Keywords: penalty method, critical time step, explicit dynamics, inertia penalty, bipenalty method.

Explicit time integration is widely used in transient finite element analysis. The major advantage is robustness: since no equilibrium iterations are performed, the convergence difficulties often encountered in implicit analysis are circumvented, which is a key issue in large-scale industrial computations. The main drawback is stability: explicit algorithms are only conditionally stable, so the time step must be kept below a critical time step to ensure a stable solution. This critical time step is proportional to (the square root of) the ratio of mass over stiffness. Thus, increased stiffness properties and increased inertia properties tend to decrease and increase the critical time step, respectively.

The penalty method, on the other hand, is a popular way to prescribe constraints. This is usually done by means of large diagonal entries in the stiffness matrices, which can be interpreted as stiff springs. The main drawback of this stiffness-based penalty method for explicit analysis is that the critical time step decreases (compared to that of the unpenalised problem), so more steps are required to complete a computation. This problem can be solved by using mass, rather than stiffness, penalties [1]: if larger diagonal entries are added to the mass matrix, which can be regarded as heavy masses, then the critical time step increases [2]. However, this gain in stability is overshadowed by a loss in accuracy compared to the more commonly used stiffness penalties [2].

A combination of the best of both penalty functions is the so-called bipenalty method, in which stiffness and inertia penalties are used simultaneously [3,4]. The use of two penalties instead of one implies that the constraint is modelled with greater accuracy. Furthermore, with an adequate ratio of stiffness to mass penalty, the critical time step of the unpenalised problem is retained. Hence, the key issue in the bipenalty method is computing this so-called "critical penalty ratio" (CPR). The CPR can be computed by solving two small eigenvalue problems: one to determine the maximum eigenfrequency of an un-penalised element, and one to find a ratio of the two penalties whereby the maximum eigenfrequency of the un-penalised element is substituted. For simple element types (such as two-node bars, two-node beams and four-node squares) it is possible to find closed-form solutions for the CPR. Numerical examples show the validity of the CPR: taken the penalty ratio smaller or equal to the CPR yields stable results, whereas penalty ratios larger than the CPR produce numerical instabilities.

The bipenalty method is superior to using stiffness penalties, since the restrictions on the time step and associated long CPU times for simulations are avoided. The bipenalty method is also superior to using inertia penalties since it offers higher accuracy.

1

S. Ilanko, "Introducing the use of positive and negative inertial functions in asymptotic modelling", Proceedings of the Royal Society A, 461, 2545-2562, 2005. doi:10.1098/rspa.2005.1472

2

J. Hetherington, H. Askes, "Penalty methods for time domain computational dynamics based on positive and negative inertia", Computers and Structures, 87, 1474-1482, 2009. doi:10.1016/j.compstruc.2009.05.011

3

N. Asano, "A penalty function type of virtual work principle for impact contact problems of two bodies", Bulletin of the Japan Society of Mechanical Engineers, 29, 3701-3709, 1986.

4

H. Askes, M. Caramés-Saddler, A. Rodríguez-Ferran, "Bipenalty method for time domain computational dynamics", Proceedings of the Royal Society A, 466, 1389-1408, 2010. doi:10.1098/rspa.2009.0350

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