Free‐surface tracking in 2D with the harmonic polynomial cell method: Two alternative strategies

Several cases of nonlinear wave propagation are studied numerically in two dimensions within the framework of potential flow. The Laplace equation is solved with the harmonic polynomial cell (HPC) method, which is a field method with high‐order accuracy. In the HPC method, the computational domain is divided into overlapping cells. Within each cell, the velocity potential is represented by a sum of harmonic polynomials. Two different methods denoted as immersed boundary (IB) and multigrid (MG) are used to track the free surface. The former treats the free surface as an IB in a fixed Cartesian background grid, while the latter uses a free‐surface fitted grid that overlaps with a Cartesian background grid. The simulated cases include several nonlinear wave mechanisms, such as high steepness and shallow‐water effects. For one of the cases, a numerical scheme to suppress local wave breaking is introduced. Such scheme can serve as a practical mean to ensure numerical stability in simulations where local breaking is not significant for the result. For all the considered cases, both the IB and MG method generally give satisfactory agreement with known reference results. Although the two free‐surface tracking methods mostly have similar performance, some differences between them are pointed out. These include aspects related to modeling of particular physical problems as well as their computational efficiency when combined with the HPC method.     

A cut‐cell non‐conforming Cartesian mesh method for compressible and incompressible flow

This paper details a multigrid‐accelerated cut‐cell non‐conforming Cartesian mesh methodology for the modelling of inviscid compressible and incompressible flow. This is done via a single equation set that describes sub‐, trans‐, and supersonic flows. Cut‐cell technology is developed to furnish body‐fitted meshes with an overlapping mesh as starting point, and in a manner which is insensitive to surface definition inconsistencies. Spatial discretization is effected via an edge‐based vertex‐centred finite volume method. An alternative dual‐mesh construction strategy, similar to the cell‐centred method, is developed. Incompressibility is dealt with via an artificial compressibility algorithm, and stabilization achieved with artificial dissipation. In compressible flow, shocks are captured via pressure switch‐activated upwinding. The solution process is accelerated with full approximation storage (FAS) multigrid where coarse meshes are generated automatically via a volume agglomeration methodology. This is the first time that the proposed discretization and solution methods are employed to solve a single compressible–incompressible equation set on cut‐cell Cartesian meshes. The developed technology is validated by numerical experiments. The standard discretization and alternative methods were found equivalent in accuracy and computational cost. The multigrid implementation achieved decreases in CPU time of up to one order of magnitude. Copyright © 2007 John Wiley & Sons, Ltd.     

Turbulent flow in an intake-manifold

Three-dimensional, turbulent fluid flow analysis with computational methods has emerged as a viable tool in the design process of engine components for passenger cars. If applied in the early stages of the component development such analysis can help reduce the product development time drastically, and may support the design engineer to evaluate several competing design aspects before product completion. However the grid generation for complex geometries still poses a difficult and time consuming taks which strongly influences the accuracy of the numerical solution. The present paper studies the numerical solution of the flow in an inlet-manifold of a five-cylinder engine with two intake-ports for each cylinder. A fast and reliable grid generation technique is discussed in detail. The finite volume based Navier-Stokes solver with schemes of first and second order accuracy for the convective terms is used. Numerical results at several mass flow rates and different boundary conditions at the intake ports are compared with measurements. Finally, the computational results are discussed with respect to their applicability to support the design process of an inlet-manifold.     

Evaluation of three unstructured multigrid methods on 3D finite element problems in solid mechanics

Multigrid has been a popular solver method for finite element and finite difference problems with regular grids for over 20 years. The application of multigrid to unstructured grid problems, in which it is often difficult or impossible for the application to provide coarse grids, is not as well understood. In particular, methods that are designed to require only data that are easily available in most finite element applications (i.e. fine grid data), constructing the grid transfer operators and coarse grid operators internally, are of practical interest. We investigate three unstructured multigrid methods that show promise for challenging problems in 3D elasticity: (1) non‐nested geometric multigrid, (2) smoothed aggregation, and (3) plain aggregation algebraic multigrid. This paper evaluates the effectiveness of these three methods on several unstructured grid problems in 3D elasticity with up to 76 million degrees of freedom. Published in 2002 by John Wiley & Sons, Ltd.     

