Multi-connected boundary conditions in solid mechanics and surgery theory. (15th July 2021) - Ren, Huilong; Zhuang, Xiaoying; Anitescu, Cosmin; Rabczuk, Timon
Highlights: A framework for multi-element boundary/domain conditions is proposed. The weak form for multi-element boundary conditions is proposed. An efficient numerical method based on the nonlocal operator method is developed. The method allows point mapping between multi-element boundary/domain conditions. Abstract: 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.