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

Nikolaos Karaliolios, Dimitiros L. Karabalis

TL;DR

The paper presents a weak-form, piecewise polynomial numerical method for the SDOF problem, using Bernstein-based bases to achieve high-order convergence while conserving energy. It develops a fully discrete, stepwise algorithm across subintervals, with detailed treatment of the $p=3$ case and extensions to damped and undamped dynamics. A thorough error analysis combines density results, projection errors, and subspace misalignment, delivering bounds and convergence behavior dependent on the regularity of the force and damping. The approach aims to overcome limitations of classical damping and stability theorems, offering a potentially spectrally accurate, energy-preserving alternative with favorable error propagation characteristics. The authors indicate that numerical evidence and broader MDOF extensions are planned for Part III.

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.

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

TL;DR

The paper presents a weak-form, piecewise polynomial numerical method for the SDOF problem, using Bernstein-based bases to achieve high-order convergence while conserving energy. It develops a fully discrete, stepwise algorithm across subintervals, with detailed treatment of the case and extensions to damped and undamped dynamics. A thorough error analysis combines density results, projection errors, and subspace misalignment, delivering bounds and convergence behavior dependent on the regularity of the force and damping. The approach aims to overcome limitations of classical damping and stability theorems, offering a potentially spectrally accurate, energy-preserving alternative with favorable error propagation characteristics. The authors indicate that numerical evidence and broader MDOF extensions are planned for Part III.

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.
Paper Structure (27 sections, 29 theorems, 182 equations, 43 figures)

This paper contains 27 sections, 29 theorems, 182 equations, 43 figures.

Table of Contents

  1. Application of the weak formulation to piece-wise polynomial interpolation
  2. Choice of functions
  3. Some useful properties of Bernstein polynomials
  4. The functions and their derivatives
  5. Integrals and $L^{2}$ products
  6. Polynomial approximation of functions in Sobolev spaces
  7. From Legendre to Bernstein basis
  8. From $[0,h]$ to $[0,\bar{T}]$, the space $\mathcal{P}_{h,p}$
  9. The first step of the algorithm, $j=0$
  10. The remaining steps
  11. The case $p=3$
  12. The undamped case
  13. The damped case
  14. Error estimates
  15. Some approximation properties
  16. ...and 12 more sections

Key Result

Lemma 3.1

The following formula holds true for $i = 1,\cdots , p-1$:

Figures (43)

  • Figure 1: The exponent expressing the size of the highest order coefficient of a Bernstein polynomial written in the Legendre basis as a power of $p$.
  • Figure 2: The exponent expressing the size of the second highest order coefficient of a Bernstein polynomial written in the Legendre basis as a power of $m=p$.
  • Figure 3: The exponent expressing the size of a middle order coefficient of a Bernstein polynomial written in the Legendre basis as a power of $m=p$.
  • Figure 4: The base-$p$ exponent expressing the error of projection of the force corresponding the homogeneous problem. This is the quantity $s_{h}(p)$ in the established notation.
  • Figure 5: The range of base-$p$ exponents expressing the error of projection of the force corresponding the homogeneous problem. The quantities $s_{h}(p)$, $\bar{s}_{h}(p)$, and $\underline{s}_{h}(p)$ in the established notation are plotted, representing the exponent for the worst-case angle, the mean angle and the best-case angle.
  • ...and 38 more figures

Theorems & Definitions (41)

  • Lemma 3.1
  • proof
  • Lemma 3.2
  • Corollary 3.3
  • proof
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • proof
  • Lemma 3.6
  • ...and 31 more