Multigrid approach to predictive wave-front reconstruction in adaptive optical systems

A computationally efficient approach, based on the principles of multigrid methods, to predictive wave-front reconstruction in adaptive optical systems is described. Local predictive estimators are computed by use of recursive least squares on multiple grids. Each grid is increasingly coarse, allowing for temporal prediction of the behavior of both high- and low-spatial-frequency aberrations. Example numerical simulation results are given, showing that implementing the recursive least-squares algorithm for predictive estimation in a multigrid fashion greatly accelerates convergence to the steady-state optimal estimator condition. By implementation of the multigrid predictive reconstructor in parallel, the computational cost of implementing a predictive wave-front reconstruction scheme that uses recursive least squares for each processor at each cycle can be reduced from [symbol: see text](m2) to [symbol: see text](2m), where m is the number of actuators.     

基于网格/无网格的三维超音速流场数值模拟

该文针对绕三维复杂外形流动的数值模拟,提出了一套基于笛卡尔网格/无网格的混合算法。该算法采用计算效率高的笛卡尔网格覆盖全场,而在计算物体表面及周围邻近区域布置无网格离散点。通过对重叠区域的划分和信息传递,建立各计算区域之间的耦合关系,形成完整的计算系统。为验证混合算法数值模拟的适用性和准确性,该文对超音速圆球绕流进行了计算,将激波的位置、形状与理论分析结果进行了对比。并进一步采用该算法对Ma=2.5、0°~10°攻角下的B1AC2R标准导弹模型进行了计算,获得的气动力系数与实验值基本吻合,表明该文提出的方法可以进行实际工程应用。     

A sub‐grid scale finite element agglomeration multigrid method with application to the Boltzmann transport equation

This article describes a new element agglomeration multigrid method for solving partial differential equations discretised through a sub‐grid scale finite element formulation. The sub‐grid scale discretisation resolves solution variables through their separate coarse and fine scales, and these are mapped between the multigrid levels using a dual set of transfer operators. The sub‐grid scale multigrid method forms coarse linear systems, possessing the same sub‐grid scale structure as the original discretisation, that can be resolved without them being stored in memory. This is necessary for the application of this article in resolving the Boltzmann transport equation as the linear systems become extremely large. The novelty of this article is therefore a matrix‐free multigrid scheme that is integrated within its own sub‐grid scale discretisation using dual transfer operators and applied to the Boltzmann transport equation. The numerical examples presented are designed to show the method's preconditioning capabilities for a Krylov space‐based solver. The problems range in difficulty, geometry and discretisation type, and comparisons made with established methods show this new approach to perform consistently well. Smoothing operators are also analysed and this includes using the generalized minimal residual method. Here, it is shown that an adaptation to the preconditioned Krylov space is necessary for it to work efficiently. Copyright © 2012 John Wiley & Sons, Ltd.     

A local grid refinement scheme for B-spline material point method

A local grid refinement scheme for the material point method with B-spline basis functions (BSMPM) is developed based on the concept of bridging domain approach. The fine grid is defined in the local large-deformation regions to accurately capture the complex material responses, whereas other spatial domains are discritized by coarse grids. In the overlapping domain between the fine and coarse grids, the constraint of particle displacements obtained with different grids is enforced using the Lagrange multiplier method to eliminate the spurious stress reflection at the fine/coarse grid interface. Representative numerical examples have shown that the BSMPM simulations with the proposed local grid refinement scheme could provide the solutions in good agreement with those obtained with the uniformly fine grid, and that no significant spurious stress reflection is induced at the fine/coarse grid interface, even for the bridging domain size as small as the cell size of the fine grid. It is also found that the proposed local grid refinement method can significantly reduce the BSMPM computational time compared with the cases for uniformly fine grids. A multitime-step algorithm is presented and shown to considerably enhance the efficiency of the present local grid refinement scheme with no compromise in accuracy.     

