非匹配网格上广义Stokes问题的区域分裂解法及事后误差估算

发展了一种广义Stokes问题的无覆盖区域分裂解法。子域交界面上的约束条件是通过引入一Lagrange乘子而得到弱满足的,在有限元离散子域的交界处网格可以是非匹配的。应用Petrov Galerkin方法解每个子域上的广义Stokes问题,而交界面上的Lagrange乘子则通过共轭梯度法迭代求解,各变量均由线性函数离散。对上述区域分裂解法,还构造了基于求解当地问题的误差事后估算方法。各变量的当地误差估算器定义在二阶非连续鼓包(bump)函数的空间中。最后给出了基于事后误差估算值的自适应网格上的数值结果。     

Enriched multi-point flux approximation for general grids

It is well known that the two-point flux approximation, a numerical scheme used in most commercial reservoir simulators, has O(1) error when grids are not K-orthogonal. In the last decade, the multi-point flux approximations have been developed as a remedy. However, non-physical oscillations can appear when the anisotropy is really strong. We found out the oscillations are closely related to the poor approximation of pressure gradient in the flux computation.In this paper, we propose the control volume enriched multi-point flux approximation (EMPFA) for general diffusion problems on polygonal and polyhedral meshes. Non-physical oscillations are not observed for realistic and strongly anisotropic heterogeneous material properties described by a full tensor. Exact linear solutions are recovered for grids with non-planar interfaces, and a first and second order convergence are achieved for the flux and scalar unknowns, respectively.     

Anti-diffusion method for interface steepening in two-phase incompressible flow

In this paper, we present a method for obtaining sharp interfaces in two-phase incompressible flows by an anti-diffusion correction, that is applicable in a straight-forward fashion for the improvement of two-phase flow solution schemes typically employed in practical applications. The underlying discretization is based on the volume-of-fluid (VOF) interface-capturing method on unstructured meshes. The key idea is to steepen the interface, independently of the underlying volume-fraction transport equation, by solving a diffusion equation with reverse time, i.e. an anti-diffusion equation, after each advection time step of the volume fraction. As the solution of the anti-diffusion equation requires regularization, a limiter based on the directional derivative is developed for calculating the gradient of the volume fraction. This limiter ensures the boundedness of the volume fraction. In order to control the amount of anti-diffusion introduced by the correction algorithm we propose a suitable stopping criterion for interface steepening. The formulation of the limiter and the algorithm for solving the anti-diffusion equation are applicable to 3-dimensional unstructured meshes. Validation computations are performed for passive advection of an interface, for 2-dimensional and 3-dimensional rising-bubbles, and for a rising drop in a periodically constricted channel. The results demonstrate that sharp interfaces can be recovered reliably. They show that the accuracy is similar to or even better than that of level-set methods using comparable discretizations for the flow and the level-set evolution. Also, we observe a good agreement with experimental results for the rising drop where proper interface evolution requires accurate mass conservation.     

Adaptively deformed mesh based interface method for elliptic equations with discontinuous coefficients

Mesh deformation methods are a versatile strategy for solving partial differential equations (PDEs) with a vast variety of practical applications. However, these methods break down for elliptic PDEs with discontinuous coefficients, namely, elliptic interface problems. For this class of problems, the additional interface jump conditions are required to maintain the well-posedness of the governing equation. Consequently, in order to achieve high accuracy and high order convergence, additional numerical algorithms are required to enforce the interface jump conditions in solving elliptic interface problems. The present work introduces an interface technique based adaptively deformed mesh strategy for resolving elliptic interface problems. We take the advantages of the high accuracy, flexibility and robustness of the matched interface and boundary (MIB) method to construct an adaptively deformed mesh based interface method for elliptic equations with discontinuous coefficients. The proposed method generates deformed meshes in the physical domain and solves the transformed governed equations in the computational domain, which maintains regular Cartesian meshes. The mesh deformation is realized by a mesh transformation PDE, which controls the mesh redistribution by a source term. The source term consists of a monitor function, which builds in mesh contraction rules. Both interface geometry based deformed meshes and solution gradient based deformed meshes are constructed to reduce the L(∞) and L(2) errors in solving elliptic interface problems. The proposed adaptively deformed mesh based interface method is extensively validated by many numerical experiments. Numerical results indicate that the adaptively deformed mesh based interface method outperforms the original MIB method for dealing with elliptic interface problems.     

