A meshless algorithm with moving least square approximations for elliptic Signorini problems

Based on the moving least square(MLS) approximations and the boundary integral equations(BIEs), a meshless algorithm is presented in this paper for elliptic Signorini problems. In the algorithm, a projection operator is used to tackle the nonlinear boundary inequality conditions. The Signorini problem is then reformulated as BIEs and the unknown boundary variables are approximated by the MLS approximations. Accordingly, only a nodal data structure on the boundary of a domain is required. The convergence of the algorithm is proven. Numerical examples are given to show the high convergence rate and high computational efficiency of the presented algorithm.     

SVD–GFD scheme to simulate complex moving body problems in 3D space

The present paper presents a hybrid meshfree-and-Cartesian grid method for simulating moving body incompressible viscous flow problems in 3D space. The method combines the merits of cost-efficient and accurate conventional finite difference approximations on Cartesian grids with the geometric freedom of generalized finite difference (GFD) approximations on meshfree grids. Error minimization in GFD is carried out by singular value decomposition (SVD). The Arbitrary Lagrangian–Eulerian (ALE) form of the Navier–Stokes equations on convecting nodes is integrated by a fractional-step projection method. The present hybrid grid method employs a relatively simple mode of nodal administration. Nevertheless, it has the geometrical flexibility of unstructured mesh-based finite-volume and finite element methods. Boundary conditions are precisely implemented on boundary nodes without interpolation. The present scheme is validated by a moving patch consistency test as well as against published results for 3D moving body problems. Finally, the method is applied on low-Reynolds number flapping wing applications, where large boundary motions are involved. The present study demonstrates the potential of the present hybrid meshfree-and-Cartesian grid scheme for solving complex moving body problems in 3D.     

Closure Approximations for Passive Scalar Turbulence: A Comparative Study on an Exactly Solvable Model with Complex Features

Some standard closure approximations used in turbulence theory are analyzed by examining systematically the predictions these approximations produce for a passive scalar advection model consisting of a shear flow with a fluctuating cross sweep. This model has a general geometric structure of a jet flow with transverse disturbances, which occur in a number of contexts, and it encompasses a wide variety of possible spatio-temporal statistical structures for the velocity field, including strong long-range correlations. Even though the Eulerian and Lagrangian velocity statistics are not equal and the passive scalar statistics exhibit broader-than-Gaussian intermittency, this model is nevertheless simple enough so that many passive scalar statistics can be computed exactly and compared systematically with the predictions of the closure approximations. Our comparative study illustrates the strength and weaknesses of the closure approximations and points out the physical phenomena that these approximations are able or not able to describe properly. In particular it is shown that the direct interaction approximation (DIA), one of the most sophisticated closure approximations available, fails to reproduce adequately the statistical features of the scalar and may even lead to absdurd predictions, even though the equations it produces are rather complicated and difficult to analyze. Two alternative closure approximations, the Modified DIA (MDIA) and the Renormalized Lagrangian Approximation (RLA), with different levels of sophistication, both are simpler to use than the DIA and perform better. In particular, it is shown that both closure approximations always reproduce exactly the second order statistics for the scalar and that the MDIA is even able to capture intermittency effects.     

A splitting extrapolation for solving nonlinear elliptic equations with d-quadratic finite elements

Nonlinear elliptic partial differential equations are important to many large scale engineering and science problems. For this kind of equations, this article discusses a splitting extrapolation which possesses a high order of accuracy, a high degree of parallelism, less computational complexity and more flexibility than Richardson extrapolation. According to the problems, some domain decompositions are constructed and some independent mesh parameters are designed. Multi-parameter asymptotic expansions are proved for the errors of approximations. Based on the expansions, splitting extrapolation formulas are developed to compute approximations with high order of accuracy on a globally fine grid. Because these formulas only require us to solve a set of smaller discrete subproblems on different coarser grids in parallel instead of on the globally fine grid, a large scale multidimensional problem is turned into a set of smaller discrete subproblems. Additionally, this method is efficient for solving interface problems.     

