Tyings in linear systems of equations modelled with positive and negative penalty functions

Askes, Harm; Piercy, Stewart; Ilanko, Sinniah

doi:10.1002/cnm.1023

Cited by 7 publications

(5 citation statements)

References 8 publications

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“…As with traditional methods, the implementation is simple. The formulation of the constraint equation (2) and the equations of motion (6) describe the technique completely-except for the values of penalty parameters˛s and˛m, which are to be selected by the analyst. Because it is the magnitude of these parameters relative to the existing entries in K and M, which dictates the accuracy of the constraint imposition, penalty factors are often used to express the magnitude of the penalty, denoted by…”

Section: Bipenalty Matrix Formulationmentioning

confidence: 99%

“…Owing in part to their greater level of versatility, penalty methods have become a commonly chosen alternative. Examples include node‐to‐node tyings , interface elements/cohesive surfaces and contact conditions . Despite the fact that penalty methods only approximately impose the given constraints, they are often favoured over the Lagrange multiplier method because they are straightforward to implement, and they do not introduce any extra solution variables.…”

Section: Introductionmentioning

confidence: 99%

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The bipenalty method for arbitrary multipoint constraints

Hetherington

Ferran

Askes

2012

Numerical Meth Engineering

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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.

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Section: Bipenalty Matrix Formulationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The bipenalty method for arbitrary multipoint constraints

Hetherington

Ferran

Askes

2012

Numerical Meth Engineering

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“…A long standing criticism of the penalty approach was the lack of a procedure to choose an appropriate magnitude for the penalty parameter, which must be large enough to effect a constraint but not too large as to cause numerical instability [1,2]. This has been addressed in some recent publications, which show how negative stiffness and positive and negative inertial type penalty parameters can be used to determine and control the error associated with any violation of the constraint(s) [8][9][10][11][12][13][14][15][16][17][18][19][20][21]. Most of these papers provide the theoretical basis or simple applications for verification, but more recently negative penalty parameters have also been used in enforcing constraints in more complicated problems [3,22].…”

Section: Introductionmentioning

confidence: 99%

Bipenalty method from a frequency domain perspective

Ilanko

Monterrubio

2012

Numerical Meth Engineering

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SUMMARY In a recent paper, it was shown that for time domain analysis, the simultaneous use of inertial and stiffness type penalty parameters to enforce constraints was found to yield accurate and converging results without causing any stability problems. From a frequency domain perspective, this is somewhat unexpected because the solution converges from below when stiffness penalty parameters are used to model constraints, and the convergence is from above when inertial penalty parameters are used. The purpose of this paper is to explain the effect of the simultaneous use of stiffness and inertial penalty parameters on the natural frequencies of constrained systems. In this work, it is shown that if suitably tuned, the bipenalty approach works well for frequency domain analysis also, and that with two different tuned set of stiffness and inertial penalty parameters, bounded solutions to the natural frequencies of constrained systems may be obtained. The method is applicable for any linear eigenvalue problem. Copyright © 2012 John Wiley & Sons, Ltd.

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“…Ilanko and coworkers (Ilanko & Dickinson 1999;Ilanko 2002aIlanko ,b, 2003Ilanko , 2005aAskes & Ilanko 2006;Askes et al 2008) proposed the use of positive as well as negative values for the penalty parameters and they showed that the exact solution is bracketed by the results obtained with either positive or negative penalties. This result can be used to interpolate between results obtained with moderately large (but not very large) positive and negative penalty parameters.…”

Section: Introductionmentioning

confidence: 99%

“…In recent years research has been carried out aimed at using lower values of the penalty parameters, while maintaining accuracy of constraint imposition, or aimed at penalty formulations that have a less adverse effect on the critical time step. Ilanko and coworkers (Ilanko & Dickinson 1999;Ilanko 2002aIlanko ,b, 2003Ilanko , 2005aAskes & Ilanko 2006;Askes et al 2008) proposed the use of positive as well as negative values for the penalty parameters and they showed that the exact solution is bracketed by the results obtained with either positive or negative penalties. This result can be used to interpolate between results obtained with moderately large (but not very large) positive and negative penalty parameters.…”

Section: Introductionmentioning

confidence: 99%

Bipenalty method for time domain computational dynamics

2009

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Penalty functions can be used to add constraints to systems of equations. In computational mechanics, stiffness-type penalties are the common choice. However, in dynamic applications stiffness penalties have the disadvantage that they tend to decrease the critical time step in conditionally stable time integration schemes, leading to increased CPU times for simulations. In contrast, inertia penalties increase the critical time step. In this paper, we suggest the simultaneous use of stiffness penalties and inertia penalties, which is denoted as the bipenalty method. We demonstrate that the accuracy of the bipenalty method is at least as good as (and usually better than) using either stiffness penalties or inertia penalties. Furthermore, for a number of finite elements (bar elements, beam elements and square plane stress/plane strain elements) we have derived ratios of the two penalty parameters such that their combined effect on the critical time step is neutral. The bipenalty method is thus superior to using stiffness penalties, because the decrease in critical time step can be avoided. The bipenalty method is also superior to using inertia penalties, since the constraints are realized with higher accuracy.

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Tyings in linear systems of equations modelled with positive and negative penalty functions

Cited by 7 publications

References 8 publications

The bipenalty method for arbitrary multipoint constraints

The bipenalty method for arbitrary multipoint constraints

Bipenalty method from a frequency domain perspective

Bipenalty method for time domain computational dynamics

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