In this thesis, we introduce a new approach to mixed methods for
fourth-order problems with focus on Kirchhoff plates and Kirchhoff-Love
shells. The main achievement of this work is the derivation of new mixed
variational formulations that are only based on standard $H^1$ spaces. This
offers the possibility to apply well-known techniques for second-order
problems both regarding the discretization and the solution strategy.
In the first part, we consider the Kirchhoff plate bending problem with
mixed boundary conditions involving clamped, simply supported, and free
boundary parts. For this problem a new mixed variational formulation is
derived, which satisfies Brezzi's conditions and is equivalent to the
original problem, without additional convexity assumptions on the domain.
These important properties come at the cost of involving an appropriate
nonstandard Sobolev space for the auxiliary variable, the bending moment
tensor, which is related to the Hessian of the vertical displacement. For
the vertical displacement the standard Sobolev space $H^1$ (with appropriate
boundary conditions) is used. Based on a regular decomposition of this
nonstandard space, the fourth-order problem can be equivalently written as a
system of three (consecutively to solve) second-order elliptic problems in
standard Sobolev spaces.
This decomposition result on the continuous level leads in the discrete
setting 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 modular building blocks of
the new method.
In the second part, we consider the Kirchhoff-Love shell problem, which is a
more general fourth-order problem including beside the fourth-order
derivative term (present in the Kirchhoff-plate bending problem) additional
lower-order derivative terms. By extending the technique introduced for
plates a new mixed formulation solely based on standard $H^1$ spaces is
obtained. However, due to the additional lower-order derivative terms, the
system can no longer be solved consecutively.
Nevertheless, this allows for flexibility in the construction of
discretization spaces, e.g., standard $C^0$-coupling of multi-patch
isogeometric spaces is sufficient. In terms of solution strategies,
efficient methods for standard second-order problems like multigrid can be
used as building blocks of preconditioners for iterative solvers.
The performance of the resulting discretization methods for plates and
shells is demonstrated by numerical experiments. In case of plates, under
the assumption of a polygonal domain, a rigorous numerical analysis is
performed. All considered methods are implemented in the C++ library