Coupling of Gaussian Beam and Finite Difference Solvers for Semiclassical Schrödinger Equations

In the semiclassical regime, solutions to the time-dependent Schrödinger equation for molecular dynamics are highly oscillatory. The number of grid points required for resolving the oscillations may become very large even for simple model problems, making solution on a grid intractable. Asymptotic methods like Gaussian beams can resolve the oscillations with little effort and yield good approximations when the atomic nuclei are heavy and the potential is smooth. However, when the potential has variations on a small length-scale, quantum phenomena become important. Then asymptotic methods are less accurate. The two classes of methods perform well in different parameter regimes. This opens for hybrid methods, using Gaussian beams where we can and finite differences where we have to. We propose a new method for treating the coupling between the finite difference method and Gaussian beams. The new method reduces the needed amount of overlap regions considerably compared to previous methods, which improves the efficiency.     

A low numerical dissipation immersed interface method for the compressible Navier–Stokes equations

A numerical method to solve the compressible Navier–Stokes equations around objects of arbitrary shape using Cartesian grids is described. The approach considered here uses an embedded geometry representation of the objects and approximate the governing equations with a low numerical dissipation centered finite-difference discretization. The method is suitable for compressible flows without shocks and can be classified as an immersed interface method. The objects are sharply captured by the Cartesian mesh by appropriately adapting the discretization stencils around the irregular grid nodes, located around the boundary. In contrast with available methods, no jump conditions are used or explicitly derived from the boundary conditions, although a number of elements are adopted from previous immersed interface approaches. A new element in the present approach is the use of the summation-by-parts formalism to develop stable non-stiff first-order derivative approximations at the irregular grid points. Second-order derivative approximations, as those appearing in the transport terms, can be stiff when irregular grid points are located too close to the boundary. This is addressed using a semi-implicit time integration method. Moreover, it is shown that the resulting implicit equations can be solved explicitly in the case of constant transport properties. Convergence studies are performed for a rotating cylinder and vortex shedding behind objects of varying shapes at different Mach and Reynolds numbers.     

对称性及多群中子扩散方程数值解

在多群中子扩散方程解析解的基础上,利用方程及求解域的对称性建立了新的数值求解中子扩散方程的理论模型.该模型显著的优点是适用于各种对称区域(二维、三维区域)尤其是非正方形区域中子扩散方程的求解,它彻底避免了常规节块法应用于非正方形几何时所出现的奇异性问题,且所得的解在求解域内任意点上均满足扩散方程.以二、三维六角形几何为例建立了数学模型,并用基准问题校核了模型的正确性. 关键词： 中子扩散方程 对称群 数值解 解析     

Electromagnetics via the Taylor-Galerkin Finite Element Method on Unstructured Grids

Traditional techniques for computing electromagnetic solutions in the time domain rely on finite differences. These so-called FDTD (finite-difference time-domain) methods are usually defined only on regular lattices of points and can be too restrictive for geometrically demanding problems. Great geometric flexibility can be achieved by abandoning the regular latticework of sample points and adopting an unstructured grid. An unstructured grid allows one to place the grid points anywhere one chooses, so that curved boundaries can be fit with ease and local regions in which the field gradients are steep can be selectively resolved with a fine mesh. In this paper we present a technique for solving Maxwell's equations on an unstructured grid based on the Taylor-Galerkin finite-element method. We present several numerical examples which reveal the fundamental accuracy and adaptability of the method. Although our examples are in two dimensions, the techniques and results generalize readily to 3D.     

NS方程的各向异性笛卡尔网格法研究

将近年发展起来的用于Euler方程求解的具有局部均匀网格总体非结构特性的笛卡尔网格法推广到NS方程的求解。为了与流场的各向异性相适应、减少网格点数量,提出了一种各向异性网格加密法。另外还研究了分级笛卡尔网格对内点格式稳定性的影响和插值固体边界条件的稳定性。数值结果表明各向异性笛卡尔网格法相对于传统的各向同性网格方法能大量节省网格点数量而且与后者具有同样的精度。     

Letter: On a Semi-Numeric Method to Study the Evolution of a General-Relativistic Radiating Shell and X-Ray Bursters

A semi-numeric method was used to study the evolution of a radiating shell in its Post-Newtonian and Newtonian approximations. The solutions, show that a relativistic effect can be found in collapsed stars (X and gamma ray bursters).     

虚拟流体方法中界面处Riemann问题定义方式的改进

对RGFM中定义Riemann问题的方式进行改进,取距离界面适当远处的插值点处的状态作为Riemann问题的初值.并用数值算例对改进前后的RGFM进行比较.     

