A high-resolution method for the depth-integrated solute transport equation based on an unstructured mesh

This paper presents a high-resolution numerical method for solving mass transport problems involving advection and anisotropic diffusion in shallow water based on unstructured mesh. An alternating operator-splitting technique is adopted to advance the numerical solution with advection and diffusion terms solved separately in two steps. By introducing a new r-factor into the Total Variation Diminishing (TVD) limiter, an improved finite-volume method is developed to solve the advection term with significant reduction of numerical diffusion and oscillation errors. In addition, a coordinate transformation is introduced to simplify the diffusion term with the Green-Gauss theorem used to deal with the anisotropic effect based on unstructured mesh. The new scheme is validated against three benchmark cases with separated and combined advection and diffusion transport processes involved. Results show that the scheme performs better than existing methods in predicting the advective transport, particularly when a sharp concentration front is in presence. The model also provides a sound solution for the anisotropic diffusion phenomenon. Anisotropic diffusion has been largely neglected by existing flow models based on unstructured mesh, which usually treat the diffusion process as being isotropic for simplicity. Based on the flow field provided by the ELCIRC model, the developed transport model was successfully applied to simulate the transport of a hypothetical conservative tracer in a bay under the influence of tides.     

A fully discrete ε-uniform method for singular perturbation problems on equidistant meshes

We propose a fully discrete ε-uniform finite-difference method on an equidistant mesh for a singularly perturbed two-point boundary-value problem (BVP). We start with a fitted operator method reflecting the singular perturbation nature of the problem through a local BVP. However, to solve the local BVP, we employ an upwind method on a Shishkin mesh in local domain, instead of solving it exactly. Thus, we show that it is possible to develop a ε-uniform method, totally in the context of finite differences, without solving any differential equation exactly. We further study the convergence properties of the numerical method proposed and prove that it nodally converges to the true solution for any ε. Finally, a set of numerical experiments is carried out to validate the theoretical results computationally.     

Implicit linearity preservation type schemes for moving meshes

We are concerned with the accurate implicit approximation of compressible flows in a fixed and moving mesh context, such as piston engine flows. Geometries are commonly complex and flows compressible. Therefore, it is convenient to develop the numerical approach in the context of a space-time finite-volume formulation for unstructured meshes. The hyperbolic flux is obtained by a generalized Riemann solver taking into account the mesh motion. Using the linearity preservation property we propose a new class of stable implicit schemes developing low numerical viscosity. These schemes can be viewed as a correction of the usual MUSCL flux, induced by the time derivative and mesh motion. Accurate numerical results are obtained for transonic (shock tube) as well as low Mach number flows (diesel engine). It is numerically proved, that for large time steps, those approximations can be as accurate as some explicit schemes. The proposed schemes, due the compactness of the stencils, are well adapted for parallelization strategy.     

Adaptive computational chemotaxis based on field in bacterial foraging optimization

Bacterial foraging optimization (BFO) is predominately used to find solutions for real-world problems. One of the major characteristics of BFO is the chemotactic movement of a virtual bacterium that models a trial solution of the problems. It is pointed out that the chemotaxis employed by classical BFO usually results in sustained oscillation, especially on rough fitness landscapes, when a bacterium cell is close to the optima. In this paper we propose a novel adaptive computational chemotaxis based on the concept of field, in order to accelerate the convergence speed of the group of bacteria near the tolerance. Firstly, a simple scheme is designed for adapting the chemotactic step size of each field. Then, the scheme chooses the fields which perform better to boost further the convergence speed. Empirical simulations over several numerical benchmarks demonstrate that BFO with adaptive chemotactic operators based on field has better convergence behavior, as compared against other meta-heuristic algorithms.     

