一种改进的无单元伽辽金方法

使用单位分解积分,对传统的无单元伽辽金方法进行改进.有限覆盖和单位分解是单位分解积分的数学基础,对单位分解积分进行了严格证明,并指出使用Shepard函数作为单位分解函数是一个很好的选择.数值实例表明,使用单位分解积分进行数值求积的无单元伽辽金方法是一种真正的无网格方法,与经典的背景网格积分相比具有更高的精度.     

Free Mesh Method: fundamental conception, algorithms and accuracy study

The finite element method (FEM) has been commonly employed in a variety of fields as a computer simulation method to solve such problems as solid, fluid, electro-magnetic phenomena and so on. However, creation of a quality mesh for the problem domain is a prerequisite when using FEM, which becomes a major part of the cost of a simulation. It is natural that the concept of meshless method has evolved. The free mesh method (FMM) is among the typical meshless methods intended for particle-like finite element analysis of problems that are difficult to handle using global mesh generation, especially on parallel processors. FMM is an efficient node-based finite element method that employs a local mesh generation technique and a node-by-node algorithm for the finite element calculations. In this paper, FMM and its variation are reviewed focusing on their fundamental conception, algorithms and accuracy.     

耦合径向基函数与多项式基函数的无网格方法

耦合径向基函数和多项式基函数,形成一种新的近似函数.该近似函数对散乱分布的离散数据点进行逼近时,只需节点信息,不需要划分网格.详细描述了耦合近似函数的建立、属性、插值行为及其形函数和形函数导数的性质.最后引入修正变分原理和单位分解积分技术求解边值问题,并给出了计算实例,表明耦合径向基函数和多项式基函数是一种非常有效的方法.     

Electric and magnetic losses modeled by a stable hybrid with explicit–implicit time-stepping for Maxwell’s equations

A stable hybridization of the finite-element method (FEM) and the finite-difference time-domain (FDTD) scheme for Maxwell’s equations with electric and magnetic losses is presented for two-dimensional problems. The hybrid method combines the flexibility of the FEM with the efficiency of the FDTD scheme and it is based directly on Ampère’s and Faraday’s law. The electric and magnetic losses can be treated implicitly by the FEM on an unstructured mesh, which allows for local mesh refinement in order to resolve rapid variations in the material parameters and/or the electromagnetic field. It is also feasible to handle larger homogeneous regions with losses by the explicit FDTD scheme connected to an implicitly time-stepped and lossy FEM region. The hybrid method shows second-order convergence for smooth scatterers. The bistatic radar cross section (RCS) for a circular metal cylinder with a lossy coating converges to the analytical solution and an accuracy of 2% is achieved for about 20 points per wavelength. The monostatic RCS for an airfoil that features sharp corners yields a lower order of convergence and it is found to agree well with what can be expected for singular fields at the sharp corners. A careful convergence study with resolutions from 20 to 140 points per wavelength provides accurate extrapolated results for this non-trivial test case, which makes it possible to use as a reference problem for scattering codes that model both electric and magnetic losses.     

On the flexibility of agglomeration based physical space discontinuous Galerkin discretizations

In this work we show that the flexibility of the discontinuous Galerkin (dG) discretization can be fruitfully exploited to implement numerical solution strategies based on the use of elements with very general shapes. Thanks to the freedom in defining the mesh topology, we propose a new h-adaptive technique based on agglomeration coarsening of a fine mesh. The possibility to enhance the error distribution over the computational domain is investigated on a Poisson problem with the goal of obtaining a mesh independent discretization.The main building block of our dG method consists of defining discrete polynomial spaces directly on physical frame elements. For this purpose we orthonormalize with respect to the L²-product a set of monomials relocated in a specific element frame and we introduce an easy way to reduce the cost related to numerical integration on agglomerated meshes. To complete the dG formulation for second order problems, two extensions of the BR2 scheme to arbitrary polyhedral grids, including an estimate of the stabilization parameter ensuring the coercivity property, are here proposed.     

Discrete Torsion Phases as Topological Actions

In this paper, we show that discrete torsion phases in string orbifold partition functions, and membrane discrete torsion phases, are topological actions on the simplicial manifolds associated to orbifold group actions. For this purpose, we introduce an integration theory of smooth Deligne cohomology on a general simplicial manifold, and prove that the integration induces a well-defined paring between the smooth Deligne cohomology and the singular cycles.     

A multiscale hp-FEM for 2D photonic crystal bands

A multiscale generalised hp-finite element method (MSFEM) for time harmonic wave propagation in bands of locally periodic media of large, but finite extent, e.g., photonic crystal (PhC) bands, is presented. The method distinguishes itself by its size robustness, i.e., to achieve a prescribed error its computational effort does not depend on the number of periods. The proposed method shows this property for general incident fields, including plane waves incident at a certain angle to the infinite crystal surface, and at frequencies in and outside of the bandgap of the PhC. The proposed MSFEM is based on a precomputed problem adapted multiscale basis. This basis incorporates a set of complex Bloch modes, the eigenfunctions of the infinite PhC, which are modulated by macroscopic piecewise polynomials on a macroscopic FE mesh. The multiscale basis is shown to be efficient for finite PhC bands of any size, provided that boundary effects are resolved with a simple macroscopic boundary layer mesh. The MSFEM, constructed by combing the multiscale basis inside the crystal with some exterior discretisation, is a special case of the generalised finite element method (g-FEM). For the rapid evaluation of the matrix entries we introduce a size robust algorithm for integrals of quasi-periodic micro functions and polynomial macro functions. Size robustness of the present MSFEM in both, the number of basis functions and the computation time, is verified in extensive numerical experiments.     

