Publication
Numerical Simulation of Frictional Contact Problems using Nagata Patches in Surface Smoothing
| Summary: | All movements in the world involve contact and friction, from walking to car driving. The contact mechanics has application in many engineering problems, including the connection of structural members by bolts or screws, design of gears and bearings, sheet metal or bulk forming, rolling contact of car tyres, crash analysis of structures, as well as prosthetics in biomedical engineering. Due to the nonlinear and non-smooth nature of contact mechanics (contact area is not known a priori), such problems are currently solved using the finite element method within the field of computational contact mechanics. However, most of the commercial finite element software packages presently available are not entirely capable to solve frictional contact problems, demanding for efficient and robust methods. Therefore, the main objective of this study is the development of algorithms and numerical methods to apply in the numerical simulation of 3D frictional contact problems between bodies undergoing large deformations. The presented original developments are implemented in the in-house finite element code DD3IMP. The formulation of quasi-static frictional contact problems between bodies undergoing large deformations is firstly presented in the framework of the continuum mechanics, following the classical scheme used in solid mechanics. The kinematic description of the deformable bodies is presented adopting an updated Lagrangian formulation. The mechanical behaviour of the bodies is described by an elastoplastic constitutive law in conjunction with an associated flow rule, allowing to model a wide variety of contact problems arising in industrial applications. The frictional contact between the bodies is established by means of two conditions: the principle of impenetrability and the Coulomb’s friction law, both imposed to the contact interface. The augmented Lagrangian method is applied for solving the constrained minimization incremental problem resulting from the frictional contact inequalities, yielding a mixed functional involving both displacements and contact forces. The spatial discretization of the bodies is performed with isoparametric solid finite elements, while the discretization of the contact interface is carried out using the classical Node-to-Segment technique, preventing the slave nodes from penetrating on the master surface. The geometrical part of the contact elements, defined by a slave node and the closest master segment, is created by the contact search algorithm based on the selection of the closest point on the master surface, defined by the normal projection of the slave node. In the particular case of contact between a deformable body and a rigid obstacle, the master rigid surface can be described by smooth parameterizations typically used in CAD models. However, in the general case of contact between deformable bodies, the spatial discretization of both bodies with low order finite elements yields a piecewise bilinear representation of the master surface. This is the central source of problems in solving contact problems involving large sliding, since it leads to the discontinuity of the surface normal vector field. Thus, a surface smoothing procedure based on the Nagata patch interpolation is proposed to describe the master contact surfaces, which led to the development of the Node-to-Nagata contact element. The accuracy of the surface smoothing method using Nagata patches is evaluated by means of simple geometries. The nodal normal vectors required for the Nagata interpolation are evaluated from the CAD geometry in case of rigid master surfaces, while in case of deformable bodies they are approximated using the weighted average of the normal vectors of the neighbouring facets. The residual vectors and tangent matrices of the contact elements are derived coherently with the surface smoothing approach, allowing to obtain quadratic convergence rate in the generalized Newton method used for solving the nonlinear system of equations. The developed surface smoothing method and corresponding contact elements are validated through standard numerical examples with known analytical or semi-analytical solutions. More advanced frictional contact problems are studied, covering the contact of a deformable body with rigid obstacles and the contact between deformable bodies, including self-contact phenomena. The smoothing of the master surface improves the robustness of the computational methods and reduces strongly the non-physical oscillations in the contact force introduced by the traditional faceted description of the contact surface. The presented results are compared with numerical solutions obtained by other authors and experimental results, demonstrating the accuracy and performance of the implemented algorithms for highly nonlinear problems. |
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| Main Authors: | Neto, Diogo Mariano Simões |
| Subject: | Contact Mechanics Finite Element Method Augmented Lagrangian Method Large Sliding Node-to-Segment Discretization Surface Smoothing Nagata Patches Mecânica do Contacto Método dos Elementos Finitos Método do Lagrangeano Aumentado, Grandes Escorregamentos Node-to-Segment Suavização de Superfícies Superfícies Nagata |