A knowledge base for finite element mesh design

The finite element method (FEM) is the most successful numerical method, that is used extensively by engineers to analyse stresses and deformations in physical structures. These structures should be represented as a finite element mesh. Defining an appropriate geometric mesh model that ensures low approximation errors and avoids unnecessary computational overheads is a very difficult and time consuming task. It is the major bottleneck in the FEM analysis process. The inductive logic programming system GOLEM has been employed to construct the rules for deciding about the appropriate mesh resolution. Five cylindrical mesh models have been used as a source of training examples. The evaluation of the resulting knowledge base shows that conditions in the domain are well represented by the rules, which specify the required number of the finite elements on the edges of the structures to be analysed using FEM. A comparison between the results obtained by this knowledge base and conventional mesh generation techniques confirms that the application of inductive logic programming is an effective approach to solving the problem of mesh design.     

Upwind-Difference Potentials Method for Patlak-Keller-Segel Chemotaxis Model

We develop a novel upwind-difference potentials method for the Patlak-Keller-Segel chemotaxis model that can be used to approximate problems in complex geometries. The chemotaxis model under consideration is described by a system of two nonlinear PDEs: a convection-diffusion equation for the cell density coupled with a reaction-diffusion equation for the chemoattractant concentration. Chemotaxis is an important process in many medical and biological applications, including bacteria/cell aggregation and pattern formation mechanisms, as well as tumor growth. Furthermore modeling of real biomedical problems often has to deal with the complex structure of computational domains. There is consequently a need for accurate, fast, and computationally efficient numerical methods for different chemotaxis models that can handle arbitrary geometries. The upwind-difference potentials method proposed here handles complex domains with the use of only Cartesian meshes, and can be easily combined with fast Poisson solvers. In the numerical tests presented below we demonstrate the robustness of the proposed scheme.     

Second-Order Accurate Godunov Scheme for Multicomponent Flows on Moving Triangular Meshes

This paper presents a second-order accurate adaptive Godunov method for two-dimensional (2D) compressible multicomponent flows, which is an extension of the previous adaptive moving mesh method of Tang et al. (SIAM J. Numer. Anal. 41:487–515, 2003) to unstructured triangular meshes in place of the structured quadrangular meshes. The current algorithm solves the governing equations of 2D multicomponent flows and the finite-volume approximations of the mesh equations by a fully conservative, second-order accurate Godunov scheme and a relaxed Jacobi-type iteration, respectively. The geometry-based conservative interpolation is employed to remap the solutions from the old mesh to the newly resulting mesh, and a simple slope limiter and a new monitor function are chosen to obtain oscillation-free solutions, and track and resolve both small, local, and large solution gradients automatically. Several numerical experiments are conducted to demonstrate robustness and efficiency of the proposed method. They are a quasi-2D Riemann problem, the double-Mach reflection problem, the forward facing step problem, and two shock wave and bubble interaction problems.     

Numerical solution of potential flow equations with a predictor-corrector finite difference method

We develop a numerical solution algorithm of the nonlinear potential flow equations with the nonlinear free surface boundary condition.A finite difference method with a predictor-corrector method is applied to solve the nonlinear potential flow equations in a two-dimensional (2D) tank.The irregular tank is mapped onto a fixed square domain with rectangular cells through a proper mapping function.A staggered mesh system is adopted in a 2D tank to capture the wave elevation of the transient fluid.The finite difference method with a predictor-corrector scheme is applied to discretize the nonlinear dynamic boundary condition and nonlinear kinematic boundary condition.We present the numerical results of wave elevations from small to large amplitude waves with free oscillation motion,and the numerical solutions of wave elevation with horizontal excited motion.The beating period and the nonlinear phenomenon are very clear.The numerical solutions agree well with the analytical solutions and previously published results.     

基于无网格径向点插值方法的简谐激励下的连续体结构拓扑优化

针对用有限元法进行连续体结构拓扑优化时需不断重构网格来处理网格畸变和网格移动,且存在数值计算不稳定等问题,基于无网格径向点插值方法(Radial Point Interpolation Method,RPIM)对简谐激励下的连续体结构进行拓扑优化.选取节点的相对密度作为设计变量,以结构动柔度最小化为目标函数,基于带惩罚的各向同性固体微结构(Solid Isotropic Microstructure with Penalization,SIMP)模型建立简谐激励下的优化模型;采用伴随法求解得到目标函数的敏度分析公式;利用优化准则法求解优化模型.经典的二维连续体结构拓扑优化算例证明该方法的可行性和有效性.     