A piecewise linear finite element discretization of the diffusion equation for arbitrary polyhedral grids

We develop a piecewise linear (PWL) Galerkin finite element spatial discretization for the multi-dimensional radiation diffusion equation. It uses recently introduced piecewise linear weight and basis functions in the finite element approximation and it can be applied on arbitrary polygonal (2D) or polyhedral (3D) grids. We first demonstrate some analytical properties of the PWL method and perform a simple mode analysis to compare the PWL method with Palmer’s vertex-centered finite-volume method and with a bilinear continuous finite element method. We then show that this new PWL method gives solutions comparable to those from Palmer’s. However, since the PWL method produces a symmetric positive-definite coefficient matrix, it should be substantially more computationally efficient than Palmer’s method, which produces an asymmetric matrix. We conclude that the Galerkin PWL method is an attractive option for solving diffusion equations on unstructured grids.     

IDeC(k): A new velocity reconstruction algorithm on arbitrarily polygonal staggered meshes

We propose a novel algorithm for velocity reconstruction from staggered data on arbitrary polygonal staggered meshes. The formulation of the new algorithm is based on a constant polynomial reconstruction approach in conjunction with an iterative defect correction method and is referred to as the IDeC(k) reconstruction. The algorithm is designed for second order accuracy of the reconstructed velocity field and also leads to a consistent estimate of velocity gradients. Accuracy, convergence and robustness of the new algorithm are studied on different mesh topologies and the need for higher-order reconstruction is demonstrated. Numerical experiments for several cases including incompressible viscous flows establish the IDeC(k) reconstruction as a generic, fast, robust and higher-order accurate algorithm on arbitrary polygonal meshes.     

动态断裂力学的无网格流形方法

运用无网格流形方法求解动态断裂力学问题.该方法利用单位分解法和有限覆盖技术建立形函数，形函数的建立不受域内不连续的影响，可较好地求解裂纹问题.对于局部化问题，该方法的形函数构造较其他方法更为有效，避免了其他方法在建立试函数时没有考虑不连续尖端的缺点.由于采用有限覆盖技术建立试函数，该方法克服了不连续对试函数的影响，尤其当不连续变得复杂时，更能显示该方法在处理不连续方面的优点.在求解动态断裂力学问题时，弹性动力学积分弱形式的推导采用加权残数法，空间离散采用基于单位分解法的无网格流形方法，时间离散主要采用Newmark法.最后给出两个数值算例，将计算结果与解析解对比，说明该方法的正确性和可行性. 关键词： 有限覆盖 无网格流形方法 动态断裂力学 动态应力强度因子     

Meshless Collocation Method for Inverse Source Identification Problems

A novel meshless scheme is proposed for inverse source identification problems of Helmholtz-type equations. It is formulated by the non-singular general solutions of the Helmholtz-type equations augmented with radial basis functions. Under this meshless scheme, we can determine smooth source terms from partially accessible boundary measurements with accurate results. Numerical examples are presented to verify validity and accuracy of the present scheme. It is demonstrated that the present scheme is simple, accurate, stable and computationally efficient for inverse smooth source identification problems.     

Wavelet frame based scene reconstruction from range data

How to reconstruct the scene (a visible surface) from a set of scattered, noisy and possibly sparse range data is a challenging problem in robotic navigation and computer graphics. As most real scenes can be modeled by piecewise smooth surfaces, traditional surface fitting techniques (e.g. smoothing spline) generally can not preserve sharp discontinuities of surfaces. Based on sparse approximation of piecewise smooth functions in frame domain, we propose a new tight frame based formulation for reconstructing a piecewise smooth surface from a sparse range data set, which is robust to both additive noise and outliers. Furthermore, the resulting minimization problem from our formulation can be efficiently solved by the split Bregman method [1], [2]. The numerical experiments show that the proposed approach is capable of reconstructing a piecewise smooth surface with sharp edges from sparse range data corrupted with noise and outliers.     

A meshless model for transient heat conduction analyses of 3D axisymmetric functionally graded solids

A meshless numerical model is developed for analyzing transient heat conductions in three-dimensional （3D） axisymmetric continuously nonhomogeneous functionally graded materials （FGMs）. Axial symmetry of geometry and boundary conditions reduces the original 3D initial-boundary value problem into a two-dimensional （2D） problem. Local weak forms are derived for small polygonal sub-domains which surround nodal points distributed over the cross section. In order to simplify the treatment of the essential boundary conditions, spatial variations of the temperature and heat flux at discrete time instants are interpolated by the natural neighbor interpolation. Moreover, the using of three-node triangular finite element method （FEM） shape functions as test functions reduces the orders of integrands involved in domain integrals. The semi-discrete heat conduction equation is solved numerically with the traditional two-point difference technique in the time domain. Two numerical examples are investigated and excellent results are obtained, demonstrating the potential application of the proposed approach.     

