C0 Interior Penalty Methods for the Gao Beam Model

The dynamic Gao beam model [Gao, 1996] is governed by a nonlinear fourth-order partial differential equation containing second-order time and fourth-order space terms. The C0 interior penalty methods [Brenner & Sung, 2005] efficiently solve the biharmonic equation. The biharmonic equation generally requires C1 basis functions for the classical H2-conforming FEMs, such as the cubic splines and linear B-splines. However, the C0 interior penalty methods use continuous basis functions in a broken Sobolev space which are easier to construct than the C1 basis functions. Interior penalty methods are adopted to weakly impose the continuity of the first derivative of the basis functions. Through this research, we define the C0 interior penalty semidiscrete formulation for the Gao beam model and prove the existence of its solutions and the error estimates. Also, numerical experiments support the theoretical results and presented buckling states of the Gao beams.

In the above figure, the penalty parameter controls the height of the jumps. As the parameter becomes large, small jump discontinuities happen. In other words, larger values for the penalty parameter produce a more substantial stabilizing effect on all the jumps.

The above figure explains that although the Gao beams move up and down as time passes, they never move across the x-axis, unlike standard linear beams, which show the post-buckling state of the nonlinear beams.

• J. Ahn, S. Lee, and E.-J. Park, C0 interior penalty methods for a dynamic nonlinear beam model, Applied Mathematics and Computation, 339 (2018) 685-700.