A decomposition result for Kirchhoff plate bending problems and a new discretization approach
This provides a flexible modular discretization approach for Kirchhoff plate problems, but it is an incremental extension of prior work on biharmonic problems.
The authors extend a decomposition-based mixed variational formulation from biharmonic to Kirchhoff plate bending problems with mixed boundary conditions, enabling the solution via three consecutive second-order problems. They construct a nonstandard Sobolev space and verify Brezzi's conditions to achieve this.
A new approach is introduced for deriving a mixed variational formulation for Kirchhoff plate bending problems with mixed boundary conditions involving clamped, simply supported, and free boundary parts. Based on a regular decomposition of an appropriate nonstandard Sobolev space for the bending moments, the fourth-order problem can be equivalently written as a system of three (consecutively to solve) second-order problems in standard Sobolev spaces. This leads to new discretization methods, which are flexible in the sense, that any existing and well-working discretization method and solution strategy for standard second-order problems can be used as a modular building block of the new method. Similar results for the first biharmonic problem have been obtained in our previous work [W. Krendl, K. Rafetseder and W. Zulehner, A decomposition result for biharmonic problems and the Hellan-Herrmann-Johnson method, ETNA, 2016]. The extension to more general boundary conditions encounters several difficulties including the construction of an appropriate nonstandard Sobolev space, the verification of Brezzi's conditions, and the adaptation of the regular decomposition.