The coupling of mixed and primal finite element methods for the coupled body-plate problem

This paper considers the coupled problem of a three-dimensional elastic body and a two-dimensional plate, which are rigidly connected at their interface. The plate consists of a plane elasticity model along the plane direction and a plate bending model with Kirchhoff assumptions along the transverse direction. The Hellinger-Reissner formulation is adopted for the body by introducing the stress as an auxiliary variable, while the primal formulation is employed for the plate. The well-posedness of the new mixed weak formulation is established. This approach enables direct stress approximations and allows for non-matching meshes at the interface since the continuity condition of the displacement acts as a natural boundary condition for the body. Under certain assumptions, discrete stability and error estimates are derived for both conforming and nonconforming finite element methods. Two specific pairs of conforming and nonconforming finite elements are shown to satisfy the required assumptions, respectively. Furthermore, the problem is reduced to an interface problem based on the domain decomposition, which can be solved effectively by a conjugate gradient iteration. Numerical experiments are conducted to validate the theoretical results.