Deriving simulation models from bond graphs with algebraic loops.: The extension to multibond graph systems

The first part of this paper makes an introduction to the causal problems raised in some bond graph models when implementing causality. As a consequence, zero-order causal paths (ZCPs) provided of algebraic loops appear. Variables involved in these loops are related themselves by means of algebraic assignments. An algorithm to detect and open all the zero-order causal paths existing in any bond graph model is described. Opening of topological loops will be the employed method. Break variables will be the key to open the loops. The algorithm selects in an optimized way the number of variables necessary to open all the topological loops corresponding to the ZCPs existing in the model. The result is a set of differential-algebraic equations solved using a backward-differential formulae numerical method. How to solve multibond graph systems including zero-order causal paths is the subject of the second part of this paper. The constitutive relationships of multibond resistors, transformers and gyrators give way to a new class of zero-order causal paths whose most important characteristic is that their associated topological loops involve more than one direction. In medium–large models these relationships produce a combinatory explosion of causal paths only treatable via software. The same presented algorithms will be able to analyze and optimally choose the appropriate break variables to open all the zero-order causal paths.