Simple finite-difference algorithm for calculating waveguide modes

Abstact A simple algorithm for solving the finite-difference (FD) equations for the mode eigenvalues and field distributions of a linear waveguide is presented. By applying the discretized Helmholtz operator column-by-column to an index distribution defined on an N x N grid, a matrix whose size is only a few times N x N is obtained. This yields a reduction in computation time and space compared with the other classical FD approaches which involve an N ² x N ² matrix. Our method is tested against problems for which the exact solutions are known, and we find a high degree of accuracy. Despite the existence of fast algorithms for the treatment of the classical N ² x N ² matrix, our new algorithm presents some advantages over existing FD methods of comparable speed, including: the ability to find all the modes and associated field profiles, very high numerical stability, and no numerical approximations in the procedure. In addition, some general optimum expressions for the domain size and density of grid points which are consistent with the desired precision are provided, and apply to any FD method including ours.     

Deforming composite grids for solving fluid structure problems

We describe a mixed Eulerian–Lagrangian approach for solving fluid–structure interaction (FSI) problems. The technique, which uses deforming composite grids (DCG), is applied to FSI problems that couple high speed compressible flow with elastic solids. The fluid and solid domains are discretized with composite overlapping grids. Curvilinear grids are aligned with each interface and these grids deform as the interface evolves. The majority of grid points in the fluid domain generally belong to background Cartesian grids which do not move during a simulation. The FSI-DCG approach allows large displacements of the interfaces while retaining high quality grids. Efficiency is obtained through the use of structured grids and Cartesian grids. The governing equations in the fluid and solid domains are evolved in a partitioned approach. We solve the compressible Euler equations in the fluid domains using a high-order Godunov finite-volume scheme. We solve the linear elastodynamic equations in the solid domains using a second-order upwind scheme. We develop interface approximations based on the solution of a fluid–solid Riemann problem that results in a stable scheme even for the difficult case of light solids coupled to heavy fluids. The FSI-DCG approach is verified for three problems with known solutions, an elastic-piston problem, the superseismic shock problem and a deforming diffuser. In addition, a self convergence study is performed for an elastic shock hitting a fluid filled cavity. The overall FSI-DCG scheme is shown to be second-order accurate in the max-norm for smooth solutions, and robust and stable for problems with discontinuous solutions for a wide range of constitutive parameters.     

Numerical accuracy of “equivalent” spherical approximations for computing ensemble-averaged scattering properties of fractal soot aggregates

Numerical accuracy is quantitatively assessed in conjunction with the application of four “equivalent” spherical approximations in the computation of the optical properties of small aggregate soot particles. The approximations are based on equal volume, equal surface area, the radius of gyration, and a collection of independent spheres with the same volume and the same volume-to-projected area ratio as the original nonspherical particle. A diffusion-limited cluster-cluster aggregation algorithm is used to specify the geometries of soot particles. Furthermore, the Generalized Multi-particle Mie (GMM) method is utilized to compute the single-scattering properties of individual soot aggregate particles assumed to be randomly oriented in space. The ensemble-averaged single-scattering properties of the particles are obtained by accounting for the probability distribution functions (PDF) of the number of monomers per aggregate at two wavelengths, 0.628 and 1.1 μm. It is shown that all of the aforementioned equivalent-spherical approximations lead to large errors in the computation of the phase function.     

严谨而精密的创新——瑞利进行研究的特点

着重介绍了瑞利在测量氮气密度过程中发现惰性气体氩的研究和创新历程,以及瑞利对光学、声学等物理学的各个领域的主要贡献,指出正是求实的科学态度、锲而不舍的钻研精神和以创新为乐趣的事业追求是瑞利一生中取得丰硕的研究成果的主要原因。     

Uncertainty quantification via random domain decomposition and probabilistic collocation on sparse grids

Quantitative predictions of the behavior of many deterministic systems are uncertain due to ubiquitous heterogeneity and insufficient characterization by data. We present a computational approach to quantify predictive uncertainty in complex phenomena, which is modeled by (partial) differential equations with uncertain parameters exhibiting multi-scale variability. The approach is motivated by flow in random composites whose internal architecture (spatial arrangement of constitutive materials) and spatial variability of properties of each material are both uncertain. The proposed two-scale framework combines a random domain decomposition (RDD) and a probabilistic collocation method (PCM) on sparse grids to quantify these two sources of uncertainty, respectively. The use of sparse grid points significantly reduces the overall computational cost, especially for random processes with small correlation lengths. A series of one-, two-, and three-dimensional computational examples demonstrate that the combined RDD–PCM approach yields efficient, robust and non-intrusive approximations for the statistics of diffusion in random composites.     