H -ADAPTIVE REFINEMENT STRATEGY FOR ACOUSTIC PROBLEMS WITH A SET OF NATURAL FREQUENCIES

The finite element method has been applied to the analysis of acoustic problems with several natural frequencies and mode shapes. First, a recovery-based error estimation is performed following the well-known procedures of structural problems. Then, an h -adaptive refinement strategy is proposed that leads to a finite element mesh with the minimum number of elements and with a specified error for each of the natural frequencies included in the analysis. The procedure provides a useful numerical tool, since the computational requirements are reduced. In addition, results obtained by means of the minimum element size procedure are shown for comparison purposes. The similarity of the meshes given by the two methods is justified on the basis of the equations that lead to the element size of the mesh. The procedure has been applied to some numerical examples to illustrate its validity.     

Simulations of multiphase flows with multiple length scales using moving mesh interface tracking with adaptive meshing

In multiphase flows, the length scales of thin regions, such as thin films between nearly touching drops and thin threads formed during the interface pinch-off, are usually several orders of magnitude smaller than the size of the drops. In this paper, a number of extra length criteria for adaptive meshes are developed and implemented in the moving mesh interface tracking method to solve these multiple-length-scale problems with high fidelity. A nominal length scale based on the solutions of Laplace’s equations with the unit normal vectors of surfaces as the boundary conditions is proposed for the adaptive mesh refinement in the thin regions. For almost flat interfaces/boundaries which are near to the thin regions, the averaged length of the interior edges sharing the two nodes with the boundary edge is introduced for the mesh adaptation. The averaged length of the interfacial edges is used for the interior elements near the interfaces but outside of the thin regions. For the interior mesh away from the interfaces/boundaries, different averaged length scales based on the initial mesh are employed for the adaptive mesh refining and coarsening. Numerous cases are simulated to demonstrate the capability of the proposed schemes in handling multiple length scales, which include the relaxation and necking of an elongated droplet, droplet–droplet head-on approaching, droplet-wall interactions, and a droplet pair in a shear flow. The smallest length resolved for the thin regions is three orders of magnitude smaller than the largest characteristic length of the problem.     

Two variational approaches for domain decomposition problems solved by SGBEM with nonconforming discretizations

Two original approaches for the solution of elastic boundary value problems with domain decomposition (DDBVP) using the symmetric Galerkin boundary element method (SGBEM) are presented. Each approach is based on a variational principle, a difference between them consisting in the treatment of the coupling conditions which connect the solutions through an interface. The computer codes developed are able to deal with curved interfaces in a domain decomposition problem discretized by nonmatching meshes of linear elements along the interfaces. The effectiveness of the methods is documented by a numerically solved example. The text was submitted by the authors in English.     

Face offsetting: A unified approach for explicit moving interfaces

Dynamic moving interfaces are central to many scientific, engineering, and graphics applications. In this paper, we introduce a novel method for moving surface meshes, called the face offsetting method, based on a generalized Huygens’ principle. Our method operates directly on a Lagrangian surface mesh, without requiring an Eulerian volume mesh. Unlike traditional Lagrangian methods, which move each vertex directly along an approximate normal or user-specified direction, our method propagates faces and then reconstructs vertices through an eigenvalue analysis locally at each vertex to resolve normal and tangential motion of the interface simultaneously. The method also includes techniques for ensuring the integrity of the surface as it evolves. Face offsetting provides a unified framework for various dynamic interface problems and delivers accurate physical solutions even in the presence of singularities and large curvatures. We present the theoretical foundation of our method, and also demonstrate its accuracy, efficiency, and flexibility for a number of benchmark problems and a real-world application.     

A high-order discontinuous Galerkin method for wave propagation through coupled elastic–acoustic media

We introduce a high-order discontinuous Galerkin (dG) scheme for the numerical solution of three-dimensional (3D) wave propagation problems in coupled elastic–acoustic media. A velocity–strain formulation is used, which allows for the solution of the acoustic and elastic wave equations within the same unified framework. Careful attention is directed at the derivation of a numerical flux that preserves high-order accuracy in the presence of material discontinuities, including elastic–acoustic interfaces. Explicit expressions for the 3D upwind numerical flux, derived as an exact solution for the relevant Riemann problem, are provided. The method supports h-non-conforming meshes, which are particularly effective at allowing local adaptation of the mesh size to resolve strong contrasts in the local wavelength, as well as dynamic adaptivity to track solution features. The use of high-order elements controls numerical dispersion, enabling propagation over many wave periods. We prove consistency and stability of the proposed dG scheme. To study the numerical accuracy and convergence of the proposed method, we compare against analytical solutions for wave propagation problems with interfaces, including Rayleigh, Lamb, Scholte, and Stoneley waves as well as plane waves impinging on an elastic–acoustic interface. Spectral rates of convergence are demonstrated for these problems, which include a non-conforming mesh case. Finally, we present scalability results for a parallel implementation of the proposed high-order dG scheme for large-scale seismic wave propagation in a simplified earth model, demonstrating high parallel efficiency for strong scaling to the full size of the Jaguar Cray XT5 supercomputer.     

