The Bipenalty Method for Explicit Structural Dynamics

The penalty method is a versatile and widely used technique for imposing constraints in finite element analysis. Traditionally, it is implemented by adding artificial stiffness to the system equations. However, this leads to large eigenfrequencies being introduced, which, when used in explicit dynamics, can significantly decrease the critical time step of an analysis. This in turn vastly increases computational expense, increases the chances of instability, and generally leads to a less robust formulation. The mass penalty method, a less widely used penalty technique that operates on the mass matrix of a dynamic system, does not introduce large eigenfrequencies, but is less accurate and less versatile than the traditional stiffness penalty method. In this thesis, the two methods are combined to form the bipenalty method. A general formulation is provided that can be used to describe any number of arbitrary multipoint constraints. Mathematical proofs are developed that show that the bipenalty method, like the stiffness penalty method, introduces extra eigenfrequencies into the system, but that they can be carefully controlled by manipulation of the stiffness and mass penalty parameters. It is shown that it is a simple matter to select these parameters such that the critical time step is unaffected by the constraints. The constraint imposition accuracy of the method is analysed, and found to be comparable to that of the traditional stiffness penalty method. An algorithm is provided that describes how to select the penalty parameters for maximum accuracy. Low errors are attainable due to the fact that very large penalty parameters can be used without causing instability. The method is then applied for the first time to two problem types commonly solved using penalty methods: crack propagation (modelled using cohesive surfaces), and contact-impact. A series of examples are presented that demonstrate the stability and accuracy of the method.