Evaluation of the parameters of fluorescence-intensity decay with allowance for the energy-transfer processes

Algorithms for evaluation of the parameters of fluorescence-intensity decay with allowance for the energytransfer processes are suggested. The evaluation is accomplished using the standard methods of linear optimization that provide a high rate of data processing, an acceptable accuracy, and independence of the evaluation procedure from the initial approximations. The latter makes it possible to use these algorithms both independently and for generation of initial approximations for iteration procedures, thus making the search for the minimum of the criterion more rapid and efficient. The algorithms suggested are tested by simulation methods. Belarusian State University, 4 F. Skorina Ave., Minsk, 220002, Belarus; e-mail: novikov@rfe.bsu.unibel.by. Translated from Zhurnal Prikladnoi Spektroskopii, Vol. 66, No. 6, pp. 836–841, November–December, 1999.     

Structural,elastic, and electronic properties of topological semimetal WC-type MX family by first-principles calculation

Recently, the non-centrosymmetric WC-type materials(i.e., MoP, ZrTe, TaN, etc) have attracted extensive interest due to the discovery of their topological properties.By means of the first-principles calculations, here we have investigated the structural, thermodynamic, elastic, and electronic properties of the WC-type MX compounds(TiS, TiSe, TiTe, ZrS, ZrSe,ZrTe, HfS, HfSe, and HfTe).Among these nine compounds, five of them(TiS, ZrS, ZrSe_(0.9), ZrTe, and Hf0.92 Se) have been experimentally synthesized to crystallize in the WC-type structure and other four members have never been reported.Our calculations demonstrated that they are all structurally, thermodynamically, and dynamically stable, indicating that all of them should be possibly synthesized.We have also derived their elastic constants of single crystalline and their bulk and shear moduli in terms of the R.Hill approximations.Furthermore, in similarity to ZrTe, all these compounds have been theoretically derived to be topological semimetals.Whereas TiS is unique because of the coexistence of the Dirac nodal lines(DNLs) and sixfold degenerate nodal points(sixfold DNPs), the other eight members are revealed to exhibit coexisted Weyl nodes(WPs) and triply degenerate nodal points(TDNPs).Their electronic and topological properties have been further discussed.     

Accurate computation of Stokes flow driven by an open immersed interface

We present numerical methods for computing two-dimensional Stokes flow driven by forces singularly supported along an open, immersed interface. Two second-order accurate methods are developed: one for accurately evaluating boundary integral solutions at a point, and another for computing Stokes solution values on a rectangular mesh. We first describe a method for computing singular or nearly singular integrals, such as a double layer potential due to sources on a curve in the plane, evaluated at a point on or near the curve. To improve accuracy of the numerical quadrature, we add corrections for the errors arising from discretization, which are found by asymptotic analysis. When used to solve the Stokes equations with sources on an open, immersed interface, the method generates second-order approximations, for both the pressure and the velocity, and preserves the jumps in the solutions and their derivatives across the boundary. We then combine the method with a mesh-based solver to yield a hybrid method for computing Stokes solutions at N² grid points on a rectangular grid. Numerical results are presented which exhibit second-order accuracy. To demonstrate the applicability of the method, we use the method to simulate fluid dynamics induced by the beating motion of a cilium. The method preserves the sharp jumps in the Stokes solution and their derivatives across the immersed boundary. Model results illustrate the distinct hydrodynamic effects generated by the effective stroke and by the recovery stroke of the ciliary beat cycle.     

Asymptotic behavior of the Kohn-Sham exchange potential

The Kohn-Sham exchange potential of finite systems is shown to approach different asymptotic limits on nodal surfaces of the energetically highest-occupied orbital than in other regions. This leads to barrier-well structures in the near asymptotic region, which have a strong influence on virtual orbitals and thus on excitation energies. Common approximations for the exchange potential do not exhibit these features. These asymptotic structures, however, can be correctly described by effective exact-exchange methods. Conditions for the presence of an asymptotic barrier well in the full exchange-correlation potential are discussed.