Boundary conditions are critical to the partial differential equations (PDEs) as they constrain the PDEs ensuring a unique and well defined solution. Based on combinatorial and surgery theory of manifolds, we develop multi-element boundary conditions as the generalization of the traditional boundary conditions in classical mechanics: Dirichlet boundary conditions, Neumann boundary conditions and Robin boundary conditions. The multi-element boundary/domain conditions glue the physical quantities at several points of different boundaries or domains on the fly, where the point-to-point correspondence (point mapping) on several boundaries are established on the common local coordinate system and the interactions are realized through the “wormhole” (i.e. the constraint equations). The study on weak form shows that the general multi-element boundary conditions are inconsistent with the variational principle/weighted residual method. To circumvent this dilemma, a numerical scheme based on augmented Lagrange method and nonlocal operator method (NOM) is proposed to deal with the mechanical problem equipped with general multi-element boundary conditions. Numerical tests show that the structures have completely different deformation modes for different multi-element boundary conditions.

边界条件对于偏微分方程（PDE）至关重要，因为它们限制了PDE，从而确保了唯一且定义明确的解决方案。基于流形的组合和外科理论，我们开发了多元素边界条件，作为经典力学中传统边界条件的推广：狄利克雷边界条件，诺伊曼边界条件和罗宾边界条件。多元素边界/域条件会实时粘合不同边界或域的几个点上的物理量，其中几个边界上的点对点对应关系（点映射）是在公共局部坐标系和相互作用下建立的通过“虫洞”（即约束方程式）实现。对弱形式的研究表明，一般的多元素边界条件与变分原理/加权残差法不一致。为了解决这一难题，提出了一种基于增强拉格朗日方法和非局部算子方法（NOM）的数值方案，以解决具有一般多元素边界条件的力学问题。数值试验表明，该结构在不同的多元素边界条件下具有完全不同的变形模式。