Discontinuous subgrid formulations for transport problems

In this paper we develop two discontinuous Galerkin formulations within the framework of the two-scale subgrid method for solving advection–diffusion-reaction equations. We reformulate, using broken spaces, the nonlinear subgrid scale (NSGS) finite element model in which a nonlinear eddy viscosity term is introduced only to the subgrid scales of a finite element mesh. Here, two new subgrid formulations are built by introducing subgrid stabilized terms either at the element level or on the edges by means of the residual of the approximated resolved scale solution inside each element and the jump of the subgrid solution across interelement edges. The amount of subgrid viscosity is scaled by the resolved scale solution at the element level, yielding a self adaptive method so that no additional stabilization parameter is required. Numerical experiments are conducted in order to demonstrate the behavior of the proposed methodology in comparison with some discontinuous Galerkin methods.     

On improving the efficiency of tensor voting

This paper proposes two alternative formulations to reduce the high computational complexity of tensor voting, a robust perceptual grouping technique used to extract salient information from noisy data. The first scheme consists of numerical approximations of the votes, which have been derived from an in-depth analysis of the plate and ball voting processes. The second scheme simplifies the formulation while keeping the same perceptual meaning of the original tensor voting: The stick tensor voting and the stick component of the plate tensor voting must reinforce surfaceness, the plate components of both the plate and ball tensor voting must boost curveness, whereas junctionness must be strengthened by the ball component of the ball tensor voting. Two new parameters have been proposed for the second formulation in order to control the potentially conflictive influence of the stick component of the plate vote and the ball component of the ball vote. Results show that the proposed formulations can be used in applications where efficiency is an issue since they have a complexity of order O(1). Moreover, the second proposed formulation has been shown to be more appropriate than the original tensor voting for estimating saliencies by appropriately setting the two new parameters.     

Analysis of a stabilized finite element approximation of the transient convection-diffusion-reaction equation using orthogonal subscales

In this paper we analyze a stabilized finite element method to solve the transient convection-diffusion-reaction equation based on the decomposition of the unknowns into resolvable and subgrid scales. We start from the time-discrete form of the problem and obtain an evolution equation for both components of the decomposition. A closed-form expression is proposed for the subscales which, when inserted into the equation for the resolvable scale, leads to the stabilized formulation that we analyze. Optimal error estimates in space are provided for the first order, backward Euler time integration. Received: 31 January 2001 / Accepted: 30 September 2001     

Large eddy simulation and ALE mesh motion in Rayleigh-Taylor instability simulation

Many large eddy simulation (LES) techniques have been developed for stationary computational meshes. This study applies a single equation LES to Arbitrary Lagrangian-Eulerian (ALE) simulations of Rayleigh-Taylor instability and investigates its effects. Behavior of LES is similar for Eulerian and ALE simulations for the test problem studied. However, the motion of the mesh can be tied to the subgrid scale model in the form of a relaxation weight based on subgrid scale energy. This increases mesh resolution in areas of high subgrid scale energy.     

A parameter-free dynamic diffusion method for advection–diffusion–reaction problems

In this paper, we present a two-scale finite element formulation, named Dynamic Diffusion (DD), for advection–diffusion–reaction problems. By decomposing the velocity field in coarse and subgrid scales, the latter is used to determine the smallest amount of artificial diffusion to minimize the coarse-scale kinetic energy. This is done locally and dynamically, by imposing some constraints on the resolved scale solution, yielding a parameter-free consistent method. The subgrid scale space is defined by using bubble functions, whose degrees of freedom are locally eliminated in favor of the degrees of freedom that live on the resolved scales. Convergence tests on a two-dimensional example are reported, yielding optimal rates. In addition, numerical experiments show that DD method is robust for a wide scope of application problems.     

