Numerical integration of non‐linear elastic multi‐body systems

This paper is concerned with the modelling of nonlinear elastic multi‐body systems discretized using the finite element method. The formulation uses Cartesian co‐ordinates to represent the position of each elastic body with respect to a single inertial frame. The kinematic constraints among the various bodies of the system are enforced via the Lagrange multiplier technique. The resulting equations of motion are stiff, non‐linear, differential‐algebraic equations. The integration of these equations presents a real challenge as most available techniques are either numerically unstable, or present undesirable high frequency oscillations of a purely numerical origin. An approach is proposed in which the equations of motion are discretized so that they imply conservation of the total energy for the elastic components of the system, whereas the forces of constraint are discretized so that the work they perform vanishes exactly. The combination of these two features of the discretization guarantees the stability of the numerical integration process for non‐linear elastic multi‐body systems. Examples of the procedure are presented.