Keywords: finite element methods, constraints, penalty methods, stability, explicit dynamics, critical time step.

Constraint equations are often an essential part of numerical analysis. Constraint equations accompany the usual system of simultaneous equations that must be solved, and they account for the application of special conditions on the degrees of freedom. Constraints can involve just one degree of freedom (i.e. the so-called single point constraints) such as Dirichlet boundary conditions, or multiple degrees of freedom (i.e. the so-called multi-point constraints) such as contact conditions, tiess or interface elements.

A popular method to impose constraints that is simple to understand and to implement is the use of penalty functions. In its standard format, penalty functions operate on the stiffness of the system, and in fact penalised degrees of freedom can be understood to have springs with high stiffnesses attached to them in order to effect the constraint. The higher the stiffness of the artificial spring, the more accurately the constraint is enforced. Unfortunately, this has severe disadvantages in dynamics, since adding springs with high stiffnesses to a system affects the speed of propagation of waves. In particular, the most popular time integration technique in structural dynamics, namely explicit time integration, relies on the time step being smaller than a so-called critical time step, and it is exactly this critical time step that is reduced significantly by the introduction of stiffness-type penalties. Thus, using stiffness-type penalties forces the analyst to use much reduced time steps, by which the total simulation time increases significantly.

An alternative formulation of penalty functions affects the mass matrix, rather than the stiffness matrix, of the system. In this variant of the penalty method, the penalised degrees of freedom can be understood to have artificial inertia with large masses attached to them, so as to prohibit acceleration. Using mass-type penalties does not have the adverse effects on the critical time step that stiffness-type penalties have. However, since mass-type penalties operate on the accelerations, violation of displacement constraints is not always identified, so that significant errors may occur.

To combine the best of both types of penalties, the paper suggests the use of two types of penalties simultaneously - this is what is called the "bi-penalty method". With a controlled ratio of mass penalty to stiffness penalty, the critical time step of the penalised parts of the structure can be controlled. Furthermore, the bi-penalty method offers an accuracy that is much superior to the use of mass-type penalties alone. In this paper, the bi-penalty method for single-point and multi-point constraints is discussed, and mathematical proofs (with associated rules of thumb for use in practice) are provided on how to select the ratio of the two penalties to avoid problems with numerical stability. The bi-penalty method is tested for a range of dynamic loading conditions, including contact and crack propagation simulated with interface elements. It is shown that the accuracy of the bi-penalty method is very similar to that of stiffness-type penalties, whilst avoiding the severe reductions of applicable time steps. It is also shown that the bi-penalty method is applicable in situations where the use of mass-type penalties fails. Thus, the bi-penalty method combines the advantages of stiffness penalties and mass penalties, whereas the drawbacks are avoided.

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