A moving Kriging interpolation-based meshfree method for free vibration analysis of Kirchhoff plates

The present work aims to make a further development of a novel meshfree method for free vibration analysis of classical Kirchhoff’s plates. The deflection of plates is approximated by the moving Kriging interpolation method which possesses the Kronecker’s delta property. This thus makes the proposed method efficient and straightforward in imposing the essential boundary conditions, and no special treatment techniques are required. A standard weak form is adapted to discrete the governing partial differential equations of plates. Numerical examples with different geometric shapes are considered to demonstrate the applicability and the accuracy of the proposed method.     

Meshfree-enriched simplex elements with strain smoothing for the finite element analysis of compressible and nearly incompressible solids

This paper presents a meshfree-enriched finite element formulation for triangular and tetrahedral elements in the analysis of two and three-dimensional compressible and nearly incompressible solids. The new formulation is first established in two-dimensional case by introducing a meshfree approximation into a linear triangular finite element with an enriched node. The interpolation functions of the four-noded triangular element are constructed by the meshfree convex approximations and are completed to a polynomial of degree one. The reference mapping using the constructed interpolation functions is shown to be invertible everywhere in the element and the global element area is proven to be conserved under a standard three-point integration rule. The triangular element formulation is extendable to the tetrahedral element in three-dimensional case. To provide a locking-free analysis for the nearly incompressible materials, an area-weighted strain smoothing is developed in conjunction with the enriched interpolation functions to yield a discrete divergence-free property at the integration point. The resultant element formulation with strain smoothing is shown to pass the patch test. To introduce the smoothed strain into Galerkin formulation, a modified Hu–Washizu variational principle is adopted to formulate the discrete equations. Since the Kronecker-delta property in element interpolation is held along the element boundary using meshfree convex approximation, boundary conditions can be treated in a standard way. Several numerical benchmarks are provided to demonstrate the effectiveness and accuracy of the proposed method.     

An adaptive phase field method for the mixture of two incompressible fluids

This paper develops an adaptive moving mesh method to solve a phase field model for the mixture of two incompressible fluids. The projection method is implemented on a half-staggered, moving quadrilateral mesh to keep the velocity field divergence-free, and the conjugate gradient or multigrid method is employed to solve the discrete Poisson equations. The current algorithm is composed by two independent parts: evolution of the governing equations and mesh-redistribution. In the first part, the incompressible Navier-Stokes equations are solved on a fixed half-staggered mesh by the rotational incremental pressure-correction scheme, and the Allen-Cahn type of phase equation is approximated by a conservative, second-order accurate central difference scheme, where the Lagrangian multiplier is used to preserve the mass-conservation of the phase field. The second part is an iteration procedure. During the mesh redistribution, the phase field is remapped onto the newly resulting meshes by the high-resolution conservative interpolation, while the non-conservative interpolation algorithm is applied to the velocity field. The projection technique is used to obtain a divergence-free velocity field at the end of this part. The resultant numerical scheme is stable, mass conservative, highly efficient and fast, and capable of handling variable density and viscosity. Several numerical experiments are presented to demonstrate the efficiency and robustness of the proposed algorithm.     

Data discovering of inverse Robin boundary conditions problem in arbitrary connected domain through meshless radial point Hermite interpolation

In this paper, a suitable method is presented to treat the partial derivative equations, especially the Laplace equation having the Robin boundary conditions. These equations come from classical physics, especially the branch of thermodynamics, and have an efficient role in the field of heat and temperature. Our motivation is to reset a harmonic data obtained from Robin’s conditions in the arbitrary plane domain particularly on its boundaries. The applied method is a nodal Hermite meshless collocation technique at which it is formed of radial basis functions to get out the shape functions which is the key to construct the local bases in the neighborhoods of the nodal points. Moreover, by taking into consideration the Hermite interpolation technique, we can impose the boundary conditions directly, the named technique is called “MRPHI,” meshless radial point Hermite interpolation, and it is done on some examples so that trustworthy results are obtained.

