshes, that is according the equation solution variation to refine and coarsen grids so as to decrease total mesh number. The four space adaptive strategies are given and analyzed their advantages and disadvantages. The numerical examples show the validity of our methods.     

An adaptive Rothe method for the wave equation

The adaptive Rothe method approaches a time-dependent PDE as an ODE in function space. This ODE is solved virtually using an adaptive state-of-the-art integrator. The actual realization of each time-step requires the numerical solution of an elliptic boundary value problem, thus perturbing the virtual function space method. The admissible size of that perturbation can be computed a priori and is prescribed as a tolerance to an adaptive multilevel finite element code, which provides each time-step with an individually adapted spatial mesh. In this way, the method avoids the well-known difficulties of the method of lines in higher space dimensions. During the last few years the adaptive Rothe method has been applied successfully to various problems with infinite speed of propagation of information. The present study concerns the adaptive Rothe method for hyperbolic equations in the model situation of the wave equation. All steps of the construction are given in detail and a numerical example (diffraction at a corner) is provided for the 2D wave equation. This example clearly indicates that the adaptive Rothe method is appropriate for problems which can generally benefit from mesh adaptation. This should be even more pronounced in the 3D case because of the strong Huygens' principle. Accepted: 12 August 1997     

Multidisciplinary Simulation of the Maneuvering of an Aircraft

A computational methodology for the simulation of the transient aeroelastic response of an unrestrained and flexible aircraft during high-G maneuvers is presented. The key components of this methodology are: (a) a three-field formulation for coupled fluid/structure interaction problems; (b) a second-order time-accurate and geometrically conservative flow solver for CFD computations on unstructured dynamic meshes; (c) a corotational finite element method for the solution of geometrically nonlinear and unrestrained structural dynamics problems; (d) a robust method for updating an unrestrained and unstructured moving fluid mesh; and (e) a second-order time-accurate staggered algorithm for time-integrating the coupled fluid/structure semi-discrete equations of motion. This computational methodology is illustrated with the simulation on a parallel processor of several three-dimensional high-G pullup maneuvers of the Langley Fighter in the transonic regime, using a detailed finite element aeroelastic model.     

A systematic approach to unstructured mesh generation for ocean modelling using GMT and Terreno

A systematic approach to unstructured mesh generation for ocean modelling is presented. The method optimises unstructured meshes to approximate bathymetry to a user specified accuracy which may be defined as a function of longitude, latitude and bathymetry. GMT (Generic Mapping Tools) is used to perform the initial griding of the bathymetric data. Subsequently, the Terreno meshing package combines automated shoreline approximation, mesh gradation and optimisation methods to generate high-quality bathymetric meshes. The operation of Terreno is based upon clearly defined error measures and this facilitates the automation of unstructured mesh generation while minimising user intervention and the subjectivity that this can introduce.     

Discontinuous upwind residual distribution: A route to unconditional positivity and high order accuracy

This paper proposes an approach to the approximation of time-dependent hyperbolic conservation laws which is both second order accurate in space and time (for any sufficiently smooth solution profile, even one containing turning points) and free of spurious oscillations for any time-step. The numerical algorithm is based on the concept of fluctuation distribution, applied on a space-time mesh of triangular prisms, for which second order accurate schemes already exist which are oscillation-free if the time-step satisfies a CFL-type constraint. This restriction is lifted here by combining the concept of a two-layer scheme with a representation of the solution which is allowed to be discontinuous-in-time. Numerical results are presented in two space dimensions, using unstructured meshes of space-time triangular prisms, for the scalar advection equation, Burgers’ equation and the Euler equations of gas dynamics.     