二维非线性抛物方程广义差分法/有限体积法的自适应计算

给出三角网上二维非线性抛物方程广义差分法(有限体积法)的一种基于残量估计的后验误差估计,并在此基础上设计了自适应计算方案,以适应物理解在时空的大梯度变化.提出了适合发展方程自适应计算的三角网数据结构(不是树状结构)和灵活的局部粗化算法.     

Volume-of-fluid algorithm with different modified dynamic material ordering methods and their comparisons

Volume-of-fluid (VOF) interface reconstruction methods are used to define material interfaces to separate different materials in a mixed cell. These material interfaces are then used to evaluate transport flux at each cell edges in multi-material hydrodynamic calculations. Most of the VOF interface reconstruction methods and volume transport schemes rely on an accurate material order unique to each computational cell. Similarly, to achieve overshoot-free volume fractions, a non-intersecting interface reconstruction procedure has to be performed with the help of a ‘material-order list’ determined prior to interface reconstruction. It is, however, the least explored area of VOF technique especially for ‘onion-skin’ or ‘layered’ model. Also, important technical details how to prevent intersection among different material interfaces are missing in many literature. Here, we present an efficient VOF interface tracking algorithm along with modified ‘material order’ methods and different interface reconstruction methods. The relative accuracy of different methods are evaluated for sample problems. Finally, a convergence study with respect to mesh-size is performed.     

Mesh quality effects on the accuracy of CFD solutions on unstructured meshes

The order of accuracy and error magnitude of node- and cell-centered schemes are examined on representative unstructured meshes and flowfield solutions for computational fluid dynamics. Specifically, we investigate the properties of inviscid and viscous flux discretizations for isotropic and highly stretched meshes using the Method of Manufactured Solutions. Grid quality effects are studied by randomly perturbing the base meshes and cataloguing the error convergence as a function of grid size. For isotropic grids, node-centered approaches produce less error than cell-centered approaches. Moreover, a corrected node-centered scheme is shown to maintain third order accuracy for the inviscid terms on arbitrary triangular meshes. In contrast, for stretched meshes, cell-centered schemes are favored, with cell-centered prismatic approaches in particular showing the lowest levels of error. In three dimensions, simple flux integrations on non-planar control volume faces lead to first-order solution errors, while second-order accuracy is recovered by triangulation of the non-planar faces.     

The estimation of truncation error by τ-estimation revisited

The aim of this paper was to accurately estimate the local truncation error of partial differential equations, that are numerically solved using a finite difference or finite volume approach on structured and unstructured meshes. In this work, we approximated the local truncation error using the τ-estimation procedure, which aims to compare the residuals on a sequence of grids with different spacing. First, we focused the analysis on one-dimensional scalar linear and non-linear test cases to examine the accuracy of the estimation of the truncation error for both finite difference and finite volume approaches on different grid topologies. Then, we extended the analysis to two-dimensional problems: first on linear and non-linear scalar equations and finally on the Euler equations. We demonstrated that this approach yields a highly accurate estimation of the truncation error if some conditions are fulfilled. These conditions are related to the accuracy of the restriction operators, the choice of the boundary conditions, the distortion of the grids and the magnitude of the iteration error.     

A numerical scheme for the solution of axisymmetric radiative transfer problems in irregular domains filled by media with opaque and transparent diffuse and specular boundaries

This paper presents a new numerical scheme of the discrete ordinates method for the solution of axisymmetric radiative transfer problems in irregular domains filled by media with opaque and transparent diffuse and specular (Fresnel) boundaries and interfaces. New test problems of radiative transfer, which describe radiative transfer in domains with Fresnel interfaces, are proposed in this paper. These problems admit analytic solutions and can be used as benchmark ones. The proposed scheme is applied to the solution of the problems. Numerical results show that the presence of Fresnel interfaces leads to an appreciably larger error in numerical solution. This is connected with the “discontinuity” of the Fresnel reflectivity, which, through numerical diffusion, leads to the distortion of numerical solution. Modification of the scheme allows to reduce the numerical error.     

