Numerical resolution of large deflections in cantilever beams by Bernstein spectral method and a convolution quadrature

The mathematical modeling of the large deflections for the cantilever beams leads to a nonlinear differential equation with the mixed boundary conditions. Different numerical methods have been implemented by various authors for such problems. In this paper, two novel numerical techniques are investigated for the numerical simulation of the problem. The first is based on a spectral method utilizing modal Bernstein polynomial basis. This gives a polynomial expression for the beam configuration. To do so, a polynomial basis satisfying the boundary conditions is presented by using the properties of the Bernstein polynomials. In the second approach, we first transform the problem into an equivalent Volterra integral equation with a convolution kernel. Then, the second order convolution quadrature method is implemented to discretize the problem along with a finite difference approximation for the Neumann boundary condition on the free end of the beam. Comparison with the experimental data and the existing numerical and semi-analytical methods demonstrate the accuracy and efficiency of the proposed methods. Also, the numerical experiments show the Bernstein-spectral method has a spectral order of accuracy and the convolution quadrature methods equipped with a finite difference discretization gives a second order of accuracy.