Guaranteed-Quality Simplical Mesh Generation with Cell Size and Grading Control

Unstructured mesh quality, as measured geo-metrically, has long been known to influence solution accuracy and efficiency for finite-element and finite-volume simulations. Recent guaranteed-quality unstructured meshing algorithms are therefore welcome tools. However, these algorithms allow no explicit control over mesh resolution or grading. We define a geometric length scale, similar in principle to the local feature size, which allows automatic global control of mesh resolution and grading. We describe how to compute this length scale efficiently and modify Ruppert’s two-dimensional and Shewchuk’s three-dimensional meshing algorithms to produce meshes matching our length scale. We provide proofs of mesh quality, good grading, and size optimality for both two- and three-dimensions, and present examples, including comparison with existing schemes known to generate good-quality meshes.     

Application of normalized flux in pressure-based algorithm

The paper describes an implementation of normalized flux diagram (NFD) scheme into a pressure-based implicit finite-volume procedure to solve the Euler and Navier-Stokes equations on a non-orthogonal mesh with collocated finite-volume formulation. The newly developed algorithm has two new features: (i) the use of the normalized flux and space formulation (NFSF) methodology to bound the convective fluxes and (ii) the use of a high-resolution scheme in calculating interface density values to enhance the shock-capturing property of the algorithm. The procedure incorporates the k-ε eddy-viscosity turbulence model. The algorithm is first tested for inviscid flows at different Mach numbers ranging from subsonic to supersonic on a bump in channel geometry and inside a planar convergent-divergent nozzle. The results have been compared with those using the same scheme in conjunction with primitive variable limiter (NVD). Also, there has been comparison between the results and predicted data using TVD scheme on the basis of characteristic variable. After these comparison, it was found that the limiter on flux, predicted a sharper shock and there was better boundedness here than limiter on primitive variables in coarse mesh. The method is then validated against experimental data for the case of turbulent transonic flow passing via a gas turbine rotor blade cascade for which wind-tunnel experimental data exist. Findings show a remarkable quality of resolution when NFD scheme is used.     

Online Mesh Refinement for Parallel Atmospheric Models

Forecast precisions of climatological models are limited by computing power and time available for the executions. As more and faster processors are used in the computation, the resolution of the mesh adopted to represent the Earth’s atmosphere can be increased, and consequently the numerical forecast is more accurate. However, a finer mesh resolution, able to include local phenomena in a global atmosphere integration, is still not possible due to the large number of data elements to compute in this case. To overcome this situation, different mesh refinement levels can be used at the same time for different areas of the domain. Thus, our paper evaluates how mesh refinement at run time (online) can improve performance for climatological models.The online mesh refinement (OMR) increases dynamically mesh resolution in parts of a domain,when special atmosphere conditions are registered during the execution. Experimental results show that the execution of a model improved by OMR provides better resolution for the meshes, without any significant increase of execution time. The parallel performance of the simulations is also increased through the creation of threads in order to explore different levels of parallelism.     

Classifier-independent feature selection on the basis of divergence criterion

Feature selection aims to choose a feature subset that has the most discriminative information from the original feature set. In practical cases, it is preferable to select a feature subset that is universally effective for any kind of classifier because there is no underlying information about a given dataset. Such a trial is called classifier-independent feature selection. We took notice of Novovičová et al.’s study as a classifier-independent feature selection method. However, the number of features have to be selected beforehand in their method. It is more desirable to determine a feature subset size automatically so as to remove only garbage features. In this study, we propose a divergence criterion on the basis of Novovičová et al.’s method.     

