We extend the softFEM idea to isogeometric analysis (IGA) to reduce the stiffness (consequently, the condition numbers) of the IGA discretized problem. We refer to the resulting approximation technique as softIGA. We obtain the resulting discretization by first removing the IGA spectral outliers to reduce the system's stiffness. We then add high-order derivative-jump penalization terms (with negative penalty parameters) to the standard IGA bilinear forms. The penalty parameter seeks to minimize spectral/dispersion errors while maintaining the coercivity of the bilinear form. We establish dispersion errors for both outlier-free IGA (OF-IGA) and softIGA elements. We also derive analytical eigenpairs for the resulting matrix eigenvalue problems and show that the stiffness and condition numbers of the IGA systems significantly improve (reduce). We prove a superconvergent result of order $h^{2p+4}$ for eigenvalues where $h$ characterizes the mesh size and $p$ specifies the order of the B-spline basis functions. To illustrate the main idea and derive the analytical results, we focus on uniform meshes in 1D and tensor-product meshes in multiple dimensions. For the eigenfunctions, softIGA delivers the same optimal convergence rates as the standard IGA approximation. Various numerical examples demonstrate the advantages of softIGA over IGA.
翻译:我们把软FEM概念扩大到IGA离散问题的等离析分析(IGA),以减少IGA离散问题的僵硬性(随后是条件号),我们把由此产生的近似技术称为软IGA。我们通过首先去除IGA光谱外端器获得由此产生的离散性,以减少系统的僵硬性;然后在IGA双线格式标准表格中添加高阶衍生衍生物-跳跃惩罚条件(带有负惩罚参数),惩罚参数力求尽量减少光谱/散射错误,同时保持双线形式的共振性。我们为离散性IAA(OF-IGA)和软IGA元素建立分散性错误。我们还为由此产生的矩阵光谱外断分离性,以减少系统的僵硬性;我们然后在IGA系统的僵硬性和条件值标准值中添加高压值($+2peni)的超级一致结果,在双线形式中以美元标值标度和美元表示软度IGA(OA)的超值,在B-GA(IA)和SO(O)的正统值上显示一个主点的焦值的焦值值值,在SLIGA(UA)上显示一个分析结果。