     

Boundary value identification of inverse Cauchy problems in arbitrary plane domain through meshless radial point Hermite interpolation

This article presents an efficient method to solve elliptic partial differential equations which are the nucleus of several physical problems, especially in the electromagnetic and mechanics, such as the Poisson and Laplace equations, while the subject is to recover a harmonic data from the knowledge of Cauchy data on some part of the boundary of the arbitrary plane domain. This method is a local nodal meshless Hermite-type collocation technique. In this method, we use the radial-based functions to call out the shape functions that form the local base in the vicinity of the nodal points. We also take into account the Hermit interpolation technique for imposing the derivative conditions directly. The proposed technique called pseudospectral meshless radial point Hermit interpolation is applied on some illustrative examples by adding random noises on source function and reliable results are observed.

     

Flow field computation for the high voltage gas blast circuit breaker with the moving boundary

We studied the gas dynamics for the ideal gas in the simplified high voltage (HV) gas blast circuit breaker with the moving boundary. The piston and the electric contact are moving. Since the boundary is moving, it is difficult for the ordinary finite difference (FD) method or the finite element (FE) method to compute the solution. For the purpose of numerical simplicity and efficiency, we introduced an upwind meshfree scheme which is an excellent scheme for the time varying domain. Despite the low coding and computational cost, the numerical simulation is successfully conducted. Our method is even more efficient when considering a three-dimensional computation with a moving boundary.     

针对Kriging插值结果的空间查询方法

Kriging插值方法及其各种改进模型已被广泛应用,但由于其插值结果是栅格形式,因此不利于与矢量数据叠加分析。为了更加便捷地使用插值结果,在衡量Voronoi图和最小外接矩形特点的基础上,提出了适用于Kriging插值结果的数据结构及空间查询方法。查询某一点位的特征值时,先通过区域的最小外接矩形初步判断出该位置可能存在的区域,进而逐一判断点与所选区域的空间关系,根据点所在区域的属性值得到该点位的特征值。该方法实现了对Kriging插值结果的空间查询,其正确性通过某露天矿的实际运行数据得到了验证。实验结果表明,该方法的查询效率控制在毫秒级,能够满足矿区车载终端程序及类似应用的需求。     

空间插值分析算法综述

空间插值分析算法是一种应用于将离散点的测量数据转换为连续数据表面的算法,能够将连续数据曲面与其他空间现象的分布情况进行比较,它在空间信息方面具有广泛的应用场景,尤其是地理信息方面.对泰森多边形法、反距离权重插值法、样条函数插值法、克里金插值法等空间插值算法的插值原理和应用场景进行综述,对空间插值分析算法的进展和未来研究方向进行了探讨.     

基于时空变异函数的Kriging插值及实现

Kriging（克里金）算法通常用于对空间变量进行插值,但不能直接应用于时空变量,它需要进行时空扩展。以月平均气温数据为例,运用时空Kriging方法结合R统计语言进行时空插值研究及其实现。通过时序分解去除气温数据中季节变化项,在分别得到空间变异函数和时间变异函数的基础上构建一类积和式时空变异函数来描述变量的时空相关结构,并给出基于R语言的具体实现步骤。将普通Kriging方法进行时空扩展,应用于气温数据的时空插值中。验证结果表明,基于时空变异函数的Kriging方法能提供较高精度的插值效果,这为时空变量的插值预测提供了有效的途径。     

基于流体动力学的火焰实时绘制技术

为了实时有效地渲染真实的火焰,引入了基于流体动力学的气体建模方法和分形插值技术,在粗网格下采用半拉格朗日方法和隐式差分格式,直接求取火焰的纳威-斯托克斯方程,这种数值解法在粗糙网格、大的时间步下也能无条件稳定,能达到实时渲染的目的。在细网格下,为了渲染湍流火焰,采用分形插值的方法,增强湍流火焰的边缘细节。实验结果表明：该方法实现简单,仿真速度快,显示的动画效果真实,并且是元条件稳定的。     

基于Delaunay三角网的克里金并行算法优化

当采样点数据量较大时, 可以采用Delaunay三角剖分建立三角网来使用局部邻域采样点进行克里金插值. 但是该算法需要对每个插值点拟合半变异函数, 插值点规模大时造成巨大开销. 为此, 本文提出了一种以三角形为单位拟合半变异函数的克里金插值方法, 采用CPU-GPU负载均衡将部分计算优化, 充分考虑不均匀样本对克里金插值效果的影响. 结果表明, 本文算法能够保证不均匀样本集的插值效果, 提升了计算性能且能够保证较高的精度.     

