The bipenalty method for arbitrary multipoint constraints

Abstract

SUMMARYIn finite element (FE) analysis, traditional penalty methods impose constraints by adding virtual stiffness to the FE system. In dynamics, this can decrease the critical time step of the system when conditionally stable time integration schemes are used by introducing spurious modes with high eigenfrequencies. Recent studies have shown that using mass penalties alongside traditional stiffness penalties can mitigate this effect for systems with a one single-point constraint. In the present work, we extend this finding to include systems with an arbitrary set of multipoint constraints. By analysing the generalised eigenvalue problem, we show that the values of spurious eigenfrequencies may be controlled by the choice of stiffness and mass penalty parameters. The method is demonstrated using numerical examples, including a one-dimensional contact-impact formulation and a two-dimensional crack propagation analysis. The results show that constraint imposition using the bipenalty method can be employed such that the critical time step of an analysis is unaffected, whereas also displaying superiority over the mass penalty method in terms of accuracy and versatility.

Coupling of heterogeneous kinematics and Finite Element approximations applied to composite beam structures

“…Nevertheless, it does not rely on an energy principle. Recently, a MPC formulation for eigenfrequency calculation and dynamic contact problems was developed by Hetherington et al [21].…”

On stability and reflection‐transmission analysis of the bipenalty method in contact‐impact problems: A one‐dimensional, homogeneous case study

“…Moreover, optimum values of the penalty ratios-the so-called CPRs-have been derived for a number of finite elements such that the critical time step size of the penalized system remains unaffected. A new method of calculating the CPR associated with a finite element formulation was developed in Hetherington et al 10 Recently, this research was extended to include systems with an arbitrary set of multipoint constraints 11 and to cohesive zone (surface) elements. 21 The standard stiffness penalty method adds an extra term to the strain energy (14) to enforce the zero gap on the contact boundary…”

“…Owing to mathematical difficulty, it limits the classes of elements that can be taken into account. To overcome this problem, a simple relationship between the CPR of an element and its maximum unpenalized eigenfrequency was derived in Hetherington et al Thus, the multiple constraints and more complex element formulations can be directly accounted for Hetherington et al The effect of the penalty stiffness on the accuracy in contact problems has been studied in Huněk and in Hetherington et al, where also the effect of damping penalty parameters has been analyzed.…”

Studies in Numerical Stability of Explicit Contact-Impact Algorithm to the Finite Element Solution of Wave Propagation Problems

“…A new method of calculating the CPR associated with a finite element formulation was developed in [2]. Recently, this finding was extended to include systems with an arbitrary set of multipoint constraints [3]. Now, a brief description of the bipenalty method follows.…”

“…In order to overcome this problem, a simple relationship between the CPR of an element and its maximum unpenalised eigenfrequency was derived in [2]. Thus, the multiple constraints and more complex element formulations can be directly accounted for [3].…”