Lattice CP models and their large-N behaviour

Two different lattice actions reproducing the CP^N−1 model in the continuum are discussed and, in the two cases, strong and weak coupling expansions relative to different physical quantities are obtained. Large-N behaviour is analyzed by evaluating the limiting value of the two partition functions. In neither case does a smooth transition seem possible between strong and weak coupling regions for the limiting model.     

Small-scale instabilities in dynamical systems with sliding

We demonstrate with a minimal example that in Filippov systems (dynamical systems governed by discontinuous but piecewise smooth vector fields) stable periodic motion with sliding is not robust with respect to stable singular perturbations. We consider a simple dynamical system that we assume to be a quasi-static approximation of a higher-dimensional system containing a fast stable subsystem. We tune a system parameter such that a stable periodic orbit of the simple system touches the discontinuity surface: this is the so-called grazing-sliding bifurcation. The periodic orbit remains stable, and its local return map becomes piecewise linear. However, when we take into account the fast dynamics the local return map of the periodic orbit changes qualitatively, giving rise to, for example, period-adding cascades or small-scale chaos.     

一类时变时滞混沌系统的参数辨识方法

时变的未知时滞参数普遍存在于混沌系统中,它使得混沌系统同步控制变得非常困难. 针对时滞混沌系统中参数时变且未知的问题, 提出了一种新颖的辨识方法. 该方法首先将未知时变参数用分段常数函数来近似, 把求解非线性函数的问题转化为参数向量选择问题, 其中分段常数函数的高度向量成为待求解参数向量; 然后推导了目标函数对分段常数高度向量的梯度信息, 结合序列二次规划法求解得最优分段函数; 随着分段数的增加, 最优分段函数将逼近原非线性时变函数. 数值实例结果验证了该方法的有效性.     

Modeling merging and breakup in the moving mesh interface tracking method for multiphase flow simulations

The three-dimensional, moving mesh interface tracking (MMIT) method coupled with local mesh adaptations by Quan and Schmidt [S.P. Quan, D.P. Schmidt, A moving mesh interface tracking method for 3D incompressible two-phase flows, J. Comput. Phys. 221 (2007) 761–780] demonstrated the capability to accurately simulate multiphase flows, to handle large deformation, and also to perform interface pinch-off for some specific cases. However, another challenge, i.e. how to handle interface merging (such as droplet coalescence) has not been addressed. In this paper, we present a mesh combination scheme for interface connection and a more general mesh separation algorithm for interface breakup. These two schemes are based on the conversion of liquid cells in one phase to another fluid by changing the fluid properties of the cells in the combination or separation region. After the conversion, the newly created interface is usually ragged, and a local projection method is employed to smooth the interface. Extra mesh adaptation criteria are introduced to handle colliding interfaces with almost zero curvatures as the distance between the interfaces diminishes. Simulations of droplet pair collisions including both head-on and off-center coalescences show that the mesh adaptations are capable of resolving very small length scales, and the mesh combination and mesh separation schemes can handle the topological transitions in multiphase flows. The potential of our method to perform detailed investigations of droplet coalescence and breakup is also displayed.     

Computational performance of Free Mesh Method applied to continuum mechanics problems

The free mesh method (FMM) is a kind of the meshless methods intended for particle-like finite element analysis of problems that are difficult to handle using global mesh generation, or a node-based finite element method that employs a local mesh generation technique and a node-by-node algorithm. The aim of the present paper is to review some unique numerical solutions of fluid and solid mechanics by employing FMM as well as the Enriched Free Mesh Method (EFMM), which is a new version of FMM, including compressible flow and sounding mechanism in air-reed instruments as applications to fluid mechanics, and automatic remeshing for slow crack growth, dynamic behavior of solid as well as large-scale Eigen-frequency of engine block as applications to solid mechanics.     

Mimetic finite difference method for the Stokes problem on polygonal meshes

Various approaches to extend finite element methods to non-traditional elements (general polygons, pyramids, polyhedra, etc.) have been developed over the last decade. The construction of basis functions for such elements is a challenging task and may require extensive geometrical analysis. The mimetic finite difference (MFD) method works on general polygonal meshes and has many similarities with low-order finite element methods. Both schemes try to preserve the fundamental properties of the underlying physical and mathematical models. The essential difference between the two schemes is that the MFD method uses only the surface representation of discrete unknowns to build the stiffness and mass matrices. Since no extension of basis functions inside the mesh elements is required, practical implementation of the MFD method is simple for polygonal meshes that may include degenerate and non-convex elements. In this article, we present a new MFD method for the Stokes problem on arbitrary polygonal meshes and analyze its stability. The method is developed for the general case of tensor coefficients, which allows us to apply it to a linear elasticity problem, as well. Numerical experiments show, for the velocity variable, second-order convergence in a discrete L² norm and first-order convergence in a discrete H¹ norm. For the pressure variable, first-order convergence is shown in the L² norm.