A direct elimination algorithm for quasi-static and dynamic contact problems

doi:10.1016/j.finel.2014.09.001

Finite Elements in Analysis and Design

Volume 93, January 2015, Pages 107-125

https://doi.org/10.1016/j.finel.2014.09.001 Get rights and content

Highlights

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A new direct elimination algorithm for full stick and frictionless quasi-static and dynamic contact problems has been developed and implemented.
•
Conserving time stepping algorithms have been used within the framework of the new contact formulation.
•
A selected number of quasi-static and dynamic numerical examples has been chosen to show the performance of the new contact formulation.
•
The performance of the conserving time stepping algorithm has been shown for a frictionless case.

Abstract

This paper deals with the computational modeling and numerical simulation of contact problems at finite deformations using the finite element method. Quasi-static and dynamic problems are considered and two particular frictional conditions, full stick friction and frictionless cases, are addressed. Lagrange multipliers and regularized formulations of the contact problem, such as penalty or augmented Lagrangian methods, are avoided and a new direct elimination method is proposed. Conserving algorithms are also introduced for the proposed formulation for dynamic contact problems. An assessment of the performance of the resulting formulation is shown in a number of selected benchmark tests and numerical examples, including both quasi-static and dynamic contact problems under full stick friction and frictionless contact conditions. Conservation of key discrete properties exhibited by the time stepping algorithm used for dynamic contact problems is also shown in an example.

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 $2 \leq n_{\dim} \leq 3$ be the space dimension and $I : = [0, T] \subset ℝ_{+}$ the time interval of interest. Let the open sets $Ω^{(1)} \subset ℝ^{n_{\dim}}$ and $Ω^{(2)} \subset ℝ^{n_{\dim}}$ , with smooth boundaries $\partial Ω^{(1)}$ and $\partial Ω^{(2)}$ and closures ${\bar{Ω}}^{(1)} = Ω^{(1)} \cup \partial Ω^{(1)}$ and ${\bar{Ω}}^{(2)} = Ω^{(2)} \cup \partial Ω^{(2)}$ , be the reference placement of two continuum bodies $ℬ^{(1)}$ and $ℬ^{(2)}$ .

For each body $ℬ^{(i)}$ we denote by $X^{(i)} \in {\bar{Ω}}^{(i)}$ the vector position of the material particles at the reference configuration, $φ^{(i)} : {\bar{Ω}}^{(i)} \times I \to ℝ^{n_{\dim}}$ the orientation preserving deformation maps, $V^{(i)} : = \partial_{t} φ^{(i)}$ 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 $X^{(i)} \in {\bar{Ω}}_{h}^{(i)}$ , displacements $u^{(i)}$ and material velocities $V^{(i)}$ of body $ℬ^{(i)}$ , take the form $X^{(i)} : = \sum_{A = 1}^{n_{n o d e}^{(i)}} N_{A} (ξ) X_{A}^{(i)}, u^{(i)} : = \sum_{A = 1}^{n_{n o d e}^{(i)}} N_{A} (ξ) u_{A}^{(i)}, V^{(i)} : = \sum_{A = 1}^{n_{n o d e}^{(i)}} N_{A} (ξ) V_{A}^{(i)}$ where $x^{(i)} : = X^{(i)} + u^{(i)}$ gives the current placement of the particle $X^{(i)} \in {\bar{Ω}}_{h}^{(i)}$ of body $ℬ^{(i)}$ , $X_{A}^{(i)} \in {\bar{Ω}}_{h}^{(i)}$ , $u_{A}^{(i)}$

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.

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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 Design
Citation 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].
This work proposes a new approach to describe 2D frictional contact problems based on the use of coupling finite elements (CFEs) and a combined contact/friction damage constitutive model. This technique is able to establish the interaction between non-matching meshes, representing different subdomains (bodies) of a problem, which share common boundaries. The formulation is developed within a framework considering elastic bodies, small deformations and static loading conditions. Initially, the bodies are discretized in finite elements in a totally independent way. Then, CFEs are inserted to enforce the contact constraints in the context of penalty methods. As the CFEs share nodes from both sides of the non-matching meshes, an appropriate constitutive model can be easily defined to describe the interaction between the bodies on the contact interface based on the concept of a relative displacement. The proposed constitutive model is able to describe the frictional slippage and separation between the bodies. This constitutive model is integrated using an implicit–explicit scheme to avoid numerical convergence problems. The insertion of CFEs does not introduce any additional degree of freedom to the problem and these elements are introduced into the standard finite element mesh in a pre-processing stage to avoid pre-processing tasks during the analysis. The same framework is applied to deal with overlapping and non-overlapping meshes and its implementation in conventional FE programs is quite simple without drastic modifications. Five numerical analyses are performed and the results demonstrated that the strategy developed is able to represent the interaction between bodies, coherently and accurately.
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

View full text

A direct elimination algorithm for quasi-static and dynamic contact problems

Highlights

Abstract

Section snippets

Introduction, motivation and goals

Local formulation

Finite element formulation of the continuum problem without frictional contact constraints

Introduction and notation

Numerical examples

Conclusions

Acknowledgments

Int. J. Plasticity

Comput. Methods Appl. Mech. Eng.

Int. J. Plasticity

Comput. Methods Appl. Mech. Eng.

Comput. Methods Appl. Mech. Eng.

Comput. Methods Appl. Mech. Eng.

Comput. Methods Appl. Mech. Eng.

Comput. Methods Appl. Mech. Eng.

Comput. Struct.

Comput. Methods Appl. Mech. Eng.

Comput. Methods Appl. Mech. Eng.

Comput. Methods Appl. Mech. Eng.

Comput. Methods Appl. Mech. Eng.

Comput. Methods Appl. Mech. Eng.

Comput. Methods Appl. Mech. Eng.

Comput. Struct.

Comput. Methods Appl. Mech. Eng.

Comput. Methods Appl. Mech. Eng.

Comput. Methods Appl. Mech. Eng.

Comput. Struct.

Comput. Methods Appl. Mech. Eng.

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