On stability and reflection‐transmission analysis of the bipenalty method in contact‐impact problems: A one‐dimensional, homogeneous case study

Abstract

Summary The stability and reflection‐transmission properties of the bipenalty method are studied in application to explicit finite element analysis of one‐dimensional contact‐impact problems. It is known that the standard penalty method, where an additional stiffness term corresponding to contact boundary conditions is applied, attacks the stability limit of finite element model. Generally, the critical time step size rapidly decreases with increasing penalty stiffness. Recent comprehensive studies have shown that the so‐called bipenalty technique, using mass penalty together with standard stiffness penalty, preserves the critical time step size associated to contact‐free bodies. In this paper, the influence of the penalty ratio (ratio of stiffness and mass penalty parameters) on stability and reflection‐transmission properties in one‐dimensional contact‐impact problems using the same material and mesh size for both domains is studied. The paper closes with numerical examples, which demonstrate the stability and reflection‐transmission behavior of the bipenalty method in one‐dimensional contact‐impact and wave propagation problems of homogeneous materials.

Explicit dynamics in impact simulation using a NURBS contact interface

“…For 2D and 3D impact problems with penalty regularization, an estimation of the critical time step can be based on diagonalization of the penalty stiffness and the element eigenvalue inequalities, which is virtually equivalent to a 1D impact problem . The bipenalty approach, where both inertia and stiffness terms are penalized to guarantee a constant critical time step upon the increase of the penalty stiffness, is valid for 1D impact problems . Another possibility to guarantee a stable time integration is based on monitoring of energy balance or linear stability during the simulation and a possible restart with an adjusted time step.…”

“…32 The bipenalty approach, where both inertia and stiffness terms are penalized to guarantee a constant critical time step upon the increase of the penalty stiffness, is valid for 1D impact problems. 33 Another possibility to guarantee a stable time integration is based on monitoring of energy balance or linear stability during the simulation and a possible restart with an adjusted time step.…”

Estimation of Stability Limit Based on Gershgorin's Theorem for Explicit Contact-Impact Analysis Signorini Problem Using Bipenalty Approach

“…Thus, smaller safer values of critical Courant numbers are obtained in comparison with exact ones calculated from solution of eigenvalue problem. Further details of this work including extension to other one-dimensional contact models can be found in prepared paper [10].…”

Bi-Penalty Stabilized Explicit Finite Element Algorithm for One-Dimensional Contact-Impact Problems

“…In the work Kopačka (2018), the equations of motion of impacted elastic solids with respect to the bipenalty stabilization has been derived. Also the stability of the bipenalty method has been analyzed there.…”

“…And the stiffness penalty parameter can be chosen arbitrarily. The application of the bipenalty method into dynamic conctact problems have been applied in Kopačka (2018 Another trouble in modelling of dynamic contact problems is existence of spurious contact oscillations. In this paper, we suggest a methodology witch is able to avoid the both mentioned troubles in modelling of dynamic contact problems and we applied them into one-dimensional problems.…”