论文标题
线性Kirchhoff-Love壳的弱边界条件执行：配方，错误分析和验证
Weak Boundary Condition Enforcement for Linear Kirchhoff-Love Shells: Formulation, Error Analysis, and Verification
论文作者
论文摘要
薄外壳的稳定而准确的建模需要适当地执行所有类型的边界条件。不幸的是，对于Kirchhoff-love壳，由于必须应用功能和衍生边界条件，因此很难对Dirichlet边界条件进行强大的强制执行。一种流行的替代方法是采用Nitsche的方法来弱执行所有边界条件。但是，尽管文献中已经提出了许多基于Nitsche的配方，但它们缺乏全面的错误分析和验证。实际上，与常见边界条件规格一起使用时，现有的制剂在变化上是不一致的，而产量的亚最佳收敛速率。在本文中，我们提出了一种基于Nitsche的新型配方，用于线性的Kirchhoff-Love壳，该公式可证明是稳定的，并且对于一般可允许的边界条件集，并且最佳地收敛。为了达到我们的配方，我们首先提出一个框架，用于构建Nitsche的方法，以解决任何抽象的变分约束最小化问题。然后，我们将此框架应用于线性Kirchhoff-love shell，对于基于NURBS的ISOOGENOTRIC分析的特定情况，我们证明所得的配方在壳体能量标准和标准的$ l^2 $ -NORM中都能产生最佳的收敛速率。在此过程中，我们为可允许的边界条件的一般集合得出了Euler-Lagrange方程，并表明文献中通常呈现的Euler-Lagrange边界条件是不正确的。我们通过制造解决方案的新壳障碍物路线来验证我们的配方，该课程包括平坦，抛物线，双曲线和椭圆形几何形状配置。与当前的最佳实践相比，这些制造的解决方案使我们能够稳健地测量整个外壳的误差，而当前最佳实践仅在特定位置测量位移和应力误差。
Stable and accurate modeling of thin shells requires proper enforcement of all types of boundary conditions. Unfortunately, for Kirchhoff-Love shells, strong enforcement of Dirichlet boundary conditions is difficult because both functional and derivative boundary conditions must be applied. A popular alternative is to employ Nitsche's method to weakly enforce all boundary conditions. However, while many Nitsche-based formulations have been proposed in the literature, they lack comprehensive error analyses and verifications. In fact, existing formulations are variationally inconsistent and yield sub-optimal convergence rates when used with common boundary condition specifications. In this paper, we present a novel Nitsche-based formulation for the linear Kirchhoff-Love shell that is provably stable and optimally convergent for general sets of admissible boundary conditions. To arrive at our formulation, we first present a framework for constructing Nitsche's method for any abstract variational constrained minimization problem. We then apply this framework to the linear Kirchhoff-Love shell and, for the particular case of NURBS-based isogeometric analysis, we prove that the resulting formulation yields optimal convergence rates in both the shell energy norm and the standard $L^2$-norm. In the process, we derive the Euler-Lagrange equations for general sets of admissible boundary conditions and show that the Euler-Lagrange boundary conditions typically presented in the literature is incorrect. We verify our formulation by manufacturing solutions for a new shell obstacle course that encompasses flat, parabolic, hyperbolic, and elliptic geometric configurations. These manufactured solutions allow us to robustly measure the error across the entire shell in contrast with current best practices where displacement and stress errors are only measured at specific locations.
