Chapter5 Body Wise Time Integration of Multi Body Dynamic Systems Wolfgang Witteveen Abstract Several papers have been published in the past on the issue of decomposing a nonlinear system into subsystems for more efficient time integration. In this paper each body of a multi body system is considered as one subsystem. The subsystems (the bodies) are interacting via connection forces. The sources of such connection forces are constraints or directly applied forces. This contribution is restricted to constraint forces only. During a step which is named “body iteration”, those forces are considered as constant and the state of the system is computed for each body separately. This can be massively parallelized which can be an efficiency advantage in case of computational costly problems like the ones occurring in parameter estimation. During an “constraint update step” the constraints are evaluated based on the body’s current state. If the error is not small enough the interface forces are updated and the inner loop is executed once again until the error of the constraints is negligible. It turns out, that the constraints can be updated separately as well, which can be used again for parallel computing. In the paper, the theory will be outlined and implemented using an N body pendulum. Finally, the advantages and disadvantages of this approach are critically discussed. Keywords Time integration • Multi body systems • Multi body dynamics • Parallel computing Nomenclature N Number of bodies M Number of constraints a, b Numbers of bodies which are involved in a certain constraint tn Denotes the time at time step n h Step size of time integration (hDtn tn 1) ’,“,” HHT parameters qi State vector of body number i Cj Constraint equation j Pj Constraint force j aPj a,b Constraint force j acting on body a with involved bodies a and b œj Lagrange multipliers for constraint j C N j qj Constraint Jacobian for constraint j E N Identity matrix M N Mass matrix Qi Forces acting on body i J N i Jacobian of body number i ei Residuum of equation of motion for body i cj Residuum of constraint equation j pj Residuum of constraint force j W. Witteveen ( ) University of Applied Sciences Upper Austria – Wels Campus, Stelzhammerstr. 69, Wels 4600, Austria e-mail: wolfgang.witteveen@fh-wels.at © The Society for Experimental Mechanics, Inc. 2015 R. Allemang (ed.), Special Topics in Structural Dynamics, Volume 6, Conference Proceedings of the Society for Experimental Mechanics Series, DOI 10.1007/978-3-319-15048-2_5 55
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