Kriging方法在输电线路气象环境建模中的应用

由于气象测点个数有限、分布不均匀且与线路走廊不一致,无法为研究灾害引发的线路故障及其防御技术提供精确的输电线路气象环境数据模型.为此,本文分析了Kriging空间插值方法的实现原理,应用Kriging插值方法实现对输电线路气象环境数据的网格化建模,并制定插值结果校验方法及评价标准,选取江苏省级电网可获取的996个气象测点2017年某日的温度数据,通过Kriging法进行网格化插值并对结果进行分析比较,验证了适用于该区域输电线路的温度数据网格化插值的Kriging半变异函数模型的选取过程.     

空间克里金插值的时空扩展与实现

空间克里金插值常用来补充采样点不足的问题,当数据分布与时间和空间都有关系时,面向空间的方法直接应用到时空过程可能导致有价值信息在时间维的丢失,由此导致了时空克里金插值的研究。研究的目标是将空间插值模型扩展到时空领域并实现时空变异函数、时空插值和时空交叉验证。其方法是首先获得最佳变异函数模型和时空下的有效基台值、块金值与变程,然后实现时空克里金插值的扩展,最后通过时空交叉验证去验证扩展的时空克里金插值方法的有效性。验证结果表明,扩展的时空方法能为随机领域以一定的精度提供较多的信息,为不同时空环境下的预测或插值提供了一个有效的途径。     

加权最小二乘法改进遗传克里金插值方法研究

数据内插被广泛应用于地统计分析领域,克里金插值作为其中最为有效的方法之一,其原理是通过建立变异函数理论模型,得到可靠的权重值和拉格朗日系数,构成求解待测点的线性组合。为了有效地提高插值精度,文中利用加权最小二乘法优化遗传算法中的适应度函数,进而改进普通基于遗传算法优化的克里金插值方法。并且在MATLAB中利用外部工具箱确定模型参数,最后通过实例验证,将该方法与普通克里金插值以及遗传克里金插值结果进行对比,发现采用该方法,插值效果较好且误差也较小,证明了通过加权最小二乘法可以有效改进普通遗传克里金插值方法。     

Numerical solution of the shallow water equations with a fractional step method

A fractional step technique for the numerical solution of the shallow water equations is applied to study the evolution of the potential vorticity field. The height and velocity field of the shallow water equations are discretized on a fixed Eulerian grid and time-stepped with a fractional step method recently reported in [M. Shoucri, Comput. Phys. Comm. 164 (2004) 396; M. Shoucri, A. Qaddouri, M. Tanguay, J. Côté, A Fractional Steps Method for the Numerical Solution of the Shallow Water Equations, International Workshop on Solution of Partial Differential Equations, The Fields Institute, Toronto, August 2002], where the Riemann invariants of the equations are interpolated at each time step along the characteristics using a cubic spline interpolation. The potential vorticity, which develops steep gradients and evolve into thin filaments during the evolution, is nicely calculated at every time-step from the solution of the code. The method is efficient and has lower numerical diffusion than other methods, since it evolves the equations without the iterative steps involved in the multi-dimensional interpolation problem, and without the iteration associated with the intermediate step of solving a Helmholtz equation, usually associated with other methods like the semi-Lagrangian method. The absence of iterative steps in the present technique makes it very suitable for problems in which small time steps and grid sizes are required, as for instance in the present problem where steepness of the gradients and small scale structures are the main features of the potential vorticity, and more generally for problems of regional climate modeling. The simplicity of the method makes it very suitable for parallel computer.     

