A direct elimination algorithm for quasi-static and dynamic contact problems
Section snippets
Introduction, motivation and goals
Numerical analysis of contact problems has been one of the hot research topics of interest over the last decades. Contact problems arise in many applications, such as in crashworthiness, projectile impact, and material forming processes, i.e. sheet metal forming, bulk forming, casting, friction stir welding, cutting, and powder compaction. Despite the important progresses achieved in computational contact mechanics, the numerical simulation of contact problems is still nowadays a complex task,
Local formulation
Let be the space dimension and the time interval of interest. Let the open sets and , with smooth boundaries and and closures and , be the reference placement of two continuum bodies and .
For each body we denote by the vector position of the material particles at the reference configuration, the orientation preserving deformation maps, the material
Finite element formulation of the continuum problem without frictional contact constraints
Let us consider first the finite element discretization of quasi-static and dynamic continuum problem without frictional contact constraints. Using a standard finite element discretization, the material coordinates , displacements and material velocities of body , take the formwhere gives the current placement of the particle of body , ,
Introduction and notation
Within the direct elimination algorithm for contact problems proposed in this work, the restrictions arising by the contact between the bodies are introduced through the direct elimination of the displacements of the slave nodes. From a computational implementation point of view, this direct elimination method is carried out through a number of transformations made on the global tangent operator. In order to conveniently visualize those transformations, let us introduce the following notation.
Numerical examples
In this section a selection of representative quasi-static and dynamic numerical examples, that illustrate the performance of the contact formulation proposed, is shown. Three quasi-static and one dynamic numerical examples have been chosen.
First, a contact patch test is considered. An assessment of the error obtained using the direct elimination method, for different mesh sizes and different Young׳s modulus, has been performed. In the second example, a Hertzian contact problem [23] is
Conclusions
In this paper a new formulation for quasi-static and dynamic contact problems, under full stick friction and frictionless contact conditions, has been developed and implemented. The constraints arising in full stick and frictionless contact problems are imposed in a strong fashion by a direct elimination of the involved degrees of freedom of the resulting system of equations. Drawbacks inherent to the penalty method, such as the selection of suitable penalty parameters or the ill-conditioning
Acknowledgments
Authors would like to acknowledge COMPASS for providing the code RamSeries where the contact algorithms shown in this work have been implemented. Collaborations of Dr. Julio Garcia and Eng. Jaume Sagues, as well as the support of the students Massimo Angelini and Matias Bossio, are gratefully acknowledged.
References (66)
- et al.
On the formulation of coupled thermoplastic problems with phase-change
Int. J. Plasticity
(1999) - et al.
On the numerical modeling of frictional wear phenomena
Comput. Methods Appl. Mech. Eng.
(1999) - et al.
On the constitutive modeling of coupled thermomechanical phase-change problems
Int. J. Plasticity
(2001) - et al.
Formulation and analysis of conserving algorithms for frictionless dynamic contact/impact problems
Comput. Methods Appl. Mech. Eng.
(1998) - et al.
A new dissipative time-stepping algorithm for frictional contact problems: formulation and analysis
Comput. Methods Appl. Mech. Eng.
(1999) - et al.
On the formulation of high-frequency dissipative time-stepping algorithms for nonlinear dynamics. Part I: low-order methods for two model problems and nonlinear elastodynamics
Comput. Methods Appl. Mech. Eng.
(2001) - et al.
A single surface contact algorithm for the post-buckling analysis of shell structures
Comput. Methods Appl. Mech. Eng.
(1990) The discrete null space method for the energy consistent integration of constrained mechanical systems. Part I: holonomic constraints
Comput. Methods Appl. Mech. Eng.
(2005)The return mapping method for the integration of friction constitutive relations
Comput. Struct.
(1989)Exact energy-momentum conserving algorithms for general models in nonlinear elasticity
Comput. Methods Appl. Mech. Eng.
(2000)
Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement
Comput. Methods Appl. Mech. Eng.
Algorithmic symmetrization of Coulomb frictional problems using augmented Lagrangians
Comput. Methods Appl. Mech. Eng.
A new solution procedure for application of energy-conserving algorithms to general constitutive models in nonlinear elastodynamics
Comput. Methods Appl. Mech. Eng.
Energy consistent algorithms for dynamic finite deformation plasticity
Comput. Methods Appl. Mech. Eng.
Modelling unilateral frictionless contact using the null-space method and cubic B-spline interpolation
Comput. Methods Appl. Mech. Eng.
Algorithms and numerical results for finite element approximations of contact problems with non-classical friction laws
Comput. Struct.
Models and computational methods for dynamic friction phenomena
Comput. Methods Appl. Mech. Eng.
A mortar segment-to-segment contact method for large deformation solid mechanics
Comput. Methods Appl. Mech. Eng.
A mortar segment-to-segment frictional contact method for large deformations
Comput. Methods Appl. Mech. Eng.
An augmented Lagrangian treatment of contact problems involving friction
Comput. Struct.
Exact energy-momentum conserving algorithms and symplectic schemes for nonlinear dynamics
Comput. Methods Appl. Mech. Eng.
Finite element formulations of large deformation impact-contact problems with friction
Comput. Struct.
A new frictional time integration algorithm for large slip multi-body frictional contact problems
Comput. Methods Appl. Mech. Eng.
Numerical analysis of coupled thermomechanical contact problems. Computational model and applications
Arch. Comput. Methods Mech.
Contact-impact by the pinball algorithm with penalty and Lagrangian methods
Int. J. Numer. Methods. Eng.
An enhanced energy conserving time stepping algorithm for frictionless particle contacts
Int. J. Numer. Methods. Eng.
Unilateral non-linear dynamic contact of thin-walled structures using a primal-dual active set strategy
Int. J. Numer. Methods. Eng.
Energy consistent algorithms in computational contact mechanics
Energy consistent algorithms for frictional contact problems
Int. J. Numer. Methods. Eng.
On the numerical modeling of the thermomechanical contact for metal casting analysis
J. Heat Transfer
A generalized Newton method for contact problems with friction
Journal de Mécanique Théorique et Appliquée
Les Inequations en Mecanique et en Physique
Cited by (3)
Computational modeling of 2D frictional contact problems based on the use of coupling finite elements and combined contact/friction damage constitutive model
2022, Finite Elements in Analysis and DesignCitation Excerpt :In literature, there is evidence that the first studies involving contact mechanics were conducted by Hertz [10], who presented the analytical solution for frictionless contact between two elliptic elastic bodies. After the advent of computers and with more computational power available in the last decades, many efforts were dedicated to improve numerical treatment of contact problems, mainly supported by the finite element method framework (e.g. [11–18]). However, due to the complexity involved in this field of study, even today, it is still a great challenge to solve contact problems from the mathematical and the engineering point of view [18–21].
An application of isogeometric analysis and boundary integral element method for solving nonlinear contact problems
2020, Applied Sciences (Switzerland)Conserving algorithms for frictionless and full stick friction dynamic contact problems using the direct elimination method
2018, International Journal for Numerical Methods in Engineering