Variational mesh adaptation II: error estimates and monitor functions

The key to the success of a variational mesh adaptation method is to define a proper monitor function which controls mesh adaptation. In this paper we study the choice of the monitor function for the variational adaptive mesh method developed in the previous work [J. Comput. Phys. 174 (2001) 924]. Two types of monitor functions, scalar matrix and non-scalar matrix ones, are defined based on asymptotic estimates of interpolation error obtained using the interpolation theory of finite element methods. The choice of the adaptation intensity parameter is also discussed for each of these monitor functions. Asymptotic bounds on interpolation error are obtained for adaptive meshes that satisfy the regularity and equidistribution conditions. Two-dimensional numerical results are given to verify the theoretical findings.     

Heuristic a posteriori estimation of error due to dissipation in finite volume schemes and application to mesh adaptation

A heuristic method is proposed to estimate a posteriori that part of the total discretization error which is attributable to the smoothing effect of added dissipation, for finite volume discretizations of the Euler equations. This is achieved by observing variation in a functional of the solution as the level of dissipation is varied, and it is deduced for certain test-cases that the dissipation alone accounts for the majority of the functional error. Based on this result an error estimator and mesh adaptation indicator is proposed for functionals, relying on the solution of an adjoint problem. The scheme is considerably implementationally simpler and computationally cheaper than other recently proposed a posteriori error estimators for finite volume schemes, but does not account for all sources of error. In mind of this, emphasis is placed on numerical evaluation of the performance of the indicator, and it is shown to be extremely effective in both estimating and reducing error for a range of 2d and 3d flows.     

An anisotropic mesh adaptation method for the finite element solution of heterogeneous anisotropic diffusion problems

Heterogeneous anisotropic diffusion problems arise in the various areas of science and engineering including plasma physics, petroleum engineering, and image processing. Standard numerical methods can produce spurious oscillations when they are used to solve those problems. A common approach to avoid this difficulty is to design a proper numerical scheme and/or a proper mesh so that the numerical solution validates the discrete counterpart (DMP) of the maximum principle satisfied by the continuous solution. A well known mesh condition for the DMP satisfaction by the linear finite element solution of isotropic diffusion problems is the non-obtuse angle condition that requires the dihedral angles of mesh elements to be non-obtuse. In this paper, a generalization of the condition, the so-called anisotropic non-obtuse angle condition, is developed for the finite element solution of heterogeneous anisotropic diffusion problems. The new condition is essentially the same as the existing one except that the dihedral angles are now measured in a metric depending on the diffusion matrix of the underlying problem. Several variants of the new condition are obtained. Based on one of them, two metric tensors for use in anisotropic mesh generation are developed to account for DMP satisfaction and the combination of DMP satisfaction and mesh adaptivity. Numerical examples are given to demonstrate the features of the linear finite element method for anisotropic meshes generated with the metric tensors.     

Time accurate anisotropic goal-oriented mesh adaptation for unsteady flows

We present a new algorithm for combining an anisotropic goal-oriented error estimate with the mesh adaptation fixed point method for unsteady problems. The minimization of the error on a functional provides both the density and the anisotropy (stretching) of the optimal mesh. They are expressed in terms of state and adjoint. This method is used for specifying the mesh for a time sub-interval. A global fixed point iterates the re-evaluation of meshes and states over the whole time interval until convergence of the space–time mesh. Applications to unsteady blast-wave and acoustic-wave Euler flows are presented.     

Fully anisotropic goal-oriented mesh adaptation for 3D steady Euler equations

This paper studies the coupling between anisotropic mesh adaptation and goal-oriented error estimate. The former is very well suited to the control of the interpolation error. It is generally interpreted as a local geometric error estimate. On the contrary, the latter is preferred when studying approximation errors for PDEs. It generally involves non local error contributions. Consequently, a full and strong coupling between both is hard to achieve due to this apparent incompatibility. This paper shows how to achieve this coupling in three steps.First, a new a priori error estimate is proved in a formal framework adapted to goal-oriented mesh adaptation for output functionals. This estimate is based on a careful analysis of the contributions of the implicit error and of the interpolation error. Second, the error estimate is applied to the set of steady compressible Euler equations which are solved by a stabilized Galerkin finite element discretization. A goal-oriented error estimation is derived. It involves the interpolation error of the Euler fluxes weighted by the gradient of the adjoint state associated with the observed functional. Third, rewritten in the continuous mesh framework, the previous estimate is minimized on the set of continuous meshes thanks to a calculus of variations. The optimal continuous mesh is then derived analytically. Thus, it can be used as a metric tensor field to drive the mesh adaptation. From a numerical point of view, this method is completely automatic, intrinsically anisotropic, and does not depend on any a priori choice of variables to perform the adaptation.3D examples of steady flows around supersonic and transsonic jets are presented to validate the current approach and to demonstrate its efficiency.     

