A weighted finite element mass redistribution method for dynamic contact problems (original) (raw)
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ESAIM Mathematical Modelling and Numerical Analysis
This paper focuses on a one-dimensional wave equation being subjected to a unilateral boundary condition. Under appropriate regularity assumptions on the initial data, a new proof of existence and uniqueness results is proposed. The mass redistribution method, which is based on a redistribution of the body mass such that there is no inertia at the contact node, is introduced and its convergence is proved. Finally, some numerical experiments are reported.
This paper focuses on a one-dimensional elastodynamic contact problem and aims to give some new numerical results. Under appropriate regularity assumptions on the initial data, a new proof of existence and uniqueness results is proposed. An approximation of this evolutionary problem combining the nite element method as well as the mass redistribution method that consists on a redistribution of the body mass such that there is no inertia at the contact node, is introduced. Then two benchmark problems (one being new) with their analytical solutions are presented and some possible discretizations using di erent time{ integration schemes are described. Finally, numerical experiments are reported and analyzed.
Mass redistribution method for finite element contact problems in elastodynamics
European Journal of Mechanics - A/Solids, 2008
This paper is devoted to a new method dealing with the semi-discretized finite element unilateral contact problem in elastodynamics. This problem is ill-posed mainly because the nodes on the contact surface have their own inertia. We introduce a method based on an equivalent redistribution of the mass matrix such that there is no inertia on the contact boundary. This leads to a mathematically well-posed and energy conserving problem. Finally, some numerical tests are presented.
2014
Abstract. This note deals with two and three–dimensional elastodynamic contact problems. An approximated solution combining the finite element and mass redistribution methods is exhibited. The mass redistribution method consists in a redistribution of the body mass such that the inertia at the contact node vanishes. Some numerical experiments using two time–integration methods, the Crank–Nicolson as well as backward Euler methods, highlighted the convergence properties of the mass redistribution method.
A robust finite element redistribution approach for elastodynamic contact problems
Applied Numerical Mathematics, 2016
This paper deals with a one-dimensional elastodynamic contact problem and aims to highlight some new numerical results. A new proof of existence and uniqueness results is proposed. More precisely, the problem is reformulated as a differential inclusion problem, the existence result follows from some a priori estimates obtained for the regularized problem while the uniqueness result comes from a monotonicity argument. An approximation of this evolutionary problem combining the finite element method as well as the mass redistribution method which consists on a redistribution of the body mass such that there is no inertia at the contact node, is introduced. Then two benchmark problems, one being new with convenient regularity properties, together with their analytical solutions are presented and some possible discretizations using different time-integration schemes are described. Finally, numerical experiments are reported and analyzed.
Dynamic boundary integral “equation” method for unilateral contact problems
Engineering Analysis With Boundary Elements, 1991
The aim of the present paper is the extension of the method of boundary integral equations (B.I.E.) to dynamic unilateral contact problems. Using semidiscretization, with respect to time, and then the inequality constrained principle of minimum potential and the equivalent variational inequality formulation, we derive saddle point formulations for the problems using appropriate Langrangian functions. An elimination technique gives rise to a minimum 'principle' on the boundary with respect to the unknown normal displacements of the contact region, which has as parameters the velocities etc. of the previous time steps. It is also shown that the minimum problem is equivalent to a multivalued boundary integral equations problem involving symmetric operators. The theory is illustrated by numerical examples, which also treat the case of impact of the structure with its support. In order to achieve this last task, an appropriate time discretization scheme has been chosen. Numerical examples dealing with the seismic behaviour of two-dimensional structures supported by the ground are presented to illustrate the method.
Proceedings of the 7th International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering (COMPDYN 2015), 2019
This work proposes an extremely efficient and accurate time explicit solution algorithm for the simulation of contact-impact problems with finite elements. The methodology combines three existent techniques. First, for the treatment of the contact problem, a bipenalty contact-impact algorithm for explicit dynamics with stabilization of the contact constraints through the addition of stiffness and mass penalties [1]. Second, for the reduction of the contact problem to the interface, the method of localized Lagrange multipliers [2], that is able to formulate the contact problem in partitioned form, uncoupling the contact terms from the free body matrices. And finally, for an efficient construction of the free-free inverse mass matrices, the method based on localized Lagrange multipliers to generate inverse mass matrices [3] with specific projectors for the application of boundary conditions. It is also demonstrated that the resultant methodology is very well suited for parallelization in multicore machines. Different numerical experiments will be used to prove the effectiveness of the proposed formulation. 786 COMPDYN 2019 7 th ECCOMAS Thematic Conference on Computational Methods in Structural Dynamics and Earthquake Engineering M. Papadrakakis, M. Fragiadakis (eds.
A finite element method for contact/impact
Finite Elements in Analysis and Design, 1998
Ideas from the analysis of differential-algebraic equations are applied to the numerical solution of frictionless contact/impact problems in solid mechanics. Index-one and two formulations for dynamic contact-impact within the context of the finite element method are derived. The resulting equations are shown to stabilize the kinematic fields at the contact interface, at the expense of a small energy loss, which is shown to decrease consistently with mesh refinement. This energy dissipation is shown to be necessary for the establishment of persistent contact. A Newmark-type time integration scheme is derived from the proposed formulation, and shown to yield excellent results in modeling the transition to contact/impact.
A novel finite element formulation for frictionless contact problems
International Journal for Numerical Methods in Engineering, 1995
This article advocates a new methodology for the finite element solution of contact problems involving bodies that may undergo finite motions and deformations. The analysis is based on a decomposition of the two-body contact problem into two simultaneous sub-problems, and results naturally in geometrically unbiased discretization of the contacting surfaces. A proposed two-dimensional contact element is specifically designed to unconditionally allow for exact transmission of constant normal traction through interacting surfaces.
A contact-stabilized Newmark method for dynamical contact problems
International Journal for Numerical Methods in Engineering, 2008
The numerical integration of dynamical contact problems often leads to instabilities at contact boundaries caused by the non-penetration condition between bodies in contact. Even a recent energy dissipative modification due to Kane et al. (1999), which discretizes the non-penetration constraints implicitly, is not able to circumvent artificial oscillations. For this reason, the present paper suggests a contact stabilization which avoids artificial oscillations at contact interfaces and is also energy dissipative. The key idea of this contact stabilization is an additional L 2-projection at contact interfaces, which can easily be added to any existing time integration scheme. In case of a lumped mass matrix, this projection can be carried out completely locally, thus creating only negligible additional numerical cost. For the new scheme, an elementary analysis is given, which is confirmed by numerical findings in an illustrative test example (Hertzian two body contact).