具有线性滑移界面的弹性问题的一种混合高阶方法
A Hybrid High-Order Method for the Elasticity Problem with Linear Slip Interface
- University College London（伦敦大学学院）
- Nanjing Forestry University（南京林业大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出一种混合高阶方法求解线性滑移界面的弹性问题，通过局部对称应变重构和正则化界面稳定化，覆盖从完美粘接到无牵引力的全部柔度范围，并证明能量范数下h^{k+1}、L^2范数下h^{k+2}的收敛阶且无锁。
AI中文摘要：
我们设计并分析了一种用于具有线性滑移界面（即弹簧型界面条件）的线性弹性问题的混合高阶（HHO）方法，其中位移在界面上的跳跃通过柔度张量$\bK=\alpha\bI+(\beta-\alpha)\udl{n}\otimes\udl{n}$与牵引力成正比。网格贴合界面，离散未知量在网格面上为次数$k\ge 1$的多项式，在网格单元内为次数$k+1$的多项式，并允许一般多面体单元。该方法的两个新颖之处在于：局部对称应变重构（其融入了界面条件）以及界面稳定化（其由正则化界面刚度$\bS_h$构建，遵循Hansbo和Hansbo的思想\cite{HH04}）。因此，单一公式涵盖了从完美粘接界面$\alpha=\beta=0$（此时界面稳定化起到Nitsche型罚项的作用）到通过$\alpha,\beta\to+\infty$获得的无牵引力界面的整个柔度范围，且没有未知量附加到跳跃上。我们证明了离散双线性形式是强制的，并且误差在能量范数下以$h^{k+1}$的阶收敛，在$L^2$范数下以$h^{k+2}$的阶收敛，且常数与柔度参数和Lamé系数$\lambda$无关，因此该方法也是无锁的。提供了各种数值算例和比较以确认理论结果。
英文摘要：
We design and analyse a hybrid high-order (HHO) method for the linear elasticity problem with a linear slip interface, i.e.\ with interface conditions of spring type, in which the jump of the displacement across the interface is proportional to the traction through a compliancy tensor $\bK=α\bI+(β-α)\udl{n}\otimes\udl{n}$. The mesh is fitted to the interface, the discrete unknowns are polynomials of degree $k\ge 1$ on the mesh faces and of degree $k+1$ in the mesh cells, and general polytopal cells are allowed. The two novelties of the method lie in the local symmetric strain reconstruction, which incorporates the interface condition, and in the interface stabilisation, which is built from a regularised interface stiffness $\bS_h$ in the spirit of Hansbo and Hansbo \cite{HH04}. As a consequence, a single formulation covers the whole range of compliancies, from the perfectly bonded interface $α=β=0$, where the interface stabilisation acts as a Nitsche-type penalty, to the traction-free interface obtained as $α,β\to+\infty$, and no unknown is attached to the jump. We prove that the discrete bilinear form is coercive and that the errors converge as $h^{k+1}$ in the energy norm and as $h^{k+2}$ in the $L^2$ norm, with constants that are independent of the compliancy parameters and of the Lamé coefficient $λ$, so that the method is also locking free. Various numerical examples and comparisons are provided to confirm the theoretical results.
