Numerical Analysis and Computational Geometry Research of Characteristic Finite Element Methods

1.数值分析研究在特征有限元格式中，需要积分由特征映射和有限元空间的基函数组成的函数。虽然被积函数在每个单元上都不是光滑的，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。研究了数值积分公式产生的原因和适用的阶次。介绍了一个关于随机三角形的模型集成问题。研究了使平均误差为零的数值积分公式。由此得到了一个猜想，它为特征有限元格式提供了一个合适的数值积分阶数，即对于在d维空间中使用PK有限元的特征格式，(2k+d)阶公式是合理的。计算几何研究在上述数值积分中，具有迎风点的三角形是图像…必须通过特征映射找到更多的集成节点。这种类型的问题称为点定位问题。对于点定位问题，提出了基于区域梯形剖分的梯形图方法，为点定位问题构建了数据结构。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构造高效搜索树的预处理算法。相关研究内容：(1)对实际形状优化计算中有效的牵引法进行了分析；(2)对有限元的阶次进行了分析，使牵引法在有限元格式中具有更高的精度；(3)推导了Boussinesq方程的二维极限模型，即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得到的模糊图像重建精细缺陷图像的新方法。(5)通过Helmholtz分解的傅里叶变换得到SINC函数方法的新算法。较少

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