On a new mixed formulation of Kirchhoff plates on curvilinear polygonal domains

TL;DR

This paper introduces a new mixed formulation for Kirchhoff plate problems on curvilinear polygonal domains, utilizing Lagrange multipliers to overcome difficulties in extending previous methods to curved boundaries.

Contribution

It proposes an alternative mixed variational formulation using Lagrange multipliers for Kirchhoff plates on curvilinear polygons, extending prior work on polygonal domains.

Findings

Discretization method based on three second-order problems

New mixed formulation with Lagrange multipliers for curved boundaries

Potential for improved numerical solutions on complex domains

Abstract

For Kirchhoff plate bending problems on domains whose boundaries are curvilinear polygons a discretization method based on the consecutive solution of three second-order problems is presented. In Rafetseder and Zulehner (preprint, arXiv:1703.07962) a new mixed variational formulation of this problem is introduced using a nonstandard Sobolev space (and an associated regular decomposition) for the bending moments. In case of a polygonal domain the coupling condition for the two components in the decomposition can be interpreted as standard boundary conditions, which allows for an equivalent reformulation as a system of three (consecutively to solve) second-order elliptic problems. The extension of this approach to curvilinear polygonal domains poses severe difficulties. Therefore, we propose in this paper an alternative approach based on Lagrange multipliers.