Hindawi Publishing Corporation Advances in Mechanical Engineering Volume 2013, Article ID 906120, 16 pages http://dx.doi.org/10.1155/2013/906120 Research Article Assessing Numerical Error in Structural Dynamics Using Energy Balance Rabindranath Andujar, 1 Jaume Roset, 1 and Vojko Kilar 2 1 Department of Applied Physics, Polytechnic University of Catalonia, Campus Diagonal Nord, Building B5, C/Jordi Girona 1-3, 08034 Barcelona, Spain 2 Faculty of Architecture, University of Ljubljana, Zoisova cesta 12, 1000 Ljubljana, Slovenia Correspondence should be addressed to Vojko Kilar; vojko.kilar@fa.uni-lj.si Received 30 May 2013; Accepted 23 September 2013 Academic Editor: Mehdi Ahmadian Copyright © 2013 Rabindranath Andujar et al. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Tis work applies the variational principles of Lagrange and Hamilton to the assessment of numerical methods of linear structural analysis. Diferent numerical methods are used to simulate the behaviour of three structural confgurations and benchmarked in their computation of the Lagrangian action integral over time. According to the principle of energy conservation, the diference at each time step between the kinetic and the strain energies must equal the work done by the external forces. By computing this diference, the degree of accuracy of each combination of numerical methods can be assessed. Moreover, it is ofen difcult 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. Te 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 efect of four diferent dynamic loadings. 1. Introduction 1.1. Targets and Interest of Our Research. Variational mechan- ics date back as far as the Eighteenth Century, when Leibniz, Euler, Maupertuis, and Lagrange devised the calculus of vari- ations and the principles of least action. Tis methodology of treating physical phenomena is based on the notion that everything in nature tends to a state of minimal energy [1]. In structural engineering practice, there is a preference to use forces and accelerations rather than energy concepts. Unfortunately, this approach ofen 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]. In parallel we will deal with a systematic treatment of the numerical methods which have proliferated since the 1950s with the ever-increasing power of computers. Tis ceaseless growth in numbers and terminology has given place to a cumbersome mix of mathematics, physics, and computer science that is ofen difcult to grasp. We have used our work to propose a possible categorization according to the physical qualities which they represent instead of according to their mathematical properties. 1.2. Variational Mechanics. According to the principles of variational mechanics [3], the diference between kinetic energy and strain energy in a structural system equals the applied work due to external forces. In this way, by computing the energy scalars and carefully accounting for this diference at each time step, one should be able to infer the degree of accuracy of a simulation [4]. Te correct values should not in any case diverge much from zero, and deviations from this value would give us an idea about how accurate and stable a method is. 1.3. Numerical Methods for Structural Analysis. In previous works published by the authors [5, 6], it was shown how the vast amount of existing numerical methods can be grouped