Software design for finite difference schemes based on index notation

A formulation of finite difference schemes based on the index notation of tensor algebra is advocated. Finite difference operators on regular grids may be described as sparse, banded, “tensors”. Especially for higher space dimensions, it is claimed that a band tensor formulation better corresponds to the inherent problem structure than does conventional matrix notation.Tensor algebra is commonly expressed using index notation. The standard index notation is extended with the notion of index offsets, thereby allowing the common traversal of band tensor diagonals.The transition from mathematical index notation to implementation is presented. It is emphasized that efficient band tensor computations must exploit the particular problem structure, which calls for a combination of general index notation software with special-purpose band tensor routines.     

Subgrid stabilization of Galerkin approximations of linear contraction semi-groups of class C0

This paper presents a stabilized Galerkin technique for approximating linear contraction semi-groups of class C⁰ in a Hilbert space. The main result of this paper is that this technique yields optimal error estimates in the graph norm. The key idea is twofold, first it consists in introducing an approximation space that is broken up into resolved scales and subgrid scales so that the generator of the semi-group satisfies a uniform inf-sup condition with respect to this decomposition. Second, the Galerkin approximation is slightly modified by introducing an artificial diffusion acting only on the subgrid scales. Received: 30 April 1999 / Revised version: 17 June 1999     

A variational multiscale a posteriori error estimation method for mixed form of nearly incompressible elasticity

This paper presents an error estimation framework for a mixed displacement–pressure finite element method for nearly incompressible elasticity. The proposed method is based on Variational Multiscale (VMS) concepts, wherein the displacement field is decomposed into coarse scales that can be resolved by a given finite element mesh and fine scales that are beyond the resolution capacity of the mesh. Variational projection of fine scales onto the coarse-scale space via variational embedding of the fine-scale solution into the coarse-scale formulation leads to the stabilized method with two major attributes: first, it is free of volumetric locking and, second, it accommodates arbitrary combinations of interpolation functions for the displacement and pressure fields. This VMS-based stabilized method is equipped with naturally derived error estimators and offers various options for numerical computation of the error. Specifically, two error estimators are explored. The first method employs an element-based strategy and a representation of error via a fine-scale error equation defined over element interiors which is evaluated by a direct post-solution evaluation. This quantity when combined with the global pollution error results in a simple explicit error estimator. The second method involves solving the fine-scale error equation through localization to overlapping patches spread across the domain, thereby leading to an implicit calculation of the local error. This implicit calculation when combined with the global pollution error results in an implicit error estimator. The performance of the stabilized method and the error estimators is investigated through numerical convergence tests conducted for two model problems on uniform and distorted meshes. The sharpness and robustness of the estimators is shown to be consistent across the test cases employed.     

Multiscale-stabilized solutions to one-dimensional systems of conservation laws

We present a variational multiscale formulation for the numerical solution of one-dimensional systems of conservation laws. The key idea of the proposed formulation, originally presented by Hughes [Comput. Methods Appl. Mech. Engrg., 127 (1995) 387–401], is a multiple-scale decomposition into resolved grid scales and unresolved subgrid scales. Incorporating the effect of the subgrid scales onto the coarse scale problem results in a finite element method with enhanced stability properties, capable of accurately representing the sharp features of the solution. In the formulation developed herein, the multiscale split is invoked prior to any linearization of the equations. Special attention is given to the choice of the matrix of stabilizing coefficients and the discontinuity-capturing diffusion. The methodology is applied to the one-dimensional simulation of three-phase flow in porous media, and the shallow water equations. These numerical simulations clearly show the potential and applicability of the formulation for solving highly nonlinear, nearly hyperbolic systems on very coarse grids. Application of the numerical formulation to multidimensional problems is presented in a forthcoming paper.     

