The weak form of the SDOF and MDOF equation of motion, part II: A numerical method for the SDOF problem

TL;DR

This paper introduces a new numerical method for solving the SDOF problem that is more efficient, energy-conserving, and free from common limitations of existing algorithms, based on the weak form of the equation of motion.

Contribution

It presents a novel numerical approach using piece-wise polynomial interpolation that achieves high approximation rates and conserves energy, overcoming key drawbacks of previous methods.

Findings

High approximation accuracy proportional to polynomial degree

Energy conservation in the numerical solution

Elimination of numerical damping and Dahlqvist Barrier limitations

Abstract

A new, more efficient, numerical method for the SDOF problem is presented. Its construction is based on the weak form of the equation of motion, as obtained in part I of the paper, using piece-wise polynomial functions as interpolation functions. The approximation rate can be arbitrarily high, proportional to the degree of the interpolation functions, tempered only by numerical instability. Moreover, the mechanical energy of the system is conserved. Consequently, all significant drawbacks of existing algorithms, such as the limitations imposed by the Dahlqvist Barrier theorem and the need for introduction of numerical damping, have been overcome.