一种基于IMM方法的气囊折叠参数化设计的应用

为提高驾驶员侧环向折叠气囊模型的生成效率,开发了基于IMM算法的参考网格模型和映射网格模型自动生成程序。程序在VC＋＋环境中开发,对模型实现了基于气袋直径、折叠次数、折叠环间距和网格密度等参数的参数化设计,并通过算例对建模过程和映射算法进行了验证。结果表明此程序能够快速、准确地生成参考网格模型和映射网格模型。极大地缩短模型生成时间,同时保证有限元网格的均匀性和参考网格与映射网格的对应。     

A computational paradigm for multiresolution topology optimization (MTOP)

This paper presents a multiresolution topology optimization (MTOP) scheme to obtain high resolution designs with relatively low computational cost. We employ three distinct discretization levels for the topology optimization procedure: the displacement mesh (or finite element mesh) to perform the analysis, the design variable mesh to perform the optimization, and the density mesh (or density element mesh) to represent material distribution and compute the stiffness matrices. We employ a coarser discretization for finite elements and finer discretization for both density elements and design variables. A projection scheme is employed to compute the element densities from design variables and control the length scale of the material density. We demonstrate via various two- and three-dimensional numerical examples that the resolution of the design can be significantly improved without refining the finite element mesh.     

基于凸规划方法的计算机辅助结构拓扑优化设计

基于SIMP密度-刚度插值模型和移动渐近线方法,推导并建立了线弹性连续体结构刚度拓扑优化设计的数学模型．对中间密度材料进行研究,得出了惩罚因子的合理取值范围,分析了棋盘格式和网格依赖性等数值计算中存在的问题,并结合一种敏度过滤技术改善了这些问题．给出文中方法的程序流程,开发出二维结构的拓扑优化系统,通过一些经典的算例,证明了文中方法的有效性．     

Checkerboard and minimum member size control in topology optimization

Checkerboard-like material distributions are frequently encountered in topology optimization of continuum structures, especially when first order finite elements are used. It has been shown that this phenomenon is caused by errors in the finite element formulation. Minimum member size control is closely related to the problem of mesh dependency of solutions in topology optimization. With increasing mesh density, the solution of a broad class of problems tends to form an increasing number of members with decreasing size. Different approaches have been developed in recent years to overcome these numerical difficulties. However, limitations exist for those methods, either in generality or in efficiency. In this paper, a new algorithm for checkerboard and direct minimum member size control has been developed that is applicable to the general problem formulation involving multiple constraints. This method is implemented in the commercial software Altair OptiStruct. March 22, 2000     

Conservative interpolation on unstructured polyhedral meshes: An extension of the supermesh approach to cell-centered finite-volume variables

A method is presented for conservatively transferring, or remapping, cell-centered variable fields from one mesh to another with second-order accuracy. The method is generally applicable to any polyhedral source or target mesh. Like the work of Farrell et al. [1], which was designed for finite-element computations, the proposed methodology uses a logical supermesh consisting of the intersections of polyhedra from both meshes. The resulting transfer process is well-suited for finite-volume methods that rely on cell-centered variables. The accuracy and efficacy of the new remapping process is demonstrated with numerical experiments and a computational fluid dynamics test.     

Adaptive moving mesh methods for two-dimensional resistive magneto-hydrodynamic PDE models

An adaptive moving mesh technique is applied to magneto-hydrodynamics (MHD) model problem. The moving mesh strategy is based on the approach proposed in Li et al. [Li R, Tang T, Zhang P. Moving mesh methods in multiple dimensions based on harmonic maps. J Comput Phys 2001;170:562-88] to separate the mesh-moving and PDE evolution at each time step. The Magneto-hydrodynamic equations are discretized by a finite-volume method in space, and the mesh-moving part is realized by solving the Euler-Lagrange equations to minimize a certain variation with the directional splitting monitor function. A conservative interpolation is used to redistribute the numerical solutions on the new meshes. Numerical results demonstrate the accuracy and effectiveness of the proposed algorithm.     

A parallel,load-balanced MHD code on locally-adapted,unstructured grids in 3d

An efficient parallel code for the approximate solution of initial boundary value problems for hyperbolic balance laws is introduced. The method combines three modern numerical techniques: locally-adaptive upwind finite-volume methods on unstructured grids, parallelization based on non-overlapping domain decomposition, and dynamic load balancing. Key ingredient is a hierarchical mesh in three space dimensions.The proposed method is applied to the equations of compressible magnetohydrodynamics (MHD). Results for several testproblems with computable exact solution and for a realistic astrophysical simulation are shown.