| Year: | 2014 |
| Country: | Portugal |
| Document type: | doctoral thesis |
| Access type: | open access |
| Associated institution: | Universidade de Coimbra |
| Language: | English |
| Origin: | Estudo Geral - Universidade de Coimbra |
| Summary: | All movements in the world involve contact and friction, from walking to car driving. The contact mechanics has application in many engineering problems, including the connection of structural members by bolts or screws, design of gears and bearings, sheet metal or bulk forming, rolling contact of car tyres, crash analysis of structures, as well as prosthetics in biomedical engineering. Due to the nonlinear and non-smooth nature of contact mechanics (contact area is not known a priori), such problems are currently solved using the finite element method within the field of computational contact mechanics. However, most of the commercial finite element software packages presently available are not entirely capable to solve frictional contact problems, demanding for efficient and robust methods. Therefore, the main objective of this study is the development of algorithms and numerical methods to apply in the numerical simulation of 3D frictional contact problems between bodies undergoing large deformations. The presented original developments are implemented in the in-house finite element code DD3IMP. The formulation of quasi-static frictional contact problems between bodies undergoing large deformations is firstly presented in the framework of the continuum mechanics, following the classical scheme used in solid mechanics. The kinematic description of the deformable bodies is presented adopting an updated Lagrangian formulation. The mechanical behaviour of the bodies is described by an elastoplastic constitutive law in conjunction with an associated flow rule, allowing to model a wide variety of contact problems arising in industrial applications. The frictional contact between the bodies is established by means of two conditions: the principle of impenetrability and the Coulomb’s friction law, both imposed to the contact interface. The augmented Lagrangian method is applied for solving the constrained minimization incremental problem resulting from the frictional contact inequalities, yielding a mixed functional involving both displacements and contact forces. The spatial discretization of the bodies is performed with isoparametric solid finite elements, while the discretization of the contact interface is carried out using the classical Node-to-Segment technique, preventing the slave nodes from penetrating on the master surface. The geometrical part of the contact elements, defined by a slave node and the closest master segment, is created by the contact search algorithm based on the selection of the closest point on the master surface, defined by the normal projection of the slave node. In the particular case of contact between a deformable body and a rigid obstacle, the master rigid surface can be described by smooth parameterizations typically used in CAD models. However, in the general case of contact between deformable bodies, the spatial discretization of both bodies with low order finite elements yields a piecewise bilinear representation of the master surface. This is the central source of problems in solving contact problems involving large sliding, since it leads to the discontinuity of the surface normal vector field. Thus, a surface smoothing procedure based on the Nagata patch interpolation is proposed to describe the master contact surfaces, which led to the development of the Node-to-Nagata contact element. The accuracy of the surface smoothing method using Nagata patches is evaluated by means of simple geometries. The nodal normal vectors required for the Nagata interpolation are evaluated from the CAD geometry in case of rigid master surfaces, while in case of deformable bodies they are approximated using the weighted average of the normal vectors of the neighbouring facets. The residual vectors and tangent matrices of the contact elements are derived coherently with the surface smoothing approach, allowing to obtain quadratic convergence rate in the generalized Newton method used for solving the nonlinear system of equations. The developed surface smoothing method and corresponding contact elements are validated through standard numerical examples with known analytical or semi-analytical solutions. More advanced frictional contact problems are studied, covering the contact of a deformable body with rigid obstacles and the contact between deformable bodies, including self-contact phenomena. The smoothing of the master surface improves the robustness of the computational methods and reduces strongly the non-physical oscillations in the contact force introduced by the traditional faceted description of the contact surface. The presented results are compared with numerical solutions obtained by other authors and experimental results, demonstrating the accuracy and performance of the implemented algorithms for highly nonlinear problems. |
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