Isogeometric analysis of contact problems has been recently proved to yield significant advantages with respect to the use of standard Lagrange discretizations [1,2]. As NURBS geometries can attain the desired degree of continuity at the element boundaries, they possess the premises to alleviate all the problems arising particularly in sliding contact when using conventional Lagrange polynomial elements, which are only C0-continuous at the interelement nodes. Such problems have often been faced by introducing smoothing techniques, some of which involving NURBS interpolation. These procedures generally improve the performance of the contact algorithms by enhancing the continuity of the contact surfaces, however they do not increase the order of convergence as the higher-order approximation does not involve the bulk behavior of the solids. In this work, NURBS-based isogeometric analysis is adopted to model 3D large deformation frictionless contact problems. The performance of isogeometric analysis is compared to that of standard C0-continuous Lagrange finite element interpolations. The proposed contact formulation is based on a mortar approach, extended to NURBS-based interpolations, and combined with a simple integration procedure which does not involve segmentation of the contact surfaces. The augmented Lagrangian method is chosen to obtain an exact satisfaction of the contact constraints. As proposed by previous researchers, a Newton-like solution scheme is applied to solve the saddle-point problem simultaneously for the displacements and the Lagrange multipliers. The presented examples deal with both small- and large-deformation cases. The quality of the solution is examined in terms of contact stress distributions in the small-deformation examples, and in terms of global load vs. displacement behavior for the large-deformation, large-sliding examples. In both cases, the results obtained with the isogeometric analysis and with the traditional Lagrange discretizations are compared for varying resolution and order of the contact surfaces. Based on results obtained in this investigation, it can be concluded that the proposed contact mortar formulation using NURBS-based isogeometric analysis displays a significantly superior performance with respect to the same formulation using standard Lagrange polynomials. The contact pressure distributions stemming from the NURBS parameterizations are always non-negative, are practically insensitive to changes in the interpolation order, and improve monotonically as the mesh resolution increases. The respective distributions obtained from Lagrange parameterizations are highly sensitive to the interpolation order, display significant spurious oscillations and may attain large non-physical negative values. In 3D large frictionless sliding problems the time histories of the tractions obtained from the NURBS discretizations are remarkably smooth and improve in quality with increasing order of the parameterization. Conversely, the curves obtained from Lagrange parameterizations display irregular oscillations whose magnitude increases with the interpolation order and which may even prevent convergence. This superiority of the NURBS parameterization over the Lagrange one for contact modeling is a combined effect of the higher continuity achieved at the inter-element boundaries and of the inherent non-negativeness of the NURBS interpolation functions. While these two favorable features may also be individually obtained in different ways (e.g., higher geometric continuity can be pursued by means of smoothing techniques and inherent non-negativeness is possessed by other categories of shape functions), NURBS-based isogeometric analysis provides a very simple framework in which both are simultaneously and naturally achieved. The individual role played by these two factors could be isolated e.g. by using C0 Berstein polynomials, which are non-negative and possess the variation diminishing and convex hull properties like the NURBS basis. This is left as a potential topic for further investigations.

An augmented Lagrangian approach to isogeometric analysis of 3D large deformation contact problems / De Lorenzis, L.; Wriggers, P.; Zavarise, G.. - ELETTRONICO. - (2011). (Intervento presentato al convegno ICCCM11 International Conference on Computational Contact Mechanics tenutosi a Hannover nel jgiugno 2011).

An augmented Lagrangian approach to isogeometric analysis of 3D large deformation contact problems

Zavarise G.
2011

Abstract

Isogeometric analysis of contact problems has been recently proved to yield significant advantages with respect to the use of standard Lagrange discretizations [1,2]. As NURBS geometries can attain the desired degree of continuity at the element boundaries, they possess the premises to alleviate all the problems arising particularly in sliding contact when using conventional Lagrange polynomial elements, which are only C0-continuous at the interelement nodes. Such problems have often been faced by introducing smoothing techniques, some of which involving NURBS interpolation. These procedures generally improve the performance of the contact algorithms by enhancing the continuity of the contact surfaces, however they do not increase the order of convergence as the higher-order approximation does not involve the bulk behavior of the solids. In this work, NURBS-based isogeometric analysis is adopted to model 3D large deformation frictionless contact problems. The performance of isogeometric analysis is compared to that of standard C0-continuous Lagrange finite element interpolations. The proposed contact formulation is based on a mortar approach, extended to NURBS-based interpolations, and combined with a simple integration procedure which does not involve segmentation of the contact surfaces. The augmented Lagrangian method is chosen to obtain an exact satisfaction of the contact constraints. As proposed by previous researchers, a Newton-like solution scheme is applied to solve the saddle-point problem simultaneously for the displacements and the Lagrange multipliers. The presented examples deal with both small- and large-deformation cases. The quality of the solution is examined in terms of contact stress distributions in the small-deformation examples, and in terms of global load vs. displacement behavior for the large-deformation, large-sliding examples. In both cases, the results obtained with the isogeometric analysis and with the traditional Lagrange discretizations are compared for varying resolution and order of the contact surfaces. Based on results obtained in this investigation, it can be concluded that the proposed contact mortar formulation using NURBS-based isogeometric analysis displays a significantly superior performance with respect to the same formulation using standard Lagrange polynomials. The contact pressure distributions stemming from the NURBS parameterizations are always non-negative, are practically insensitive to changes in the interpolation order, and improve monotonically as the mesh resolution increases. The respective distributions obtained from Lagrange parameterizations are highly sensitive to the interpolation order, display significant spurious oscillations and may attain large non-physical negative values. In 3D large frictionless sliding problems the time histories of the tractions obtained from the NURBS discretizations are remarkably smooth and improve in quality with increasing order of the parameterization. Conversely, the curves obtained from Lagrange parameterizations display irregular oscillations whose magnitude increases with the interpolation order and which may even prevent convergence. This superiority of the NURBS parameterization over the Lagrange one for contact modeling is a combined effect of the higher continuity achieved at the inter-element boundaries and of the inherent non-negativeness of the NURBS interpolation functions. While these two favorable features may also be individually obtained in different ways (e.g., higher geometric continuity can be pursued by means of smoothing techniques and inherent non-negativeness is possessed by other categories of shape functions), NURBS-based isogeometric analysis provides a very simple framework in which both are simultaneously and naturally achieved. The individual role played by these two factors could be isolated e.g. by using C0 Berstein polynomials, which are non-negative and possess the variation diminishing and convex hull properties like the NURBS basis. This is left as a potential topic for further investigations.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11583/2706463
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