C0 Interior Penalty Methods for the Gao Beam Model

The dynamic Gao beam model [Gao, 1996] is described by a nonlinear fourth-order partial differential equation with second-order time and fourth-order spatial terms. The C0 interior penalty methods [Brenner & Sung, 2005] effectively solve the biharmonic equation. Typically, the biharmonic equation requires C1 basis functions for classical H2-conforming FEMs, such as cubic splines and linear B-splines. However, C0 interior penalty methods use continuous basis functions in a broken Sobolev space, which are easier to construct than C1 basis functions. These interior penalty methods are employed to weakly enforce the continuity of the first derivative of the basis functions. In this research, we define the C0 interior penalty semidiscrete formulation for the Gao beam model and establish the existence of solutions and error estimates. Additionally, numerical experiments support the theoretical findings and demonstrate buckling states of the Gao beams.

In the above figure, the penalty parameter determines the height of the jumps. As the parameter increases, small jump discontinuities occur. In other words, larger values for the penalty parameter create a stronger stabilizing effect on all the jumps.

The above figure shows that although the Gao beams move up and down over time, they never cross the x-axis, unlike standard linear beams, which display the post-buckling state of nonlinear beams.

Related Publications

• 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.