Numerical Analysis and Computational Geometry Research of Characteristic Finite Element Methods

1. 在特征有限元格式中，需要对由特征映射和有限元空间基函数组成的函数进行积分。虽然在每个单元上的积分不是光滑的，但经验表明，使用高阶数值积分公式可以得到更好的有限元解。研究了其产生的原因和数值积分公式的合适顺序。介绍了随机三角形上的一个模型积分问题。研究了使平均误差为零的数值积分公式。由此得到了一个为特征有限元方案提供适当数值积分阶数的猜想，即(2k+d)-阶公式对于d维空间中使用Pk有限元的特征方案是合理的。在以上的数值积分中，需要找到一个具有逆风点的三角形，该三角形是通过特征映射对积分节点进行更大的映射。这种类型的问题称为点定位问题。针对点的定位问题，提出了梯形图方法，该方法基于域的梯形细分构造了问题的数据结构。数据结构的效率取决于输入数据的顺序。提出了一种通过一次执行梯形图法构造高效搜索树的预处理算法。相关研究：(1)分析了牵引法，该方法对形状优化问题的实际计算是有效的；(2)分析了有限元的阶次，使牵引法在有限元格式中具有更好的精度；(3)推导了Boussinesq方程的二维极限模型，该方程是三维方程在厚度趋于零时的极限；(4)提出了一种从涡流检测（ECT）获得的模糊图像重建精细缺陷图像的新方法。(5)利用Helmholtz分解的傅里叶变换获得SINC函数方法的新算法。少

1. Research in Numerical analysisIn characteristic finite element schemes, it is required to integrate functions composed from characteristic mappings and base functions of finite element spaces. Although the integrands are not smooth on each element, it has been experienced that the use of higher-order numerical integration formulas gives better finite element solutions. Its reason and suitable order of the numerical integration formula are studied. A model integration problem on stochastic triangles is introduced. Numerical integration formula that makes the mean error zero is investigated. As a result, a conjecture that provides an appropriate order of numerical integration for the characteristic finite element schemes is obtained, that is, (2k+d)-th order formula is reasonable for the characteristic schemes using Pk finite element in the d-th dimensional space.2. Research in Computational GeometryIn above numerical integrations, the triangle having an upwind point, which is the ima … More ge of an integration node by the characteristic mapping, has to be found. Problem of this type is called the point location problem. For the point location problem, one has proposed the trapezoidal map method, which constructs a data structure for the problem based on trapezoidal subdivision of the domain. Efficiency of the data structure depends on the order of input data. A preprocessing algorithm in order to construct an efficient search tree by one time execution of the trapezoidal map method is proposed.3. Related researches : (1)Analysis of the traction method, which is effective for practical computation of shape optimization problems, (2) Analysis of order of a finite element, which gives the traction method better accuracy in the finite element schemes, (3) Derivation of 2D limit models of Boussinesq equation, which is limit of 3D equation as the thickness tends to zero, (4) A new method which reconstruct fine flaw image from a blurred image obtained by the eddy current testing (ECT) is proposed. (5) A new algorithm which attain the SINC function method by the Fourier transformation of Helmholtz decomposition. Less