代数多重网格法在岩体力学有限元分析中的应用

代数多重网格法具有存贮量小、收敛精度高和计算时间少等优点,将代数多重网格方法引入到岩体力学有限元计算领域,论述了基于单元聚集和能量极小意义下适于岩体力学有限元求解的代数多重网格粗化策略与插值算子,并详细描述了相应的代数多重网格算法.数值试验表明:在岩体力学与工程问题的有限元数值计算中,代数多重网格求解法是高效的、适用的,较直接法和其他常用迭代方法具有明显的优越性.     

A multigrid scheme for the implicit solution of the compressible flow equations

A multigrid scheme has been developed for the acceleration of the solution of compressible inviscid and viscous flow problems. A higher order accurate upwind conservative finite volume scheme has been used for the discretization of the Euler and the Reynolds-Averaged Navier-Stokes equations. For the multigrid implementation, the alternative point of view of the Full Approximation scheme has been employed together with a conservative restriction operator to maintain the fine grid accuracy. The present multigrid scheme has been designed to take full advantage of the implicit unfactored solution scheme of the single grid code by introducing an alternative multigrid V-cycle. The proposed method attains up to 25-fold acceleration with respect to the single grid solution for moderate size grids. Moreover, the results demonstrate that the computational time increases proportional to the number of volumes when global refinements are applied so, the present multigrid scheme is very favorable for large scale computations.This work was partially supported by the EC-programme: BRITE/ EURAM Project Aero 0018. The authors would like to thank MBB GmbH for providing NsFlex code     

A feature‐preserving volumetric technique to merge surface triangulations

Several extensions and improvements to surface merging procedures based on the extraction of iso‐surfaces from a distance map defined on an adaptive background grid are presented. The main objective is to extend the application of these algorithms to surfaces with sharp edges and corners. In order to deal with objects of different length scales, the initial background grids are created using a Delaunay triangulation method and local voxelizations. A point enrichment technique that introduces points into the background grid along detected surface features such as ridges is used to ensure that these features are preserved in the final merged surface. The surface merging methodology is extended to include other Boolean operations between surface triangulations. The iso‐surface extraction algorithms are modified to obtain the correct iso‐surface for multi‐component objects. The procedures are demonstrated with various examples, ranging from simple geometrical entities to complex engineering applications. The present algorithms allow realistic modelling of a large number of complex engineering geometries using overlapping components defined discretely, i.e. via surface triangulations. This capability is very useful for grid generation starting from data originated in measurements or images. Copyright © 2002 John Wiley & Sons, Ltd.     

A locally analytic technique applied to grid generation by elliptic equations

One technique for obtaining grids for irregular geometries is to solve sets of elliptic partial differential equations. The solution of the partial differential equations yields a grid which discretizes the physical solution domain and also a transformation for the irregular physical domain to a regular computational domain. Expressing the governing equation of interest in the computational domain requires the derivatives of the physical to computational domain transformation, i.e., the metrics. These metrics are typically determined by numerical differentiation, which is a potential source of error. The locally analytic method uses the analytic solution of the locally linearized equation to develop numerical stencils. Thus, the locally analytic method allows numerical differentiation with no loss of accuracy. In this paper, the locally analytic method is applied to the solution of the Poisson and Brackbill–Saltzman equations. Comparison with an exact solution shows the locally analytic method to be more accurate than the finite difference method, both in solving the partial differential equation and evaluating the metrics. However, it is more computationally expensive.     

Repairing unstructured triangular mesh intersections

Computational analysis and design has become a fundamental part of product research, development, and manufacturing in aerospace, automotive, and other industries. In general, the success of the specific application depends heavily on the accuracy and consistency of the computational model used. The aim of this work is to reduce the time needed to prepare geometry for volume grid generation. This will be accomplished by developing tools that semi‐automatically repair discrete data. Providing another level of automation to the process of repairing large, complex problems in discrete data will significantly accelerate the grid generation process. The developed algorithms are meant to offer a semi‐automated solution to a complicated geometrical problem — specifically discrete mesh intersection. The intersection‐repair strategy presented here focuses on repairing the intersection in‐place as opposed to rediscretizing the intersecting geometries. Combining robust, efficient methods of detecting intersections and then repairing intersecting geometries in‐place produces a significant improvement over techniques used in current literature. The result of this intersection process is a nonintersecting geometry that is free of duplicate and degenerate geometry. Results are presented showing the accuracy and consistency of the intersection repair tool. Copyright © 2012 John Wiley & Sons, Ltd.     