Efficient simulation of three-dimensional anisotropic cardiac tissue using an adaptive mesh refinement method

A recently developed space-time adaptive mesh refinement algorithm (AMRA) for simulating isotropic one- and two-dimensional excitable media is generalized to simulate three-dimensional anisotropic media. The accuracy and efficiency of the algorithm is investigated for anisotropic and inhomogeneous 2D and 3D domains using the Luo-Rudy 1 (LR1) and FitzHugh-Nagumo models. For a propagating wave in a 3D slab of tissue with LR1 membrane kinetics and rotational anisotropy comparable to that found in the human heart, factors of 50 and 30 are found, respectively, for the speedup and for the savings in memory compared to an algorithm using a uniform space-time mesh at the finest resolution of the AMRA method. For anisotropic 2D and 3D media, we find no reduction in accuracy compared to a uniform space-time mesh. These results suggest that the AMRA will be able to simulate the 3D electrical dynamics of canine ventricles quantitatively for 1 s using 32 1-GHz Alpha processors in approximately 9 h.     

A priori mesh quality estimation via direct relation between truncation error and mesh distortion

The purpose of the present work is the derivation and evaluation of a priori mesh quality indicators for structured, unstructured, as well as hybrid grids. Emphasis is placed on deriving direct relations between the indicators and mesh distortion. The work is based on use of the finite volume discretization for evaluation of first order spatial derivatives. The analytic form of the truncation error is derived and applied to elementary types of mesh distortion including typical hybrid grid interfaces. The corresponding analytic expressions provide direct relations between computational accuracy and the degree of stretching, skewness, shearing and non-alignment of the mesh.     

A new procedure for obtaining the Voigt function dependent upon the complex error function

This work proposes a new method for obtaining the differential equation of the Voigt function and, from this equation, expressing the Voigt function as dependent upon the complex error function. In addition, the integral expression of the successive derivatives of the Voigt function is given, and from this a method is generalized which permits the representation, also, of other functions depending on the complex error function. This enables us to simplify other functions which are the convolution of a Gaussian function with rational polynomial functions. Moreover, the relationship between the Lorentzian (w_L), Gaussian (w_G) and Voigt (w_V) widths at half maximum for the function is given, which is of great interest in diverse branches of physics, such as plasma spectroscopy, astrophysics, nuclear magnetic resonance, etc.     

A priori mesh quality metric error analysis applied to a high-order finite element method

Characterization of computational mesh’s quality prior to performing a numerical simulation is an important step in insuring that the result is valid. A highly distorted mesh can result in significant errors. It is therefore desirable to predict solution accuracy on a given mesh. The HiFi/SEL high-order finite element code is used to study the effects of various mesh distortions on solution quality of known analytic problems for spatial discretizations with different order of finite elements. The measured global error norms are compared to several mesh quality metrics by independently varying both the degree of the distortions and the order of the finite elements. It is found that the spatial spectral convergence rates are preserved for all considered distortion types, while the total error increases with the degree of distortion. For each distortion type, correlations between the measured solution error and the different mesh metrics are quantified, identifying the most appropriate overall mesh metric. The results show promise for future a priori computational mesh quality determination and improvement.     

从反射面检测锥体棱镜直角误差的方法

介绍了一种使用平面干涉仪检测锥体棱镜直角误差的方法。使用这一检测方法时需要制做一个角度为24.4°的夹具,将被测棱镜放置在夹具之上。这个夹具的作用是,可以使来自干涉仪的标准平行光束(测试光束)由反射面折射入棱镜内,并且垂直于被测直角棱线,经被测直角两个面的反射光仍按原光路返回。当出射光束与干涉仪的标准平行光束(参考光束)重合时产生干涉,实现在平面干涉仪上对锥体棱镜直角误差的测量。与传统的测量方法相比较,该方法具有测量准确、计算精度高的特点。适合于对高精度锥体棱镜(α≤3″)直角误差的测量。     

噪声环境下基于最大后验非线性变换的隐马尔可夫模型自适应算法

由于训练环境和识别环境的失配,识别系统的性能会严重下降。为此,提出了基于最大后验概率非线性变换的环境自适应算法,可以减小由于环境的失配所引起的系统性能的下降。在本算法中,利用分段线性回归近似非线性变换将训练环境下隐马尔可夫模型(HMM)的均值向量变换到识别环境,减小环境的失配,变换参数的估计采用了最大后验概率估计(MAP)。数字语音识别实验证明:该环境自适应算法的识别性能优于MLST,MAPLR和MLLR等算法。     

A new algorithm for anisotropic solutions

We establish a new algorithm that generates a new solution to the Einstein field equations, with an anisotropic matter distribution, from a seed isotropic solution. The new solution is expressed in terms of integrals of an isotropic gravitational potential; and the integration can be completed exactly for particular isotropic seed metrics. A good feature of our approach is that the anisotropic solutions necessarily have an isotropic limit. We find two examples of anisotropic solutions which generalise the isothermal sphere and the Schwarzschild interior sphere. Both examples are expressed in closed form involving elementary functions only.     