High-order curvilinear meshing using a thermo-elastic analogy

With high-order methods becoming increasingly popular in both academia and industry, generating curvilinear meshes that align with the boundaries of complex geometries continues to present a significant challenge. Whereas traditional low-order methods use planar-faced elements, high-order methods introduce curvature into elements that may, if added naively, cause the element to self-intersect. Over the last few years, several curvilinear mesh generation techniques have been designed to tackle this issue, utilizing mesh deformation to move the interior nodes of the mesh in order to accommodate curvature at the boundary. Many of these are based on elastic models, where the mesh is treated as a solid body and deformed according to a linear or non-linear stress tensor. However, such methods typically have no explicit control over the validity of the elements in the resulting mesh. In this article, we present an extension of this elastic formulation, whereby a thermal stress term is introduced to ‘heat’ or ‘cool’ elements as they deform. We outline a proof-of-concept implementation and show that the adoption of a thermo-elastic analogy leads to an additional degree of robustness, by considering examples in both two and three dimensions.     

A study on using hierarchical basis error estimates in anisotropic mesh adaptation for the finite element method

A common approach for generating an anisotropic mesh is the M-uniform mesh approach where an adaptive mesh is generated as a uniform one in the metric specified by a given tensor M. A key component is the determination of an appropriate metric, which is often based on some type of Hessian recovery. Recently, the use of a global hierarchical basis error estimator was proposed for the development of an anisotropic metric tensor for the adaptive finite element solution. This study discusses the use of this method for a selection of different applications. Numerical results show that the method performs well and is comparable with existing metric tensors based on Hessian recovery. Also, it can provide even better adaptation to the solution if applied to problems with gradient jumps and steep boundary layers. For the Poisson problem in a domain with a corner singularity, the new method provides meshes that are fully comparable to the theoretically optimal meshes.     

A comparison of second- and sixth-order methods for large-eddy simulations

Large-eddy simulations of spatially developing planar turbulent jets are performed using a compact finite-difference scheme of sixth-order and an advective upstream splitting method-based method of second-order accuracy. The applicability of these solution schemes with different subgrid scale models and their performance for realistic turbulent flow problems are investigated. Solutions of the turbulent channel flow are used as an inflow condition for the turbulent jets. The results compare well with each other and with analytical and experimental data. For both solution schemes, however, the influence of the subgrid scale model on the time averaged turbulence statistics is small. This is known to be the case for upwind schemes with a dissipative truncation error, but here it is also observed for the high-order compact scheme. The reason is found to be the application of a compact high-frequency filter, which has to be used with strongly stretched computational grids to suppress high-frequency oscillations. The comparison of the results of the two schemes shows hardly any difference in the quality of the solutions. The second-order scheme, however, is computationally more efficient.     

Stabilized finite element approximation of transient incompressible flows using orthogonal subscales

In this paper we present a stabilized finite element method to solve the transient Navier–Stokes equations based on the decomposition of the unknowns into resolvable and subgrid scales. The latter are approximately accounted for, so as to end up with a stable finite element problem which, in particular, allows to deal with convection-dominated flows and the use of equal velocity–pressure interpolations. Three main issues are addressed. The first is a method to estimate the behavior of the stabilization parameters based on a Fourier analysis of the problem for the subscales. Secondly, the way to deal with transient problems discretized using a finite difference scheme is discussed. Finally, the treatment of the nonlinear term is also analyzed. A very important feature of this work is that the subgrid scales are taken as orthogonal to the finite element space. In the transient case, this simplifies considerably the numerical scheme.     

Subgrid modeling for convection-diffusion-reaction in one space dimension using a Haar Multiresolution analysis

In this paper we propose and study a subgrid model for linear convection-diffusion-reaction equations with fractal rough coefficients. The subgrid model is based on scale extrapolation of a modeling residual from coarser scales using a computed solution on a finest scale as reference. We show in experiments that a solution with subgrid model on a scale h in most cases corresponds to a solution without subgrid model on a scale less than h/4. We also present error estimates for the modeling error in terms of modeling residuals.     

