Calculation of energy during the resolution

by L. Idoux, EDF R&D / AMA

The energy balance can be a very useful information. First, it provides information on the quality of results in the same way that a residue of equilibrium in the Newton algorithm, and secondly it helps to identify the physical phenomena involved and so meet the following questions: what is the energy brought to my structure and what becomes of it?

Up to now, the calculation of energy is available in Code_Aster, but not for every case. For example, it is not possible to know the energy dissipated by damping. In addition, the access to the energy balance is only possible afterwards by post-processing, which raises some difficulties.

• To get an accurate calculation, it is necessary to store a lot of time steps. Indeed, calculating energies requires a temporal integration performed by the trapezoidal method, whose accuracy is strongly dependent on the fineness of the time discretization.
• In the presence of many calculation steps, the CPU cost devoted to the reading of the results and the various arithmetic operations can be high. This is particularly unfortunate that some of the work has been performed during the computation but must be repeated.

With this in mind, a new calculation option was developed. It calculates the energy balance of the full model during the calculation. This option is activated by the keyword ENERGIE in the commands STAT_NON_LINE U4.51.03, DYNA_NON_LINE U4.53.01 and DYNA_VIBRA U4.53.02 ; for the latter, in the case of calculations on physical basis. The presence of this keyword triggers the calculation of the energy balance as follows:

where :

• W_ext is the work of external forces,
• W_int is the total strain energy of the system,
• E_cin is the kinetic energy of the system,
• W_liai is the energy dissipated by the connections (friction for example),
• W_amor is the energy dissipated by damping.

The results are displayed in the message file and saved in a table at the end of the calculation for recovery. For theoretical precisions, please see the new documentation R4.09.01 (automatic translation).

To illustrate this development, the calculation of the energy balance is enabled in the test case sdlv120a (automatic translation). It consists of the absorption of a compression wave in an elastic bar, with absorbing boundary elements. The blue curve represents the energy supplied from the outside, as an imposed displacement at the end of the bar. This energy propagates along the bar in the form of kinetic energy and total strain energy, and is fully dissipated at the other end of the bar through the absorbing boundary elements.

Available in the 11.1.7 version and future releases