HP-Multigrid as Smoother algorithm for higher order discontinuous Galerkin discretizations of advection dominated flows. Part II: Optimization of the Runge–Kutta smoother

Using a detailed multilevel analysis of the complete hp-Multigrid as Smoother algorithm accurate predictions are obtained of the spectral radius and operator norms of the multigrid error transformation operator. This multilevel analysis is used to optimize the coefficients in the semi-implicit Runge–Kutta smoother, such that the spectral radius of the multigrid error transformation operator is minimal under properly chosen constraints. The Runge–Kutta coefficients for a wide range of cell Reynolds numbers and a detailed analysis of the performance of the hp-MGS algorithm are presented. In addition, the computational complexity of the hp-MGS algorithm is investigated. The hp-MGS algorithm is tested on a fourth order accurate space–time discontinuous Galerkin finite element discretization of the advection–diffusion equation for a number of model problems, which include thin boundary layers and highly stretched meshes, and a non-constant advection velocity. For all test cases excellent multigrid convergence is obtained.     

Electrical characterization of surface and interface potentials on SiC

Contact free vibrating capacitor results have shown that the SiC surface is more stable, compared to Si, and it is possible to identify the different (Si or C) planes on SiC substrates. The surface charge density seems to be higher after compression welding process. Electrostatic (corona) charge on the surface results in accumulation and depletion, and probably avalanche breakdown instead of equilibrium inversion. However, the equilibrium Q–V curve still can be measured starting from the inversion region.Among C–V methods the capabilities of V–Q and mercury C–V have been investigated, as two major electrical measurement techniques for SiC qualification. SiC–silicon-dioxide interfaces and SiC epitaxial layers were characterized with HF/LF C–V and V–Q measurement techniques. These methods were developed basically for Si measurements, but they could easily be adapted for measuring SiC too.     

Analysis of interface layers by spectroscopic ellipsometry

We investigate the relative validity of the Bruggeman effective-medium approximation and several alloy models to describe interfaces in the analysis of spectroscopic ellipsometric data of laminar samples, using data obtained on an Al_xGa_1−xAs multilayer sample fabricated specifically for this purpose. The investigation highlights the types of errors that result from the use of inappropriate models. Optimum results are obtained with the alloy model where the graded-composition regions are approximated with multilayer stacks.     

Examination of high-throughput hybrid calculations using coarser reciprocal space meshes

High-throughput density functional theory calculations have been typically performed with reduced accuracy and notable error in the band gap. Here we suggest several approaches to calculate the optoelectronic properties by using coarser k-point meshes for the Fock exchange potential. In our benchmark calculations, we were able to obtain the optical properties of zinc-blende and wurtzite materials with reasonable accuracy. We also propose an approach of high-throughput calculations using a pre-converged wavefunction by the reduced k-point meshes for the Fock exchange and performing the subsequent non-self-consistent-field calculations.     

Modeling material responses by arbitrary Lagrangian Eulerian formulation and adaptive mesh refinement method

In this paper we report an efficient numerical method combining a staggered arbitrary Lagrangian Eulerian (ALE) formulation with the adaptive mesh refinement (AMR) method for materials modeling including elastic–plastic flows, material failure, and fragmentation predictions. Unlike traditional AMR applied on fixed domains, our investigation focuses on the application to moving and deforming meshes resulting from Lagrangian motion. We give details of this numerical method with a capability to simulate elastic–plastic flows and predict material failure and fragmentation, and our main focus of this paper is to create an efficient method which combines ALE and AMR methods to simulate the dynamics of material responses with deformation and failure mechanisms. The interlevel operators and boundary conditions for these problems in AMR meshes have been investigated, and error indicators to locate material deformation and failure regions are studied. The method has been applied on several test problems, and the solutions of the problems obtained with the ALE–AMR method are reported. Parallel performance and software design for the ALE–AMR method are also discussed.     

Comparison of multimesh hp-FEM to interpolation and projection methods for spatial coupling of thermal and neutron diffusion calculations

Multiphysics solution challenges are legion within the field of nuclear reactor design and analysis. One major issue concerns the coupling between heat and neutron flow (neutronics) within the reactor assembly. These phenomena are usually very tightly interdependent, as large amounts of heat are quickly produced with an increase in fission events within the fuel, which raises the temperature that affects the neutron cross section of the fuel. Furthermore, there typically is a large diversity of time and spatial scales between mathematical models of heat and neutronics. Indeed, the different spatial resolution requirements often lead to the use of very different meshes for the two phenomena. As the equations are coupled, one must take care in exchanging solution data between them, or significant error can be introduced into the coupled problem. We propose a novel approach to the discretization of the coupled problem on different meshes based on an adaptive multimesh higher-order finite element method (hp-FEM), and compare it to popular interpolation and projection methods. We show that the multimesh hp-FEM method is significantly more accurate than the interpolation and projection approaches considered in this study.