An analysis of the petrov—galerkin finite element method

Petrov-Galerkin finite element methods, using different test and trial functions, are applied to the solution of an unsymmetric two-point boundary value problem intended to simulate certain aspects of convection-diffusion problems. For a specified space of trial functions we utilise an energy error bound to optimize this class of methods over a family of test spaces. The optimized method performs well provided the asymmetry in the differential operator does not lead to boundary layers in the solution. Following an analysis of the boundary layer behaviour of the continuous problem, L-splines are introduced, and, by studying their behaviour for coarse meshes, we are able to modify the original schemes to produce so-called “disconnected” finite element methods. Even for coarse meshes, when no nodes occur in the boundary layer, the accuracy at all nodal points is good. This would make them good candidates for application in more general situations.     

A Spectral Stochastic Semi-Lagrangian Method for Convection-Diffusion Equations with Uncertainty

Real life convection-diffusion problems are characterized by their inherent or externally induced uncertainties in the design parameters. This paper presents a spectral stochastic finite element semi-Lagrangian method for numerical solution of convection-diffusion equations with uncertainty. Using the spectral decomposition, the stochastic variational problem is reformulated to a set of deterministic variational problems to be solved for each Wiener polynomial chaos. To obtain the chaos coefficients in the corresponding deterministic convection-diffusion equations, we implement a semi-Lagrangian method in the finite element framework. Once this representation is computed, statistics of the numerical solution can be easily evaluated. These numerical techniques associate the geometrical flexibility of the finite element method with the ability offered by the semi-Lagrangian method to solve convection-dominated problems using time steps larger than its Eulerian counterpart. Numerical results are shown for a convection-diffusion problem driven with stochastic velocity and for an incompressible viscous flow problem with a random force. In both examples, the proposed method demonstrates its ability to better maintain the shape of the solution in the presence of uncertainties and steep gradients.     

Numerical investigation on convergence of boundary knot method in the analysis of homogeneous Helmholtz, modified Helmholtz, and convection-diffusion problems

This paper concerns a numerical study of convergence properties of the boundary knot method (BKM) applied to the solution of 2D and 3D homogeneous Helmholtz, modified Helmholtz, and convection-diffusion problems. The BKM is a new boundary-type, meshfree radial function basis collocation technique. The method differentiates from the method of fundamental solutions (MFS) in that it does not need the controversial artificial boundary outside physical domain due to the use of non-singular general solutions instead of the singular fundamental solutions. The BKM is also generally applicable to a variety of inhomogeneous problems in conjunction with the dual reciprocity method (DRM). Therefore, when applied to inhomogeneous problems, the error of the DRM confounds the BKM accuracy in approximation of homogeneous solution, while the latter essentially distinguishes the BKM, MFS, and boundary element method. In order to avoid the interference of the DRM, this study focuses on the investigation of the convergence property of the BKM for homogeneous problems. The given numerical experiments reveal rapid convergence, high accuracy and efficiency, mathematical simplicity of the BKM.     

A high-order Lagrangian-decoupling method for the incompressible Navier-Stokes equations

In this paper we present a high-order Lagrangian-decoupling method for the unsteady convection diffusion and incompressible Navier-Stokes equations. The method is based upon Lagrangian variational forms that reduce the convection-diffusion equation to a symmetric initial value problem, implicit high-order backward-differentiation finite difference schemes for integration along characteristics, finite element or spectral element spatial discretizations and mesh-invariance procedures and high-order explicit time-stepping schemes for deducing function values at convected space-time points. The method improves upon previous finite element characteristic methods through the systematic and efficient extension to high-order accuracy and the introduction of a simple structure-preserving characteristic-foot calculation procedure which is readily implemented on modern architectures. The new method is significantly more efficient than explicit-convection schemes for the Navier-Stokes equations due to the decoupling of the convection and Stokes operators and the attendant increase in temporal stability. Numerous numerical examples are given for the convection-diffusion and Navier-Stokes equations for the particular case of a spectral element spatial discretization.     

Numerical construction of optimal adaptive grids in two spatial dimensions

This article concerns a new procedure to generate a solution-adaptive grid for convection dominated problems in two spatial dimensions based on finite element approximations. The procedure extends a one-dimensional equidistribution principle which minimizes the interpolation error in appropriate norms. The idea in extending such a technique to two spatial dimensions is to select two directions which can reflect the physics of the problems, and then the one-dimensional equidistribution principle is applied to the chosen directions. The final grids generated are connected through a sweep-line based unstructured grid technique. Model problems considered are the two-dimensional convection-diffusion problems where boundary and interior layers occur. Numerical results of model problems illustrating the efficiency of the proposed scheme are presented. Comparisons of the solutions with an existing scheme will also be provided.     

An efficient parallel algorithm with application to computational fluid dynamics

When solving time-dependent partial differential equations on parallel computers using the nonoverlapping domain decomposition method, one often needs numerical boundary conditions on the boundaries between subdomains. These numerical boundary conditions can significantly affect the stability and accuracy of the final algorithm.In this paper, a stability and accuracy analysis of the existing methods for generating numerical boundary conditions will be presented, and a new approach based on explicit predictors and implicit correctors will be used to solve convection-diffusion equations on parallel computers, with application to aerospace engineering for the solution of Euler equations in computational fluid dynamics simulations. Both theoretical analyses and numerical results demonstrate significant improvement in stability and accuracy by using the new approach.     