A second-order 3D electromagnetics algorithm for curved interfaces between anisotropic dielectrics on a Yee mesh

A new frequency-domain electromagnetics algorithm is developed for simulating curved interfaces between anisotropic dielectrics embedded in a Yee mesh with second-order error in resonant frequencies. The algorithm is systematically derived using the finite integration formulation of Maxwell’s equations on the Yee mesh. Second-order convergence of the error in resonant frequencies is achieved by guaranteeing first-order error on dielectric boundaries and second-order error in bulk (possibly anisotropic) regions. Convergence studies, conducted for an analytically solvable problem and for a photonic crystal of ellipsoids with anisotropic dielectric constant, both show second-order convergence of frequency error; the convergence is sufficiently smooth that Richardson extrapolation yields roughly third-order convergence. The convergence of electric fields near the dielectric interface for the analytic problem is also presented.     

基于光流分层方法的平面3D运动估测

无人机自主着舰末端视觉导引中舰机间相对位姿的估测,可以看作机载摄像机对甲板平面3D运动的估测。提出了一种光流分层方法:首先利用已知焦距的机载摄像机拍摄着舰靶标区域的图像序列,并采用Lucas方法计算相邻两帧图像的光流场;而后通过分层模型,将由光流场进行3D运动检测的非线性问题转化为了两个线性问题。该方法无需图像间的特征匹配,可线性解算出着舰靶标区域相对于无人机的三维运动参数,进而得到舰机间的相对位姿信息。计算机合成图和摄像机实拍图像的实验结果验证了该算法的正确性和有效性。     

Dynamic waveband grooming based on hierarchical integrated grooming auxiliary graph in multi-domain mesh optical networks

With the number of wavelengths in fibers increasing, the number of optical switching ports in conventional Optical Cross-Connect (OXC) keeps enhancing, so that waveband grooming technique is proposed to save the switching ports in OXC. Most of previous works focused on waveband grooming in single-domain optical network. Since the current optical backbone is actually divided to multiple domains according to the different network providers, it is necessary to study the waveband grooming in multi-domain optical networks. However, waveband grooming in multi-domain optical networks is more challenging than that in single domain networks because of the routing scalability and security issues. Therefore, in this paper, we propose a new heuristic Hierarchical Multi-domain Waveband Grooming (HMWG) algorithm based on Hierarchical Integrated Grooming Auxiliary Graph (H-IGAG) to reduce the total number of optical switching ports. The H-IGAG is compared of the Intra-domain Waveband Integrated Auxiliary Graph (Intra-WIAG) and the Inter-domain waveband Virtual Topology Graph (Inter-VTG). For the demand in single-domain, HMWG directly computes the route from the source node to destination node in the single-domain with waveband grooming on Intra-WIAG. For the demand spanning different domains, HMWG first computes the route from the source node to the selected border node in source domain and computes the route from the selected border node to the destination node in destination domain with waveband grooming on Intra-WIAG, respectively. Then, HMWG computes the route from the selected border node in source domain to the selected border node in destination domain with waveband grooming on Inter-VTG. Simulation results show that, compared with previous grooming algorithm, HMWG can obtain better performance.     

基于小波神经网络的光纤陀螺误差补偿方法

为了提高光纤陀螺的测量精度,提出了一种基于小波神经网络的误差补偿方法。首先使用小波分析中的Mallat分解算法提取出陀螺信号中的主趋势项,对其误差余项进行重构。然后将重构信号作为小波神经网络的目标输出,将原始陀螺信号作为训练样本。为了提高小波神经网络的训练速度同时防止其陷入局部极小值,采用增加动量因子和自适应调整学习速率的方法来改进训练方法。训练后建立的神经网络模型对光纤陀螺误差具有良好的估计能力。结果表明,经过小波神经网络方法补偿后,光纤陀螺的输出精度达到了0. 019 4°/s,光纤陀螺的测量性能得到了提高。     

空间各向异性两体势向列相液晶形变研究

基于分子两体势研究向列相液晶的形变.该两体势是空间各向异性的并且依赖于液晶的弹性常数.理论处理中假定具有理想向列序，这意味着分子长轴取向方向与液晶指向矢是重合的，而总自由能等于总相互作用能.以解析形式研究了三种基本的Fréedericksz 转变并对混合排列向列相液晶盒中的指向矢分布进行了数值计算.检查了文献中最近提出的两种从弹性能到两体作用势的映射方案，发现只有一种方案给出的结果与连续体理论一致. 关键词： 空间各向异性两体势 理想向列序 液晶形变 Fréedericksz 转变