Multilevel preconditioners for a quadrature Galerkin solution of a biharmonic problem

Efficient multilevel preconditioners are developed and analyzed for the quadrature finite element Galerkin approximation of the biharmonic Dirichlet problem. The quadrature scheme is formulated using the Bogner–Fox–Schmit rectangular element and the product two‐point Gaussian quadrature. The proposed additive and multiplicative preconditioners are uniformly spectrally equivalent to the operator of the quadrature scheme. The preconditioners are implemented by optimal algorithms, and they are used to accelerate convergence of the preconditioned conjugate gradient method. Numerical results are presented demonstrating efficiency of the preconditioners. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2006     

A quadrature finite element Galerkin scheme for a biharmonic problem on a rectangular polygon

A quadrature Galerkin scheme with the Bogner–Fox–Schmit element for a biharmonic problem on a rectangular polygon is analyzed for existence, uniqueness, and convergence of the discrete solution. It is known that a product Gaussian quadrature with at least three‐points is required to guarantee optimal order convergence in Sobolev norms. In this article, optimal order error estimates are proved for a scheme based on the product two‐point Gaussian quadrature by establishing a relation with an underdetermined orthogonal spline collocation scheme. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

气体团簇离子束两步能量修形法的平坦化效应

本文提出采用气体团簇离子束的两步能量修形法来改善4H-SiC(1000)晶片表面形貌.先用15 keV的高能Ar团簇离子进行整体修形,再用5 keV的低能团簇离子优化表面.结果表明,在相同的团簇离子剂量下,与单一15 keV的高能团簇处理相比,两步法修形后的表面具有更低的均方根粗糙度,两者分别为1.05 nm和0.78 nm.本文还以原子级平坦表面为研究对象,揭示了载能团簇引起的半球形离子损伤(弧坑)与团簇能量的关系,及两步能量修形法在弧坑修复中的优势.在原子力显微镜表征的基础上,引入了二维功率谱密度函数,以直观全面地给出材料的表面形貌特征及其随波长(频率)的分布.结果表明,经任何能量的团簇离子轰击的表面,在0.05—0.20μm波长范围内,团簇轰击都能有效地降低粗糙度,而在0.02—0.05μm范围内,则出现了粗化效应,这是由于形成了半球形离子损伤,但第二步更低能量的团簇离子处理可以削弱这种粗化效应.     

An error analysis and the mesh independence principle for a nonlinear collocation problem

A nonlinear Dirichlet boundary value problem is approximated by an orthogonal spline collocation scheme using piecewise Hermite bicubic functions. Existence, local uniqueness, and error analysis of the collocation solution and convergence of Newton's method are studied. The mesh independence principle for the collocation problem is proved and used to develop an efficient multilevel solution method. Simple techniques are applied for estimating certain discretization and iteration constants that are used in the formulation of a mesh refinement strategy and an efficient multilevel method. Several mesh refinement strategies for solving a test problem are compared numerically. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2006     

气体团簇离子束装置的设计及其在表面平坦化、自组装纳米结构中的应用

根据超声膨胀原理,n(10-10^4)个气体原子可以绝热冷却后凝聚在一起形成团簇,经过离化后,形成带一个电荷量的团簇离子,比如Arn^+.当团簇离子与固体材料相互作用时,由于平均每个原子携带的能量(~eV)较低,仅作用于材料浅表面区域,因此,气体团簇离子束是材料表面改性的优良选择.本文介绍了一台由武汉大学加速器实验室自主研制的气体团簇离子束装置,包括整体构造、工作原理及实验应用.中性团簇束由金属锥形喷嘴(F=65-135μm,q=14°)形成,平均尺寸为3000 atoms/cluster,经离化后,其离子束流达到了50μA.Ar团簇离子因其反应活性较低,本文运用Ar团簇离子(平均尺寸为1000 atoms/cluster)进行了平坦化和自组装纳米结构的研究.单晶硅片经Ar团簇离子束处理后,均方根粗糙度由初始的1.92 nm降低到0.5 nm,同时观察到了束流的清洁效应.利用Ar团簇离子束的倾斜(30°-60°)轰击,在宽大平坦的单晶ZnO基片上形成了纳米波纹,而在ZnO纳米棒表面则形成了有序的纳米台阶,同时,利用二维功率谱密度函数分析了纳米结构在基片上的表面形貌和特征分布,并计算了纳米波纹的尺寸和数量.