Navier-Stokes Solution by New Compact Scheme for Incompressible Flows

A new high spectral accuracy compact difference scheme is proposed here. This has been obtained by constrained optimization of error in spectral space for discretizing first derivative for problems with non-periodic boundary condition. This produces a scheme with the highest spectral accuracy among all known compact schemes, although this is formally only second-order accurate. Solution of Navier-Stokes equation for incompressible flows are reported here using this scheme to solve two fluid flow instability problems that are difficult to solve using explicit schemes. The first problem investigates the effect of wind-shear past bluff-body and the second problem involves predicting a vortex-induced instability.     

Enriched Spectral Method for Stiff Convection-Dominated Equations

A novel and simple numerical method for stiff convection-dominated problems is studied in presence of boundary or interior layers. A version of the spectral Chevyshev-collocation method enriched with the so-called corrector functions is investigated. The corrector functions here are designed to capture the stiffness of the layers (see the Appendix), and the proposed method does not rely on the adaptive grid points. The extensive numerical results demonstrate that the enriched spectral methods are very accurate with low computational cost.     

Domain Decomposition Spectral Method for Mixed Inhomogeneous Boundary Value Problems of High Order Differential Equations on Unbounded Domains

In this paper, we develop domain decomposition spectral method for mixed inhomogeneous boundary value problems of high order differential equations defined on unbounded domains. We introduce an orthogonal family of new generalized Laguerre functions, with the weight function x ^?? , ?? being any real number. The corresponding quasi-orthogonal approximation and Gauss-Radau type interpolation are investigated, which play important roles in the related spectral and collocation methods. As examples of applications, we propose the domain decomposition spectral methods for two fourth order problems, and the spectral method with essential imposition of boundary conditions. The spectral accuracy is proved. Numerical results demonstrate the effectiveness of suggested algorithms.     

Incompressible Navier–Stokes solutions by a triangular spectral/p element projection method

The application of the fractional step projection method recently proposed by Guermond and Quartapelle to the numerical approximation of unsteady Navier–Stokes solutions by means of a spectral/p element method is considered. In particular we illustrate the second-order pressure correction technique and evaluate its accuracy properties in some test cases. Stability with respect to the compatibility condition between the approximation spaces for velocity and pressure is also addressed. The high (spectral) accuracy in space and the second-order accuracy in time are verified by two simple test cases with analytical solution. A more interesting problem is solved showing the ability of the method to produce very accurate results also for problems in complex geometries.     

Spectral projective newton-methods for quasilinear elliptic boundary value problems

We consider Newton-like methods for the solution of quasilinear elliptic boundary value problems. The quasilinear problems are linearized by a Newton-method and the linear problems are approximately solved by a spectral projection method (e.g., the Ritz-Galerkin or the collocation method). convergence results are derived that show the spectral accuracy of this method. The results are of a local type which means that we assume the starting approximation to be sufficiently near to the exact solution.     

Partial semi-coarsening multigrid method based on the HOC scheme on nonuniform grids for the convection–diffusion problems

A partial semi-coarsening multigrid method based on the high-order compact (HOC) difference scheme on nonuniform grids is developed to solve the 2D convection–diffusion problems with boundary or internal layers. The significance of this study is that the multigrid method allows different number of grid points along different coordinate directions on nonuniform grids. Numerical experiments on some convection–diffusion problems with boundary or internal layers are conducted. They demonstrate that the partial semi-coarsening multigrid method combined with the HOC scheme on nonuniform grids, without losing the high-order accuracy, is very efficient and effective to decrease the computational cost by reducing the number of grid points along the direction which does not contain boundary or internal layers.     

结合语义边界信息的道路环境语义分割方法

语义分割是实现道路语义环境解释的重要方法,深度学习语义分割由于卷积、池化及反卷积的作用使分割边界模糊、不连续以及小目标漏分错分,影响了分割效果,降低了分割精度。针对上述问题,提出了一种结合语义边界信息的新的语义分割方法,首先在语义分割深度模型中构建了一个语义边界检测子网,利用网络中的特征共享层将语义边界检测子网络学习到的语义边界信息传递给语义分割网络;然后结合语义边界检测任务和语义分割任务定义了新的模型代价函数,同时完成语义边界检测和语义分割两个任务,提升语义分割网络对物体边界的描述能力,提高语义分割质量。最后在Cityscapes数据集上进行一系列实验证明,结合语义边界信息的语义分割方法在准确率上比已有的语义分割网络SegNet提升了2.9%,比ENet提升了1.3%。所提方法可以改善语义分割中出现的分割不连续、物体边界不清晰、小目标错分漏分、分割精度不高等问题。     

Efficient Chebyshev–Petrov–Galerkin Method for Solving Second-Order Equations

A new efficient Chebyshev–Petrov–Galerkin (CPG) direct solver is presented for the second order elliptic problems in square domain where the Dirichlet and Neumann boundary conditions are considered. The CPG method is based on the orthogonality property of the kth-derivative of the Chebyshev polynomials. The algorithm differs from other spectral solvers by the high sparsity of the coefficient matrices: the stiffness and mass matrices are reduced to special banded matrices with two and four nonzero diagonals respectively. The efficiency and the spectral accuracy of CPG method have been validated.     

