Methodologies for non-linear dynamic simulations in product development

In this thesis the efficient numerical simulation of non-linear dynamic systems is addressed through the use of reduced models. The problem of reducing simulation time with marginal loss of accuracy has been studied for many decades, with the purpose of accelerating the design phase and allowing the use of more accurate virtual prototypes. The process of transforming an original model and describing a complex physical system into a less computational demanding one, is generically defined as model order reduction or model reduction. The resulting model is therefore known as reduced model. Despite decades of attempts and several successfully applied methods, this topic still represents an open point, especially for what concerns complex non-linear systems. The aim of this thesis is to develop methodologies which exploit the linear modal analysis as a reliable and consolidated tool in reducing the computational cost of non-linear systems. Formulations which retains the non-linear behaviour while exploiting well established linear analyses are sought. Non-linearities in non-linear systems can then be retained or linearised around linearisation points. After a review of the literature, in Chapter 2, both approaches are examined. First, a reduced model which dedefines the non-linearities in a cubic form is implemented (Chapter 3). Then, a novel reduction method based on the linearisation in the configurations space is proposed in Chapter 4 and 5. Chapter 4 discusses the linearisation procedure, with the use of a specific base for each linearisation point, so that the non-linear system is globally approximated by a piecewise linear system, described through a set of linear ones. Interactions between them are then used to retain the non-linear properties, with the local linearised systems named subsystems. The reduction of the model is discussed in Chapter 5, with a focus on the mode selection procedure in generating reduced linear subsystems, while in Chapter 6, after an application to a simple lumped system, two categories of examples are proposed, defining two possible interaction methods regarding the set of subsystems. In the first category a discrete interaction is used, with a subsystem replacing the previous one, while in the second category a continuous interaction is implemented, with more reduced linear subsystems evolving simultaneously. For each category single and multi-parameters examples are proposed, with an analysis of the performance included. The method discussed in Chapter 3 is implemented, developing a non-linear beam element and testing the reduction on both numerical and experimental cases. Good agreement in reproducing the reference data is proven for the considered examples. The novel method developed in Chapter 4 and 5 is described, discussed and applied to several numerical examples. This method proves effective in reducing the computational time while maintaining a good approximation. An energy-based mode selection algorithm is introduced and applied, showing positive effects on the model reduction method performance.