A two-steps time discretization scheme using the SPH method for shock wave propagation

The Smoothed Particle Hydrodynamics (SPH) is a meshfree method which has been applied to a wide range of problems. In the present work, a new time integration algorithm using a corrected SPH spatial discretization for small deformations is applied to solve the propagation of shock waves in viscoplastic continua. In the method presented herein the equations are formulated in terms of stress and velocity. A corrected Lagrangian kernel is employed and two different sets of particles are used for the time discretization. Numerical instabilities are not present when using this new SPH formulation. The method proposed here has been proved to be efficient and it provides solutions of good accuracy.     

薄板样条函数在空间数据插值中的应用

薄板样条函数是空间数据插值中一种重要的方法,介绍了该方法的基本原理,并以珠江河道地形数据为例,借助地理信息系统的二次开发功能,将薄板样条函数应用于空间插值,通过与测试样本点以及克里金插值在最大值、标准误等方面的比较分析,证明薄板样条函数是一种有效的空间数据插值方法。     

DRM multidomain mass conservative interpolation approach for the BEM solution of the two-dimensional navier-stokes equations

The dual reciprocity method (DRM) is a technique to transform the domain integrals that appear in the boundary element method into equivalent boundary integrals. In this approach, the nonlinear terms are usually approximated by an interpolation applied to the convective terms of the Navier-Stokes equations. In this paper, we introduce a radial basis function interpolation scheme for the velocity field, that satisfies the continuity equation (mass conservative). The proposed method performs better than the classical interpolation used in the DRM approach to represent such a field. The new scheme together with a subdomain variation of the dual reciprocity method allows better approximation of the nonlinear terms in the Navier-Stokes equations.     

Robust optimization of structural dynamic characteristics based on adaptive Kriging model and CNSGA

Dynamic characteristics greatly influence the comprehensive performance of a structure. But they are rarely included as objectives in traditional robust optimization of structures. In this study, a robust optimization model including both means and standard deviations of dynamic characteristic indices in the objective and constraint functions is constructed for improving the structural dynamic characteristics and reducing their fluctuations under uncertainty. Adaptive Kriging models are employed for the efficient computation of dynamic characteristics. An intelligent resampling technology is proposed to save computational costs and accelerate convergence of Kriging models, which takes full advantage of the test points for precision verification, the sample points within the local region of the biggest relative maximum absolute error and the near-optimal point to improve the global and local precision of Krigings. The high efficiency of proposed intelligent resampling technology is demonstrated by a numerical example. Finally, an efficient algorithm integrating adaptive Kriging models, Monte Carlo (MC) method, constrained non-dominated sorting genetic algorithm (CNSGA) is proposed to solve the robust optimization model of structural dynamic characteristics. Kriging models are interfaced with MC method to efficiently compute the fitness of individuals during CNSGA. The implementation of proposed methodology is explained in detail and highlighted by the robust optimization of a cone ring fixture with lots of circumferentially distributed holes in a large turbo generator aimed at moving its natural frequencies away from the exciting one. The comparison of the optimized design with the initial one demonstrates that the proposed methodology is feasible and applicable in engineering practice.     

Adaptive space–time finite element methods for the wave equation on unbounded domains

Comprehensive adaptive procedures with efficient solution algorithms for the time-discontinuous Galerkin space–time finite element method (DGFEM) including high-order accurate nonreflecting boundary conditions (NRBC) for unbounded wave problems are developed. Sparse multi-level iterative schemes based on the Gauss–Seidel method are developed to solve the resulting fully-discrete system equations for the interior hyperbolic equations coupled with the first-order temporal equations associated with auxiliary functions in the NRBC. Due to the local nature of wave propagation, the iterative strategy requires only a few iterations per time step to resolve the solution to high accuracy. Further cost savings are obtained by diagonalizing the mass and boundary damping matrices. In this case the algebraic structure decouples the diagonal block matrices giving rise to an explicit multi-corrector method. An h-adaptive space–time strategy is employed based on the Zienkiewicz–Zhu spatial error estimate using the superconvergent patch recovery (SPR) technique, together with a temporal error estimate arising from the discontinuous jump between time steps of both the interior field solutions and auxiliary boundary functions. For accurate data transfer between meshes, a new enhanced interpolation (EI) method is developed and compared to standard interpolation and projection. Numerical studies of transient radiation and scattering demonstrate the accuracy, reliability and efficiency gained from the adaptive strategy.