Accelerated and robust finite volume Navier-Stokes solver for all speeds

The paper describes the results obtained from a multigrid accelerated Navier-Stokes solver. The method is based on 2-D explicit cell-centred finite volume Reynolds averaged Navier-Stokes (N-S) flow solver for speeds from near-incompressible Mach numbers to high hypersonic Mach numbers including flows at high angles of attack. The time integration is done using a hybrid 5-stage Runge-Kutta local time stepping scheme. With the help of a simple technique, the capability of the Jameson-Schmidt-Turkel numerical dissipation scheme has been enhanced to include hypersonic flows. The iterative procedure is accelerated significantly by incorporating a multigrid technique which has been used in all computations up to about supersonic speeds. Systematic numerical experiments were conducted to evolve guidelines to generate airfoil grid that could offer reliable flow simulations. The computed results are in very good agreement with experimental data where available, especially from the point of view of predicting large suction peaks and shock locations where considerable departures are often seen in the literature. Further, the highly accelerated computations make this code a useful tool of practical interest in preliminary aerodynamic design.     

Multigrid preconditioned conjugate-gradient method for large-scale wave-front reconstruction

We introduce a multigrid preconditioned conjugate-gradient (MGCG) iterative scheme for computing open-loop wave-front reconstructors for extreme adaptive optics systems. We present numerical simulations for a 17-m class telescope with n = 48756 sensor measurement grid points within the aperture, which indicate that our MGCG method has a rapid convergence rate for a wide range of subaperture average slope measurement signal-to-noise ratios. The total computational cost is of order n log n. Hence our scheme provides for fast wave-front simulation and control in large-scale adaptive optics systems.     

Vibration of functionally graded carbon nanotube reinforced quadrilateral plates using geometric transformation discrete singular convolution method

Free vibration characteristics of thick skew plates reinforced by functionally graded carbon nanotubes (CNTs) reinforced composite are presented. Discrete singular convolution (DSC) method is used for the numerical solution of vibration problems via geometric mapping technique. Using the geometric transformation via a four-node element, the straight-sided quadrilateral physical domain is mapped into a square domain in the computational space. Then the method of discrete singular convolution with some singular kernels such as Regularized Shannon's delta (RSD) and Lagrange's delta kernels (LDK) have been used for spatial discretizing of the resulting governing equation of motion. Calculated results have been presented in order to show the effects of volume fraction of CNT, skew angles, CNT distribution types, plate aspect ratio and length-to-thickness ratio on the frequency of CNT reinforced skew plate. The current results are compared with the related results available in the literature and obtained by different methods. It is shown that reasonable accurate results are obtained for free vibration of nanocomposite plates with less computational effort for higher modes. Several test examples for different plate have been selected to demonstrate the convergence properties, accuracy, and simplicity in numerical implementation of DSC procedures. This approach has verified the accuracy and applicability of DSC method to the class of problem considered in this study. Furthermore, in the numerical examples in the scope of the study, the results obtained with DSC method using a coarser grid are more accurate than the values obtained by finite elements and differential quadrature (DQ) methods. It is also revealed that the method of discrete singular convolution is a promising and potential approach for computational mechanics of nonrectangular plates with nanocomposite reinforced.     

An X‐FEM and level set computational approach for image‐based modelling: Application to homogenization

The advances in material characterization by means of imaging techniques require powerful computational methods for numerical analysis. The present contribution focuses on highlighting the advantages of coupling the extended finite elements method and the level sets method, applied to solve microstructures with complex geometries. The process of obtaining the level set data starting from a digital image of a material structure and its input into an extended finite element framework is presented. The coupled method is validated using reference examples and applied to obtain homogenized properties for heterogeneous structures. Although the computational applications presented here are mainly two‐dimensional, the method is equally applicable for three‐dimensional problems. Copyright © 2010 John Wiley & Sons, Ltd.