Study of subsonic�Csupersonic gas flow through micro/nanoscale nozzles using unstructured DSMC solver

We use an extended direct simulation Monte Carlo (DSMC) method, applicable to unstructured meshes, to numerically simulate a wide range of rarefaction regimes from subsonic to supersonic flows through micro/nanoscale converging–diverging nozzles. Our unstructured DSMC method considers a uniform distribution of particles, employs proper subcell geometry, and follows an appropriate particle tracking algorithm. Using the unstructured DSMC, we study the effects of back pressure, gas/surface interactions (diffuse/specular reflections), and Knudsen number on the flow field in micro/nanoscale nozzles. If we apply the back pressure at the nozzle outlet, a boundary layer separation occurs before the outlet and a region with reverse flow appears inside the boundary layer. Meanwhile, the core region of inviscid flow experiences multiple shock-expansion waves. In order to accurately simulate the outflow, we extend a buffer zone at the nozzle outlet. We show that a high viscous force creation in the wall boundary layer prevents any supersonic flow formation in the divergent part of the nozzle if the Knudsen number exceeds a moderate magnitude. We also show that the wall boundary layer prevents forming any normal shock in the divergent part. In reality, Mach cores would appear at the nozzle center followed by bow shocks and expansion region. We compare the current DSMC results with the solution of the Navier–Stokes equations subject to the velocity slip and temperature jump boundary conditions. We use OpenFOAM as a compressible flow solver to treat the Navier–Stokes equations.     

Parallel Unstructured Dynamic Grid Direct Monte Carlo Simulation of Molecular Gas Dynamics and Its Applications

A new parallel dynamic unstructured grid DSMC method is presented in this paper. The code developed has been applied to the simulation of thin film deposition over microstructures. Surface deformation in such cases poses a challenge for accurate evaluation of gas flow due to the fact that the deposited film thickness is comparable to the feature size. In this study a method is developed to move the mesh at run time. Since in parallel simulation each partition moves independently of the others, a parallel version of moving mesh is proposed to synchronize the displacement of the neighboring partitions, so that there is a smooth transition from one partition to another. An efficient tool for tracking particles during simulation is also presented. Furthermore, the influence of parameters, such as sticking coefficient and aspect ratio on step coverage for a 1m wide trench by sputter deposition was studied. The results showed that the step coverage deteriorated with increasing sticking coefficient and aspect ratio.     

多介质拉氏自适应流体动力学软件LAD2D研制及其应用

拉氏方法是内爆动力学过程数值模拟的主要方法.针对高温、高压、多介质和大变形等内爆问题,采用非结构任意多边形网格底层管理、计算过程中网格邻域可变技术,以及拉氏自适应网格加密方法和层次化、模块化程序设计思想,自主研发非结构拉氏自适应网格流体动力学软件LAD2D.从物理模型、计算方法、程序设计、程序验证与确认、大变形问题数值模拟等方面系统地介绍LAD2D.LAD2D对多介质爆轰弹塑性流体大变形问题有很强的适应能力.     

网格缩减的自适应hp伪谱法

针对控制变量不连续的最优控制问题,本文提出一种自适应更新的忉伪谱法,这种方法在(Legendre Gauss Radau,LGR)点处取配点,能够以较小的网格规模获得较高的精度.通过计算相对误差估计,判断网格规模是增加还是缩减,若相对容许误差大于给定值,则增加网格区间数或网格配点数提高解的精度,反之则合并网格或减小网格配点数缩减网格规模提高计算效率.将hp伪谱法应用于最优控制问题,仿真验证了hp伪谱法的优越性.     

An appropriate mesh representation for tetrahedral finite volume computations

The amounts of information used for the solution of three-dimensional partial differential equations in engineering applications is formidable. Part of the problem is that in unstructured meshes (adequate for advanced numerical techniques) the information is local in nature. An alternate representation of an unstructured tetrahedral mesh is proposed. Besides having some storage advantages over representations commonly used for finite volume computations, we show that preprocessing complexity is lower than in the conventional representation. The resulting simplified structure may result in significant gains for applications that go through a number of mesh refinement cycles.     

