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Project

Model Reduction of Contact Problems in Flexible Multibody Dynamics

The research presented in this dissertation is concerned with the efficient solution of dynamic contact problems by means of distributed-parameter methods. Such problems are encountered in the computer-aided design of mechanical systems, where contact interactions are ubiquitous and often vital to the proper functioning of the system. With weight reduction and increased operating frequencies among the primary objectives of present-day engineering, a dynamic description of these problems becomes increasingly important.

Distributed-parameter methods such as the Finite Element (FE) method are often combined with the Flexible Multibody Dynamics (FMBD) approach to construct mathematical models of the interacting components when their geometries are complex and/or deformable. These models however come at a high computational cost due to three main reasons: the high number of Degrees Of Freedom (DOFs) involved, the large number of time increments needed to numerically solve the time-dependent equilibrium equations, and finally the high number of Floating Point Operations (FLOPs) required to evaluate the nonlinear contact forces.

In this dissertation an attempt is made to mitigate the computational burden of dynamic contact problems by tackling all three of these issues. While the methods and procedures discussed in this dissertation apply to a broad class of contact problems, the dynamic gear contact problem is selected as a common theme throughout the text.

The first and central contribution of this dissertation is the development of a novel Model-Order Reduction (MOR) method that is specifically tailored to the solution of FE-based contact problems in FMBD. The method belongs to the class of projection-based MOR methods and can be seen as a parametric variant of the Component Mode Synthesis (CMS) approach. By judiciously choosing the ingredients of the reduction basis, the method is shown to reduce the number of DOFs in the full-order model with several orders of magnitude, far beyond what is attainable using classical CMS. In comparison to the former, the computational complexity of the method is shown to be two orders of magnitude smaller while maintaining similar levels of accuracy. The method is first developed and validated for the perfectly-aligned gear contact problem, and then extended to the case of spatially misaligned gear pairs and multi-mesh gear trains.

The second contribution of this dissertation is the systematic investigation of numerical integration schemes for dynamic contact problems in light of the novel MOR method. Owing to the significant dimensional reduction that is enabled by this method, both explicit and implicit integrators are brought to the fore. The optimal integration scheme is selected by comparing the performances of a number of well-established adaptive integrators in solving the reduced-order gear contact problem. A number of the method’s distinguishing features are used to further optimize the selected integration scheme, including the availability of an analytical generalized contact force Jacobian and the clustering of the quasi-static contact deformations into a comparatively small subset of shape vectors. Compared to the fixed-increment central difference integrator that is used as a reference integrator in this dissertation, the selected integration schemes enable a reduction of overall computing time with a factor of 20 and more.

Having reduced the number of DOFs and optimized the numerical integration scheme, up to 98% of the computing time is spent on evaluating the FE-based contact forces. The cost of this operations scales quadratically with the number of finite elements used to discretize the contact surfaces, thereby partially jeopardizing the order reduction that is brought about by the novel MOR method. In order to remove the dependence of the ROM’s complexity on the dimensions of the underlying FE model, the application of two well-established hyperreduction schemes to the analysis of dynamic contact problems is investigated. Particular challenges that are dealt with are the development of efficient hyperreduction schemes in the context of sliding contact interactions, and the preservation of Newton’s law of action and reaction, which is vital to the robust solution of dynamic contact problems. Using the extended hyperreduction schemes, overall computational gains of roughly one order of magnitude as compared to the non-hyperreduced ROM are observed.

By lowering the number of DOFs, optimizing the numerical integration scheme and reducing the number of FLOPs in the contact force evaluation, this dissertation reduces the overall simultion cost of dynamic contact simulations with three to four orders of magnitude as compared to standard reduction techniques, without jeopardizing the first-principles approach of FE modelling. While the real-time simulation of complex contact interactions remains a challenge yet to be met, the current work moves these kinds of simulations from institutional supercomputing clusters to the mechanical engineer’s personal computer.

Date:2 Jul 2012 →  14 Sep 2018
Keywords:Multibody dynamics, Model reduction, Contact mechanics
Disciplines:Control systems, robotics and automation, Design theories and methods, Mechatronics and robotics, Computer theory
Project type:PhD project