A new numerical method for singularly perturbed turning point problems with two boundary layers based on reproducing kernel method

In this paper, a simple numerical method is proposed for solving singularly perturbed boundary layers problems exhibiting twin boundary layers. The method avoids the choice of fitted meshes. Firstly the original problem is transformed into a new boundary value problem whose solution does not change rapidly by a proper variable transformation; then the transformed problem is solved by using the reproducing kernel method. Two numerical examples are given to show the effectiveness of the present method.     

Stabilization of the Spectral Element Method in Convection Dominated Flows by Recovery of Skew-Symmetry

We investigate stability properties of the spectral element method for advection dominated incompressible flows. In particular, properties of the widely used convective form of the nonlinear term are studied. We remark that problems which are usually associated with the nonlinearity of the governing Navier–Stokes equations also arise in linear scalar transport problems, which implicates advection rather than nonlinearity as a source of difficulty. Thus, errors arising from insufficient quadrature of the convective term, commonly referred to as ‘aliasing errors’, destroy the skew-symmetric properties of the convection operator. Recovery of skew-symmetry can be efficiently achieved by the use of over-integration. Moreover, we demonstrate that the stability problems are not simply connected to underresolution. We combine theory with analysis of the linear advection-diffusion equation in 2D and simulations of the incompressible Navier–Stokes equations in 2D of thin shear layers at a very high Reynolds number and in 3D of turbulent and transitional channel flow at moderate Reynolds number. For the Navier–Stokes equations, where the divergence-free constraint needs to be enforced iteratively to a certain accuracy, small divergence errors can be detrimental to the stability of the method and it is therefore advised to use additional stabilization (e.g. so-called filter-based stabilization, spectral vanishing viscosity or entropy viscosity) in order to assure a stable spectral element method.     

Analytical integral algorithm applied to boundary layer effect and thin body effect in BEM for anisotropic potential problems

A new completely analytical integral algorithm is proposed and applied to the evaluation of nearly singular integrals in boundary element method (BEM) for two-dimensional anisotropic potential problems. The boundary layer effect and thin body effect are dealt with. The completely analytical integral formulas are suitable for the linear and non-isoparametric quadratic elements. The present algorithm applies the analytical formulas to treat nearly singular integrals. The potentials and fluxes at the interior points very close to boundary are evaluated. The unknown potentials and fluxes at boundary nodes for thin body problems with the thickness-to-length ratios from 1E−1 to 1E−8 are accurately calculated by the present algorithm. Numerical examples on heat conduction demonstrate that the present algorithm can effectively handle nearly singular integrals occurring in boundary layer effect and thin body effect in BEM. Furthermore, the present linear BEM is especially accurate and efficient for the numerical analysis of thin body problems.     

基于轻量型U-net的钢材金相图像晶界分割方法

在金相组织的晶粒度自动化评估工作中,对晶粒边界识别的精准与否直接影响着金相组织晶粒度等级的评估准确度。针对钢材金相图像中晶粒边界密集程度高、边缘复杂且晶粒边界识别准确性低的问题,提出一种基于轻量型U-net卷积神经网络的金相图像晶界分割方法,该轻量型网络模型将浅层特征层用跳跃连接的方式拼接在上采样过程中,使网络学习到更多的有效特征信息;减少了网络层数并在特征提取过程中添加了一次卷积过程,减少了网络参数量并提高了对晶界的预测速度和准确率;实验结果表明,该方法在117张金相图像测试集上像素准确率达到93.91%、特异度为96.73%、灵敏度为81.6%。与传统U-net网络相比,像素准确率提高了0.2%,网络参数量相对减少了61.5%。本方法对金相晶界分割具有有效性和优越性。     

Spectral methods for the computation of discontinuous solutions

The extension of spectral methods to the computation of discontinuous weak solutions of hyperbolic equations is considered. A new postprocessing method that produces a close to spectral accuracy is introduced. Conservation of moments with spectral accuracy is proved for the linear case. Model problems demonstrate the theoretical results. Numerical results are also included.     

Momentum-time flux conservation method for one-dimensional wave equations

We present a conservation element and solution element method in time and momentum space. Several paradigmatic wave problems including simple wave equation, convection-diffusion equation, driven harmonic oscillating charge and nonlinear Korteweg-de Vries (KdV) equation are solved with this method and calibrated with known solutions to demonstrate its use. With this method, time marching scheme is explicit, and the nonreflecting boundary condition is automatically fulfilled. Compared to other solution methods in coordinate space, this method preserves the complete information of the wave during time evolution which is an useful feature especially for scattering problems.