The penalty method is widely used for the numerical solution of contact problems, due to its simplicity, clear physical meaning and flexibility. However, its application satisfies exactly the contact constraints only in the limit as the penalty parameter tends to infinity. The solution can be improved using augmentation schemes. However such schemes perform often unsatisfactorily, as they display a poor rate of convergence to the exact solution and their efficiency is also strongly affected by the magnitude of the penalty parameter. This paper follows up to a previous work [1], where a new method was proposed to perform the augmentations based on estimated values of the augmented Lagrangians. In this method, at each augmentation the converged state is used to extract some data. This piece of information updates a database which is then used for the prediction of the Lagrangians at the next augmentation. The estimation is primarily based on the evolution of the constraint violation with respect to the evolution of the contact forces. In the first version of the method, presented in [1], the estimation of the augmented Lagrangians at each augmentation was performed via the construction of an hyperplane interpolating the available database of contact forces. The method was shown to exhibit a noticeable efficiency in detecting nearly exact contact forces already at the first augmentation, and to achieve superlinear convergence for the subsequent minimisation of the residual of constraints. Remarkably, the method was also found relatively insensitive to the penalty parameter. This allows a solution that fulfils the constraints very rapidly, even when using very small penalty values. In this paper, the original method is applied with a significant modification in the procedure used for estimation of the augmented Lagrangians. In place of the aforementioned hyperplane interpolation, techniques of linear regression analysis are used to predict the updated augmented Lagrangians at each subsequent augmentation. The main objective is to regularize the convergence path to the exact solution, and thus to further reduce the number of augmentations required for convergence. To this aim, several linear regression techniques are analyzed, evaluating their relative performance in the context of the proposed method. The effect of the most significant parameters controlling the management of the Lagrangian database is also investigated. The method is shown to perform remarkably well, independently from the value of the penalty parameter.

On fast augmentation of penalty-based contact / Zavarise, G.; De Lorenzis, L.. - STAMPA. - (2010). (Intervento presentato al convegno 7th Medeting on Unilateral Problems in Structural Analysis tenutosi a Palmanova (Udine) - Italia nel giugno 2010).

### On fast augmentation of penalty-based contact

#### Abstract

The penalty method is widely used for the numerical solution of contact problems, due to its simplicity, clear physical meaning and flexibility. However, its application satisfies exactly the contact constraints only in the limit as the penalty parameter tends to infinity. The solution can be improved using augmentation schemes. However such schemes perform often unsatisfactorily, as they display a poor rate of convergence to the exact solution and their efficiency is also strongly affected by the magnitude of the penalty parameter. This paper follows up to a previous work [1], where a new method was proposed to perform the augmentations based on estimated values of the augmented Lagrangians. In this method, at each augmentation the converged state is used to extract some data. This piece of information updates a database which is then used for the prediction of the Lagrangians at the next augmentation. The estimation is primarily based on the evolution of the constraint violation with respect to the evolution of the contact forces. In the first version of the method, presented in [1], the estimation of the augmented Lagrangians at each augmentation was performed via the construction of an hyperplane interpolating the available database of contact forces. The method was shown to exhibit a noticeable efficiency in detecting nearly exact contact forces already at the first augmentation, and to achieve superlinear convergence for the subsequent minimisation of the residual of constraints. Remarkably, the method was also found relatively insensitive to the penalty parameter. This allows a solution that fulfils the constraints very rapidly, even when using very small penalty values. In this paper, the original method is applied with a significant modification in the procedure used for estimation of the augmented Lagrangians. In place of the aforementioned hyperplane interpolation, techniques of linear regression analysis are used to predict the updated augmented Lagrangians at each subsequent augmentation. The main objective is to regularize the convergence path to the exact solution, and thus to further reduce the number of augmentations required for convergence. To this aim, several linear regression techniques are analyzed, evaluating their relative performance in the context of the proposed method. The effect of the most significant parameters controlling the management of the Lagrangian database is also investigated. The method is shown to perform remarkably well, independently from the value of the penalty parameter.
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2010
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11583/2706443`
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