Abstract
This work applies the variational principles of Lagrange and Hamilton to the assessment of numerical methods of linear structural analysis. Different numerical methods are used to simulate the behaviour of three structural configurations and benchmarked in their computation of the Lagrangian action integral over time. According to the principle of energy conservation, the difference at each time step between the kinetic and the strain energies must equal the work done by the external forces. By computing this difference, the degree of accuracy of each combination of numerical methods can be assessed. Moreover, it is often difficult to perceive numerical instabilities due to the inherent complexities of the modelled structures. By means of the proposed procedure, these complexities can be globally controlled and visualized in a straightforward way. The paper presents the variational principles to be considered for the collection and computation of the energy-related parameters (kinetic, strain, dissipative, and external work). It then introduces a systematic framework within which the numerical methods can be compared in a qualitative as well as in a quantitative manner. Finally, a series of numerical experiments is conducted using three simple 2D models subjected to the effect of four different dynamic loadings.
Highlights
For the constraint integration we will limit ourselves to the constraint reduction (CR) technique, whereas, in the case of time integration we will study the Newmark Beta (NB), Hilber-Hughes-Taylor (HHT), Chung-Hulbert’s generalizedalpha (CH), and Wilson Theta (WTH) methods
In this chapter we provide the results of our numerical experiments, where several combinations of methods were used in diverse simulations
As opposed to the analyst’s intuition, in spite of dealing with linear models we obtained curves that vary significantly from one method to another. As expected, this divergence is more pronounced with larger time steps and increases with the complexity of the model
Summary
Variational mechanics date back as far as the Eighteenth Century, when Leibniz, Euler, Maupertuis, and Lagrange devised the calculus of variations and the principles of least action. This methodology of treating physical phenomena is based on the notion that everything in nature tends to a state of minimal energy [1]. There is a preference to use forces and accelerations rather than energy concepts This approach often limits our understanding of the phenomena, as, for example, in the case of earthquakes, damage is a function of the square of the velocity, and not so much of the acceleration [2]
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