Grid method: Application to the characterization of cracks

In this paper we present an application of the grid method to the assessment of crack initiation and growth in brittle materials. It is shown that a grid deposited on the surface of a structure can be utilized to characterize the crack location and opening. Furthermore, one single picture only is required to carry out a control over a predefined area. First, the principle of the method is introduced, showing that a periodical pattern encoding a cracked surface can be analyzed by signal processing. The phase modulation of the grid can be calculated very accurately for any small deformation of the structure. Discontinuities due to cracks are easily detected on the phase field and the crack width is proportional to the amplitude of the discontinuity. Secondly, the performances of the method are presented, mainly depending on the spatial frequency of both the pattern and the sampling. It is shown that good results can generally be obtained with few setting difficulties and equipment investment. Finally, we sum up some basic recommendations for the practical use of the technique.     

Multigrid preconditioned Krylov subspace methods for three-dimensional numerical solutions of the incompressible Navier–Stokes equations

We consider numerical methods for the incompressible Reynolds averaged Navier–Stokes equations discretized by finite difference techniques on non-staggered grids in body-fitted coordinates. A segregated approach is used to solve the pressure–velocity coupling problem. Several iterative pressure linear solvers including Krylov subspace and multigrid methods and their combination have been developed to compare the efficiency of each method and to design a robust solver. Three-dimensional numerical experiments carried out on scalar and vector machines and performed on different fluid flow problems show that a combination of multigrid and Krylov subspace methods is a robust and efficient pressure solver. This revised version was published online in June 2006 with corrections to the Cover Date.     

A Cartesian‐grid collocation method based on radial‐basis‐function networks for solving PDEs in irregular domains

This paper reports a new Cartesian‐grid collocation method based on radial‐basis‐function networks (RBFNs) for numerically solving elliptic partial differential equations in irregular domains. The domain of interest is embedded in a Cartesian grid, and the governing equation is discretized by using a collocation approach. The new features here are (a) one‐dimensional integrated RBFNs are employed to represent the variable along each line of the grid, resulting in a significant improvement of computational efficiency, (b) the present method does not require complicated interpolation techniques for the treatment of Dirichlet boundary conditions in order to achieve a high level of accuracy, and (c) normal derivative boundary conditions are imposed by means of integration constants. The method is verified through the solution of second‐ and fourth‐order PDEs; accurate results and fast convergence rates are obtained. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Governing equations, experiments, and the experimentalist

It is argued that the most important test of the understanding of any experiment is whether or not the results are consistent with the equations and boundary conditions believed to govern the flow. If they are not, then either the measurements are incorrect, the equations or boundary conditions are wrong, or the experiment performed was not the one believed to have been done. It is suggested that many apparently contradictory experimental results are, in fact, the result of the latter and have not been recognized as such because of a failure to apply the governing equations to the data. The arguments are supported by examples from turbulent shear flow experiments.     

A two‐grid stabilization method for solving the steady‐state Navier‐Stokes equations

We formulate a subgrid eddy viscosity method for solving the steady‐state incompressible flow problem. The eddy viscosity does not act on the large flow structures. Optimal error estimates are obtained for velocity and pressure. The numerical illustrations agree completely with the theoretical results. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

A higher-order hybrid spline difference method on adaptive mesh for solving singularly perturbed parabolic reaction–diffusion problems with robin-boundary conditions

We propose a hybrid numerical scheme to discretize a class of singularly perturbed parabolic reaction–diffusion problems with robin-boundary conditions on an equidistributed grid. The hybrid difference scheme is developed by using a modified backward difference scheme in time, a combination of the cubic spline and exponential spline difference scheme in space. The proposed scheme uses a cubic spline difference scheme for the discretization of robin-boundary conditions. For the time discretization of the problem, we use the standard uniform mesh while a layer adapted equidistributed grid is generated for the spatial discretization. By equidistributing a curvature-based monitor function, the spatial adaptive grid is able to capture the presence of parabolic boundary layers without using any prior information about the solution. Parameter uniform error estimates are derived to illustrate an optimal convergence of first-order in time and second-order in space for the proposed discretization. The accuracy of the proposed scheme is confirmed by the numerical experiments that underpin the theoretical analysis.     

New Sixth-Order Compact Schemes for Poisson/Helmholtz Equations

Some new sixth-order compact finite difference schemes for Poisson/Helmholtz equations on rectangular domains in both two- and three-dimensions are developed and analyzed. Different from a few sixth-order compact finite difference schemes in the literature, the finite difference and weight coefficients of the new methods have analytic simple expressions. One of the new ideas is to use a weighted combination of the source term at staggered grid points which is important for grid points near the boundary and avoids partial derivatives of the source term. Furthermore, the new compact schemes are exact for 2D and 3D Poisson equations if the solution is a polynomial less than or equal to 6. The coefficient matrices of the new schemes are $M$-matrices for Helmholtz equations with wave number $K≤0,$ which guarantee the discrete maximum principle and lead to the convergence of the new sixth-order compact schemes. Numerical examples in both 2D and 3D are presented to verify the effectiveness of the proposed schemes.     

二维振荡叶栅非定常粘性流动数值模拟

采用显式四步Runge-Kutta格式，结合Baldwon-Lomax紊流模型求解Navier-Stokes方程，借助运动网格技术，完成了对二维振荡叶栅非定常粘性流动的数值模拟。为了加速求解过程，引入了变系数隐式残差光顺方法，取得了较好效果。数值结果与已公布的数据有很好的一致性。     

面向星敏感器的星模式识别算法

介绍了到目前为止出现的所有面向星敏感器的星模式识别算法,它们是概率统计算法、三角形算法、匹配组算法、网格算法、奇异值分解算法、神经网络算法和遗传算法,并分两组比较了它们的性能,给出了具有指导意义的结论。     

The normal contact of rough surfaces with fine discretization

Summary The numerical treatment of a 3-d frictionless normal contact of rough bodies made of a linear–elastic, isotropic material is considered. First, the governing equations and their discretized forms are presented. Due to the order of the problem, the computation time rapidly increases with the grid refinement. Therefore, common computer programs have to be limited to 50×50 contact points. Several numerical techniques (multigrid, iterative solver, distributive relaxation) are used. Cell-centered grid transformation is applied and grid operators have been established for this purpose. In order to handle upto 1000 × 1000 cells the program CON3D has been developed.