Coupled solid (FVM)–fluid (DSMC) simulation of micro-nozzle with unstructured-grid

The flow field and temperature distributions of free molecular micro-electro-thermal resist jet (FMMR) were studied resorting to DSMC–FVM coupled method. Direct simulation Monte Carlo (DSMC) method is the most useful tool to simulate the flow field of FMMR and unstructured grid is suitable for the flow simulation in a complicated region with tilted wall surface. DSMC code based on unstructured grid system was developed and the result was compared with that of structured grid and analytical solution to validate the reliability of the developed code. The DSMC method was then used to simulate the fluid flow in the micro-nozzle (Kn > 0.01) and the temperature distribution in the nozzle wall was obtained by the finite volume method (FVM). The Dirichlet–Neumann method was used to couple the wall heat flux and temperature between flow field and solid area. The effect of different income pressure was studied in detail and the results showed that the temperature of solid area changed drastically at different income pressure, so the commonly-adopted method of pre-setting boundary temperature before simulation was unreasonable.     

Development of a parallel Poisson's equation solver with adaptive mesh refinement and its application in field emission prediction

A parallel electrostatic Poisson's equation solver coupled with parallel adaptive mesh refinement (PAMR) is developed in this paper. The three-dimensional Poisson's equation is discretized using the Galerkin finite element method using a tetrahedral mesh. The resulting matrix equation is then solved through the parallel conjugate gradient method using the non-overlapping subdomain-by-subdomain scheme. A PAMR module is coupled with this parallel Poisson's equation solver to adaptively refine the mesh where the variation of potentials is large. The parallel performance of the parallel Poisson's equation is studied by simulating the potential distribution of a CNT-based triode-type field emitter. Results with ∼100 000 nodes show that a parallel efficiency of 84.2% is achieved in 32 processors of a PC-cluster system. The field emission properties of a single CNT triode- and tetrode-type field emitter in a periodic cell are computed to demonstrate their potential application in field emission prediction.     

Parallel adaptive solution of 3D boundary value problems by Hessian recovery

A parallel technique for an adaptive solution of 3D boundary value problems is described. It incorporates a parallel mesh generation and a parallel iterative solution of the corresponding discrete problem. Both generation and solution are problem independent and may be considered as black-boxes.     

Subdomain generation for non-convex parallel finite element domains

In this paper the Subdomain Generation Method (SGM), originally formulated in Khan & Topping (1993; Khan, A. I. & Topping, B. H. V., Subdomain generation for parallel finite element analysis. Comput. Syst. Engng, 1993, 4(4/6), 473–488) for convex finite element domains, is generalized for arbitrary shaped domains. Modifications to the original SGM are described which allow partitioning of non-convex domains. These modifications have been made to the formulation of the optimization module and the predictive module. The examples presented in Khan & Topping (1993) have been re-worked and two more examples have been added which demonstrate the application of the method to arbitrary shaped domains. It is shown with the aid of the examples that the method provides well-balanced subdomains very efficiently and allows parallel adaptive mesh generation. The method in its present form may be used to partition unstructured graphs in two or three dimensions. Since the computational cost for the mesh partitioning with this method depends solely upon the initial coarse mesh, hence the computational cost does not increase with the increase in the mesh density of the final mesh. The method in its present form is unsuitable for relatively coarse grained parallel computers, however the modifications which would impart a greater degree of scalability to this method are discussed.     

A posteriori error estimation and adaptive mesh refinement for combined thermal-stress finite element analysis

Local and global error estimators and an associated h-based adaptive mesh refinement schemes are proposed for coupled thermal-stress problems. The error estimators are based on the “flux smoothing” technique of Zienkiewicz and Zhu with important modifications to improve convergence performance and computational efficiency. Adaptive mesh refinement is based on the concept of adaptive accuracy criteria, previously presented by the authors for stress-based problems and extended here for coupled thermal-stress problems. Three methods of mesh refinement are presented and numerical results indicate that the proposed method is the most efficient in terms of number of adaptive mesh refinements required for convergence in both the thermal and stress solutions. Also, the proposed method required a smaller number of active degrees of freedom to obtain an accurate solution.