海量断层网格数据的并行简化

利用Marching Cube算法重建的网格数据通常存在三角面片数量庞大的特点,必须对其进行一定程度的简化才能够方便地使用。对于海量断层网格数据可以将其分成连续的若干段,然后将各段在不同的计算节点上进行简化操作以达到并行的效果,但是这样会丢失各段连接处的拓扑信息,因而不利于后续的网格操作。采取了一种新的网格数据存储格式,并基于此提出了相应的合并算法。结果表明该算法能够很好地保持各段之间的拓扑关系,从而实现了断层网格数据的分布式并行简化。     

Classical and variational multiscale LES of the flow around a circular cylinder on unstructured grids

The effects of numerical viscosity, subgrid scale (SGS) viscosity and grid resolution are investigated in LES and VMS-LES simulations of the flow around a circular cylinder at Re=3900 on unstructured grids. The separation between the largest and the smallest resolved scales in the VMS formulation is obtained through a variational projection operator and finite-volume cell agglomeration. Three different non-dynamic eddy-viscosity SGS models are used both in classical and in VMS-LES. The so-called small-small formulation is used in VMS-LES, i.e. the SGS viscosity is computed as a function of the smallest resolved scales. Two different grid resolutions are considered. It is found that, for each considered SGS model, the amount of SGS viscosity introduced in the VMS-LES formulation is significantly lower than in classical LES. This, together with the fact that in the VMS formulation the SGS viscosity only acts on the smallest resolved scales, has a strong impact on the results. However, a significant sensitivity of the results to the considered SGS model remains also in the VMS-LES formulation. Moreover, passing from classical LES to VMS-LES does not systematically lead to an improvement of the quality of the numerical predictions.     

水平集网格简化算法在遗址场景中的应用

将水平集方法引入到三维模型网格简化中,构造符号距离函数,函数的零集定义为初始曲面;引入一个能量泛涵,通过对其极小化诱导出一个水平集形式的二阶几何偏微分方程,从而将网格简化过程转化为隐式模型的体素扩散过程。该方法目前已经用于文化遗产数字化的大场景和文物的模型简化中。对水平集网格简化算法和现常用的基于点对收缩的网格简化算法在视觉质量和几何误差方面做了比较和分析,实验表明该方法适用于任意拓扑形状的网格模型,使得模型大规模简化后,在保持较低误差的同时,仍然能够保持相当多的重要几何特征和较好的整体视觉效果。     

Dynamically adapted mesh refinement for combustion front tracking

A strategy of self-adaptive mesh refinement is developed to perform a multi-resolution approach for combustion problems. The objective is to track and resolve the small length scales arising in the thickness of a flame front. The discretisation is based on a series of nested grids of increasing accuracy. Every nested grid can slide on the grid immediately coarser, as far as certain mesh constraints are satisfied. We implement a finite volume formulation which considers a set of non-overlapping sub-domains and three different procedures for solution reconnection are compared: the two first ones envisage a multi-domain approach and the third one builds the composite grid assembling. The three methods have recourse to multi-grid technique in the sense that they exploit grids of coarser levels—actually unused in the formulation—in order to compute solution on each sub-domain. Efficiency of methods is discussed.A test of accuracy versus the exact solution of a stiff elliptic problem is presented: precision in reconnection, refinement consistency, gain in accuracy and CPU time feasibility are checked. Finally, we present qualitative results of feasibility with respect to the tracking of a premixed flame subjected to diffusive-thermal instability which wrinkles the combustion front.     

用户控制的纹理合成

提出一种基于用户控制的纹理合成算法．该算法适用于任意二维平面和任意拓扑的三维网格．可方便地控制纹理合成时方向和尺度的连续变化．对于任意平面区域需剖分成较均匀的三角网格，以剖分得到的二角形作为基本的合成单元来进行合成．根据用户在此三角网格上指定表示纹理方向和大小的矢量来插值生成矢量场，用以控制合成纹理的变化．该算法可以自然扩展到三维三角网格，以三角面片作为合成单元，合成后直接输出每个顶点的纹理坐标．该算法对二维和三维纹理合成给出了统一实现的框架．实验结果表明，该算法可以在任意目标区域根据用户的交互生成令人满意的纹理合成效果．