论文标题
边界惩罚技术，可从张量产品网格上删除异常值分析中的离群值
A boundary penalization technique to remove outliers from isogeometric analysis on tensor-product meshes
论文作者
论文摘要
我们引入了一项边界惩罚技术，以改善同几何分析（IGA）的光谱近似。当使用$ c^{p-1}，p $ -th（$ p \ ge3 $）订单isogeomementric elements时，该技术将删除在近似光谱的高频区域中出现的异常值。我们专注于1D中的经典拉普拉斯（Dirichlet）特征值问题，以说明这个想法，然后使用张量产生结构来生成刚度和质量矩阵，以解决多个维度问题。为了删除离群值，我们从域边界处的解决方案和测试空间惩罚了高阶导数的乘积。直观地，我们通过弱施加精确解决方案的特征来构建更好的近似值。有效地，我们以最小的额外计算成本为边界上的变异配方添加了术语。然后，我们概括将异常值分析删除的想法删除Neumann特征值问题（对于$ p \ ge2 $）。边界惩罚不会改变测试和解决方案空间。在罚款到无穷大的限制案例中，我们对dirichlet eigenvalue问题的$ c^2 $立方元素进行分散分析，以及$ c^1 $ $ c^1 $ quadratic Elements neumann eigenvalue问题。我们获得了所得矩阵特征值问题的分析本特征。数值实验显示了离散操作员特征值和特征功能的最佳收敛速率。
We introduce a boundary penalization technique to improve the spectral approximation of isogeometric analysis (IGA). The technique removes the outliers appearing in the high-frequency region of the approximate spectrum when using the $C^{p-1}, p$-th ($p\ge3$) order isogeometric elements. We focus on the classical Laplacian (Dirichlet) eigenvalue problem in 1D to illustrate the idea and then use the tensor-product structure to generate the stiffness and mass matrices for multiple dimensional problems. To remove the outliers, we penalize the product of the higher-order derivatives from both the solution and test spaces at the domain boundary. Intuitively, we construct a better approximation by weakly imposing features of the exact solution. Effectively, we add terms to the variational formulation at the boundaries with minimal extra computational cost. We then generalize the idea to remove the outliers for the isogeometric analysis to the Neumann eigenvalue problem (for $p\ge2$). The boundary penalization does not change the test and solution spaces. In the limiting case when the penalty goes to infinity, we perform the dispersion analysis of $C^2$ cubic elements for Dirichlet eigenvalue problem and $C^1$ quadratic elements for Neumann eigenvalue problem. We obtain the analytical eigenpairs for the resulting matrix eigenvalue problems. Numerical experiments show optimal convergence rates for the eigenvalues and eigenfunctions of the discrete operator.
