# A New Approach to Mixed Methods for Kirchhoff-Love Plates and Shells

## Katharina Rafetseder

**July 13, 2018, 10:30 a.m. S2 416-1**

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.

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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.

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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.

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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 G+Smo.