C0 interior penalty methods for the Gao beam model
One of my research interests was applying the C0 interior penalty FEMs to the Gao beam model. The dynamic Gao beam model (Gao, 1996) is governed by a nonlinear fourth-order partial differential equation, which contains second-order time and fourth-order space derivative terms. The C0 interior penalty methods (Brenner & Sung, 2005), a type of discontinuous Galekin method, efficiently handle the fourth-order space derivative. In general, the fourth-order derivative 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 need continuous interior penalty basis functions which are easier to construct than the C1 basis functions. Through this research, I have defined the C0 interior penalty semidiscrete formulation for the Gao beam model. I have proved the existence of its solutions and the error estimates. Also, numerical experiments supported the theoretical results and presented buckling states of the Gao 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.
The penalty parameter σ controls the height of the jumps. As the parameter becomes bigger, lower jump discontinuities happen. In other words, larger values of σ produce better stabilizing effects on all the jumps.
Numerical approximations with the different choices of the penalty parameter
Although the Gao beams move up and down as time passes, they never across the x-axis unlike standard linear beams, which show the post buckling state of the nonlinear beams.
The post buckling state of the nonlinear beams